Philippe David

Valérie Guisset

Keywords: as analyzed through spatial correlations . . . . . . . . . . . . . . . . . . . adsorption problems and binary alloys . . . . . . . 15 2.2.2 Magnetic (or structural) frustration . . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Commensurate, ordered, solid-like phases . . . . . . . . liquid-like phases . . . . . . . . .

gave support on the UHV system used in Grenoble for the sample fabrication and STM imaging. In particular, they repaired the setup when it did not work, they mounted the molecular evaporator I used for sample fabrication and explained me how to use the UHV setup. They also helped, together with assistance from Eric Eyraud, to set up our low-temperature experiment. Valérie Guisset prepared the STM tips.

Ana Cristina Gómez Herrero taught me how to prepare the molecular samples, from the introduction of the copper crystal in the setup to the molecular evaporation.

Sylvie Godey performed the first measurements of the C 60 /Cu(111) samples at 77 K, in Lille. She also taught me how to image using STM and nc-AFM at 5 K, and continue to measure the samples when I left Lille. Maxime Berthe helped us to perform scanning tunneling spectroscopy measurements. Hugo Therssen kept an eye on the UHV/STM setup the evenings we needed to left the lab early. l'équipe informatique (grand merci à Patrick pour le dépannage et pour me parler de son tour du monde)... Quand on arrive dans un laboratoire comme l'Institut Néel on n'aurait pas de repères sans votre accompagnement.

It is time to acknowledge the non-permanent members of the spin ice consortium: Vojta, Ondrej, Lucas, Lucien and Marek. Thank you for the rich discussions, questions and exchanges. Being able to discuss with other "less-experienced" researchers was very important for me. Un grand merci à Lucas et Lucien, pour partager avec moi les questionnements existentiels, les repas de midi et les ras-le-bol.

Merci aux co-bureaux incroyables du K207 : Yann et Tristan, les glaces (surtout celles qui ne sont pas de spin!) ont un meilleur goût avec vous. Ainsi que les bières, les après-midi ensoleillées et la vie de thésarde. Merci beaucoup à tous les deux pour avoir partagé les doutes, pour nous avoir soutenus les uns les autres, pour avoir appris et grandi en parallèle. Je remercie aussi à mon favori du bureau D216, Clément, pour tes blagues (que je vais éviter de qualifier) et ta douceur. Loïc et Guillaume, quoi dire ? Merci d'être très doués en jeux de mots débiles, très forts en traquenards, des grands danseurs et des magnifiques hôtes. Mélanie, merci de ta sororité et ton amitié. C'était un beau cadeau.

Merci à Miren et Florie, physiciennes au regard pétillant, intelligent, incisif, magique quoi ! Vous avez été des piliers dans cette histoire, du début du master à la fin de la thèse.

Je remercie aussi à mes colocs, pour leur compagnie, pour leur partage du quotidien, pour m'avoir appris des chouettes choses qui n'étaient pas de la physique, pour m'avoir sorti de mes routines, m'avoir nourri à la fin du processus d'écriture. . . Merci à Jean, Kenza, Sofia, Albin, Clément et Hugo. Et merci à Ida et Bastien, qui sont partis à temps pour ne pas supporter une thèse mais qui m'ont quand même accompagné ces années.

Antoine, merci pour être un contrepoids de sensibilité dans ce monde de scientifiques un peu trop cartésiens.

Gracias a toda la familia lingüística por hacerme sentir menos lejos de casa:

The code

Benjamin Canals developed the Monte Carlo (MC) simulations that have been intensively used during this project (chapter IV and chapter V). Nicolas Rougemaille ran an important part of the MC simulations to look for the parameters of the Hamiltonian which allow to find good agreement with the experiments.

Yannick Dappe performed all the density functional (DFT) calculations presented in this manuscript (chapter III), which include tens of different calculations and took hundreds of thousands of computing hours. César González simulated the scanning tunneling microscopy (STM) images from the DFT relaxed structures presented in Figure III. [START_REF] Cepek | Chemisorption and fragmentation of on Pt(111) and Ni(110)[END_REF].

Alexandre Artaud shared with me his Python code to correct the hysteresis of piezoelectric actuators of the STM tip on the STM images measured at room temperature.

Johann Coraux wrote two Python codes, one used to identify molecular positions in STM images in a semi-automatic way, the other used to measure the relevant molecule-substrate distances from the DFT generated files.

Funding

The Nanosciences Foundation, hosted by the Université Grenoble Alpes Foundation, funded my Ph.D. grant.

The Groupement de Recherche NS-CPU (Nanosciences en champ proche sous ultravide) funded accommodation fees during the measurement campaign in Lille.

Acknowledgements, remerciements or agradecimientos

I hope the colleagues who participated in this project feel acknowledged enough for their scientific work. I mentioned them in the previous section Collaborators and I would like to remember that this work would have not been possible without their help. The richness of the project (which I hope is reflected in the next pages) is the best proof of their scientific quality and engagement. I thank all of them for that.

I would also like to acknowledge the members of my jury. I thank them for their kind and smart questions during the defense. It was a great pleasure to discuss with all of you: I really enjoyed the question session (and I am not sure if it is often the case!). I would also like to thank you for your remarks, comments and corrections. I think you highlighted interesting points which improved the final version of the manuscript. I particularly acknowledge Fréderic's and André's work. When I read their reports about my manuscript I was pleasantly surprised by how they reformulate my main conclusions in a different, clever way. Finally, I thank all of them for their encouragements for the next steps of my scientific career.

I would like now to acknowledge the non-scientific support (provided by scientific and non-scientific people), because I firmly think that finishing a PhD project (in covid period!) without a strong, stable net of people to take care of us is very difficult, or impossible.

Johann, Nico et Benjamin, quelle fine équipe! Je vous remercie d'abord en tant que groupe. Je pense que tous les trois vous avez toujours essayé de me guider au mieux pendant ces trois années et demi. Chacun d'entre vous a su apporter sa manière de voir la science, l'organisation du travail et la vie. Merci pour le temps et l'énergie que vous avez investi dans ce projet, et pour vous rendre toujours disponibles pour répondre à mes questions (plus ou moins scientifiques). Merci aussi pour prendre votre métier, pas seulement celui de chercheurs, mais surtout celui de directeurs de thèse comme une tâche importante dans laquelle s'engager. Et un grand merci pour votre honnêteté intellectuelle et votre curiosité, faire des sciences avec des passionnés de leurs sujets est un plaisir. Johann, j'admire ta capacité à rester diplomatique, calme et à faire la médiation entre nous sans émettre un jugement. Je ne pense pas être la première à te le dire, mais ces habilités rendent le travail en équipe beaucoup plus simple et agréable. Dans tout un autre domaine, je crois que je ne te l'ai jamais dit avant, mais j'ai vécu des grands moments d'excitation scientifique en discutant avec toi sur comment ces foutues molécules pouvaient bien s'organiser. Ce fut un plaisir.

Nico, je voudrais te remercier pour ta sincérité, pour cette façon dont tu abordes les choses de face et pour partager le constat que nos colères peuvent être des puissantes sources d'énergie. Cependant, je ne pense pas qu'on aurait survécu à tout ça sans le tact de Johann. Je te remercie aussi pour tes métaphores et tes images, ce fut chouette de discuter de trains iii avec toi. Et pour finir, un aveu-non-avoué-avant pour toi aussi : je pense que je n'aurais jamais fait le stage avec vous si je n'avais pas senti l'entrain, l'élan, avec lequel tu parlais du projet pendant notre premier entretien.

Benjamin, en étant scientifiquement plus loin, tu as peut-être été celui qui était personnellement le plus proche. Merci de tes patientes explications en science, j'ai beaucoup appris et je me suis rendu compte que c'est génial d'avoir "son théoricien de confiance". Mais surtout, merci de ton attention et ta tendresse envers moi, et merci de ce jour où tu m'as dis de faire comme je le sentais tant que ça me rendait heureuse.

Philippe et Valérie, merci à tous les deux pour le support technique. Je pense que j'aurais mesuré un tiers de mes données si vous n'étiez pas là en permanence pour nous accompagner. Philippe, merci de ta bienveillance et de ta curiosité. Valérie, je te remercie à l'infini pour toutes nos discussions. J'espère qu'on continuera à en avoir plein dans les années à venir.

Je voudrais aussi remercier les échanges avec nos collaborateurs, en dehors de la science. Merci Yannick pour ta patience, je pense que répondre à toutes mes demandes n'était pas une tâche facile, mais tu t'es toujours rendu disponible. Je voudrais aussi remercier Sylvie, pour me montrer ta manip, pour toute l'énergie et le temps que tu as investi dans les mesures et pour les discussions non-scientifiques. C'était fort sympa de travailler avec toi, et ce n'est pas fini ! Merci aussi pour tes encouragements et conseils. Merci aussi à Thierry, malgré le fait qu'on a moins eu l'occasion de discuter j'ai apprécié ta manière calme de t'intéresser au projet et de trouver du temps dans ton agenda pour avancer avec nous.

Elsa et Olivier, je vous suis reconnaissante du temps que vous avez passé à m'écouter, me conseiller et m'encourager, pour la science et pour mon projet d'avenir. Les discussion avec vous ont toujours été un plaisir.

Vincent, je te trouve un modèle d'enseignant à suivre et à admirer. Ton engagement auprès des étudiants, ainsi que ta motivation, ta capacité d'adaptation et ton esprit critique m'ont énormément inspiré et encouragé pour mes heures d'enseignement.

Florence et Marie-Aude, merci pour m'avoir suivi de loin dans cette aventure, pour votre intérêt et votre soutien, toujours très bienveillant. Laetitia et Nedjma, merci pour ce côté punchy avec lequel vous avez été là pour instaurer un rapport de force avec d'autres collègues (et avec lequel vous m'avez bien fait rigoler à des moments). Merci à toutes les quatre pour me montrer que vous croyez dans l'avenir (d'excellentes, brillantes et épanouies) physiciennes.

Laurent, je te remercie pour m'avoir introduit au labo, à l'équipe MNM et pratiquement au monde de la recherche. Merci aussi au reste de l'équipe MNM pour leurs questions en réunion et pour m'avoir accueilli.

Je voudrais aussi remercier toutes les personnes qui font fonctionner le laboratoire sans pourtant avoir participé au projet de recherche : l'équipe gestion financière (avec une mention spéciale pour David), l'équipe administrative (Sabine, Anne-Laure, Florence et Florence), These exotic behaviors do not exclusively emerge in magnetic systems. The first experimental example of frustration was actually water ice, in which oxygen atoms form a crystal, whereas hydrogen atoms do not have a unique way to bond to oxygen. In fact, there is great number of possible configurations to distribute the hydrogen atoms around the oxygens, leading to a highly-degenerate phase with a nonzero, extensive entropy at 0K.

In this thesis, we study a two-dimensional, non-magnetic structural system and demonstrate that it behaves as a frustrated magnet. The experimental system we focus on consists of assemblies of C 60 molecules deposited on an atomically flat copper surface. In certain conditions, the C 60 assembly is disordered and develops an unusual corrugation. Based on this observation, we build an analogy between the molecular structural degree of freedom (the molecular height in the corrugated arrangement) and an Ising variable. Thanks to a statistical analysis initially developed in frustrated magnetism, we are able to quantitatively demonstrate our C 60 assemblies can be described with frustrated Ising spin Hamiltonians. Our methodology is in fact well-suited to explore a variety of other on-surface systems, such as organic molecules on metal surfaces and two-dimensional binary alloys, under the point of view of magnetic frustration (despite the fact that magnetism is totally absent in these systems).

Besides, we unveil the atomic-scale details of the way C 60 molecules bond to the copper surface. In particular, we develop a conceptual framework to describe other ordered and disordered C 60 arrangements that are found to (surprisingly) coexist. We combine experimental high-resolution scanning probe microscopy and ab initio calculations to resolve the structure of these distinct molecular phases, finely characterizing several degrees of freedom (height, orientation and lateral position of the molecules). We introduce a competing-interaction model predicting the observed structures and their relative stability. From this model, we derive a rich energy landscape featuring several metastable phases and generalize its applicability to other C 60 /metal interfaces. vi Résumé L'apparition de formes de désordre sur des arrangements de la matière par ailleurs ordonnés est un sujet de recherche actif en physique de la matière condensée. Ce thème est notamment au coeur des préoccupations de communautés explorant des phénomènes de frustration magnétique, c'est à dire dans des situations où les constituants individuels du système ne peuvent minimiser simultanément toutes les interactions auxquelles ils sont soumis de la part des autres constituants. Dans ces situations, des propriétés inhabituelles sont susceptibles de se manifester :

• l'entropie ne tend pas nécessairement vers zéro à l'approche du zéro absolu (ce qui contredit le troisième principe de la thermodynamique tel qu'il est souvent enseigné), • de façon contre-intuitive, le désordre peut se caractériser par une longueur de corrélation infinie, comme dans un système ordonné, • parfois, l'état désordonné accueille des excitations qui se comportent comme des analogues classiques de monopoles magnétiques (c'est à dire, des quasi-particules dont l'existence est interdite par les équations de Maxwell).

In this thesis we investigate monolayer assemblies of fullerenes (C 60 molecules) on the (111) surface of Cu. When these molecules are deposited on metal substrates they selfassemble, forming two-dimensional lattices. The samples we investigate exhibit a puzzling coexistence of phases, some of them being ordered lattices, while others are disordered. Moreover, the observed disorder is not a trivial one (random, uncorrelated) but a correlated disorder. The questions we try to answer during this thesis are related to this phase coexistence and the existence of correlated disorder. We could resume the context of our work (in a very general sentence) as the study of ordering of microscopic matter. We asked ourselves why several phases, exhibiting different molecular arrangements, can coexist on the same system at a fixed temperature; which is the nature of the disorder we observe in our samples; what are the interactions at the origin of ordered and disordered phases. All these questions concerned a specific, particularly rich, experimental example, but they are many times addressed in other kinds of systems encountered in condensed matter physics and chemistry.

In this introduction, we start by evoking how physicists came to consider atoms and molecules as the microscopic units constituting gases, liquids and solids, and how the behavior of these microscopic particles influenced the macroscopic properties of matter. We remind the first examples of order and disorder phenomena studied in statistical physics: ideal gases and crystalline solids 1 . Next, we introduce, step by step, more complex systems exhibiting non conventional order. Later, we specifically describe how competing interactions, i.e. interactions promoting individually different kinds of ordering within a lattice, leads to the formation of exotic disordered (but correlated) phases. Two different types of models are considered: so-called continuous adsorption models for atoms on a surface and magnetically frustrated on-lattice models. Although they seem unrelated systems at first sight, we will discuss some striking similarities. We will also illustrate how both models are linked to the experimental system we studied, C 60 molecules on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface. Finally, an overview of the structure of this manuscript is presented.

1 Order and disorder in microscopic matter: solids, liquids and gases

Microscopic matter and the first studies of gases 1.Historical point of view

The idea of an indivisible unit (that cannot be cut in different parts) was already present in Ancient Greek. The word atom was, at the origin, α (a-), meaning "not", and τ εµνω (témnō), meaning "to cut". This idea remained a philosophical concept without scientific basis or empirical evidence during many centuries. Although, the existence of atoms is at the center of our current understanding of matter.

John Dalton was one of the first scientists to conceive that matter was actually formed In the first decade of the 1800's, he remarked that the well-defined ratios of different chemical elements participating in the formation of molecules could be understood by introducing the concept of atom [33]. Indeed, the proportion of different elements in molecules is always a ratio between natural numbers 3 . Considering small units that combine together in defined proportions is a simple way of explaining the origin of these ratios. Atoms were then defined as the basic unit of matter exhibiting the characteristic properties of different chemical elements, and the combination of atoms gives the formation of molecules. Dalton represented atoms as solid balls that can assemble together to form molecules (see 2 The atomic theory developed by Dalton is based on the law of definite proportions, formulated by Joseph Proust in 1797, and based in Proust's own work and in previous experiments performed by Joseph Priestley and Antoine Lavoisier. 3 An example of the reasoning developed by Dalton can be illustrated considering the formation of iron oxides. It was found that two different types of oxides exist: ferrous (black powder) and ferric oxide (red powder). The ferrous oxide forms for 100 g of iron and 28 g of oxygen, whereas the ferric oxide forms for 100 g of iron and 42 g of oxygen. For the same quantity of iron two different quantities of oxygen are involved in the formation of each oxide and the ratio between them is of 2:3 (28 g:42 g). Thus the chemical formula of ferrous and ferric oxide are Fe x O 2 and Fe x O 3 , respectively. In occurrence, x = 2 for these two compounds, but the previous reasoning does not allow to deduce this information. Similar examples exist concerning the compounds SnO and SnO 2 and N 2 O, NO and NO 2 . In his book A New System of Chemical Philosophy [33] Dalton cites a discussion published by Newton in his Philosophiae naturalis principia mathematica [START_REF] Newton | The Principia: mathematical principles of natural philosophy[END_REF] related to the behavior of fluids. Newton wrote:

" If the density of a fluid which is made up of mutually repulsive particles is proportional to the pressure, the forces between the particles are reciprocally proportional to the distance between their centers. And vice versa, mutually repulsive particles, the forces between which are reciprocally proportional to the distance between their centers, will make up an elastic fluid the density of which is proportional to the pressure.

[...] Whether elastic fluids do really consist of particles so repelling each other is a physical question. We have here demonstrated mathematically the property of fluids consisting of particles of this kind, that hence philosophers may take occasion to discuss that question.

"

From the first part of the quote it is worth to highlight that the description of matter using point-like particles, in particular to describe interactions in a gas and its relation with macroscopic pressure, emerged in physics in the 18 th century, many years before the first chemists set the basis of the atomic theory. From the second part we note that this description, which is indeed at the base of statistical physics, is considered as a purely mathematical description! For Newton, the existence of atoms remained an open question.

Probably following the ideas of Newton, Daniel Bernoulli (the author of Hydrodynamica) presented a scheme of a gas formed by point-like particles inside a cylinder [START_REF] Bernoulli | Hydrodynamica: sive de viribus et motibus fluidorum commentarii (Sumptibus Johannis Reinholdi Dulseckeri[END_REF] (see Figure I.1d) and proposed a relation between the motion of the particles and the pressure exerted by the gas. He wrote: " [...] let the cavity contain very minute corpuscles, which are driven hither and thither with a very rapid motion; so that these corpuscles, when they strike against the piston and sustain it by their repeated impacts, form an elastic fluid which will expand of itself if the weight is removed or diminished. . .

"

Indeed, he was the first known scientist to propose motion of particles as the key ingredient to understand gases and he was setting the basis of the kinetic theory of gases. In the 1820's, a time when the atomic theory of Dalton was already published, John Herapath [START_REF] Herapath | A mathematical inquiry into the causes, laws and principal phenomenae of heat, gases, gravitation, etc[END_REF] derived the mathematical relation between the velocity of the particles in a gas and the macroscopic pressure. In 1851 John James Waterston [START_REF] Waterston | On the Physics of media that are composed of free and perfectly elastic molecules in a state of motion[END_REF] proposed the link between mass, average velocity of microscopic particles and the macroscopic properties of a gas: temperature, density and pressure. Nevertheless, his work remained unpublished until 1982, when he was finally recognized as a pioneer of the kinetic gas theory [START_REF] Waterston | On the physics of media that are composed of free and perfectly elastic molecules in a state of motion[END_REF].

Finally, it was more than a century after the first proposal by Bernouilli that kinetic theory of gases started to be accepted. In 1851, Joule [START_REF] Joule | XXV. Some remarks on heat, and the constitution of elastic fluids: To the editors of[END_REF] used the arguments by Herapaths to compute the velocity of hydrogen atoms. In 1856 Krönig [START_REF] Krönig | Grundzüge einer Theorie der Gase[END_REF] consider the translation of particles to build a mathematical kinetic gas model. In 1857 Clausius [START_REF] Clausius | On the nature of the motion which we call heat[END_REF] developed the kinetic theory of gases mathematically, including translation, rotation and vibration of the particles. Finally in the 1860's Maxwell [START_REF] Brush | The Kinetic Theory Of Gases: An Anthology Of Classic Papers With Historical Commentary[END_REF] introduced the mathematical concepts, including a probability theory, necessary to establish the basis of statistical physics. Boltzmann concluded to set the basis of classical statistical mechanics with his description of the probability distribution of the velocity of molecules in a gas, the Maxwell-Boltzmann distribution [START_REF] Boltzmann | Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen[END_REF].

Summary: atoms and positional degrees of freedom

Our current understanding of matter, based on the existence of microscopic units such as molecules and atoms that compose all matter around us is relatively recent. In fact, before the 18 th century, whether these elementary units existed was a philosophical question. During the 18 th and 19 th centuries, different theories, applied both to chemistry and physics, assumed the existence of atoms and molecules. Nevertheless, it seems that, for physicists, considering elementary particles as the building blocks of matter was primarily a powerful approach to building mathematical models, and not an actual scientific evidence. Surprisingly, none of the pioneers on kinetic gas theory, which is based on the existence of molecules, addressed the nature of these molecules! The difference between this point of view and the kind of study developed in this thesis is very important. Nowadays, we are not only convinced of the existence of atoms, molecules, and even smaller particles, but we can experimentally "see" 4 them. Note the conceptual jump from this statistical models, in which molecules are almost a hypothesis, to works of the kind presented in this manuscript, in which ordering of molecules is characterized by high-resolution imaging in real space to deduce the statistical properties of the molecular configurations.

The other important message of the previous paragraphs concerns the nature of statistical physics and its purposes. Statistical physics studies microscopic degrees of freedom of atoms and molecules, to infer the macroscopic properties of matter. The case of kinetic theory of gases illustrates which type of degree of freedom was first considered in statistical physics to describe gases: microscopic positions and particle velocity, i.e. positional degrees of freedom. Statistical physics does not aim to know the exact position (⃗ r i ) and velocity (⃗ v i ) of each particle in a macroscopic system. What could we deduce from such a collection of [⃗ r i , ⃗ v i ]?

1. Order and disorder in microscopic matter: solids, liquids and gases And, how could we compute such quantities in a realistic system with a number of particles of the order of magnitude of the Avogadro's number (N A ∼ 10 23 )? Statistical physics deals with statistical quantities, such as averaged velocity and probability distributions in the case of the gas theory.

Keeping in mind these ideas, we will now introduce the microscopic characteristics, from a statistical point of view, of different states of matter. In particular, we will describe gases and solid phases, to illustrate the most common examples of order and disorder in the study of matter.

Disordered gases and ordered solids

Let us start our review of different states of matter with a citation by Clausius [START_REF] Clausius | On the nature of the motion which we call heat[END_REF]:

" A motion of the molecules takes place in all three states of aggregation.

"

so again the considered degree of freedom is positional. He continued:

" In the solid state, the motion is such that the molecules move about certain positions of equilibrium without ever forsaking the same, unless acted upon by foreign forces. In solid bodies, therefore, the motion may be characterized as a vibrating one, which may, however, be of a very complicated kind. [...] In the liquid state the molecules have no longer any definite position of equilibrium. They can turn completely around their centres of gravity ; and the latter, too, may be moved completely out of its place. The separating action of the motion is not, however, sufficiently strong, in comparison to the mutual attraction between the molecules, to be able to separate the latter entirely. [...] In liquids, therefore, an oscillatory, a rotatory, and a translatory motion of the molecules takes place, but in such a manner that these molecules are not thereby separated from each other, but, even in the absence of external forces, remain within a certain volume. Lastly, in the gaseous state the motion of the molecules entirely transports them beyond the spheres of their mutual attraction, causing them to recede in right lines according to the ordinary laws of motion. If two such molecules come into collision during their motion, they will in general fly asunder again with the same vehemence with which they moved towards each other; and this will the more readily occur, since a molecule will be attracted with much less force by another single molecule than by all the molecules which in the liquid or solid state surround it.

"

We first focus on the cases of gases and solids. In subsubsection 1.1.1 we have mentioned the case of perfect or ideal gases, which are "ideal" because no interaction between particles is considered and particle collisions are modeled as instantaneous collisions between rigid spheres. This picture corresponds to molecules or atoms moving around with an uniform rectilinear motion, with important velocities and without interacting with other particles (except during the infinitely small time of a collision).

The description of solids is also based on some "ideal" hypothesis. Solids are considered as being crystalline, i.e. the equilibrium positions of molecules are arranged in a periodic lattice that respects translation symmetry. Or to say it differently, a crystal is an infinite repetition of identical groups of atoms. Ideal crystals are at the basis of solid state physics, and allows to study many physical problems (conduction of electrons, optical properties, etc.) taking into account the symmetry of the structure. Such lattice models will be important in this manuscript, which uses the notion of regular two-dimensional lattices to address ordering of molecules.

The reason why we choose to discuss solid and gases is that they are the paradigm of perfect order and random disorder in matter, respectively. They are the two limit cases of the same problem. In a perfect gas the positions of molecules are constantly changing, whereas in a solid molecular positions are fixed and only small displacements around equilibrium positions are allowed. Low density of particles in the ideal gas framework makes that particles can be considered as being completely independent, i.e. they do not interact with each other. In solids, high density of particles leads to substantial interaction between particles. While in a gas kinetic energy (associated to thermal energy) dominates, in solids potential energy (due to interaction between molecules) rules the physics. This qualitative comparison is summed-up in Figure I.2.

In crystalline solids knowing the position of one particle implies that the position of any atom in the lattice can be known, even at very long distances. In terms of theory information, the entropy as defined by Shannon, i.e. the degree of uncertainty, is zero. This postulate is equivalent to say that the entropy of the system in the Boltzmann's definition, i.e. the number of possible microstates, is very small. We can understand solids as being a deterministic system with long range order. Gases are exactly the opposite: a priori, it is impossible to know the position of an individual particle and only average quantities (such as average velocity) can be used to characterize the system. This means that the uncertainty on the position of the particles is maximal, thus Shannon's entropy. Moreover, the number of possible configurations for a perfect gas increases with the number of particles and is then very high, and so is the entropy. We can say that perfect gases are randomly disordered.

The ideal gas and the ideal crystal can be used as starting points to build more complex models for real gases and solids. On the contrary, they cannot be used to study the case of liquids. Indeed, liquids are almost never mentioned in undergraduate courses, and they are not addressed in most of the basic books of statistical physics, as Landau et Lifchitz wrote in their Statistical Physics book [107]:

1. Order and disorder in microscopic matter: solids, liquids and gases " In gases interaction between molecules is very weak. As this interaction increases, the properties of the gas differ more and more from those of ideal gases, and finally the gas condenses into a liquid.

In the latter, the interaction between molecules is strong, and the properties of this interaction (and therefore those of the liquid) depend considerably on the particular liquid concerned. For this reason it is, as already mentioned, impossible to derive any general formulae giving a quantitative description of the properties of a liquid

"

We now enumerate some of the difficulties to understand this "intermediate" state of matter.

Liquid: ordered or disordered phase of matter, as analyzed through spatial correlations

Is a liquid an ordered or disordered state of matter? Can we imagine a liquid as a gas with strong interactions between particles? Or is it more likely a solid with a certain degree of disorder in molecular positions? If we use the qualitative criteria used in the previous section, and schematized in Figure I.2, to characterize gases and solids it is not trivial to drive a clear conclusion about the nature of liquids.

Figure I.2:

Qualitative, schematic representation of some of the properties of perfect gases and crystalline solids. Comparison with the case of a liquid state, in which the properties are intermediate and can be more similar to the case of a solid or to the case of a gas.

For a long time liquids have been considered as being disordered states (as in the previous citation by Clausius), so the mental image associated with a liquid would correspond to a gas with strong interactions between particles. In the early 1930's the first X-ray studies of liquids demonstrated that it is not the case [START_REF] Zernicke | Die Beugung von Röntgenstrahlen in Flüssigkeiten als Effekt der Molekülanordnung[END_REF][START_REF] Debye | Study of molecular order in liquids by means of X-rays[END_REF]. Liquids have an internal structure, i.e. they are partially ordered. Liquids are definitely not long-range-ordered like solids, but the position of molecules is not completely random as in gases. The idea of short range order is essential to understand liquids. It will also be a key concept in this manuscript.

On the other hand, the density of liquids is very similar to those of solid compounds, which suggests that the typical distances between particles should be the same in both states of matter. The same applies to the strength of intermolecular interactions. In this picture liquids are seen as quasi-crystalline structures with some freedom in the displacement of the particles [START_REF] Frenkel | Kinetic Theory of Liquids[END_REF].

Finally, in the 1970's describing liquids via molecular distribution functions or correlation functions emerged as a standard basis to understand their structure and deduce their macroscopic properties [START_REF] Fisher | Statistical theory of liquids[END_REF]. These correlation functions correspond to the probability of finding some particular configurations for molecules separated at a certain distance, considering groups with two, three, etc. particles. Importantly, correlation functions can be experimentally measured from scattering experiments (such as neutron scattering) and the structure of liquids can be revealed [START_REF] Egelstaff | An introduction to the liquid state[END_REF]. Spatial correlations in liquids decay fast with the distance r in most of the cases. In usual liquids the correlation decay with the distance is exponential (∝ e -r ). In this thesis we will treat some particular (non-structural) liquids in which correlations can have longer range and an algebraic decay (∝ r -α with α a real number). In this case the correlation length diverges, thus we speak about critical liquids. The equivalent of this behavior in a structural liquid is the phenomenon of critical opalescence [START_REF] Einstein | Theorie der Opaleszenz von homogenen Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des kritischen Zustandes[END_REF].

Note that correlation functions are not exclusively used in the study of structural liquids. These quantities are used, by both theorists and experimentalists, to reveal complex order in completely different condensed matter systems, such as frustrated magnets [START_REF] Ramirez | Chapter 4 Geometrical frustration[END_REF][START_REF] Lacroix | Introduction to Frustrated Magnetism[END_REF][START_REF] Diep | Frustrated spin systems[END_REF]. As we will see later, these frustrated magnets are closely related to the studies presented in this manuscript. The powerful concept of structural correlations, different from order or disorder, will be an important analysis tool.

A note on dimensionality

In this section we do not intend to make an exhaustive review on the role of dimensionality in order/disorder phenomena and related properties of matter. Indeed, the dimensions of the physical problems can substantially modify the characteristic of phase transitions, the behavior of thermodynamic properties and the nature of order in a physical system [START_REF] Domb | Phase Transitions and Critical Phenomena[END_REF]. The main message to keep in mind here is that dimensionality matters.

We illustrate the importance of dimensionality with the Hohenberg-Mermin-Wagner theorem. In our daily three-dimensional life we are used to phases of matter exhibiting long range order. It is the case of solids, in which atoms or molecules are perfectly ordered forming crystalline structures. But, what would happen if we lived in a two-dimensional or one-dimensional world? Would we find solid materials around us? What the Hohenberg-Mermin-Wagner theorem [START_REF] Mermin | Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models[END_REF][START_REF] Hohenberg | Existence of long-range order in one and two dimensions[END_REF][START_REF] Mermin | Absence of ordering in certain classical systems[END_REF] tells to us is that a two-dimensional world would be completely different. Indeed, the theorem proves that, if the interactions in the twodimensional system are short range and the degree of freedom under consideration is continuous, no molecular long range order exist at finite temperature. Under the previous conditions, crystalline solids cannot exist in two-dimensions as long as T ̸ = 0. A ordered lattice of spins with isotropic three-dimensional orientation in short-range interaction cannot either. The same is true for the one-dimensional case.

In this manuscript, the system we address is quasi-two-dimensional: a two-dimensional sub-monolayer of molecules adsorbed on a three-dimensional substrate. It would become purely two-dimensional if the interaction between the substrate and the molecules was vanishing, which is in practice not the case for C 60 /Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. The models we used to describe the system are based on the construction of two-dimensional lattices, as we will detail later. In the first part of our analysis (chapter III) the degree of freedom we consider is positional (thus continuous). In the second part (chapter IV), we address Ising variables (thus discrete).

Complex states in presence of competing interactions

In subsection 1.3 we have addressed the structural liquid state by introducing the concept of short range order and correlation. Liquid phases are intermediate states between gases and solids, in which neither long range order or uncorrelated disorder appear. Indeed, this situation is frequent in physics. Physical systems are not always in an ideal ordered or disordered limit. In many cases, intermediate states are due to competing interactions which tend to favor different, incompatible, orders in a system.

Here we introduce two different examples of competing interactions, corresponding to two different physical systems. Both of them are two-dimensional systems. The first system consists on atoms (or molecules) adsorbed on a crystalline, two-dimensional surface (subsection 2.1). The degree of freedom is, like in the previous examples of gases, liquids and solids, positional and continuous. In this case the interactions between adsorbed atoms (or molecules) and between the atoms and the surface can compete. The second case we present is different with respect to the previous one in three aspects, we consider: 1 -two-dimensional lattice problems in which particles are at fixed positions; 2 -the degree of freedom is internal (such as spins); and 3 -discrete. Competition of interactions in this second case can have two different origins: non-compatible interactions between particles or geometry, as we explain in subsection 2.2. Consequences of competing interaction on the formation of ordered and disordered phases are discussed for both systems. Finally, we compare the two cases mentioned above.

On-surface monolayers with positional degrees of freedom: atom-atom vs. atom-substrate interactions

The first case in which we consider competing interactions is the adsorption of an atomic monolayer on a solid surface. The situation is the following: let us consider the crystalline surface of a solid with lattice parameter a subs and an adsorbed monolayer that would have, at equilibrium, a lattice parameter a 0 ads 5 . We also suppose that the substrate structure keeps its order and lattice parameter, because interaction between atoms in the solid are the strongest. What kind of order develops in the monolayer adsorbed on this surface? Would atoms of the monolayer adopt their lattice parameter a 0 ads , or would the layer deform to match the periodicity of the crystalline substrate?

Frenkel Kontorova model

This type of problem have been modeled by Frank der Merwe [START_REF] Frank | One-dimensional dislocations . II . Misfitting monolayers and oriented overgrowth[END_REF][START_REF] Frank | One-dimensional dislocations. I. Static theory[END_REF] and by Frenkel and Kontorova in one-dimension [START_REF] Kontorova | On the theory of plastic deformation and twinning[END_REF]. The model considers a chain of atoms corresponding to the adsorbed monolayer as balls connected by springs with an equilibrium distance equal to the lattice parameter a 0 ads . The substrate is modeled as a periodic potential, usually sinusoidal, that interacts with atoms in the monolayer (Figure I.3). How will the positions of each adsorbed atom arrange? Note that, as in the previous discussions for solids, liquids and gases, we consider here a positional degree of freedom.

This simplified one-dimensional model presents two main advantages. First, its Hamiltonian can be written as a sum of oscillator terms accounting for the positions of adsorbed atoms submitted to a periodic potential: H = K oscillator +U potential . Second, and more important for our discussion, it features the competition of two interactions: the elastic interaction within the monolayer and the interaction potential from the substrate.

Commensurate, incommensurate and chaotic phases

In this model the competing interactions lead to the formation of three different types of phases, schematized in Figure I.3.

If interactions of adsorbed atoms with each other and with the substrate are comparable the ground state is a commensurate phase. Such phase is ordered with respect to both the adsorbed and substrate lattices, i.e. a certain periodicity exists for the adsorbed atoms and it coincides with a periodicity of the substrate, so a ads a subs = r q is a rational number. In some occasions these phases are called modulated phases, because the distance between adsorbed atoms is modulated by the substrate lattice parameter. It is possible to define a common lattice parameter or a superperiodicity, a sup = qa ads = ra subs . Commensurate phases can be of first, second,..., n th order, e.g. a phase with a sup = a ads = 2a subs is a first order one and a phase with a sup = 9a ads = 19a subs a 9 th order one,(the order index does not straightforwardly relate with a ads , which varies by only 5% in the previous example).

When the interaction between atoms completely dominates over the interaction with the substrate ideal floating phases with the substrate form. In this case the adsorbed atoms behave like a crystalline solid , with their lattice parameter the natural one for the lattice (a ads = a 0 ads ) 6 . For these phases no common period between the adsorbed lattice and the Commensurate phases with small order ("C", a abs = a subs ) and high order ("HC", 5a abs = 4a subs ). Incommensurate phases, a floating phase with the lattice parameter of the adsorbed atoms corresponding to the natural lattice parameter ("F", a abs = a 0 abs ) and the formation of solitons between commensurate regions (S). Chaotic phase in which adsorbed atoms are randomly distributed but always pinned on minima of the substrate potential. Figure adapted from [4] (for commensurate and incommensurate phases) and [START_REF] Bak | Commensurate phases, incommensurate phases and the devil's staircase[END_REF] (for chaotic phases). substrate can be found, i.e. the ratio a ads a subs is an irrational number. These phases are called floating phases, because the adsorbed layer can be seen as an independent layer floating above the substrate.

Finally, if the interaction between adsorbed atoms is small enough, the adsorbed atoms are pinned at the positions of the minimum of the potential representing the substrate 7 . For this phase the set of positions x i is disordered and the phase is called an Aubry [5,6] phase or a chaotic state 8 . No order, thus no periodicity, is observed for the adsorbed atoms. This simple one-dimensional model shows how complex order can exist in an adsorbed layer. Depending on the competition between atom-atom and atom-substrate interactions several kinds of phases can form. In fact, the number of high ordered commensurate phases that can be obtained depending on the adsorbed layer's atomic density is infinite, as well as the number of chaotic phases. The ordered phases can host characteristic defects, called solitons (also called interfacial misfit dislocations) corresponding to jumps (kinks) in the order cannot exist in truly 2D systems. Nevertheless, here we consider a perfect floating phase with a ads = a 0 ads lattice parameter as a theoretical object, useful to capture the idea behind the Frenkel-Kontorova model. 7 These phases are out-of-equilibrium phases, in which the structure is not the ground state of the onsurface problem but the positions of the adsorbed atoms or molecules are trapped in local minima of the energy. 8 The chaotic character is not directly obvious by simple observation of the atom's positions, it is put forward in specific representations of the system's structural phase space parameter describing the layer-on-substrate atomic stacking. These solitons are topological defects, and the number of kinks a topological invariant (which is defined from the Burger's vector) characterizing the phase. During a phase transition from a commensurate phase to another, this topological invariant changes. Moreover, we can ask ourselves if these phases are ordered or disordered, or said differently whether they are more solid-like or gas-like. Chaotic phases are definitely disordered, while commensurate phases are ordered, but an infinity of different orders exists! Adsorbed atoms in ideal floating phases without formation of solitons are ordered in the layer, but there is no order with respect to the substrate. When solitons form, they may appear with a certain degree of order, but the adsorbed atoms are not simply ordered in a crystalline lattice.

The model can be extended to two dimensions [START_REF] Braun | Nonlinear dynamics of the Frenkel-Kontorova model[END_REF][START_REF] Tekić | Dynamic phases in the two-dimensional underdamped driven Frenkel-Kontorova model[END_REF], and the corresponding phases are even more numerous and complex. In the two-dimensional case, the adsorbed layer can not only be compressed or extended, but it can rotates to better match the substrate. This new orientational degree of freedom gives rise to new incommensurate phases in which a new type of defects (called static distortion waves (SDW)) exist [START_REF] Novaco | Orientational Epitaxy -the Orientational Ordering of Incommensurate Structures[END_REF]. Other studies [START_REF] Kivshar | Zigzag kinks in the Frenkel-Kontorova model with a transversal degree of freedom[END_REF][START_REF] Braun | Ground state of the Frenkel-Kontorova model with a transverse degree of freedom[END_REF][START_REF] Braun | Frenkel-Kontorova model with a nonconvex transverse degree of freedom: A model for reconstructive surface growth[END_REF][START_REF] Braun | Nonlinear dynamics of the Frenkel-Kontorova model[END_REF] have addressed the case of atoms on one-dimensional potentials and considered and additional transverse degree of freedom for the adsorbed atoms, which mimics the twodimensional character of the problem. This approach leads to other families of commensurate and incommensurate phases, where the transverse degree of freedom of the atoms is also modulated by the substrate potential. ) in which each Xe atom is visible as a small dot. The commensurate regions here appear bright, while dark regions forming hexagons are incommensurate domain walls (the contrast is the opposite to the one in (a)). (c) Schematics representing the positions of Xe atoms (darker circles) and carbon atoms in graphite (smaller, lighter circles). Three commensurate structures (A, B, C) exist and they are separated by incommensurate domain walls. All the images are adapted from [START_REF] Grimm | Physisorbed Rare-Gas Monolayers: Evidence for Domain-Wall Tilting[END_REF].

In Figure I.4 we show an example of another type of incommensurate structure in twodimensions. It forms when the interaction with the substrate is significant. This incom-mensurate structure is not liquid-like (floating phases) but ordered. In these structures the lattice of adsorbed atoms is formed by several, mixed commensurate and incommensurate regions, which do not form a lattice extending over long distances. Commensurate regions are separated by domain walls. This type of structure usually forms when the elastic energy in a commensurate region is too large, thus the strain is relaxed in the form of solitons. This kind of structure has been experimentally observed on two-dimensional systems of gases adsorbed on graphite, such as argon [START_REF] Shaw | Observation of Orientational Ordering of Incommensurate Argon Monolayers on Graphite[END_REF] and xenon [START_REF] Grimm | Physisorbed Rare-Gas Monolayers: Evidence for Domain-Wall Tilting[END_REF] (the one show in Figure I.4). In this context incommensurate indicates, by opposition with commensurate phases, that the ratio a ads a subs is an irrational number. These incommensurate phases in which domain walls form are actually a more realistic picture of what happens in two-dimensional phases, in which floating regions with a constant lattice parameters exist only at short range and can be separated by defects.

On-lattice models with internal degrees of freedom: interaction and geometrical frustration

We now present a different category of models. In these on-lattice models, the position of the particles is fixed and the degree of freedom of the particles is an internal one. Some examples of internal degrees of freedom are spin for magnetic materials and orientation for molecules.

In these models we address the ordering of the internal degrees of freedom. We illustrate how competition of interactions between particles, or geometry of the lattice, can produce the formation of complex, unconventional states of matter. In particular, we address the case of magnetic and structural frustration.

Spins, adsorption problems and binary alloys

We now choose a model in which frustration arises, it is not the only one but it is the one we address in chapter IV and chapter V. We consider on-lattice two-dimensional models with internal Ising degrees of freedom. Ising degrees of freedom are by definition bivalued variables, but these variables can be of different nature. We can consider up or down spins in magnetic systems; the presence or absence of a given type of particles adsorbed on a lattice; or the chemical nature of atoms in a binary alloy (Figure I.5). To build a model, we need to take into account interactions between particles. Depending on the physical situation, interactions have different origins. For example, for magnetic compounds we can consider ferromagnetic or antiferromagnetic exchange interactions, dipolar couplings between magnetic moments, etc. For adsorbed gases and binary alloys we can take into account repulsive interactions at short distances; Coulomb or electric dipolar interactions if particles are charged; attractive van der Waals interactions... For the moment, we will simplify the problem and consider only interactions between nearest (J 1 ) and next nearest neighbors (J 2 ), independently of their nature. We will call interactions ferromagnetic (FM) when they favor neighbors to have equal Ising states, and antiferromagnetic (AF) when they favor different states. Two lattices in two dimensions can be build combining triangles: the triangular lattice (triangular plaquettes sharing their edges) and the Kagomé lattice (triangular plaquettes sharing their corners). They are called triangular Ising antiferromagnet (TIAF) and Kagomé Ising antiferromagnet (KIAF), respectively.

Magnetic (or structural) frustration

The word frustration was first proposed by Toulouse [START_REF] Toulouse | Theory of the frustration effect in spin glasses: I[END_REF][START_REF] Liebmann | Statistical mechanics of periodic frustated Ising systems[END_REF], who, according to his paper The frustration model in [START_REF] Pekalski | Modern Trends in the Theory of Condensed Matter[END_REF], was inspired by P. W. Anderson. Toulouse wrote: " One last anecdote; in the summer of 1976, I attended a lecture by P. W. Anderson on spin glasses in the patio of the Aspen Center of Physics ; I was not actively working in this field, things appeared pretty esoteric to me, but one sentence which remained as graffiti in one corner of the blackboard, struck my imagination : « the name of the game is frustration ». When I looked for a word which would evoke both the effect of contradiction and an analogy with percolation, it came back to my mind.

"

More explicitly, frustration, and in particular magnetic frustration, refers to a situation in which spins (or simply Ising variables for us) cannot simultaneously satisfy all the interactions in a lattice. In Figure I.6 we show four models in which magnetic frustration have been studied.

Two of them, the Villain's [START_REF] Villain | Spin glass with no random interactions[END_REF] and anisotropic Ising model with competing nearest-and next-nearest-neighbor interactions (ANNNI) models [START_REF] Selke | Two-dimensional Ising models with competing interactiona Monte Carlo study[END_REF][START_REF] Villain | Two-Dimensional Ising Model With Competing Interactions: Floating Phase, Walls and Dislocations[END_REF], correspond to frustration due to competing interactions, while the two others correspond to geometrically frustrated models. In Villain's model a square lattice with ferromagnetic or antiferromagnetic interactions that depends on the bond is considered. In the ANNNI model all the nearest neighbors bonds are ferromagnetic, whereas in one direction of the lattice interaction between next nearest neighbors are antiferromagnetic. In these two cases the competition between ferromagnetic and antiferromagnetic interactions makes impossible to find a configuration that satisfies all the constraints. In the case of lattices based on triangular plaquettes with only antiferromagnetic nearest neighbor interactions, frustration is due to the geometry of the lattice. As soon as we consider antiferromagnetic couplings in a unit cell with an odd number of corners (pentagon, heptagon, etc.) the system is frustrated.

Frustration due to competing interactions

In Villain's model, the ground state depends on the ratio of the ferromagnetic and antiferromagentic interactions. When the ferromagnetic interaction is much larger than the antiferromagnetic one, the resulting phase is a ferromagnetic phase. When the two interactions have the same strength they compete. Villain [START_REF] Villain | Spin glass with no random interactions[END_REF] demonstrated that in this case, no transition to an ordered state occurs. The ground state is a highly degenerate phase and its entropy is proportional to the number of sites in the lattice. The entropy per site is non zero, which violates Planck's formulation of the third law of thermodynamics. Frustration leads to the impossibility of finding an order for the internal degrees of freedom in the lattice, even at zero temperature.

The ANNNI model in two-dimensions has been studied by Villain and Bak [START_REF] Villain | Two-Dimensional Ising Model With Competing Interactions: Floating Phase, Walls and Dislocations[END_REF] who analytically showed that, for a regime in which nearest neighbor ferromagnetic (J 1 ) and next nearest neighbor antiferromagnetic (J 2 ) couplings compete, the ground state is a periodic phase with antiferromagnetic order in which double stripes of the two Ising states alternate ('• • • • ••' structures). The period of this structure is commensurate with that of the lattice. At intermediate temperatures (thermal energy competing wit the other interactions), Villain and Bak found a floating phase, in which the periodicity of the spin lattice goes from 2a at low temperature (commensurate phase with a the lattice parameter of the square lattice, so 2a corresponds to '••••••' phase) to infinity at high temperature (incommensurate phase in which no periodicity of the spin arrangement can be defined with respect to the underlying lattice). In this floating phase the spatial correlations decay algebraically (∝ r -τ , with τ depending on the periodicity of the spin pattern). This type of decay for spatial correlations is the one of some critical liquid phases, as we will mention in the next paragraphs. At high temperatures they predicted what they called a paramagnetic phase, which usually means that spins are randomly disordered. Nevertheless, they found oscillating correlations for this phase, which decay with an exponential behavior, so the term "paramagnetic" does not have the usual meaning we consider in this manuscript. They suggested that this phase forms when many dislocations appear in the floating phase and restore the broken symmetry. Surprisingly, they found that at some point of the phase diagram in which the competition between ferromagnetic and antiferromagnetic interactions is very important

(|J 1 | = 2|J 2 |),
this phase exists at zero temperature. Thus, for this relation of the interactions no phase transition to an ordered state is expected.

Frustration due to geometry

For the triangular Ising antiferromagnet (TIAF) [START_REF] Wannier | Antiferromagnetism. The Triangular Ising Net[END_REF][START_REF] Houtappel | Order-disorder in hexagonal lattices[END_REF] and the Kagomé Ising antiferromagnet (KIAF) [START_REF] Syôzi | Statistics of kagomé lattice[END_REF][START_REF] Kanô | Antiferromagnetism. the kagomé ising net[END_REF] it has been proved in the early 1950's that the ground states exhibit unusual properties. The ground states are highly degenerate, even at zero temperature. This implies that no transition to an ordered phase takes place in such systems. Moreover, the number of possible configurations at zero temperature increases as a power of the number of sites in the lattice (# conf igurations ∝ C N ), which translates into an extensive entropy. This very peculiar state of matter is disordered at any temperature. At high temperature the disorder is uncorrelated (a paramagnetic phase in magnetic terms), so comparable to a gas. At low temperature the phase is disordered and highly fluctuating but the disorder is correlated. In these systems a local order is respected: in each triangular plaquette two antiferromagnetics bonds are always satisfied and the third one is frustrated, i.e. there are two Ising state of the same type and a third one of the opposite type. The fact that this rule is respected can be detected in the correlations. Here, correlations measure the probability of finding an equal or different Ising state at a certain distance. For the TIAF, spatial correlations decay following an algebraic law (∝ r -2 ) [START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF][START_REF] Stephenson | Ising-Model Spin Correlations on the Triangular Lattice. III. Isotropic Antiferromagnetic Lattice[END_REF] at zero temperature. For the KIAF, the decay is exponential (∝ e -r ) [START_REF] Barry | Exact solutions for correlations in the Kagomé Ising Antiferromagnet[END_REF]. This type of fast decay for the spatial correlations is characteristic of liquid phases, hence the name of spin liquids used to describe the TIAF and KIAF ground states.

A historical, structural, frustrated system: water ice

The models we have presented until now were developed to study magnetic systems, so the internal Ising degrees of freedom were spins. Here we want to point-out that the first frustrated system to be discovered was not magnetic but structural, in this sense much alike the experimental molecular samples we analyze in this manuscript. Furthermore, this structural system and the exotic properties of the ground state were discovered many years before the birth of the word frustration. This system is water ice. In the early 1930's, measurements of the entropy of water ice showed a discrepancy with calculated values [START_REF] Mecke | Das Rotationsschwingungsspektrum des Wasserdampfes. I[END_REF][START_REF] Freudenberg | Das Rotationsschwingungsspektrum des Wasserdampfes. III[END_REF][START_REF] Giauque | Molecular rotation in ice at 10k. free energy of formation and entropy of water[END_REF][START_REF] Gordon | The calculation of thermodynamic quantities from spectroscopic data for polyatomic molecules; the free energy, entropy and heat capacity of steam[END_REF][START_REF] Giauque | The Entropy of Water and the Third Law of Thermodynamics. The Heat Capacity of Ice from 15 to 273 • K[END_REF]. In 1935 Pauling [START_REF] Pauling | The Structure and Entropy of Ice and of Other Crystals with Some Randomness of[END_REF] explained the origin of this discrepancy by considering the possible internal configurations for the crystal structures of water ice.

Water ice is a hexagonal three-dimensional crystal formed by H 2 O molecules. Each molecule is surrounded by four neighbor molecules in a tetrahedral arrangement, and hydrogen atoms of one molecule are oriented towards two of the neighbor molecules. The position of molecules is fixed, so we have an on-lattice problem. The internal degree of freedom is the distance between hydrogen and oxygen atoms, i.e. the bond length. Each oxygen atom in the lattice is surrounded by four hydrogen atoms, two "belonging" to the H 2 O molecule ("short" bonds) and two belonging to neighbor H 2 O molecules ("long" bonds). The structural degree of freedom is bivalued, so an Ising variable can be used to model the ordering of H atoms. Pauling considered each tetrahedron independently and found that 16 configurations exist, depending on how the "long" and "short" bonds are distributed around oxygen atoms. This hypothesis in which two protons are near the oxygen and two far away is known as the ice rule [START_REF] Bernal | A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions[END_REF][START_REF] Fowler | Note on the pseudo-crystalline structure of water[END_REF], and it has been adapted to the study of magnetic compounds and sometimes formulated as "two in, two out" to make reference to the directions of spins. Imposing this local constraint and computing the number of possible different configurations, Pauling concluded that the ground state of water ice is highly degenerate. He also computed the value of entropy at low temperature, demonstrating that it is extensive and successfully explaining the previous experimental measurements.

Since the publication of Pauling's paper, many other systems exhibiting extensive entropy at low temperature have been studied, in particular in magnetic systems. By analogy with water ice, they are usually called spin ices [START_REF] Harris | Geometrical Frustration in the Ferromagnetic Pyrochlore Ho 2 Ti 2 O 7[END_REF].

On-surface monolayers and frustrated magnetic lattices: a common framework

We have discussed the existence of exotic, unusual phases of matter due to competition of interactions in two different contexts.

The first one is the problem of the organisation of atomic or molecular monolayers on crystalline substrates, the so-called on-surface problems. The models are continuous and the degrees of freedom of the particles are positional. In this case, competition arises because the substrate and the adsorbed monolayer have different lattice parameters.

The second class deals with on-lattice problems in which particles have an internal (Ising) degree of freedom. The exotic frustrated phases of matter can be generated by either the effect of competing interactions or the effect of the lattices' geometry. These systems are intensively studied in the community of highly frustrated magnets, but they are also relevant for non-magnetic, structural problems. These two problems can seem, at first glance, as completely different and non-related. However, some authors [START_REF] Villain | Two-Dimensional Ising Model With Competing Interactions: Floating Phase, Walls and Dislocations[END_REF][START_REF] Bak | Commensurate phases, incommensurate phases and the devil's staircase[END_REF] have tried to describe magnetic frustrated systems, in particular the ANNNI model, using terms (commensurate, floating phases) originally used in the Table I.1: Comparison of different types of orders in the case of on-surface monolayers problems (Frenkel-Kontorova model) and on-lattice magnetic systems (two-dimensional Ising spin models). A comparison with generic states of matter is also proposed. Phases indicated in light blue are predicted at thermodynamic equilibrium, while phases in red are out-of-equilibrium. * a a Algebraic correlations are characteristic of critical liquids, in which the correlation length diverges, but their strength is weak at much smaller distances than in the case of solids.

Order

On-surface monolayer On-lattice magnetic system description of adsorbed monolayers. Is this example an "accident" or it is a sign of a deep link between the two classes of models? We are for the moment unable to rigorously answer this question, but we would like to analyze the similarities and the differences between the two problems.

Concerning the nature of the order, we have presented commensurate phases, incommensurate phases and chaotic phases in the case of adsorption problems. We now compare these phases with the ones in on-lattice systems with internal degrees of freedom, (Table I.1). To simplify the discussion we will focus on the case of Ising spins and use the vocabulary for magnetic phases, but we recall that this magnetic problem is equivalent to the ordering of particles adsorbed on a lattice and to binary alloys. In the first column of Table I.1 we give a basic description of the type of orders we have encountered in previous paragraphs.

Commensurate, ordered, solid-like phases

Let us define a as the lattice parameter of the substrate (for on-surface monolayers) and of the spin lattice (for magnetic Ising model). Now, we can define a ′ as the lattice parameter of the modulated order of the monolayer or as the lattice parameter of the pattern formed by spins. In Figure I.7 we show several examples of ordered phases, using the Frenkel-Kontorova picture and spin chains. For all these phases we can define a superstructure with lattice parameter a sup = qa ′ = ra and r and q natural numbers. This definition is the one of a commensurate phase. Thus, defining commensurate phases is equivalent for both on-surface and spin problems. These ordered phases have long range correlations, like in the case of a crystalline solid.

Incommensurate, liquid-like phases

Incommensurate phases are cases in which the ratio q/r is not rational, thus no common periodicity of a and a ′ can be found. These phases include floating regions, in which the adsorbed layer can be seen as independent of the surface below. In the particular case of the ANNNI model Villain and Bak showed that incommensurate phases exist, that exhibit short range correlations with an exponential or algebraic decay. This type of order within the adsorbed layer reminds the one found in usual (exponential decay) and critical (algebraic decay) liquids. We wonder whether incommensurate phases can be related to spin liquid phases, at least regarding correlation length. Proving the equivalence of the incommensurate floating and spin liquid phases will need a careful analysis of the problem that is beyond our qualitative comparison.

Finally, chaotic phases have been defined for on-substrate monolayers. These phases are also called pinned phases because the adsorbed atoms are trapped in the minimum of the substrate potential. Such phases are out-of-equilibrium phases that forms, for example, when a system is quenched (i.e. cooled down very quickly) and it does not have time to relax into its low energy state. The correlation length and strength in these phases remains an open question for us, probably dependent on the quench velocity. We wonder if these phases are more liquid-like or gas-like, even if both gases and liquids are phases at thermodynamic equilibrium, which is a substantial difference. We will not go further in the comparison, but highlight that we have presented two different physical systems in which competing interactions exist and produce unusual orders of matter. Even if we cannot prove the equivalence of both systems, we suggest that some bridges can be built between the problem of atoms adsorbed on a substrate with a positional degree of freedom and frustrated systems with internal degrees of freedom.

Two complementary viewpoints on ordered and disordered phases of C 60 molecules on a copper surface

In our work, we analyze order and disorder in an experimental non-magnetic system formed by two-dimensional lattices of C 60 molecules deposited onto a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface. In our samples we observe a striking coexistence of phases consisting of different ordered and disordered structures.

This experimental system is obviously an on-substrate monolayer system. Fullerenes have an equilibrium lattice parameter of ∼ 10Å, while the copper substrate exhibits a triangular lattice surface, with lattice parameter 2.56Å. The C 60 molecules also form close-packed (triangular) lattices where at least part of the molecules tend to match the positions of copper atoms. In-plane order of the molecules on the copper surface can be thus described using the language that is relevant for the Frenkel Kontorova model. In addition, previous studies of these systems have proposed that competition between molecule-molecule and molecule-substrate interactions plays a key role in the way the fullerenes assemble [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. In chapter III of this thesis we prove that it is indeed the case, and we use a semi-quantitative model to describe the in-plane ordering of the system. We conclude that commensurate (and incommensurate?) phases coexist in our samples.

Moreover, fullerenes on a copper surface have internal degrees of freedom, such as molecular orientation and molecular height with respect to the copper surface. In chapter IV we focus on the ordering of molecular heights. Molecular heights can be understood as well as an out-of-plane positional degree of freedom. We show that considering them as an Ising internal degree of freedom is a very good approximation. We indeed demonstrate that molecular heights experimentally observed are not a continuous variable but rather a bivalued one. The two preferred heights correspond to C 60 molecules adsorbed on the copper surfaces ("high" molecules) or to molecules on top of a vacancy of the copper ("low" molecules). Thus, we can model (some of) the observed phases using an Ising variable corresponding to high/low fullerenes on a triangular lattice. We also prove that the interaction between molecules is repulsive, i.e. antiferromagnetic-like, and that the system exhibits geometrical frustration! Our non-magnetic molecular system is a very rich experimental platform to study complex order and disorder states of matter combining two different, often considered as independent, approaches.

Overview of the thesis

The structure of this thesis could seem unusual to the reader. It is direct consequence of our two-viewpoint approach of the problem, which borrows concepts and descriptions to two communities. Indeed, different experimental, theoretical and analysis tools were needed to simultaneously understand the molecular samples 1 -from an atomic-scale point of view and 2 -using effective, frustrated, magnetic-like models. We can almost imagine this thesis as two independent projects, one performed by a surface scientist with a physico-chemistry approach and the other by a physicist working on frustrated magnets. What we believe is original in this work is that the two researchers (or the two scientific communities) have decided to discuss together! This division appears in chapter II, in which we present the tools used during this thesis. The experimental and simulation tools usually employed in surface science, including microscopy techniques and ab initio calculations are presented in section 1 and section 2, respectively. The statistical quantities, analysis and simulations that are more familiar to the community of highly frustrated magnets are described in section 3.

The two chapters presenting our results and their discussion, chapter III and chapter IV, also correspond to the above-mentioned division. They intend to be independent stories, as self-consistent as possible. Both have a similar structure including three different types of section: a review of the literature; a description, analysis and discussion of our results (which can be divided in several sections); and a generalization of our analysis of C 60 molecules on copper to other metal substrates.

The structure of the literature review in both chapters (section 1 in chapter III and section 1 in chapter IV) is organized according to an analytic structure, i.e. we present a detailed description of previous results in the literature, divided in different categories and progressively increasing the complexity of the ingredients we take into account. Our goal is two-fold: first, to introduce the vocabulary (part of the vocabulary and acronyms used in this manuscript are also presented in Appendix A) and concepts that we use to analyze our results; second, to present the studies that have been published in the literature (both theoretical and experimental), which we compare to our own results. We can hence understand these literature reviews as the background needed to follow the ensuing analysis.

In section 2 of chapter III we present our experimental observations of the fullerenes on copper assemblies using microscopy techniques. This first part is mainly descriptive: we describe the observed structures in term of their geometrical parameters. In section 3 of the same chapter we show the results of our ab initio calculations, which allow to best interpret the experimental observations and to estimate the molecule-molecule and molecule-substrate interactions at play in the C 60 molecular assemblies. Finally, in section 4 we address the origin of order and disorder in coexisting phases in our samples and estimate their energy. We use a model based on our previous results, both the experimental observations of the structure and the theoretical results on the interactions in the system. A summary and conclusions are presented in section 5. At the end of the chapter, in section 6, we suggest how the model we developed can be generalized to analyze the formation of different structural phases of C 60 molecules on metal surfaces. We discuss in which other metals we might expect complex, rich behavior as the one observed in fullerenes on copper.

In section 2 and section 3 of chapter IV we present the analysis of two of the molecular phases using Ising frustrated models. We demonstrate that one of the phases (section 2) is an experimental realization of a spin liquid phase near the transition to an ordered ground state. The other phase (section 3) can be analyzed using the same model, but a slight difference in the interactions in the system implies that the resulting phase is a realization of an Ising phase near a point of the phase diagram which is characterized by coexistence of two ordered phases. Finally, in section 5, we propose how the Ising model used to analyze the two phases mentioned above can be extended to other cases. First, we show that all the different phases of C 60 on copper can be described using a family of Ising Hamiltonians, in which the relative value of the interactions vary. Second, we propose how the previous analyses can be generalized to understand molecular phases formed by fullerenes on other metal surfaces.

The last chapter of this thesis, chapter V, gives new insights in previously published results, using the analysis developed in chapter IV. This chapter can be understood as an attempt to initiate a dialogue between surface science and magnetic frustrated communities. We chose some experimental data (microscopy images) characterizing the adsorption of fullerenes on metals and of binary atomic alloys that have been published in the literature. In earlier papers these experimental systems have been proposed as experimental realizations of the TIAF model. By a statistical analysis of the images, we determine whether it is indeed the case or not, and we propose an Ising Hamiltonian with corresponding couplings at a defined temperature that reproduces the experimental data. In a sense, this chapter may be viewed as a review of the literature, but as long as it employs all the tools progressively introduced throughout the manuscript, we preferred to present it at the end of this manuscript. We decided to treat it as a proof of concept of the power of our analysis and to propose a new path to study experimental frustrated non-magnetic systems. During this thesis a variety of experimental, analysis and simulation techniques have been employed to study the arrangement of C 60 molecules on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface. Indeed, addressing both the microscopic physics of organic molecules on metal surfaces and the statistical properties of such molecular assemblies requires to combine different tools. We can divide the tools in three groups, depending on the goal we aimed to.

Chapter II

Toolbox

The first objective of this thesis was to fabricate two-dimensional molecular assemblies of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and to image molecular configurations in real space. Fabrication of samples was made using ultra high vacuum (UHV) techniques, usually employed in surface science for the study of materials. Imaging samples with molecular resolution is possible using scanning tunneling microscope (STM) at room temperature, which is an standard, well-established, experimental technique. Careful analysis of the molecular structure observed in our STM measurements was also performed, taking into account the existence of experimental artifacts that we correct. To obtain sub-molecular resolution and resolve fine details of the molecular structure, we also used more sophisticated techniques, including STM at low temperature (5 K) and non-contact atomic force microscopy (nc-AFM). The tools related to sample fabrication and observation of the lattices are presented in section 1.

Our second goal was to describe the microscopic structure of molecules we observed in our STM images and, in particular, to disentangle structural and electronic effects in our measurements. To do so we used density functional theory calculations (DFT). Moreover, DFT calculations were used to address the interactions in the molecular system: Are C 60 molecules interacting between them? Is the interaction attractive or repulsive? Which role plays the copper substrate in the stabilization of molecular assemblies? Working principle of DFT calculations and particularities of our algorithm are explained in section 2.

Third, we aimed to describe order and, above all, disorder on C 60 on Cu(111) from a statistical point of view. The tools needed to achieve this goal are completely different from the previous ones. We did not consider microscopic details of the lattice, instead we used an effective description of molecular assemblies, based on an Ising Hamiltonian. This Hamiltonian is an archetype model in statistical physics, which describes magnetic and adsorption problems. We used Monte Carlo simulations (MC) to simulate our experimental data and define statistical quantities that describe order and disorder. In particular, we considered correlations, which inform us about how two variables are related, and quantify the strength of the relationship. The two variables can be of many different natures: spinspin correlations in magnetic systems, molecular heights correlations in our experimental system, etc. Correlations can be measured both in MC simulations and in experiments, thus compared in the two cases. We addressed the study of correlations in real and reciprocal space and we defined the equivalent of the magnetic structure factor (MSF) for our molecular assemblies. All these tools are defined in section 3 and how they are applied to the molecular system is described.

Observing molecules in real space with (sub)-molecular resolution

During this thesis 12 different samples of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] were prepared and observed by scanning tunneling microscope (STM). Ten of them were prepared in Grenoble, at Institut Néel, and two in Lille at Institut d'électronique, de microélectronique et de nanotechnologie (IEMN).

Sample preparation

In Grenoble, experiments were performed in a ultra high vacuum system (base pressure around 1 × 10 -10 mbar) formed by a tunnel, three preparation chambers and one separated STM chamber. For the experiments presented in this thesis we used one of the preparation chambers and the STM. Details about the experimental set-up are shown in Figure II.1

The Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] crystal was cleaned by repeated sputtering with argon (P=2 × 10 -5 mbar, ion energy 0.8-1 kV) and annealing (700 K). After this treatment, the surface was nicely reconstructed, exhibiting well-defined diffraction streaks in reflection high-energy electron diffraction (Dr. Gassler, Electron Devices, RHEED PCS 30 or RHEED CREATEC) and flat terraces, typically 100 nm-wide and separated by atomic step-edges as observed with STM. Submonolayers of C 60 (99.9% purity, purchased from Sigma Aldrich) were deposited from a commercial resistively heated evaporator (HE-SC-2-cell, Kentax GmbH) in the preparation chamber. The deposition rate was estimated using a quartz microbalance. We used fluxes from 0.5 angstrom/min to 1.7 angstrom/min, depending on the sample. We did not observe any significant dependence of the experimental observations on the deposition flux. To obtain these evaporation rates, the evaporator temperature was kept between 623K and 673K, depending on the desired flux. Only the crucible filled with fullerenes was heated, while the rest of the evaporator was water-cooled. We use two crucibles in which we evaporated C 60 molecules, the choice of the crucible changed the angle between the molecular evaporator and the Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] crystal, thus the deposition rate.

We prepared three types of samples depending on the substrate temperature during evaporation.

• Samples deposited with the Cu(111) crystal held at room temperature (from 293 K to 303 K). The samples presented in this thesis correspond to this cases, unless indicated.

• Samples deposited with the Cu(111) crystal kept at 333-343 K, which changes radically the adsorption of C 60 molecules on the surface, as explained in subsubsection 2.1.1. Results were the same when samples were deposited at room temperature and annealed after the deposition. In both cases the heater in the UHV system was used and temperature was controlled using a nickel-alumel thermocouple. In Lille, experimental conditions used in Grenoble were reproduced to obtain the same type of samples. Experiments were performed under ultra high vacuum with base pressures in the low 1 × 10 -10 mbar range. Copper single-crystals were prepared by repeated cycles of Ar + ion sputtering (0.8 keV) and moderate temperature annealing (670K). Fullerene were thoroughly out-gassed under ultra-high vacuum, within a Kentax evaporator. Evaporation was performed at 623K on clean Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] at room temperature.

Figure II.2:

Scheme of the cooling system used to cool the Cu(111) substrate for the preparation of C 60 samples, as presented in the manual user supplied by VG Scienta Omicron. Gas nitrogen is introduced in the coil circuit, the coil is immersed in the dewar filled with liquid nitrogen and the condensed nitrogen flows into a dedicated pipe (right part of the picture). The pipe goes into the growth chamber and liquid nitrogen is in thermal contact with the sample holder. The sample is placed at the end of the high conductivity braid (left bottom of the scheme).

Scanning probe microscopy 1.2.1 Scanning Tunneling Microscope (STM)

Scanning tunneling microscopy was performed with two different instruments, one in Grenoble, the other in Lille, both connected under ultrahigh vacuum to the growth chambers. For room temperature measurements in Grenoble a STM-1 Omicron microscope was used, with electrochemically-etched W metal probes.

For low temperature measurements in Lille, experiments were carried with a Joule-Thomson microscope (SPECS, Berlin). The STM/nc-AFM tip was a commercial lengthextensional resonator (Kolibri, SPECS Berlin) with W tip prepared by Ar sputtering and substrate conditioning. A first run of STM measurements was done at 77K. A second run of measurements including STM, non contact atomic force microscopy (nc-AFM), Kelvin probe force microscopy (KPFM) and scanning tunneling spectroscopy (STS) measurements was performed at 5 K. KPFM and STS measurements will not be discussed in this thesis.

In both cases STM measurements were performed in the constant current mode, i.e. the current passing from the sample to the STM tip is measured, and a feedback-loop is used to keep constant this value by modifying the position of the tip on the z-axis [START_REF] Voigtländer | Scanning probe microscopy: Atomic force microscopy and scanning tunneling microscopy[END_REF]. This distance in the z-axis is then saved and it gives a convolution of the topography of the sample surface and its electronic properties. In the case of STM measurements at low temperature, the STM tip was oscillating (because a combined mode of nc-AFM and STM was used, see subsubsection 1.2.3 for details) and the current signal averaged over the period of the oscillation.

Correction of distortions in STM images and data analysis

In chapter III, we use our STM measurements to characterize the structure of molecular assemblies. Two different types of informations are deduced from our images: 1 -the inplane structure of close-packed molecular lattices, including orientation with respect to the copper close-packed rows and molecular lattice parameter; 2 -the molecular height of the fullerenes. Here, we describe how to measure the in-plane structure. For details about how we can determine molecular height see subsubsection 1.2.3 and section 2.

First, we note that we cannot measure orientation of molecular lattices on raw STM measurements at room temperature. For a molecular close-packed lattice, like the one formed by fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface, the angle between unit vectors of the molecular lattice is expecte to be 120°. Our measurements exhibit always different angles between the unit vectors, that can be 5 -10% bigger or smaller than 120°. This difference evidence the existence of measurement artifacts that we need to correct in our STM images.

To determine molecular lattice parameters, we need to perform a careful analysis of our samples, specially for STM measurements at room temperature. According to previous studies [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] we expect lattice parameters a C60 ∼ 10 Å. These distances can be measured with STM operating at room temperature with a good accuracy. Nevertheless, fullerenes' lattices can adapt their lattice parameter depending on the coincidence with the substrate, and small variations of the lattice parameter (< 5%) are expected. Measuring this differences requires to be able to measure intermolecular distances with a precision smaller than 0.5Å. On top of that, if we want to make the difference between different adsorption sites on the copper surface, we need to measure distances smaller than the copper lattice parameter (a Cu(111) = 2.56 Å), typically of ∼ 1 Å.

Measuring intermolecular distances on periodic molecular lattices can be done in several manners. In all the cases we took advantage of the periodicity of the lattice to increase the precision of our measurements. We can:

• Extract profiles through molecular rows on the fullerene's lattices, and divide the length of the profile by the number of molecules.

• Superpose periodic lattices to the STM images, search for the best agreement at long distances and deduce the unit vectors of the lattice. This type of measurements are implemented in the software Gwyddion (http://gwyddion.net/) [START_REF] Nečas | Gwyddion: an open-source software for SPM data analysis[END_REF], that we used for part of our analysis. This tool also permits to determine molecular orientation of fullerenes lattices.

• Compute the Fourier transform of molecular lattices and deduce the lattice unit vectors 1. Observing molecules in real space with (sub)-molecular resolution in reciprocal space, thus in real space. Calculation of the fast Fourier transform (FFT) is also implemented in Gwyddion [START_REF] Nečas | Gwyddion: an open-source software for SPM data analysis[END_REF]. Like in the previous case, we can also determine molecular orientation of fullerenes lattices using this type of analysis.

To decide if the STM measurements at room temperature allow us to determine intermolecular distances, we performed series of measurements on different molecular regions, for which the same lattice parameters were expected (not shown). We found variations of the lattices parameters of ∼ 10%, independently of the measurement method that was employed. These differences appeared for molecular regions in the same STM image but located in different parts of the scan, for molecular regions on different STM images and even for the same molecular assembly when the size of the STM scan was changed. Thus, we concluded that raw STM images at room temperature cannot be used to determine intermolecular distances. Indeed, we need to identify artifacts in our measurements and correct them.

To correct our STM images at room temperature we consider, and correct, two different effects.

First, we corrected the hysteresis effects on the piezoelectric manipulator of the STM tip. Indeed, even in the so-called linear regime of piezoelectrics, a small hysteresis appears in the behavior of the piezoelectrics that control the position of the tip [START_REF] Nishikawa | Piezoelectric and electrostrictive ceramics for STM[END_REF][START_REF] Yothers | Real-space post-processing correction of thermal drift and piezoelectric actuator nonlinearities in scanning tunneling microscope images[END_REF]. Thus, during the scan of the surface, the speed of the tip is not constant and small distortions on the images arise. We can evidence this hysteresis by comparing forward and backward directions of the scan in the fast scan direction (x-axis for our measurements). In Figure II.3a we show the two directions of the scan. In Figure II.3b we show a scheme of the hysteresis (right panel), that depends on the forward/backward direction, and the difference between the forward and backward STM images of the same molecular region (left panel). In Figure II.3c we show the forward and backward profiles in (a). Note the existence of a lateral shift between the two profiles (top panel), which we manually corrected by aligning both profiles in the edge of the STM image (bottom panel). Even when the shift is corrected, profiles do not coincide, because of the hysteresis on piezoelectric manipulators. We considered a parabolic correction (Figure II.3d, top panel) that we deduce from the comparison of forward and backward profiles for each STM image. Finally, we verified that alignement of the profiles is correct after the correction (Figure II.3d, bottom panel). This correction was applied to each STM image measured at room temperature. We used a Python home-made code developed by Alexandre Artaud in which the lateral shift is chosen by the user and the best parabolic correction automatically found. When we have determined the second order polynomial distortion needed to correct the STM image, we apply the distortion using the software Gwyddion [START_REF] Nečas | Gwyddion: an open-source software for SPM data analysis[END_REF]. We made the same type of correction in the slow scan direction (y-axis) using upward and downward scans, and found that the effect of hysteresis in the y-axis is negligible with respect to the effect of thermal drift, detailed in the next paragraph. Moreover, to avoid the possible artifacts present in the y-axis, we measure intermolecular distances along the slow scan direction.

Second, we took into account the presence of thermal drift in our STM measurements. Being at room temperature implies that the measurement conditions are not perfectly stable and the position of the STM tip and/or the sample drift during the measurements [START_REF] Lyding | Variabletemperature scanning tunneling microscope[END_REF][START_REF] Yothers | Real-space post-processing correction of thermal drift and piezoelectric actuator nonlinearities in scanning tunneling microscope images[END_REF]. A schematics of this effect is shown in Figure II.4, in which we show how a square scan of the sample surface is indeed deformed. Drift depends on the temperature and is not a systematic error in our measurements, i.e. we cannot apply the same correction to all the STM images, but each image needs to be corrected individually. To correct drift we took advantage of the molecular lattice: we know that it is a close-packed lattice with unit vectors forming a 120°angle and the same length. We applied a deformation of the image that allows us to find the perfect close-packed lattice we expect. We only need to determine the initial vectors before correction, and the expected ones after drift correction. Using this information we can use the tool implemented in Gwyddion [START_REF] Nečas | Gwyddion: an open-source software for SPM data analysis[END_REF] (called Distortion/Affine) to correct each one of our STM images. In conclusion, each one of the STM images at room temperature shown in this manuscript was corrected. First, we correct hysteresis of the piezoelectrics in the x-axis. Next, we correct drift effects. After correction, we were able to measure intermolecular distances and orientation of the molecular lattices, as we detail in chapter III and chapter IV. The manner in which we performed these measurements are detailed in the text. Depending on the precision needed and the information we wanted to extract from the images (orientation of the molecular lattice, intermolecular distances in different molecular phases, presence of local distortions in the lattice, etc.), we used different tools: we extracted molecular profiles and determined position of the peaks, superposed periodic lattices on the images, computed the FFT, etc.

Non contact atomic force microscopy (nc-AFM)

We have mentioned before that STM measures together topographic and electronic properties of the molecular lattice. In order to make the difference between topographic and electronic signatures, we used nc-AFM techniques to observe fullerenes assemblies. While in STM we measure current passing through molecules, nc-AFM probes the forces between an oscillating tip and the sample surface [START_REF] Voigtländer | Scanning probe microscopy: Atomic force microscopy and scanning tunneling microscopy[END_REF]. When an oscillating tip is close (but not in contact) to a surface, it interacts with the sample and is submitted to forces at the (sub)nanometric scale. This tip-surface interactions modify the probe oscillation frequency, which is then a measurement of forces. Depending on the distance between the tip and the sample, forces are attractive, such as electrostatic and van der Waals forces (at larger z-distances); or repulsive, such as chemical forces (at short z-distances). Attractive forces increase the resonance frequency of the oscillating tip, whereas repulsive forces decrease the frequency.

The working principle of our nc-AFM measurements, in the frequency modulation mode First, the resonance frequency (f 0 ) and quality factor (Q) of the oscillating tip are determined. This frequency is chosen as the operation frequency. When the tip feels any interaction with the sample surface, the oscillation frequency is shifted by ∆f = f -f 0 . Oscillating tips used for nc-AFM measurements have very large stiffness factors (k = 1080 kN/m for Kolibri sensors, SPECS Berlin) that allow low amplitude measurements in the repulsive forces regime (avoiding jump-to-contact issue).

The high resolution nc-AFM measurements that we show in chapter III, were measured without using Z-feedback loop for the frequency shift. In our case, the measurements directly show the frequency shift ∆f that is related to the force gradient felt by the tip. In our analysis we will consider that the force strength is directly related to the distance between the tip and the molecules on the surface, i.e. to the topography of the sample. This assumption is a good first approximation and it has been used in other studies of fullerenes on metal surfaces [START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF]. 1These measurements were performed at IEMN in Lille at 5K. The microscope and tip were the same that we used for low temperature STM measurements, as specified in subsubsection 1.2.1. More details about the experiments could be found in Ref. [START_REF] Turek | High resolution nc-AFM and KPFM imaging of molecular assemblies at low temperature[END_REF].

Structural properties and interactions in molecular assemblies: density functional theory calculations

As mentioned in the introduction, density functional theory (DFT) calculations were used to address the microscopic structure of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface. The goal of DFT study was three-fold.

First, we wanted to clarify and complete the experimental data measured with STM. STM measurements are a convolution of topography of the surface and electronic properties of the molecular assemblies, that are entangled and difficult to distinguish experimentally. DFT is a powerful tool to interpret STM contrasts and understand the molecular structure and electronic effects [START_REF] Tersoff | Theory and Application for the Scanning Tunneling Microscope[END_REF][START_REF] Blanco | Firstprinciples simulations of STM images: From tunneling to the contact regime[END_REF][START_REF] Repp | Molecules on insulating films: scanning-tunneling microscopy imaging of individual molecular orbitals[END_REF]. In section 3 of chapter III, we used DFT calculations to simulate the current measured in STM and the corresponding images, depending on the structure of fullerenes on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface.

Second, we addressed the role of different interactions in the stabilization of fullerenes assemblies. The objective was to understand the role of molecule-substrate and moleculemolecule interactions and to be able to estimate the relevant energies in fullerene assemblies. DFT calculations were massively performed to investigate the structure and interaction in the case of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], for about twenty different configurations. These results are presented in section 3 of chapter III. We also addressed the case of C 60 molecules interacting between them in a two-dimensional network, taking into account van der Waals interactions that are beyond the usual DFT algorithms. In subsection 5.2 of chapter III, we extended our analysis to the case of other close-packed metals (M(111)), which allows us to revisit the literature and give a general picture of different mechanisms at play in C 60 /M(111) samples. We considered five different metal surfaces and in total about twenty more molecular configurations were addressed, even if not all of them are presented in this manuscript. In total our calculations include about fifty different configurations, which represents hundred of thousands of computing hours.

Finally, in section 3 in chapter III we used our DFT calculations to investigate the charge transfer on the C 60 /Cu( 111) system, because it has been reported that for fullerenes on metal surfaces a charge transfer from the metal to the molecules takes place at the metal/molecule interface [START_REF] Tsuei | Photoemission and photoabsorption study of adsorption on Cu(111) surfaces[END_REF][START_REF] Hoogenboom | Charge transfer and doping-dependent hybridization of C 60 on noble metals[END_REF][START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. We determine if this charge transfer could play an important role in the stabilization of molecular phases, via dipolar interaction between charged molecules. The study of charge transfer is particularly challenging because it strongly depends in the approximations used by each algorithm, and in particular in the basis function used to compute the charges.

In next paragraphs, we explain the basic principles of DFT and detail how our calculations were performed.

Schrödinger equation, Born-Oppenheimer approximation and Hohenberg-Kohn theorems

In physics and chemistry ab initio calculations make reference to calculations exclusively based on physical principles, i.e. no empirical parameters or experimental data are needed a priori. DFT is one of the existing and most currently used ab initio methods. This technique is used to compute the electronic structure of the ground state of a given physical system. From the electronic structure of the ground state many properties can be computed: structural, electronic transport, mechanical, magnetic, etc. DFT can be applied to many different systems, ranging from atoms and molecules to solids or interfaces between solids.

In a N-body system in condensed matter the Schrödinger equation [START_REF] Schrödinger | An undulatory theory of the mechanics of atoms and molecules[END_REF] describing the system of N particles can be written using the wave function (Ψ) and the Hamiltonian:

ĤΨ(⃗ r i ) = E i Ψ(⃗ r i ) (II.1)
where E i are the energies and ⃗ r i the spatial coordinates of each N particle. Depending on the nature of the particles the relation between the global function Ψ(⃗ r 1 , ..., ⃗ r N ) and the partial functions Ψ i (⃗ r i ) can be different, for example for indiscernible fermions Ψ(⃗ r i ) is a Slater determinant. The spin degree of freedom can also be included as a σ i coordinate in the wave function, but here we will disregard it to simplify the argumentation. In the case of a solid, we can separate the N particles in m nuclei and n electrons (N = m + n), thus the Hamiltonian writes:

Ĥ = Knuc + Kel + Ûel-nuc + Ûnuc-nuc + Ûel-el (II.2)
where K denotes kinetic energy and U makes reference to interaction energy. Each one of the previous terms write:

Knuc = - 1 2 m j=1 ∇ 2 j (II.3)
with ∇ 2 j the laplacian operator applied to the spatial coordinates of the particle j;

Kel = - 1 2 n i=1 ∇ 2 i (II.4) Ûel-nuc = - n i=1 m j=1 Z j r ij (II.5)
Z j the charge of the nuclei j and r ij the distance between j and i particles;

Ûel-el = n i=1 n k>i 1 r ik (II.6) Ûnuc-nuc = m j=1 m l>j Z j Z l r jl (II.7)
The interaction term between electrons and nuclei is attractive while interaction terms between electrons/nuclei are repulsive. All the equations are expresed in atomic units. Finding the exact solution of this equation for a large number of particles N interacting with each other is a hard and time-consuming task.

The first simplification that can be used to solve the equation is the Born-Oppenheimer approximation [START_REF] Born | Zur Quantentheorie der Molekeln[END_REF]. It neglects the kinetic term for nuclei, because nuclei's masses are much larger than those of electron. Thus, they move much slower and their spatial coordinates are roughly constant. Interaction between fixed nuclei can also be neglected because it will only contribute to the potential energy with a constant term. The Hamiltonian is then reduced to an electronic Hamiltonian:

Ĥel = Kel + Ûel-nuc + Ûel-el (II.8)
Even if the equation is simpler now, finding the wave function for n electrons interacting with each other and with m nuclei remains a hard task.

To tackle this problem DFT computations do not consider the wave function of each particle but the electronic density ρ(⃗ r), where ⃗ r is a coordinate in three-dimensional space. The electronic density is the probability of finding an electron at position ⃗ r. This electronic density is built from the wave function of the n electrons as:

ρ(⃗ r) = i=1,...,n ∥Ψ(⃗ r i = ⃗ r)∥ 2
(II.9)

The two Hohenberg-Kohn theorems [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF][START_REF] Sham | One-Particle Properties of an Inhomogeneous Interacting Electron Gas[END_REF] prove that the electronic density is the appropriate quantity to rewrite the Hamiltonian. The first theorem states that the electron density uniquely determines the Hamiltonian, or said differently, the energy of the system can be rewritten as a functional of the electronic density:

E[ρ] = K[ρ] + U [ρ] + ρ(⃗ r)V nuc d⃗ r (II.10)
with the first term giving the kinetic energy of electrons, the second term corresponding to interaction between electrons and the third one representing the interaction between electrons and the external potential created by nuclei. The second Hohenberg-Kohn theorem states that the functional that delivers the energy ground state of the system necessarily corresponds to the electronic density of the ground state.

DFT faces a huge challenge, indeed the terms K[ρ] and U [ρ] are not known, so well-chosen approximations are needed. For the kinetic term Kohn and Sham proposed to describe the n interacting electrons as a non-interacting gas under the effect of an external effective potential. Then the kinetic term can be re-written explicitly and no longer depends on the coordinates of all electrons but on the wave function of each individual electron:

K 0 = - 1 2 N i ⟨Ψ i |∇ 2 |Ψ i ⟩ (II.11)
However, the term is not equal to the true energy of the system, and the difference is "hidden" in the exchange-correlation term U xc .

The term accounting for the interaction between electrons U [ρ] can be divided in two parts. The first one is the classical Coulomb potential or Hartree energy:

U H [ρ] = 1 2 ρ(⃗ r)ρ(⃗ r ′ ) |⃗ r -⃗ r ′ | (II.12)
The second one is a non-classical contribution also hidden in the exchange-correlation functional U xc .

Finally, the U xc includes self-interaction correlation, exchange and Coulomb correlations between many particles, i.e. all the terms not taken into account in previous approximations:

U xc = K -K 0 + U -U H (II.13)
Kohn-Sham equations can be rewritten in the following form:

E[ρ] = K 0 + U H + ρ(⃗ r)V nuc d⃗ r + U xc (II.14)
where the only term that is not known is the exchange-correlation one.

Choosing the exchange-correlation term: LDA and GGA

Here we describe the most usual approximations for the exchange-correlation functional: local density approximation (LDA) and generalized gradient approximation (GGA).

LDA considers that the U xc functional depends only on the density at the spatial coordinate in which the functional is evaluated, so

U LDA xc [ρ(⃗ r)] ∝ ρ(⃗ r)
This is equivalent to consider a uniform electron gas, for which the exchange term can be computed [START_REF] Bloch | Bemerkung zur Elektronentheorie des Ferromagnetismus und der elektrischen Leitfähigkeit[END_REF][START_REF] Dirac | Note on Exchange Phenomena in the Thomas Atom[END_REF]. The correlation term has been numerically simulated using Quantum Monte Carlo [START_REF] Ceperley | Ground State of the Electron Gas by a Stochastic Method[END_REF] and functionals can be built by fitting it [START_REF] Vosko | Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis[END_REF][START_REF] Perdew | Self-interaction correction to density-functional approximations for many-electron systems[END_REF][START_REF] Perdew | Accurate and simple analytic representation of the electron-gas correlation energy[END_REF]. LDA approximation tends to overestimate correlation energy, while exchange energy is accurate within 10%. LDA can give very good results for bond lengths but usually overestimates binding energies between atoms.

GGA goes a step further and takes into account the electronic density and its gradient to build the energy functional:

U GGA xc [ρ(⃗ r)] ∝ f ρ(⃗ r), ∇ρ(⃗ r)
GGA significantly reduces the error in calculated energies.

Nevertheless, both approaches cannot properly solve situations in which electronic density are strongly inhomogeneous. In these cases other approximations and techniques have been developed for each specific problem. Indeed, finding a global approximation for the energy is not trivial and depending on the physical system (molecules, solids, metals, insulators, etc.) and the properties that are investigated (total energy, structural properties, charge density, etc.) different functionals should be used.

Linear combination of atomic orbitals (LCAO) and basis sets

Using Kohn-Sham equation requires to define wave functions of electrons present in the system, which are used to build the electronic density. To write these wave functions different approches can be employed. The most common one is the linear combination of atomic orbitals (LCAO), in which wave functions are written as a linear combination of predefined basis functions. Doing so, the equations can be linearized and rewritten using standard algebra, which simplifies the problem.

The functions used as basis set can be of many different types depending on the system considered. For instance: plane waves; Slater-type-orbitals (STO) that are the eigenfunctions of the hydrogen atoms; Gaussian-type-orbitals (GTO) used in quantum chemistry; contracted Gaussian functions (CGF) formed by linear combinations of Gaussian functions; etc.

Self-consistent solution and iterative minimization of the energy

As previously shown, Kohn-Sham equation gives the energy of the system as a function of the energy density. To find the electronic structure corresponding to the ground state the goal is to minimize the total energy of the system using an iterative process (summarized in Figure II.5). Before starting the computation an initial configuration is chosen. It consists of the positions of the atoms and molecules with a guess for their electronic density. The next steps are repeated when the atomic/molecular positions are moved, until convergence is reached:

• The potentials (U ef f = U H + ρ(⃗ r)V nuc d⃗ r + U xc ) are computed from the electronic density, within one of the possible approximations for the correlation-exchange potential.

• The Kohn-Sham equation is solved

- ∇ 2 2 + U ef f (⃗ r) Ψ i (⃗ r) = E i Ψ i (⃗ r)
using the chosen combination of atomic orbitals.

• The new charge density is computed from the lowest energy eigenstates.

• The electronic density is compared with the initial one, if it is different the process restarts. When changes in energy are below a certain threshold, the iterations stops.

Our calculations: DFT and Intermolecular Perturbation Theory using Fireball code

The DFT calculations presented in this thesis were perfomed by Yannick Dappe at Service de Physique de l'Etat Condensé (SPEC, CEA) in Saclay (France), using Fireball code. The simulation of STM images was performed by César González Pascual at Universidad Autónoma de Madrid (UAM) in Madrid (Spain). A detailed description of the code and its application to carbon systems, with corresponding references can be found in [START_REF] Dappe | Intermolecular interaction in density functional theory: Application to carbon nanotubes and fullerenes[END_REF][START_REF] Dappe | Weak Chemical Interaction and van der Waals Forces: A Combined Density Functional and Intermolecular Perturbation Theory-Application to Graphite and Graphitic Systems[END_REF]. Here we sum-up the main characteristic of the code.

The Fireball code was used to compute both structure and interaction energies between C 60 molecules and metal substrates. Our results are based on a slightly different flavor of DFT in which the functionals are not built from the electronic density but from the orbital occupation numbers, using the linear combination of atomic orbital-orbital occupancy (LCAO-OO) method [START_REF] Sankey | Ab Initio Multicenter Tight-Binding Model for Molecular-Dynamics Simulations and Other Applications in Covalent Systems[END_REF][START_REF] Lewis | Advances and Applications in the FIREBALL Ab Initio Tight-Binding Molecular-Dynamics Formalism[END_REF][START_REF] Jelınek | Multicenter Approach to the Exchange-Correlation Interactions in Ab Initio Tight-Binding Methods[END_REF]. These simulations consider periodic boundary conditions. The substrate is modeled with a five-layer stack of Cu atoms, the bottom two layers simulate bulk layers, thus their positions are fixed. The basis set is an optimization of the orbitals of carbon systems with covalent bonding. It consist of optimized minimal basis sets of localized orbitals: sp 3 for carbon atoms and sp 3 d 5 for copper atoms, with respective cutoff radii (in atomic units) 4.5 (s), 5.7 (p), 3.7 (d), for Cu, and 4.5 (s), 4.5 (p), for C, following a previous work on C 60 /Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Schull | Charge injection through single and double carbon bonds[END_REF]. Also, an sp 3 d 5 basis set of carbon orbitals has been used for a better description of charge transfer and electronic dipoles. Different sizes of supercells have been considered in this work to simulate different intermolecular distances. All the structures have been fully optimized, using a set of 8×8×1 k-points in the Brillouin zone. The convergence criteria was that forces remain below 0.1 eV/Å.

Repulsive weak interactions between overlapping orbitals of type s were taken into account using an s 2 expansion (theory perturbation) to obtain the interaction energy. Van der Waals interactions were also considered in the computations for fullerenes without substrate. This term can be seen as a long-range electronic correlation effect, proportional to the Coulomb interaction in the system. The interaction was computed like a pairwise interaction between atoms belonging to two different systems (two C 60 molecules, for instance). This interaction was evaluated within a dipolar approximation, which is legitimate given the typical distance between carbon atoms belonging to different molecules.

The tunneling current in the STM simulations was calculated using the non-equilibrium Green Function formalism developed by Keldysh [START_REF] Keldysh | Diagram Technique for Nonequilibrium Processes[END_REF][START_REF] Mingo | Theory of the Scanning Tunneling Microscope: Xe on Ni and Al[END_REF]. First, a Cu-tip formed by a pyramid of four atoms was considered, and joined to four layers of Cu(111) using the DFT procedure. Second, this tip samples the surface obtained in the DFT simulations. Consequently, the atomic and electronic properties of both tip and sample were estimated with the same DFT code. The tip is placed 5 Å, above the upmost C 60 -atoms, avoiding the multiple scattering effects. A complete description of the methodology can be found in Ref. [START_REF] González | Theoretical Study of Carbon-Based Tips For Scanning Tunnelling Microscopy[END_REF].

Order and disorder in molecular assemblies: statistical analysis

Here we present the tools we used to characterize order and disorder in our molecular samples. We start by defining the statistical quantities that allow us to describe order and disorder: correlations in real and reciprocal space. Next, we explain how we compare these quantities measured in the experiment with simulations of spin Ising Hamiltonians. In particular, we detail how to determine if the experimental observations correspond to configurations of the simulated model at or out of thermodynamic equilibrium. Finally, we describe the basic principle of Monte Carlo simulations and specify the way in which we performed our simulations.

Pairwise correlations in real space

Before introducing the pairwise correlations for spins and molecular heights, that we will use in this thesis, we illustrate the definition and meaning of spatial correlations in an easy example. Let's consider a chain of atoms in one-dimension and describe the spatial correlations in this system, depending on the ordering of atoms. A schematic cartoon is shown in Figure II.6. We define spatial correlations C(r) as the probability of finding an atom at a distance r from a reference atom in r = 0. Which would be C(r) for different states of matter? First, we can imagine that atoms in the chain behaves like in a perfect crystal.

In this case, we always find an atom if the distance is a multiple of the lattice parameter of the crystal a. The correlation function C(r) has a period equal to a, and its value is equal to 1 if r = na with n a natural number, and 0 otherwise. Second, we can consider atoms in the chain behaving like in a liquid. In this situation atoms are spaced by an average distance a with nearest neighbors, but their position is independent of the one of further neighbors. Thus, the correlation decays very fast with distance. Finally, if we consider an ideal gas in which particles are completely independent, we will find that correlation value is zero at any distance. This easy example shows how correlations in real space permit to describe order and disorder in matter. Indeed, long range correlations with periodic behavior distinguish crystalline solids; correlations decaying fast with the distance, usually exponential decay, algebraic in some particular cases, are characteristic of liquids; and absence of correlations is the signature of gases.

Figure II.6:

Ideal correlation functions C(r) for atoms in a chain, when the atoms are ordered like in a perfect crystalline solid, in a liquid phase or in an ideal gas. In the liquid case the dotted line corresponds to the oscillating C(r) behavior, the full line indicates the average decay of the correlations.

In this thesis we will use pairwise correlations in real space, defined for the case of particles in lattice models, i.e. the positions of particles in the (x, y) plane are fixed. These particles have an Ising degree of freedom σ = ±1. In our case, pairwise correlations are related to the probability of finding an equal or different Ising state for the particles at positions ⃗ r k , where k is the order of the neighbor (k = 1 are nearest neighbors, k = 2 second neighbors, etc.). The definition of pairwise correlations for N Ising particles on a lattice writes:

C k = ⟨σ(⃗ r i )σ(⃗ r i + ⃗ r k )⟩ = 1 2N 1 N k N i N k k σ(⃗ r i )σ(⃗ r i + ⃗ r k ) (II.15)
where N k is the number of neighbors of type k for each particle i. The factor 1/2 takes into account the fact that each bond is counted twice. Considering k neighbors at different distances, we can determine the length of correlations, and, like in the example of the atomic chain, decide if the Ising degree of freedom we are interested in behaves like in a crystalline, liquid or gas phase. The Ising degree of freedom, σ = ±1, accounts for different situations in this thesis. In chapter IV we simulated Ising spin Hamiltonians, so σ accounts for the up or down state of spins and C k inform us about the ordering of spins. Negative values of C k translate antiferromagnetic ordering of spins, whereas positive values correspond to ferromagnetic ordering. Also in chapter IV, we compared the results in spin systems with the ones in our molecular lattices. In this case the Ising degree of freedom is no longer magnetic, but structural: σ is an out-of-plane positional degree of freedom, i.e. a molecular height that can take "high" and "low" values. Thus, C k is a measurement of how molecular heights arrange. They tend to alternate if C k < 0 or to create molecular clusters with the same height if C k > 0. Finally, in chapter V, we extended our analysis to other surface systems, in particular to alloys formed by two chemical elements A and B. In these cases, the Ising degree of freedom σ is the nature of the element: σ = +1 corresponds to the element of type A, σ = -1 to the element of type B.

In this thesis, we computed pairwise correlations from experimental microscopy images (in the case of molecular assemblies and alloys) and also for MC configurations (in the case of Ising spin Hamiltonians). The experimental images (ours and others, previously reported in the literature) are converted in maps of an Ising variable, i.e. the molecular heights (or chemical nature) are discretized (chapter IV and chapter V). This procedure has been applied to individual images, and also to collections of images. In the latter case, we derived averaged correlations and represented, in our graphs showing correlations versus the neighbor type k, the standard deviations as error bars. Pairwise correlations extracted from MC simulated spin lattices are directly computed as an output of MC simulations. These simulations were performed as function of temperature, and the correlations were systematically calculated as function of both k and T .

The calculation for experimental images was implemented using Python, using the numpy library for the calculation and matplotlib for representation. For Monte Carlo simulations the code was developed in C++ by Benjamin Canals (see subsection 3.4). From left to right: ordered phase characterized by Bragg peaks with intensity diverging with the number of sites in the lattice, localized at specific positions in the q-space; disordered paramagnetic phase showing a very small intensity almost equally distributed on the q-space; disordered correlated phase exhibiting structured diffuse scattering (dark violet regions interconnecting the main peaks), while the main peaks' intensity remains finite (unlike Bragg peaks).

"Magnetic" structure factor

We have described how pairwise correlations in real space can be used to characterize the arrangement of Ising degrees of freedom. However, an open question remains: which distance (or equivalently which k neighbor) do we need to consider to properly characterize a phase? Is it enough to take into account the three nearest neighbors, five, ten, etc. ? To avoid this question we can define the Fourier transform of pairwise correlations, which is a visualization of all the pairwise correlations, from short to infinite range, in reciprocal space.

In the case of magnetic systems, this quantity is called magnetic structure factor (MSF). The MSF can be experimentally measured using diffraction techniques, such as neutron scattering, and is very useful to characterize order and disorder in highly frustrated magnets [START_REF] Rodríguez-Carvajal | Magnetic structures[END_REF][START_REF] Ramirez | Chapter 4 Geometrical frustration[END_REF][START_REF] Lacroix | Introduction to Frustrated Magnetism[END_REF][START_REF] Rougemaille | The magnetic structure factor of the square ice: A phenomenological description[END_REF]. Coming back to our analogy with ordering of atoms in matter, the MSF is the equivalent of diffraction patterns used to determine crystalline structure, for example using X-ray diffraction. But, instead of probing the position of atoms or molecules, it probes the magnetic degree of freedom. Thus, both ordering of atoms or of magnetic moments exhibit similar signatures in reciprocal space: a lattice of Bragg peaks appears, i.e. intensity is localized at specific locations, where it increases as ∝ N 2 , where N is the number of particles in the system. In the absence of order, magnetic structure factor does not show Bragg peaks but diffuse scattering. There is interesting information in this diffuse scattering, which is present both when disorder is random and when it is correlated. Indeed, characterizing diffuse scattering allows to make the difference between a random disordered phase, i.e. a paramagnet, and correlated disorder, for example in spin liquids. While paramagnets exhibit an almost flat distribution of diffuse scattering, correlated disorder is characterized by structured diffuse scattering. One of the most studied features in MSF for correlated disorder are the famous pinch-points (see subsection 1.1 in chapter IV and Appendix C), which are points in the q-space in which non-analytical divergence of the intensity is observed. These pinch points are the signature of an algebraic decay of spin pairwise correlations in real space, which are found in liquid phases.

In conclusion, analyzing pairwise correlations in reciprocal space, i.e. the MSF, makes possible to detect the presence of order (Bragg peaks) and to distinguish between random (non-structured diffuse scattering) and correlated disorder (structured diffuse scattering). Moreover, the MSF sums up in one single plot all the information contained in correlations in real space for short and long range. Some examples of MSFs of ordered and disordered phases are presented in Figure II.8. In this thesis, the equivalent of MSF is defined for our Ising systems, and it is used to detect tendencies to order and to study the nature of disorder in experimental non-magnetic systems. In chapter IV, MSF are used to make the link between molecular configurations observed on C 60 molecules on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface and the simulated configurations for Ising spin Hamiltonians. In chapter V MSF are computed from STM images previously published in the literature (of fullerenes assemblies and binary alloys), and again compared with MSF for simulated Ising spin Hamiltonians.

In all these cases, MSF were computed from real space images as follows. First, we convert the experimental images into perfect lattices (triangular in our case) with, at each site, a discrete Ising variable. This variable accounts, depending on the system we address, for the states up/down of spins, high/low of molecules or A/B of atoms.

The definition of the MSF is:

I(⃗ q) = 1 N ij σ i σ j e i⃗ q ⃗ R ij (II.16)
where ⃗ q is a reciprocal space vector (in practice we compute ⃗ q maps of I), N is the number of sites in the lattice (where molecules or atoms sit) and ⃗ R ij = ⃗ r j -⃗ r i is the vector connecting two sites i, j of positions ⃗ r i , ⃗ r j .

To avoid computing time-consuming sums over both i and j we simplify the previous expression and obtain the norm of a complex vector:

I(⃗ q) = 1 N ∥⃗ v q ∥ 2 = 1 N ⃗ v q • ⃗ v q * (II.17) with: ⃗ v q = i∈[1,N ]
σ i e i⃗ q⃗ r i (II.18)

For all the experimental images we symmetrize the MSF taking into account the C 3 symmetry of the triangular lattice. We use this symmetrization to easily compare the experimental configurations, which are usually non-symmetric, with MC simulated MSF which usually show the symmetry of the lattice. In fact, MC simulated MSF are symmetric because many configurations are averaged together, and even if each one is not symmetric (like our experimental configurations) the average is. The only cases in which MC MSFs are not symmetric correspond to very low temperatures, at which the ground state are ordered phases with breaking symmetry.

When possible and to improve the signal-to-noise ratio in the experimental MSF, an average was calculated over a series of (real space) images. The calculation was implemented using Python, using the numpy library for the calculation and matplotlib for representation.

Definition of an effective temperature

In our statistical analysis of non-magnetic molecular phases in chapter IV, we used correlations, both in real and reciprocal space, to decide if the configurations observed by STM can be compared with the ones predicted for Ising models. We proposed to interpret the experimental observations as arrested configurations2 of a given Ising Hamiltonian. To establish this comparison we proceed in two steps. First, we need to define the model, i.e. an Ising Hamiltonian with its corresponding couplings. To define this model we compute the experimental MSF and try to recognize the phase that can correspond to the observed intensity. Do we observe Bragg peaks or diffuse scattering, i.e. is it an ordered or disordered phase? Do we know a model which exhibits a similar MSF? Once these questions are answered, it is necessary to verify if the experimental observations are comparable with the simulated ones at a defined temperature. This step is which we call definition of an effective temperature for the experimental configurations. Defining an effective temperature has strong implications for the system under study. If all the experimental observations are in agreement with the ones of a model at a fixed temperature, we can consider that the experimental system is a realization of the model at thermodynamic equilibrium. On the contrary, if the experiment exhibits properties that cannot be described by the model at a fixed temperature, the experimental system could be a realization of the model out-of-equilibrium.

Here we detail how experimental and MC simulated pairwise correlations can be com- (T )) are always zero or positive. In this case the chosen model does not explain the experimental observations. Right panel is an example of a model in which the correlations from second to fourth neighbors allow to define an effective temperature, while the first correlation would correspond to a different effective temperature (indicated by the black arrow). In this case we would say that the chosen model explains the experimental observations in an out-of-equilibrium situation.

As we previously said, we first identify an appropriate model using comparison of the experimental MSF to simulated ones. Next, we compare the MC correlations for different kind (k) of neighbors as a function of temperature C M C k (T ) with the experimental correlations C exp k . We finally search for the temperature at which the simulated and experimental correlation values agree best, i.e. we find T that minimizes the so-called sum of squared residuals (SSR):

SSR = n k C M C k (T ) -C exp k 2 (II.19)
This temperature value is what we call the effective temperature of the experimental system, T ef f . If all the experimental correlation values are in good agreement with the simulated ones, i.e. the residual is small, an effective temperature can indeed be defined. Thus, the experimental observations may be considered as representative of arrested configurations of the model at thermodynamic equilibrium. Thus the existence of an effective temperature has two crucial implications: • the experimental system can be interpreted as a realization of the chosen Ising Hamiltonian;

• and in this framework the experimental system is at thermodynamic equilibrium.

In some cases, we used a different observable to quantify the agreement between simulated and experimental pairwise correlations, including the standard deviations of the MC simulated configurations. This quantity is a weighted residual (wSSR), which is not and absolute quantity like SSR, but measures if the experimental points fall in the standard deviation of MC simulation. wSSR writes:

wSSR = 1 n n k C M C k (T ) -C exp k 2 ∆C M C k (T ) 2
where ∆C M C k (T ) is the errorbar of each MC point at a given temperature, i.e. the standard deviation of the correlation C k averaged over many different microstates. wSSR varies from zero to infinity, and if wSSR < 1 the experimental points are, in average, inside the errorbars. This definition gives a quantitative criterion to decide if the agreement between model and experiment is good for a given temperature.

Monte Carlo simulations

Basic principle

Monte Carlo (MC) simulations are used in statistical physics to numerically compute the thermodynamic properties of systems that cannot easily be solved analytically. Assuming a Hamiltonian of the model (H), the goal is to compute the partition function of the system (Z), from which the physical observables and properties can be deduced. In order to know the partition function it is necessary to know all the possible configurations (α) of the Hamiltonian and their corresponding probability of occurrence, that are given by the Boltzmann factor:

Z = α P (α) = α exp - E α k B T
For an Ising Hamiltonian like the one we address in this thesis, the number of possible configurations is 2 N with N the number of particles, which makes it difficult to find an analtycal solution of the problem. Only some exceptions exist, for example the one-dimensional case [START_REF] Ising | Beitrag zur Theorie des Ferromagnetismus[END_REF] and the analytical solution of the ferromagnetic model without external field using transfer matrices [START_REF] Onsager | Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition[END_REF]. In most of the cases using numerical calculations is the only way to explore the behavior of Ising models.

MC simulations are based on the idea that analysis of all possible configurations is not necessary, and focusing on a representative set of states is sufficient. This is a strong hypothesis, which in some cases is indeed relevant. MC techniques sample the microstates space in an iterative way. In each MC step one microstate is generated. From these states, a number of physical quantities can be computed, without explicitly writing the partition function. The simplest MC techniques randomly sample the microstates, which is equivalent to consider a uniform probability of existence for each one of them. However, we know that it will not be the case in Ising systems, in which some states will be of much lower energy, thus much more probable. Selective sampling methods are used in this case. The one used in this thesis is the Metropolis algorithm, that will be detailed now and is schematized in The Metropolis algorithm [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF] consists in sampling the microstates following Boltzmann probabilities given by the form of the Hamiltonian. At each MC step a new configuration is computed from the previous one using local updates of Ising variables. In this way, the Metropolis algorithm builds a Markov chain, i.e. the new configuration only depends on the previous one, for which the probability of transition from one initial state (α) to a new state (α ′ ) needs to be defined. To do so, ergodicity can be used and it is considered that the probability (W α→α ′ ) of transiting from the α state to the α ′ state or vice-versa will be the same, thus:

W α→α ′ P (α) = W α ′ →α P (α ′ )
From this property we have:

W α→α ′ W α ′ →α = P (α ′ ) P (α) = exp - E α ′ -E α k B T
and a form for transition probability that respects the previous condition can be chosen.

To start the simulation a random configuration, α, is chosen and its energy is computed using the Hamiltonian. To make the system evolve single spin flips are used and the iteration process during the simulation is as follows (grey recntangle in Figure II.10):

• A single spin randomly chosen is flipped giving rise to a new configuration α ′ .

• The energy of the configuration (E α ′ ) is calculated, as well as the corresponding transition probability W α→α ′ .

• A random number R ∈ [0, 1] is generated to decide if the updated configuration α ′ is accepted.

• If R < W α→α ′ the flip is accepted and the process restarts with the α ′ state now the initial one. If R > W α→α ′ a new configuration (α ′′ ) is generated by flipping one spin and the iterative process restarts.

In principle, for each accepted state the interesting physical properties (such as energy and magnetization) are computed and stored, as well as the corresponding microstates. The iterative part can be repeated as long as needed to have relevant statistical samples at a certain temperature. The same process can be repeated for different temperatures, starting from highest ones and progressively decreasing to the lowest ones.

In practice, only a part of the microstates are used to compute observables and thermodynamic properties. To be sure that the system is at thermal equilibrium it is necessary to thermalize it, i.e. to make the system evolve through several MC steps before measuring physical quantities (red rectangle in Figure II.10). How to choose the number of thermalization steps before measuring is a complex problem that we will not discuss here. In addition, to find relevant averages for statistical quantities we need to find microstates that are uncorrelated, i.e. different enough. To ensure this, configurations calculated from consecutive MC steps are never averaged together. During the measurement part of the simulation (blue rectangle in Figure II.10) a certain number of MC steps are performed between two stored microstates. This number of steps has to be at least of the same order of magnitude as the number of spins in the system N . Nevertheless, at low temperature transition probabilities are usually small and, even after N MC steps, a small part of the spins in the system have flipped. In this case, the number of steps can be corrected taking into account the ratio of acceptance r = N conf accepted N conf tested of new configurations at a given temperature. At low temperature the algorithm follows three steps:

• It considers a given number of MC steps to evaluate the acceptance ratio r. Once this ratio is known, its inverse gives the number of MC steps to ensure that on average, each spin is flipped. Thus the new modified MC step is equal to 1 r elementary MC steps.

• The system under study is thermalized during a given number of modified MC steps.

• After thermalization, the measurements can be performed.

Acceptance ratio and freezing of the dynamics

It is clear that, as the acceptance ratio r decreases when decreasing the temperature and the number of decorrelation steps N decorr increases, the evolution of the configurations at low temperature is slower. This phenomenon is usually called freezing of the dynamics. This means that finding new accepted configurations of low energy is much more timeconsuming at low temperatures. At some point, dynamical freezing becomes so important that simulations are unable to reach lower temperatures in reasonable timescales. This problem is particularly important in the case of Metropolis algorithm based on single spin flips. Thus, one way of avoiding dynamical freezing is to use well-chosen global updates. These updates consist in flipping groups of spins, for example clusters or loops around the lattice, following rules that help to minimize the energy of the system. Building this type of algorithms is a specific problem for each Hamiltonian, because it does not exist a global rule that allows to minimize the energy in any model. However, in this thesis only local updates were used, because we consider that dynamics in our experimental molecular system is better captured by local updates.

Limitations of Metropolis algorithm on topological systems

Another fundamental difference between local and global updates is that, in some particular models, the whole phase space cannot be sampled using local updates. It is the case when the phase space is divided in disconnected topological sectors, a schematic representation is shown in Figure II.11. Topological sectors are regions of the phase space that can have the same order parameter, for example the magnetization is the same, but are characterized by different topological invariants. Topological invariants are usually scalar numbers or sets of numbers, that are determined taking into account global properties of the phases. For example, the ground state manifold of the triangular Ising Hamiltonian with antiferromagnetic couplings can be labeled by a pair of topological winding numbers W = (W 1 , W 2 ) [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF]. Indeed, local updates would never allow the transition between configurations with different winding numbers, so global updates, like the worm algorithm developed by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] are needed.

We will not go further in the development of global updates, but we will keep in mind that our Metropolis algorithm has two intrinsic limitations: dynamical freezing and impossibility of explore different topological sectors.

Monte Carlo (MC) simulations presented in this thesis were developed on C++ language by Benjamin Canals at Néel Institute in Grenoble (France). Running of the simulations and choice of MC and model parameters were done by Nicolas Rougemaille and María Alfonso Moro, also at Néel Institute in Grenoble (France). In the following sections we describe the parameters we used to perform simulations presented in chapter IV and in chapter V.

MC parameters

To control the evolution of MC simulations we use the following parameters:

• T i , the temperature from which simulation starts. This temperature needs to be much larger than the largest coupling in the system, to ensure that we start from a randomly disordered paramagnetic phase.

• T f , the smallest temperature to simulate. We need a temperature small enough to reach the ground state of the model.

• T ratio , the geometrical ratio that defines the decreasing step of the temperature.

We usually want a ratio small enough to have continuous curves for the thermodynamic quantities (specific heat, magnetic susceptibility) but not too small to optimize computation time. In chapter IV and chapter V the ratio is typically T ratio = 1.01 or 1.02.

• N therm , number of MC steps for each temperature used to thermalize the phase before starting measurements. In chapter IV and chapter V typical values were N therm = 10 4 .

• N meas , number of MC steps at each temperature for which measurements are averaged. In chapter IV and chapter V typical values are N meas = 10 4 .

• N decorr , it determines the number of steps during the measurement process between two stored configurations. In our simulations it was always fixed to 1, meaning that a modified MC step is equal to

N accepted N tested × N M C steps .
• r crit , critical ratio of acceptance to continue the simulation. We recall that a modified MC step is 1 r ×(MC steps). If the ratio r is too small, then 1 r will be large and the simulation can take forever. We therefore define a critical ratio r crit to ensure that when the dynamics slows down the simulation does not take too long. In order for the simulation to be meaningful, one has to force decorrelation. Therefore, if this is not longer possible, the program stops at the current temperature for which it has computed the last acceptance ratio. This parameter allows to stop simulation if T f inal is not reached but the dynamics is frozen. Typical values in chapter IV and chapter V were r crit = 10 -4 .

• [Q x,min , Q x,max ], [Q y,min , Q y,
max ] and q points . When magnetic structure factor is computed the limits of the reciprocal space are defined in x and y directions by

[Q x,min , Q x,max ] and [Q y,min , Q y,max ].
Reciprocal space is discretized and the number of q-points considered is N q = q 2 points . Typically we used q points = 81 and

[Q x,min , Q x,max ] = [Q y,min , Q y,max ] = [-2π, 2π]

Model parameters

MC simulations used in this thesis were always for a triangular Ising model. It was possible to choose values of the first (J 1 ), second (J 2 ) and third (J 3 ) neighbor couplings. An external field (h ext ) can also be included. The Hamiltonian writes:

H = -J 1 ⟨ij⟩ σ i σ j -J 2 ⟨⟨ij⟩⟩ σ i σ j -J 3 ⟨⟨⟨ij⟩⟩⟩ σ i σ j + h ext i σ i σ j
The simulated lattices have the same lateral dimensions (n) in x and y directions, so N = n × n sites. Boundary conditions are always periodic in our simulations, which implies to properly choose the size of our simulations. For example, if the ground state for a given set J 1 , J 2 , J 3 , h ext is an ordered phase with unit cell 3 × 3 spins we need to choose a whole size of the lattice compatible with this cell, i.e. n needs to be a multiple of 3. For the calculation of thermodynamic quantities we considered lattices with N = 20 × 20 = 400 spins. For the calculation of MSF we used bigger lattices of N = 80 × 80 = 6400 spins.

MC output: thermodynamic quantities

During the measurement steps each accepted configuration α in real space is stored and used to compute the following observables:

• Average magnetization,

⟨M ⟩ = 1 N meas Nmeas α M (α)
where

M (α) = N + -N - N + + N -
is the magnetization in a given configuration with N + /N -the number of Ising variables with value +1/ -1.

• Average of the square magnetization,

⟨M 2 ⟩ = 1 N meas Nmeas α M 2 (α) • Average energy, ⟨E⟩ = ⟨ N ij/i̸ =j H i (α)⟩ = 1 N meas Nmeas α N ij/i̸ =j H i (α)
where H is the Ising Hamiltonian defined from the couplings and the external field and sums runs over pairs of i, j spins in a given configuration α.

• Square of the average energy,

⟨E 2 ⟩ = ⟨ N ij/i̸ =j H i (α) 2 ⟩
From these observables different thermodynamic quantities can be computed:

• Specific heat, C(T ) = ⟨E 2 ⟩ -⟨E⟩ 2 k B T • Magnetic susceptibility, χ(T ) = ⟨M 2 ⟩ -⟨M ⟩ 2 k B T
Finally, the entropy can be computed from the specific heat by numerical integration:

S(T ) = C(T ) T dT + S(T → ∞)
where S(T → ∞) = N ln (2) in the Ising model.

MC outputs: correlations

Spin pairwise correlations and MSF are also computed in our MC simulations. Spin pairwise correlations are computed for the neighbors from the first to the twentieth one, the calculation is straightforward from each MC configuration following the formulas given in subsection 3.1. Once the correlations have been computed for each configuration at a given temperature, their average and standard deviation are calculated. The MSF is computed for a relevant set of configurations at each temperature, following the definition presented in subsection 3.2. Average values and standard deviations at each ⃗ q value within a given two-dimensional range are computed for a given temperature.

Chapter In this chapter we describe the experimental results obtained on assemblies of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. We study the surface crystallography of our molecular system, which forms crystals, both ordered and partially disordered. To do so, we first scrutinize the crystals' motifs, focusing on the physical chemistry of the interface between a single molecule and its substrate, down to the atomic scale. Second, we focus on the crystals' lattices, to describe the epitaxy at the organic/metal interface. We show that fullerenes on Cu(111) are a complex system characterized by a large number of structural degrees of freedom. These degrees of freedom include orientation and position of individual molecules, that correspond to different binding configurations with the substrate, and the tendency of the latter to form vacancy islands beneath (some of) the molecules. Combined with the competition between molecule-substrate and molecule-molecule interactions, this gives rise to a remarkably complex problem of hetero-epitaxy. Here, the out-of-plane corrugation of the molecular assemblies and their local lattice distortions play a crucial role. Moreover, a striking phase coexistence is observed. For molecular samples synthesized at room temperature, as much as ten phases with different structure are observed. We introduce a systematic analysis of the thermodynamics of the system, considering the multitude of C 60 /Cu(111) epitaxy configurations that can be envisaged taking into account the different degrees of freedom. This way we come up with an energy landscape, sorting the different phases according to their internal energy. Considering, additionally, entropy effects, our thermodynamics viewpoint accounts for the observed coexistence of phases.

The chapter is divided in five sections. Section 1 reviews previous results from the literature for fullerenes deposited on close-packed metals (a rather active research topic ever since the 1990s and still today). On this occasion we will introduce key concepts used to analyze the molecule-metal epitaxy, relative to the lattice mismatch and the detailed nature of the interface. In section 2 we describe our experimental results on C 60 /Cu(111) samples prepared at different deposition temperatures and observed by STM and nc-AFM techniques. The structure of fullerene monolayers on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] is analyzed, including the structural description of ten coexisting phases. In section 3 we present results from DFT calculations that allowed us to resolve the intrinsic ambiguities of STM imaging. We revisit the analysis of the structure of different phases of fullerenes on copper. Besides, we derive two kinds of interactions energies, between molecules, and between a molecule and its substrate. In section 4, using energetic arguments and semi-quantitative estimates from the DFT calculations, we present an energy landscape of the system, and discuss possible entropy effects. In section 5 we conclude. Finally, in section 6 we propose how our coincidence model could be generalized to other C 60 assemblies on close-packed metal surfaces.

Structural properties of C 60 molecules on metal surfaces: brief review of the literature

Growth of fullerenes on metallic substrates has been extensively studied because of its potential applications in micro/nano-electronics, in new generation of devices whose buildingblocks (e.g., electrodes, conductive channel) would be molecules instead of traditional metals and inorganic semiconductors [START_REF] Guldi | Fullerene for organic electronics[END_REF]. Knowing the details about the interaction and interface between C 60 molecules and metals is a key aspect to control electrode-to-molecule charge injection. As a consequence, many structural and electronic studies of single C 60 on metallic substrates and monocrystalline layers of C 60 have been performed since the nineties [START_REF] Wilson | Imaging C 60 clusters on a surface using a scanning tunnelling microscope[END_REF]2,[START_REF] Sakurai | Scanning Microscopy of Fullerenes Study[END_REF].

The applications are not the purpose of our discussion, but the review of the literature is an important starting point for our study on C 60 /Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] samples. We will see that, despite decades of research on these model systems, several open questions remain, and that common features are observed with different metal substrates. Different arguments have been proposed, yet, to explain the observed C 60 ordering phenomena. We address the case of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] with more detail, to clearly give the state of affairs before our works and to define the important vocabulary for our analysis.

In our discussion of the literature, we chose an artificial division to present the information. In the following paragraphs we treat different structural parameters as being independent, when they are in fact entangled. First, we address the degrees of freedom for single fullerenes, the motifs in the molecular crystal, and we separately consider each one of them (molecular orientation, binding sites on the substrate and formation of vacancies). Second, we describe the structure of molecular assemblies, i.e. the properties of the crystal lattice. At the end of each subsection, we sum up the fundamental results for C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] that have been reported in the literature. This way of presenting the results in the literature aims to simplify the discussion for the neophyte reader. We hope it will be the case.

Single C 60 molecules on metal surfaces

Fullerenes are organic molecules with a football-like structure. With 60 carbon atoms they have 20 hexagonal and 12 pentagonal faces. Carbon bonds are single in pentagonal faces and half of them are double in hexagonal faces. This altogether makes them high-symmetry molecules. They present rotation symmetries (5-fold for pentagonal faces, 3-fold for hexagonal faces) and mirror symmetries (for the bond between two hexagons), see Figure III.1. Taking into account all the possible transformations, we can find 120 symmetry operations, C 60 molecules belong to the icosahedral symmetry group. Combining these symmetries with those of close-packed metal surfaces produces numerous inequivalent binding configurations, allowed by several degrees of freedom: orientation of the molecule with respect to the surface (which face or bond faces the metal surface?), azimutal angle with respect to the symmetry axis and binding sites of the fullerene on the close-packed surface (that can have a 3-fold rotation symmetry or a mirror one) [START_REF] Shi | Survey of structural and electronic properties of C 60 on close-packed metal surfaces[END_REF]. Moreover, fullerenes on close-packed surfaces induce reconstruction of the metal surfaces, in the form of arrays of vacancy islands (one-atom-deep) [START_REF] Stengel | Adatom-Vacancy Mechanisms for the C 60 /Al(111)-(6× 6) Reconstruction[END_REF][START_REF] Felici | X-ray-diffraction characterization of Pt(111) surface nanopatterning induced by C 60 adsorption[END_REF][START_REF] Lin | Anomalous Surface Diffusion of C 60 and Anisotropic Growth of Nano Islands on Ni(111)[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. In the following subsections we discuss these aspects in more details.

Molecular orientation and appearance in STM images

Fullerene molecules can sit on metal surfaces in different manners, each characterized by a specific molecular orientation. We call this different possibilities "orientation of the molecule". on a C-C bond between a pentagon and a hexagon, and with a C atom pointing downwards (so-called apex configuration). High resolution STM images of fullerenes on metal substrate can indirectly reveal the orientation of the molecules. The atomic structure is not directly observed in standard STM measurements, and only the spatial variations of the local electronic density of molecular orbitals are resolved. Simulations of STM images based on DFT calculations showed that these orbitals exhibit symmetries that are reminiscent of the structural symmetries. In other words, the C backbone facing the STM tip, which is a mirror image of the part of the molecule facing the metal surface, can be deduced from the symmetry of the orbitals observed in STM [START_REF] Altman | Determination of the orientation of C 60 adsorbed on Au(111) and Ag(111)[END_REF][START_REF] Sakurai | Adsorption of fullerenes on Cu(111) and Ag (111) surfaces[END_REF][START_REF] Maruyama | Electronic structures of C 60 and C 70 adsorbed on the Cu(111) surface and intramolecular STM images[END_REF][START_REF] Hashizume | Fullerenes adsorption on Cu(111) and Ag(111) surfaces[END_REF][START_REF] Lu | Charge transfer and screening in individual C 60 molecules on metal substrates: A scanning tunneling spectroscopy and theoretical study[END_REF][START_REF] Menech | Energy-resolved STM mapping of C 60 on metal surfaces: A theoretical study[END_REF][START_REF] Schull | Orientationally Ordered (7 x 7) Superstructure of C 60 on Au(111)[END_REF][START_REF] Pinardi | Tailoring on-surface chemistry of (hetero) aromatics on transition metal surfaces[END_REF][START_REF] Jung | Atomically resolved orientational ordering of C 60 molecules on epitaxial graphene on Cu(111)[END_REF][START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF][START_REF] Leaf | A Combined Monte Carlo and Hückel Theory Simulation of Orientational Ordering in C 60 Assemblies[END_REF][START_REF] Chandler | Li@C 60 as a multi-state molecular switch[END_REF][START_REF] Wang | Commensurateincommensurate transition and strain relief patterns in monolayer C 60 on Cd[END_REF].

Besides the orientation, an azimutal degree of freedom exists for each molecule. This azimutal degree of freedom consists on a rotation of the fullerene around the z-axis (the axis perpendicular to the metal surface), as shown in Figure III.1. The azimutal orientation can also be determined from STM contrasts. Surprisingly, there is not one single molecular orientation and azimutal angle observed for fullerenes deposited on each metal, but in most of the cases several configurations were detected by STM. DFT calculations have been used to elucidate the lowest energy orientation for different metals. Results indicate that:

1. -binding energies do not only depend on the orientation of fullerenes but also on the binding site on the metal surface, and both degrees of freedom are entangled; 2. different orientations can have close-by energies and the hierarchy may depend on the metal surfaces.

Next, we will not address all possible metal surfaces but rather focus on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], our substrate of interest.

Molecular orientations and azimutal angles on Cu(111)

Motai [START_REF] Motai | C 60 Grown on the Cu(111)1x1 surface[END_REF], Sakurai and Hashizume [START_REF] Sakurai | Adsorption of fullerenes on Cu(111) and Ag (111) surfaces[END_REF][START_REF] Hashizume | Fullerenes adsorption on Cu(111) and Ag(111) surfaces[END_REF] proposed that fullerenes are adsorbed on hexagonal faces. These studies were based on STM images acquired with different energies and the comparison with calculated local density of states. Fasel et al. [START_REF] Fasel | Orientation of adsorbed C 60 molecules determined via X-Ray photoelectron diffraction[END_REF] confirmed this view using X-ray photoemission diffraction.

Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] later reported two different orientations using STM, one in which fullerenes are adsorbed on hexagonal faces and an undetermined one. They also inferred two different azimutal angles for the molecules, using symmetry arguments and low energy electron diffraction (LEED) measurements [START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF].

Conflicting reports claim, based on DFT calculations, that molecules facing a hexagon to Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] on different substrate sites (top, hcp, fcc and bridge sites, see subsubsection 1.1.2), have two configurations, azimuthaly rotated by 180°with very similar [START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF] or significantly different (0.3-1.0 eV) [START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF] energies.

More recent studies based on STM and non-contact atomic force microscopy (nc-AFM) include other molecular structures, in which molecular assemblies have different epitaxial relations with the Cu(111) lattice. They found coexistence of several molecular orientations [START_REF] Xu | Construction of a Molecular Switch Based on Two Metastable States of Fullerene on Cu(111)[END_REF][START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF], with fullerenes sitting on hexagonal, pentagonal faces, 6-6 bond and apex configuration. Comparing their STM images with STM simulations published by Wang et al. [START_REF] Wang | Commensurateincommensurate transition and strain relief patterns in monolayer C 60 on Cd[END_REF] we suggest that the STM contrast observed in these studies and assigned to the 6:6 bond attached to Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] could in fact correspond to an apex configuration. These studies also showed three different azimutal angles (rotated by 60°) for fullerenes with 6-6 bond (or apex) orientations and two azimutal angles (rotated by 180°) for fullerenes facing a hexagon to Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF].

Summary of literature results for molecular orientations on Cu(111)

To sum up, fullerenes adsorbed on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surfaces exhibit different orientations and azimutal angles, that can be inferred from STM measurements. First studies during the 90's suggested that only fullerenes with a carbon hexagon ring facing the surface were present in the samples. More recent studies have demonstrated that fullerenes can also sit with 6-6 bonds (or carbon apex, according to our interpretation) and carbon pentagon rings facing the Cu(111) surface. For hexagonal and apex configurations several azimutal angles coexist, rotated by 60°and 180°, respectively. Calculated binding energies for different configurations strongly depends on the binding site on the substrate, and no consensus has been reached in the literature. In this thesis we will conclude about the orientation of fullerenes on different epitaxial phases (subsubsection 2.2.2) and present an exhaustive DFT analysis of binding energies, as a function of both molecular orientation and binding sites (subsection 3.1). Determining the binding sites for a molecule is a difficult task, both in experiments and ab initio calculations. For instance, when microscopy techniques are used to measure molecular lattices the metal surface underneath is not visible. We could imagine a situation in which the raw metal surface is observed near a molecular region, but in this case it is very hard to find experimental parameters that allow to simultaneously image both the molecules and the metal atoms. Indeed, the sizes of fullerenes and metal atoms are very different, as well as the needed resolution. In principle, it is possible to deduce the binding sites of molecules measuring diffraction patterns and searching the best fit with a model. However, diffraction patterns average the properties of the lattice on large surfaces and no local information can be accessed. DFT calculations can be used to compute binding energies of fullerenes on different configurations. These configurations can be compared to determine the one with the lowest energy. Nevertheless, DFT calculations have limitations: different approximations for the exchange-correlation term gives different energies, intermolecular interactions such as van der Waals interactions cannot easily be taken into account, etc. Some authors [START_REF] Shi | Survey of structural and electronic properties of C 60 on close-packed metal surfaces[END_REF] have proposed symmetry arguments to decide which binding sites may be preferred for fullerenes on close-packed metals (Figure III.3b, c). It has been proposed that the symmetry axis of the molecules might align with the ones of the metal surface, which depend on the binding site. For example, fullerenes sitting on top or hollow sites (C 3 symmetry) would adsorb with a hexagonal face (also C 3 symmetry) towards the surface, and the symmetry axis would be aligned. Another favored configuration (according to this argument) would be a fullerene sitting in a bridge site (C 2 symmetry) with a 6-6 bond facing the surface (also C 2 symmetry). Nevertheless, these symmetry arguments should apply to all the close-packed metal surfaces, but experiments and DFT calculations have proposed different combinations of binding sites and molecular orientations, depending on the considered metal substrate. More careful analysis of the geometrical structure for each metal is necessary to conclude.

Binding sites of fullerenes on M(111) surfaces

Binding sites on Cu(111)

Figure III.4:

Schematics representing a fullerene adsorbed on hollow sites on the closepacked surface of Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Three layers of copper are represented by white, gray, black dots. The blue hexagon represents a hexagonal face of the molecule facing the metal surface. Note that copper lattice parameter is very close to the distance between carbon atoms on the fullerene hexagonal face.

In subsubsection 1.1.1 we have explained that first studies of fullerenes on Cu(111) re-1. Structural properties of C 60 molecules on metal surfaces: brief review of the literature ported hexagonal orientations [START_REF] Motai | C 60 Grown on the Cu(111)1x1 surface[END_REF][START_REF] Sakurai | Adsorption of fullerenes on Cu(111) and Ag (111) surfaces[END_REF][START_REF] Hashizume | Fullerenes adsorption on Cu(111) and Ag(111) surfaces[END_REF][START_REF] Fasel | Orientation of adsorbed C 60 molecules determined via X-Ray photoelectron diffraction[END_REF]. Using the same symmetry arguments cited before, i.e. the symmetry group of the molecule is the same than the one of the binding site and the symmetry axis of both are aligned (see Figure III.3), these authors proposed that fullerenes are sitting on top or hollow sites.

Moreover, geometrical arguments have been invoked for the particular case of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. The lattice parameter of Cu( 111) is a Cu = 2.556 Å which is very similar to the distance between carbon atoms on hexagonal faces of fullerenes. Thus, when a hexagonal face is placed on a hollow site three carbon atoms of the molecule coincide with three copper atoms of the surface, and strong binding between them should be favored. This type of configuration is shown in Figure III.4.

Nevertheless, these symmetry and geometry arguments are not enough to predict or explain the possible binding sites of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. During the nineties, structural analysis of STM images was used to determine binding sites, but no clear conclusion emerges [START_REF] Motai | C 60 Grown on the Cu(111)1x1 surface[END_REF][START_REF] Sakurai | Adsorption of fullerenes on Cu(111) and Ag (111) surfaces[END_REF][START_REF] Hashizume | Fullerenes adsorption on Cu(111) and Ag(111) surfaces[END_REF][START_REF] Fasel | Orientation of adsorbed C 60 molecules determined via X-Ray photoelectron diffraction[END_REF]. Motai et al. [START_REF] Motai | C 60 Grown on the Cu(111)1x1 surface[END_REF], Hashizume et al. [START_REF] Hashizume | Intramolecular Structures of C 60 Molecules Adsorbed on the Cu(111)-(1x1) Surface[END_REF] and Sakurai et al. [START_REF] Sakurai | Adsorption of fullerenes on Cu(111) and Ag (111) surfaces[END_REF] suggested that fullerenes are centered on both hcp and fcc sites and have two possible azimutal angles, altogether, four configurations exist. According to Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] only one of the two hollow sites, hcp or fcc (which of these two was not determined at the time), is occupied by fullerenes. All these studies addressed the position of fullerenes in molecular assemblies, i.e. their conclusions do not concern one single fullerene but a lattice of C 60 molecules in which intermolecular interactions should be present. Another limit of these studies is that they did not consider a priori the presence of other molecular orientations on the samples.

Larsson et al. [START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF] analyzed the case of single fullerenes on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface. They combined STM measurements at low temperature, in which they moved fullerenes using the STM tip and analyzed the binding positions, and DFT calculations. They experimentally proved that, at low temperature: 1 -C 60 molecules sit on the substrate with a hexagonal face towards the surface; 2 -two different binding sites coexist, but one of the two sites is strongly preferred (probability of occurrence of 9/10 with respect to the other site); 3 -2 different azimutal angles are observed depending on the site. Their DFT calculations were in contradiction with experiments and they were only able to conclude that the two sites can be top, hcp or fcc sites and their azimutal angles differ by 180°.

Other DFT studies have addressed the binding energy of fullerenes on different sites, when the molecules form lattices [START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. They concluded that the hcp site is the lowest energy one. According to Wang et al., bridge sites and fcc sites have the same binding energy and top sites are higher in energy. Even if hcp site has lower energy than the others, the energy difference between hcp site and fcc (or bridge) site is ∼ 0.02eV, thus smaller than thermal energy at room temperature, which could explain why previous studies found a coexistence of binding sites.

More recent studies found completely different results. In 2012 [START_REF] Xu | Detailed low-energy electron diffraction analysis of the (4×4) surface structure of C 60 on Cu(111): Seven-atom-vacancy reconstruction[END_REF], Xu et al. used LEED data to conclude that the preferred binding site is a top site. None of the previous studies took into account the possibility of finding fullerenes with other orientations, but in 2020 and 2021 [START_REF] Xu | Construction of a Molecular Switch Based on Two Metastable States of Fullerene on Cu(111)[END_REF][START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF] other molecular orientations, such as a pentagonal face, a 6-6 bond or a C atom (apex) facing the copper substrate, have been detected by STM. No DFT calculations exist for these orientations, so the binding energies of fullerenes on different sites when the orientation is one of the previous ones are not known.

Summary of literature results for binding sites of fullerenes on Cu(111)

No consensus was obtained in the literature, i.e. all high-symmetry binding sites (top, hcp, fcc and bridge) were argued to be the lowest-energy ones. One limitation of previous DFT studies is that only one of the possible orientations of fullerenes was considered: a hexagonal face towards the substrate. Recent STM studies have demonstrated that this orientation is not the only existing one, so new DFT calculations are needed. In addition, it is important to distinguish the case of single fullerenes from the case of C 60 assemblies. In the first case, only the interaction with the substrate drives the physics; whereas in the second one intermolecular interactions are expected to play a role. Taking into account various experimental observations and ab initio calculations, we suggest that the most probable scenario is a coexistence of different binding sites in fullerenes assemblies on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. In section 2 we discuss in detail which binding sites are compatible with our experimental STM images. In section 3 we present an exhaustive DFT analysis of binding energies for different molecular configurations. The interaction between the C atoms of the fullerene molecules and the surface metal atoms can be very strong. As a consequence, the lowest-energy binding configuration can be strongly distorted, compared to the case of the isolated molecule and metal surface. Moreover, some metal substrate are "alive" at the temperatures used to deposit the molecules, in the sense that the energy barriers to the formation of isolated metal adatoms are below thermal energy. Altogether, this opens the door to a complete reconstruction of the metal surface in presence of the C 60 molecules, in particular when the metal substrate is at room temperature or above. Experiments have demonstrated that substrate reconstruction is a thermally activated process. When samples are annealed the number of fullerenes sitting on reconstructed sites increases, it was the case for C 60 assemblies on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Motai | C 60 Grown on the Cu(111)1x1 surface[END_REF][START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF], Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Wang | Reconstructions of C 60 on the Ag(111)-1x1 surface[END_REF] and Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Maxwell | Covalent bonding and surface reconstruction[END_REF].

Surface reconstruction and molecular height

Substrate reconstruction consist on the formation of one-atom-deep vacancies created underneath the fullerene molecules. These vacancies increase the number and strength of C-metal bonds, thus the system energy can be decreased significantly. Vacancies can be of two types, as shown in Figure III.5, consisting of one or seven missing atoms. We illustrate how in these cases the molecules can form a larger number of bonds with the metal surface. While this number is typically 3 on the unreconstructed surface, it increases to 6 in the case of the one-atom-vacancy and even 9 in the case of the seven-atoms-vacancy (see metal atoms highlighted with a red contour in Figure III.5).

Formation of one-atom vacancies was established for C 60 molecules deposited on Pt(111) [START_REF] Felici | X-ray-diffraction characterization of Pt(111) surface nanopatterning induced by C 60 adsorption[END_REF], Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Li | Surface Geometry of C 60 on Ag(111)[END_REF] and Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] substrates [START_REF] Stengel | Adatom-Vacancy Mechanisms for the C 60 /Al(111)-(6× 6) Reconstruction[END_REF]. For Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], seven-atom vacancies are the favored reconstruction [START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF], and coexistence with one-atom vacancies has been recently proposed [START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF]. Seven-atom vacancies also form in Ru(0001) [START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF], Co(0001) [START_REF] Fourmental | Evidence of a C 60 /Co interface reconstruction and its influence on magnetic properties[END_REF] and Ni(111) [START_REF] Pang | Manipulating Magnetism at Organic/Ferromagnetic Interfaces by Molecule-Induced Surface Reconstruction[END_REF]. For Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] both one-atom and seven-atom vacancies have been proposed in the literature [START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF][START_REF] Tang | Complex orientational ordering of C 60 molecules on Au(111)[END_REF][START_REF] Tang | Probing the buried C 60 /Au(111) interface with atoms[END_REF]. The mechanisms leading to the formation of one-atom vacancies instead of seven-atom vacancies (or vice-versa) have been discussed by Shi et al. [START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF], and we will analyze the case of vacancies on Cu(111) in subsubsection 2.2.3.

Other types of vacancies with 3 or 6 missing metal atoms have been studied by DFT in the cases of Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Tang | Two bonding configurations for individually adsorbed C 60 molecules on Au(111)[END_REF][START_REF] Kaiser | Vacancy patterning and patterning vacancies: Controlled self-assembly of fullerenes on metal surfaces[END_REF], Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF], Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Kaiser | Vacancy patterning and patterning vacancies: Controlled self-assembly of fullerenes on metal surfaces[END_REF], Be(0001) [START_REF] Kaiser | Vacancy patterning and patterning vacancies: Controlled self-assembly of fullerenes on metal surfaces[END_REF], Co(0001) [START_REF] Fourmental | Evidence of a C 60 /Co interface reconstruction and its influence on magnetic properties[END_REF] but they seem to be less stable than one and seven-atom vacancies. Moreover, to our knowledge, it does not exist any experimental evidence of such structures. Thus, we do not consider them in the following discussion.

Fullerenes sitting on vacancies can exhibit a different molecular height than C 60 molecules on non-reconstructed sites. In particular, in the case of seven-atom vacancies the size of the vacancy allows C 60 molecules to "sink" in the vacancy and bond with metal atoms of the second metal layer. We could expect a very different molecular height in STM measurements, but even if the presence of vacancies can be detected, interpreting molecular heights is not as obvious as one may think, due to the contribution of electronic effects to the STM signal. Indeed, molecules within a vacancy or on an unreconstructed metal surface a priori possess different electron densities of states. It is hence not straightforward to tell, from mere inspection of an STM image, whether a molecule sits within a vacancy or simply binds in a specific way onto an unreconstructed surface, and if it sits within a vacancy, which kind of vacancy it is.

Vacancies and molecular heights on Cu(111)

For Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] apparent molecular height in STM changes by ∼ 2 Å [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] when seven-atom vacancies are present below fullerenes. This distance is almost the value of the Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] lattice parameter (a Cu = 2.56 Å). Another STM study [START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF] measures a height lower by 1.7 Å for fullerenes on seven-atom vacancies and an intermediate height, interpreted as one-atom vacancies, lower by 1.2 Å. The same measurement using nc-AFM gives height differences of 1.7 Å and 1.0 Å. We will compare these results with our measurements of molecular heights in subsubsection 2.2.2 and with our DFT calculations for Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] in section 3. Our analysis allows us to make the difference between electronic and topographic features. We also conclude that only seven-atom vacancies form in our samples.

Fullerenes assemblies on close-packed metal surfaces

In the previous subsection, we discussed the degrees of freedom associated to each individual fullerene molecule bonding on a metal surface. Given the absence of consensus in the literature, it is reasonable to assume that this is actually a complex problem, and that several low-energy fullerene-on-metal configurations, close in energy, might coexist. The complexity of the system goes beyond this state of affairs. Indeed, experiments show the formation of molecular assemblies, or lattices, with different structures on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. In this sense, there is a question of epitaxy, and the notion of commensurability and its relation with order/disorder are crucial here.

Commensurate phases of C 60 on M(111)

Commensurate phases are characteristic of hetero-epitaxial systems, in which a layer of a deposited material (fullerenes for us) grows on top of another material with different lattice parameter (close-packed metal substrate). In the case of commensurate structures a supercell (or superstructure) can be defined. It comprises the smallest number of unit cells of both the substrate and the deposited lattices [START_REF] Hillier | Epitaxial interactions between molecular overlayers and ordered substrates[END_REF][START_REF] Hooks | Epitaxy and Molecular Organization on Solid Substrates[END_REF]3]. In the case of C 60 molecules on metal surfaces the commensurability condition between two-dimensional lattices can be written in a matrix form:

⃗ a S1 ⃗ a S2 = i j -j i -j ⃗ a C 60 1 ⃗ a C 60 2 = m n -n m -n ⃗ a M 1 ⃗ a M 2
where the subscript S is for superstructure, C 60 for fullerenes lattice and M for metal lattice. The values i, j, m, n are natural numbers.

Alternatively, the commensurability condition can be described using Wood's notation (p×p)Rφ where p is the ratio of the two lattice constants and φ the relative lattice orientation. We will use the notation (p × p)Rφ for the relation between the fullerene and metal lattices, and (p ′ × p ′ )Rφ ′ for the relation between the superstructure and fullerene lattices. It can be shown that [3]:

p = a C 60 a M = m 2 + n 2 -mn i 2 + j 2 -ij p ′ = a S a C 60 = i 2 + j 2 -ij φ = ∠(⃗ a C 60 , ⃗ a M ) = arctan (in -jm) √ 3 2(im + jn) -(in + jm)
1. Structural properties of C 60 molecules on metal surfaces: brief review of the literature Most of the phases observed for fullerenes on close-packed metal surfaces have been determined as commensurate structures. In Table B.2, in Appendix B, we summarize phases reported for Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. In the four first columns the metal lattice parameter, the Wood's notations (p × p)Rφ and (p ′ × p ′ )Rφ ′ , and the molecular lattice parameter (a C60 ) are given. Last column of the table specifies the experimental technique that was used to observe the experimental phase in the literature and corresponding references.

φ ′ = ∠(⃗ a S , ⃗ a C 60 ) When φ = φ ′ = 0, the problem is one-dimensional,
Diffraction experiments are appropriate techniques to reveal the epitaxial relation between two layers, here the molecular layer and the metal surface lattice. This is less straightforward for microscopy techniques, because measuring simultaneously the molecular and the metal lattices is usually not possible. Indeed, characteristic sizes of molecules and metal atoms are very different, and the experimental conditions needed to measure each lattice are different. When diffraction data are not available (or are too difficult to interpret, due for instance to the coexistence of various kinds of superstructures, as it is the case for fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]), the two-dimensional commensurability condition can be deduced from real space measurements. Scanning probe techniques such as STM can be used to determine the angles φ, φ ′ and lattice parameter ratios p, p ′ of the Woods' notation. Note that, STM images of molecular lattices with several hundreds of molecules, where all molecular units are resolved, can be Fourier transformed to perform a sort of local diffraction experiment. This treatment allows to obtain a high precision on the determination of φ, φ ′ , p, p ′ . Nevertheless, using this kind of Fourier transforming is not common in the literature. In most of the cases, when no diffraction experiment is performed, commensurability relation is deduced having only a part of the information. In particular, the angle φ, that is typically measured with a precision of ∼ 1°, and the parameters of the superstructure φ ′ , p ′ . The parameters corresponding to the superstructure appear as a periodic molecular contrast in STM images, from which φ ′ , p ′ can be deduced (more details are presented in subsubsection 1.2.2). In our analyses (and in many cases in the literature) a supplemental hypothesis is made: only low-order commensurate superstructures form (in the one-dimensional case, this means that the situation represented in Figure III.6b does not occur). For such commensurabilities, the amount of positional coincidences between the two lattices is maximal, which is sought to provide the greatest stabilitsation to the system (formation of more interfacial bonds) [START_REF] Grimmer | Coincidence-Site Lattices and Complete Pattern-Shift Lattices in Cubic Crystals[END_REF][START_REF] Smith | Bollmann's 0-lattice theory; a geometrical approach to interface structure[END_REF][START_REF] King | Generalizing the coincidence site lattice model to non-cubic materials[END_REF][START_REF] Therrien | Matching crystal structures atom-to-atom[END_REF]. This so-called coincidence-site lattice theory (or 0-lattice theory) is a simplification of the old heteroepitaxial problem studied by Frank and van der Merwe in one-dimension [START_REF] Frank | One-dimensional dislocations . II . Misfitting monolayers and oriented overgrowth[END_REF][START_REF] Frank | One-dimensional dislocations. I. Static theory[END_REF].

Thus, except when diffraction experiments or careful analysis of images in real space using the Fourier transform are performed, the commensurate phases shown in Table B.2 are reasonable guesses about the epitaxial relation between C 60 molecules and M(111) surfaces. When a star accompanies Woods' notation in the table, the commensurate phase is our interpretation taken into account data in the literature. When no star is shown, we simply reproduce the guess proposed in the literature.

All the commensurate phases have molecular lattice parameters around the fullerite (3D lattice of fullerene) lattice parameter of ∼10 Å [START_REF] Girifalco | Interaction Potential for C 60 Molecules[END_REF][START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF]. These values are also in good agreement with the equilibrium lattice parameter for a two-dimensional lattice that we obtained using DFT calculations (see subsection 3.3): a C60-eq = 10.05 Å. We found, for different commensurate phases reported in previous publications (see Table B.2 in Appendix B), lattices parameters between 9.60 Å and 10.22 Å, corresponding to small (biaxial) strains between -4.5% and +1.7%.

The lattice constants of the superstructures range from one to seven times the molecular lattice parameter, from 9.60 Å to 70.14 Å. Supercells with the largest unit cells seem less common, only 2 out of the 19 reported phases having a lattice constant exceeding 3 × a C60 . For some of the superstructures, the unit cell is revealed by an ordered height pattern for the molecules, which translates the presence and absence of metal vacancy islands underneath some of the molecules. For these phases periodic or quasiperiodic height patterns form, revealing the presence/absence of substrate reconstruction, different binding sites and molecular orientations depending on the binding sites. In this cases it is easier to correctly determine the epitaxial relation of the molecular lattice and the metal surface.

Commensurate phases on Cu(111)

For fullerenes deposited on Cu(111) five commensurate phases have been reported in the literature.

When samples are deposited with the copper substrate at temperatures above room temperature (60 K or 70 K above) or post-annealed after deposition, one single commensurate phase is observed, with Woods' notation (p × p)Rφ = (4 × 4)R0° [START_REF] Motai | C 60 Grown on the Cu(111)1x1 surface[END_REF][START_REF] Sakurai | Adsorption of fullerenes on Cu(111) and Ag (111) surfaces[END_REF][START_REF] Hashizume | Fullerenes adsorption on Cu(111) and Ag(111) surfaces[END_REF][START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF][START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF][START_REF] Yamada | Electronic modification of C 60 monolayers via metal substrates[END_REF]. This phase is shown in Figure III.7a.

In samples deposited at room temperature, the (4 × 4)R0°phase and other four commensurate phases coexist [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF][START_REF] Yamada | Electronic modification of C 60 monolayers via metal substrates[END_REF]. These phases are shown in Figure III.7b-e. These molecular phases were identified by their φ angle, ranging from 0°to 26°. They have varying strain levels, ranging from 2.2% to -3.2%.

Phase coexistence on the same STM image is also shown in Figure III.8c. Such a phase coexistence was also reported on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Sakurai | Adsorption of fullerenes on Cu(111) and Ag (111) surfaces[END_REF][START_REF] Hashizume | Fullerenes adsorption on Cu(111) and Ag(111) surfaces[END_REF][START_REF] Wang | Reconstructions of C 60 on the Ag(111)-1x1 surface[END_REF] and Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Guo | Thermal diffusion of C 60 molecules and clusters on Au(111)[END_REF][START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF][START_REF] Tang | Complex orientational ordering of C 60 molecules on Au(111)[END_REF][START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF]

Corrugation and height patterns in fullerenes assemblies

We previously explained that fullerenes deposition on metals tends to reconstruct the substrate and in particular to promote the formation of vacancies. Here, we describe how vacancies form in the case of C 60 molecular assemblies (instead of single fullerenes). In particular, we describe how vacancies form ordered (or non-ordered) lattices below the fullerene layer.

The presence of vacancies can be directly detected with STM, keeping in mind that not only height variations are revealed with this technique, but also variations of electronic properties associated to varying orientations of the molecules for instance. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF][START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF]. Last columns of Table B.2 (Appendix B) summarizes the nature of the vacancy patterns reported in the literature on various metal surfaces, in terms of their lattice and type of vacancy (1 or 7 atoms).

Molecular height patterns on Cu(111)

The case of fullerenes on Cu( 111) is specially rich. The five phases reported in the literature all exhibit specific height patterns. Three of them are uniform or periodic and vacancies organize in ordered lattices. The two others are not ordered and approximately half of C 60 molecules sit on vacancies. Non-ordered patterns have also been observed for other metals, in particular for Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] (see Figure III.8) but the case of Cu( 111) is unique. Non-ordered height patterns for fullerenes on silver and gold seem randomly disordered, and vacancies seem to cluster under extended regions of the molecular lattices (clusters of typically 10 molecules and more). On Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] two different non-ordered height patterns exist. For one of the phases, lines of vacancies form (Figure III.7d). For the second one (Figure III.7e), vacancies seem to avoid to form under neighboring molecules, i.e. we do not observe clusters of neighboring fullerenes with the same height. These two phases seem to be disordered in a non random way (thus correlated), which is actually very unusual and intriguing, and will be at the heart of our work presented in the next chapters.

Incommensurate, aperiodic phases and local distortions of C 60 on metal surfaces

Formation of commensurate phases seems to be the rule for C 60 molecules deposited on metal surfaces. Nevertheless, few studies suggest that more complex, non-commensurate, structures can form [START_REF] Pai | Origin of peculiar STM molecular contrast in C 60 /Ag(111)[END_REF][START_REF] Pai | Ordering of an incommensurate molecular layer with adsorbate-induced reconstruction: C 60 /Ag(100)[END_REF][START_REF] Hsu | Aperiodic incommensurate phase of a C 60 monolayer on Ag(100)[END_REF][START_REF] Wang | Commensurateincommensurate transition and strain relief patterns in monolayer C 60 on Cd[END_REF].

Fullerenes on Ag(100) organize in a close-packed structure which is incommensurate with the metallic lattice underneath [START_REF] Pai | Ordering of an incommensurate molecular layer with adsorbate-induced reconstruction: C 60 /Ag(100)[END_REF][START_REF] Hsu | Aperiodic incommensurate phase of a C 60 monolayer on Ag(100)[END_REF]. The in-plane structure of the molecular lattice cannot be described using simple relations with the substrate. Formation of vacancies has been observed in these samples, which form patterns with short-range order. The repetition of the short-range patterns is aperiodic, meaning that at long distances height patterns are uncorrelated. These patterns have been interpreted as a combination of the reconstruction of the metal surface, together with lateral displacements of the fullerenes with respect to the positions in a perfect close-packed triangular lattice. The authors propose that this peculiar behavior can be explained by a strong competition between substrate-molecule and molecule-molecule interactions.

Local distortions with respect to a perfect triangular lattice of fullerenes have also been proposed in the case of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. In this case, fullerenes could slightly move to occupy more favorable binding sites than the ones corresponding to the perfect commensurate lattice. Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] proposed that these displacements are periodic and correlated to the ordered height patterns of the superstructures.

For fullerenes on Cd(0001), a transition from a commensurate phase with long range order to an incommensurate phase has been reported [START_REF] Wang | Commensurateincommensurate transition and strain relief patterns in monolayer C 60 on Cd[END_REF]. Upon thermal annealing, several structural changes occur:

1. a periodic lattice of vacancies form; 2. the whole lattice rotates; 3. fullerenes locally move from the commensurate lattice positions.

Moreover, local displacements seem to be coupled to changes of the fullerene orientations.

The final incommensurate phase strongly recalls the formation of static distortion waves predicted by Novaco and McTague [START_REF] Novaco | Orientational Epitaxy -the Orientational Ordering of Incommensurate Structures[END_REF] and recently observed in other molecular systems [START_REF] Meissner | Flexible 2D Crystals of Polycyclic Aromatics Stabilized by Static Distortion Waves[END_REF][START_REF] Ward | Soft Crystals in Flatland: Unraveling Epitaxial Growth[END_REF]. Like in the previous example on Ag [START_REF] Kontorova | On the theory of plastic deformation and twinning[END_REF], the incommensurate phase would relax strain in the molecular layer.

Even though incommensurate phases are not very frequent, these two examples illustrate the rich behavior of fullerenes deposited on metals. Numerous mechanisms play a role in this type of systems: coincidence with the substrate, reconstruction by forming vacancies, strain and intermolecular interactions. Competition between these ingredients leads to the formation of many different phases, some of them not being trivially ordered or disordered but exhibiting complex local order, aperiodicty or incommensurate phases.

Experimental observations on C 60 on Cu(111)

Binding of molecules in the (4x4)R0°"uniform" phase

Most of the experimental studies of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] address the molecular phase selectively formed when samples are annealed or deposited above room temperature. This phase is commensurate with (4 × 4)R0°unit cell of the substrate matching a (1 × 1)R0°molecular unit cell [START_REF] Motai | C 60 Grown on the Cu(111)1x1 surface[END_REF][START_REF] Fasel | Orientation of adsorbed C 60 molecules determined via X-Ray photoelectron diffraction[END_REF][START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF][START_REF] Xu | Detailed low-energy electron diffraction analysis of the (4×4) surface structure of C 60 on Cu(111): Seven-atom-vacancy reconstruction[END_REF][START_REF] Xu | Construction of a Molecular Switch Based on Two Metastable States of Fullerene on Cu(111)[END_REF]. Upon annealing, this phase features a reconstruction of the metal surface, with an array of seven-atom vacancies below each C 60 molecule, and molecular apparent height measured by STM decreases by ∼ 2 Å [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF].

The (4 × 4)R0°commensurate phase is also observed at room temperature. In this case, vacancies form under part of the molecules only, and domains with molecules all sitting in vacancies coexist with domains with molecules all sitting on the unreconstructed surface [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. At room temperature the (4 × 4)R0°phase coexist with other commensurate phases, presented in subsection 2.2.

Here we present our results for this phase. We confirm the conclusions from previous reports in the literature, and shed new light on the so far unsolved question about the nature of the binding sites for the molecules in this phase.

Overview of the molecular deposits performed at high temperature

We deposited a low amount of fullerene molecules on clean Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], which lead to the formation of compact moleclular islands consisting of one layer of molecules, not fully covering the surface (sub-monolayer deposit). The Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface was held at 330-370 K during deposition. In Figure III.9 we show some of the STM images acquired, at room temperature. Similar results were obtained when the substrate was post-annealed at 383 K after molecular deposition.

Most of the islands consists of fullerene molecules with a uniform height. Some of the molecules within these domains appear brighter. Height profiles show that the apparent height of the molecules with respect to the copper surface is either between 520 pm and 590 pm (profiles a and b1 in Figure III.9) or around 740 pm (profile b2). We recall that STM only measures apparent heights and not necessarily the true topography, due to electronic effects. We measure a height difference of 210 pm, in good agreement with the reported difference of 200 pm [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. Pai et al. concluded that the height difference corresponds to the presence (520 pm-590 pm) and absence (740 pm) of vacancies underneath the molecules. To simplify our notation in this manuscript, we will use the names "low"/"high" molecules for the presence/absence of vacancies.

In Figure III.9a, b, molecular rows are horizontal, which is precisely the [1-10] crystallographic direction of Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] (measured on a clean region of the copper crystal using STM), i.e. the dense-packed rows of the surface. Thus, the orientation of the molecular lattice is φ = 0°. In these images resolution is not good enough to clearly distinguish molecular orbitals and determine the orientation of each fullerene in the lattice. However, some molecules show different appearances. Three-fold symmetry and circular apperance observed in the inset of Figure III.9b suggest that the corresponding molecules sit on hexagonal and pentagonal faces, respectively (see Figure III.2 for comparison). Moreover, a detailed analysis of the height of these molecules (not shown) gives apparent height variations of ∼ 0.4 Å, that are probably due to different orientations of C 60 molecules.

Yellow arrows in the STM images indicate the presence of domain walls (DW) between molecular assemblies. These DW are grain boundaries probably formed during growth, for instance upon coalescence of molecular islands having nucleated at different sites. They inform us about the relative position of the fullerenes on adjacent domains. An analysis of this DW allows us to deduce which type of binding sites can be occupied by fullerenes, see subsubsection 2.1.2.

To sum up, our measurements are in agreement with previous observations reported in literature. At temperatures above 330 K we observe a molecular phase:

1. aligned with the copper substrate; 2. exhibiting a uniform height contrast; 3. with heights in agreement with the formation of seven-atom vacancies; 4. with signatures of different molecular orientations.

In Figure III.9(c,d) the angle formed by the molecular assemblies is slightly rotated with respect to the copper lattice (from 3°to 9°), but the points 2, 3 and 4 on the previous paragraph keep the same. Even at high temperatures, we found the coexistence of different commensurate phases, which was never reported in the literature. Further analysis is detailed in next subsection 2.2.

New insights on the binding sites with STM

As explained in subsubsection 1.1.2, several binding sites have been proposed in the literature for the (4x4)R0°phase, but no consensus has emerged so far. Coexistence of sites have been proposed both in experiments and DFT calculations, in particular for the non-reconstructed (4 × 4)R0°phase observed at room temperature. Here, we analyze binding sites in this phase by comparing the relative positions of the molecular lattices in different domains separated by DWs (Figure III.9). Note that we are interested here in measuring small distances with STM. Doing that with data acquired at room temperature requires careful correction of imaging distortion artifacts (see subsubsection 1.2.2).

In Figure III.10 we show an example of the type of analysis we performed. In this case we compare the positions of molecular domains in which fullerenes sit within vacancies (apparent height of ∼ 550 pm) or on the unreconstructed Cu(111) surface (200 pm higher). An example We assumed that molecules within vacancies are centered on top sites (blue circle). Green circles account for positions of fullerenes on nonreconstructed sites. Both bridge and hollow sites seem to be occupied by molecules sitting on the unreconstructed surface. Our analysis points out to coexistence of different binding sites for adjacent molecular domains, but we are not able to precisely determine the nature of the sites, both hollow sites and bridge sites being compatible with the results presented in Figure III.10c.

The same type of analysis was performed on STM images in which molecular domains with the same molecular height were observed (not shown). In this case, our data also suggested that not only one binding site exists for this molecular phase. Note that the uncertainities, associated with the positioning of the regular grids in our analysis, amount typically 0.3 Å-0.7 Å, i.e. hardly sufficient to distinguish neighbor high symmetry sites (separated by typically 0.7 Å-1 Å).

Phase coexistence at room temperature

Fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] samples prepared at room temperature exhibit coexistence of different commensurate phases. In samples prepared at room temperature, Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] reported the coexistence of five different commensurate phases, each characterized by a specific crystallographic orientation, strain, molecular height pattern, see Figure III.7. Nomenclatures Table III.1: Correspondence between the different nomenclatures that can be used to name the various observed phases of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Our notation simply gives the molecular lattice constant (a C60 ) and the angle (φ) between the dense molecular rows and the dense Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] rows. The Wood's notation uses the ratio between the phases and Cu lattice constants (along two 120 • -rotated directions) and the angle between the orientation of the two lattices. The strain being always biaxial the notation (p×p) is simplified to (p). The matrix notation gives the transformation between the unit vectors of the phase and those of Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [3]. The nomenclature describing only the molecules' height patterns, used by Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF], gives the ratio between the phases' lattice constants and the shortest intermolecular distance. The last nomenclature uses qualifiers to describe the height patterns [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. The last column indicates the number of molecule per unit cell.

a C60 Å-φ • Wood's (p)
(3 √ 7/2)R19.1 • 6 9 -9 -3 (2×2)R0 • 4 9.68 Å- 22.4 • ( 43/3)R22.4 • 1 7 -7 -6 ( √ 3 × √ 3)R30 • Disordered maze 3 9.98 Å- 26.3 • ( √ 61/2)R26.3 • 5 9 -9 -4 (2×2)R0 • Linear- wall maze 4 10.33 Å-30 • ( √ 588/6)R30 • 7 7 -7 0 ( √ 3 × √ 3)R30 • 3
for each phase are given in Table III.1. Phase coexistence in epitaxial systems is not very common. In most of the cases one phase has a significantly lower energy and is thus the only observed phase, provided that growth is not performed too far from equilibrium (e.g. at too low temperature or with too fast deposition rates). Observing metastable states is rare, not to mention several metastable states, like here. We wonder whether these metastable phases have similar energies caused by the competition between the substrate-molecule and molecule-molecule interactions. Attempting to model this complex problem has rarely been reported to our knowledge, and although some of the possible key ingredients needed to envisage such modeling have been invoked by Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF], the full set of relevant ingredients involves a large number of degrees of freedom, hence implying a complex multi-dimensional phase space. By making a number of strong hypotheses, we propose a simplified, but predictive approach to this question. • determine the commensurability relation of each phase with the substrate, which we translate in terms of a crystallographic orientation and a strain value;

• resolve the origin of different molecular heights (topography or molecular orientation);

• and propose a structural model for each phase.

This analysis allows us to propose ten commensurate phases corresponding to our observations and to identify the ingredients needed to model the energetics of the system.

Figure III.12:

Zoom on different STM images showing all the commensurate phases we found on the surface. We use angle and height contrast to make the difference between the phases and found: 0°and uniform "high" or "low" height; 5-6°and uniform "low" contrast with occasionally "high" molecules; 8-10°and uniform "low" contrast with some "high" defects; 14-16°with two different contrasts 1 -"low" uniform with "high" defects, 2periodic lattice following a √ 7 × √ 7 supercell; ∼18°with 2 × 2 periodic contrast; ∼23°with non-ordered height contrast with half of the molecules "high" and half "down"; ∼26°with non-ordered height contrast but formation of local ordered lines of "high" and "low" molecules; ∼30°with non-ordered height contrast but formation of local "zigzag" patterns. Colored picture frames will be used as color-legend for each phase in the following figures.

In-plane structure: rotational ordering, strain and coincidence with the substrate

In this section we address the in-plane structure of the coexisting molecular phases. Which angle do molecular phases form with the copper lattice? Can we find evidence of strain on the fullerenes' layers? Considering this two parameters we propose a commensurate structure and show the type of coincidence for each phase with the substrate. This way we revisit the literature, clarifying some of the previous claims and proposing alternative interpretations, and we carefully characterize the new phases we have discovered.

Orientation angle of fullerenes lattices on Cu(111) phases

First, we develop a statistical analysis of the rotation angle in different phases. To do so, we zoom on corrected STM images and measure the lattice vectors of the molecular phase for each molecular region, in each STM image (see subsubsection 1.2.2 for details). The most discriminating parameter between phases turns out to be their orientation angle, while the variations in lattice parameter are small, smaller than 5%, hence difficult to determine with precision based on room temperature STM imaging. A careful analysis of the lattice parameters is presented later, but we start addressing the orientation that is easier to measure.

In Table III.2 we indicate the average value of the angles we measure, with their standard deviation. We verify that the structure of each phase is defined by a characteristic angle, measured both in real and reciprocal space (see Figure III.15 and Figure III.16). These angles range from 0°to 30°and take well-defined values, i.e. rotation of the molecular layer on the substrate is not a continuous variable. If incommensurate phases were possible, we would expect a continuous variation of the angle. This is not the case here, indicating that coincidence with the substrate plays a major role. Thus, we expect well defined lattice parameters for each phase, given by the φ angle and the coincidence with the substrate.

From this hypothesis and the calculation of possible commensurate phases we can propose structural parameters for each phase. In subsubsection 1.2.1 we explained how commensurate phases are determined by the matricial relation between the substrate and the molecular lattice vectors. By taking the unit vectors of the copper lattice and sweeping different values of the indices that determine the relation matrices (i, j, m, n), we can compute the angles and lattice parameters of all possible commensurate phases. As explained in subsubsection 1.2.1, we generate all the commensurate phases having reasonable strains and short superstructure period. Figure III.13 shows the value of the superstructure lattice parameter, as a function of the orientation of the molecular layer. Colorscale indicates strain levels in the molecular layer, with respect to the reference value for the equilibrium distance on a two-dimensional fullerene layer a C60 = 10.05 Å (see subsection 3.3 for details). We predict 22 phases, 9 of which (colored circles in Figure III.13) correspond to the experimental STM observations, and 2 of which (colored squares with ? symbol) probably as well (the resolution in the STM data does not allow an unambiguous conclusion, yet). We can compare this graph with phases shown in Table III.2.

Table III.2:

Average angle between the fullerene and Cu dense rows measured for different phases. The standard deviation is given by the errorbars. Counts (#) indicate the number of measurements for each phase (observed on a collection of STM images). NB: This number of measurements indicates the statistical validity of our analysis, but it should not be used as a measurement of the population of each phase on the surface. Indeed, this statistic is biased by the way in which we develop the experiments, because our initial goal was to characterize the 9.68 Å-22.4°phase. Phase names with a ? symbol indicate that we were unable (for the moment) to choose which of the two commensurate phases correspond to our experimental data. Now that the commensurate structures are determined, we can produce cartoons of the atomic (copper) and molecular lattices. This kind of representation allows us to visualize to which extent the atomic and molecular positions coincide, for each phase, which should be related to the relative stability of the different phases (more coincidence implying, at least at first thought, stronger bonding). In Figure III.14, fullerenes are colored depending on their coincidence with the copper atoms. Molecules positioned on high-symmetry sites are represented with a colored circle, while fullerenes sitting on intermediate, low-symmetry binding sites are represented with black circles. For each phase we only represent molecules within one unit cell of the superstructure. We observe phases with one, three, four and seven molecules per cell. We needed to make a choice for the position of the corners of the unit cell with respect to copper atoms. Indeed, we could build many other commensurate phases by shifting the molecular layer with respect to the substrate. The representation we chose is the "most symmetric" one, meaning that we maximize the number of molecules inside the unit cell that are sitting on high-symmetry sites of the substrate.

Phase

Strain and lattice parameter of fullerenes on Cu(111) phases

We have presented an analysis of the orientation of the phases that allows us to identify commensurate phases and to determine the degree of coincidence of each phase with the copper substrate. Now, we discuss whether the experimental lattice parameters of the phases are in agreement with the structures we propose. This is not an easy information to get, at 111) depending on their orientation and supercell lattice parameter, considering strains < 4% and superstructure lattice constants < 30 Å. Black squares correspond to calculated phases, not observed experimentally. Colored circles correspond to calculated phases matching the experimental phases. Color indicates the strain in the molecular lattice. The two colored squares correspond to the two possible commensurate phases that could explain the orientation we observe at 8.4 ± 0.8°. Curves correspond to the relation between orientation and superstructure lattice parameter for incommensurate structures, for which the fullerenes layer can rotate continuously on the copper surface.

the limit of what can be discriminated with STM measurements at room temperature. Here we take advantage of the periodicity of the lattice to improve the accuracy when measuring distances. In particular, we observe characteristic distances for different phases thanks to the Fourier transform. The analysis was done according to the following steps:

• We choose one STM image in which phase coexistence is observed and we correct experimental hysteresis and drift (see subsubsection 1.2.2 for details about correction on the images).

• We zoom in regions of the STM image in which several phases coexist and compute the Fourier transform of the image.

• We filter the STM image considering separately each component on the Fourier transform. This way we can produce the equivalent of dark-field images highlighting each phase. Next, we determine the orientation of the phases, and extract radial scans intercepting the first-order harmonics of each phase in the two-dimensional Fourier transforms.

• We compare the radial scans for different orientations and classify the phases according to their lattice parameter. This analysis is difficult for three reasons. First, STM images showing coexisting phases III.2, putting in evidence the local positional coincidences between the molecules (represented with large circles) and the substrate atoms (gray circles). When molecules are black, the binding site do not correspond to a highly symmetric site.

are usually large (100 nm-200 nm), hence a single global correction of the hysteresis and drift effects is sometimes impossible. Second, the area occupied by a particular phase on an STM image may be small compared to the full STM image. In such case their Fourier harmonics can have low intensity, sometimes hardly discernable, compared to a strong background level. Third, some phases have very similar orientations (for example the ones at 19°, 23°and 26°) so that their Fourier harmonics overlap. We try to limit the importance of these effects by working on small-size regions of interest in the STM images that always contain a reference phase (4 × 4)R0°. We repeated the same type of analysis on a different STM image in which coexistence was observed (Figure III.16). Filtering of the image using the Fourier transform also gives values of the angles in agreement with our previous analysis. By cropping the image to increase the signal to noise ratio, we detect five components in reciprocal space, corresponding to phases at 0°, 8°, 17°, 23°, and 31°. Inspecting the radial scans in the Fourier transforms we can classify the phases with decreasing lattice parameter, following the sequence: (0°& 30°), (17°& 8°), 23°. It seems that for this STM image, the analysis does not allow to make the difference between the lattice parameters of the 10.22 Å-0°and 10.33 Å-30°phases. The 10.3 Å-8.2°structure does not seem a good guess for the observed 8°-rotated phase, whose lattice parameter is significantly lower and would better correspond to the 9.8 Å-9.28°s tructure. The 17°phase can be interpreted as a 10.09 Å-16.4°phase.

Out-of-plane structure: molecular heights and orientations

In the previous sections we revealed the coexistence of different commensurate phases of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] samples deposited at room temperature. We were able to describe the in-plane structure for each phase, including the orientation and the strain of the phase. So far, we have essentially forgotten about molecular heights and height patterns described in the subsubsection 1.2.2 of this chapter. To fully characterize the phases we now address these aspects, in relation with the molecular orientation.

STM apparent heights and molecular orientations

We start with a statistical analysis of the molecular heights, using apparent height profiles (few hundreds) extracted from STM images. As a reference for the heights we chose molecules sitting within seven-atom vacancies (rather than the Cu surface, whose imaging is unstable at room temperature due to the presence of Cu adatoms). We unambiguously recognize these molecules as they are the lowest ones in all the observed phases and have a characteristic height with respect to the copper surface of ∼-4.3 Å, which is smaller than the covalent diameter of fullerenes (∼7 Å). We found four characteristic molecular heights (names h 1 to h 4 ). Some of them are only found on some of the commensurate phases. We compute the average and standard deviation of all our measurements for each category of height in each phase, which we show in Figure III.17a. We indicate the measured height as a function of the orientation of the molecular layer. For phases with orientations between 5°and 15°we consider all the profiles together and found a common value for each height. This approach is justified because height patterns of these phases are very similar, as we detail later.

The key question here is the following: are these different apparent heights a topographic feature or are they caused by different molecular orientations? To answer this question we need to inspect the appearance of the molecular orbitals in STM. Doing so in STM measurements at room temperature is not possible. Thus we performed STM measurements at low temperature (5K on a setup based in Lille, see subsubsection 1.2.1 in chapter II for more information). We need to be cautious about these measurements. Fullerene phases on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] could undergo structural reorganization upon cooling down below the growth temperature, because of compression effects of the copper lattice due to temperature. We hence will not base our analysis on the structural properties of the molecular layers at low temperature, but only verify height differences and look at orbital contrasts in high resolution STM images. Some of these images are shown in Figure III.17c-f. We choose different molecular regions in which the local height patterns are the same as those we observed at room temperature and identify the observed heights. We indeed observe a correlation between the apparent height and the orientation of the fullerenes. For h 1 we can see molecular orbitals with a circular shape, indicating adsorption on a pentagonal face (see Figure III.2 for comparison). For h 2 the orbitals exhibit a two-fold symmetry with two different lobes. This contrast corresponds to the adsorption on a 6-6 bond or on an apex carbon atom. Moreover, C 60 molecules on this configuration can be rotated along their azimutal angle by 60°. For h 3 a three lobe structure is found, that we identify as fullerenes with a hexagonal face towards the surface. These molecules appear with two different azimutal angles, differing by 180°. Finally, molecules at height h 4 appear only on the phase with 0°orientation (the one observed at high temperature) and present a circular orbital, associated to the adsorption on a pentagonal face. This orientation is the same as the one for the lowest fullerenes (h 1 ) but the heights differ by 2 Å, in agreement with previous height measurements.

We have thus demonstrated a correlation between the apparent heights and the molecular orientations. However, height differences as large as 2 Å are necessarily due to reconstruction of the copper substrate. To complete our study we used nc-AFM measurements (also at low temperature) which reveals mainly topographic features (see subsubsection 1.2.3 for more information). These measurements allow us to make the difference between non-reconstructed and reconstructed sites on the substrate.

Height in nc-AFM measurements

In Figure III.18 four different nc-AFM images are shown. Each one corresponds to a different region, in which STM contrasts were comparable to the ones observed at room temperature on different commensurate phases. First we acquired a STM image, allowing us to determine the height (h 1 , h 2 , h 3 , h 4 ) and the orientation of each fullerene molecule. Second, we measured the tunneling current and the frequency shift of the tip with the Z-feedback loop open. To obtain these images we needed to progressively decrease the distance between the tip and the surface. When this distance is very small the interaction between the tip and the fullerenes is repulsive, and the frequency shift of the resonator is > 0 (white contrast on the images). For larger distances, the tip interacts in an attractive way with the molecules, yielding a negative frequency shift of the resonator (dark contrast on the images). (A-C) images were taken in the attractive regime, while (D) was taken in the repulsive regime, which explains the inverted contrast. Current images are measured simultaneously with nc-AFM, which makes them useful to control the position of the tip on the sample.

In Figure III.18A we detect molecules with two different heights h 3 and h 4 , for which we can distinguish orbital contrasts on STM images. When looking at the nc-AFM, we observed very similar contrasts for both heights, except that the fullerenes appear with a different shape with the two imaging modes. For three-lobes orbitals in STM, we found a hexagonal shape with nc-AFM; for circular orbitals the nc-AFM reveals a pentagonal symmetry. We conclude that the difference between heights h 3 and h 4 has an electronic origin, i.e. directly related to the orientation of the molecules and the orbital they present to the scanning tip of the microscope. We do not expect different reconstructed interfaces for these two heights.

In Figure III.18B and D we distinguish three different apparent heights on STM images: h 1 , h 2 and h 3 . Both in attractive and repulsive regimes in nc-AFM measurements we observe three different contrasts. Thus we expect a topographic difference. Our data are in good agreement with those in a recent report [START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF], which also detects three different topographic contrasts for fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], measured with nc-AFM and ∆f (z) spectroscopy. Molecules at height h 1 are sitting on seven-atom vacancies, following previous studies that combine STM, LEED and DFT [START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. Fullerenes at height h 3 are expected to bind to non-reconstructed sites of the copper surface. Remaining molecules at height h 2 exhibit a different topographic contrast, intermediate between the case of non-reconstructed surface and formation of seven-atom vacancies. Forcieri et al. [START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF] have proposed that these molecules sit on one-atom vacancies. In this one-atom vacancies fullerenes cannot sink like in the case of seven-atom vacancies and the observed height is an intermediate one. However, there is no proof of this scenario. We propose an alternative scenario: fullerenes on some particular binding sites on the copper substrate form stronger bonds, leading to this inter- mediate height. In conclusion, we confirm the topographic origin of heights h 1 (formation of seven-atom vacancies) and h 3 (fullerenes on non-reconstructed sites). We also detect a topographic origin for the height h 2 , but for the moment we are unable to conclude about the structure at the fullerene-copper interface in this particular case. This structure will be deduced using our DFT calculations in subsection 3.1.

In Figure III.18C we distinguish a new feature for molecular heights. On the STM image we observe fullerenes with height h 1 and h 2 . Closer inspection of the image shows that two different heights h 2 appear, which we refer to as h a 2 and h b 2 . This observation was only found in regions in which height patterns form lines of "high" and "low" molecules. This height pattern is characteristic of the phase 9.98 Å-26.3°, see Figure III.12. Using nc-AFM we were able to detect a small difference between these two heights, meaning that our observations are not only an electronic effect. Careful inspection of one high resolution STM image at low temperature allows us to quantify the height difference between both heights: h a 2 -h b 2 = 23pm. At room temperature we measured h 2 = 87 ± 14, meaning that these two heights h a 2 and h b 2 are "hidden" in the standard deviation of our measurement. We propose that these two heights correspond to fullerenes sitting on the same copper surface (reconstructed or not, but with the same base level) but binding to different sites. We will go deeper in this idea in the next section, in which we will reconsider our commensurate phases taking into account the observed height patterns.

A more detailed analysis of the nc-AFM images would be possible if we quantitatively take into account the variations on the resonator's frequency and compare them with ab initio calculations. We could also complete the study by using KPFM measurements. This type of analysis is however out of the scope of this thesis. From our analysis of molecular heights combining STM and nc-AFM, we conclude that there are four different apparent heights. These heights correspond to three different to-pographic levels of molecules on the copper surface: non-reconstructed sites, seven-atom vacancies and intermediate height due to formation of one-atom vacancies or strongest binding with the substrate. Moreover, slight variations in nc-AFM contrast reveal that molecules with very similar contrast in STM images and the same orientation are probably sitting on different binding sites. A summary of these ideas is shown in Figure III.19.

Summary of molecular heights and molecular orientations

Relation between in-plane and out-of-plane structures: formation of vacancies, lattice distortions and steric effects

In the previous sections we have separately studied the in-plane and out-of-plane structure of molecular phases. We determined the commensurate structures that correspond to experimental observations. We also explained the origin of the different molecular heights. But, what is the relationship between the in-plane and out-of-plane degrees of freedom -is there any? Phases with different commensurabilities, hence orientations exhibit different height patterns. In Figure III.12 the correlation between orientation and molecular height patterns can be seen. Here, we combine the information from the two previous sections and investigate how surface reconstruction occurs on commensurate phases. We will see if coincidence with the substrate is sufficient to explain the observed height patterns.

When looking at the different commensurate phases we remark that most of fullerenes on the surface are sitting on seven-atom vacancies. Depending on phases we found sevenatom vacancies below each fullerene (10.22 Å-0°phase), below > 6 7 of the molecules (the five phases with orientations between 5°and 17°), below 3 4 of fullerenes (10.14 Å-19.1°phase) or below 1 2 (9.68 Å-22.4°and 9.80 Å-26.3°phases). Which are the requirements needed to form vacancies? As detailed in subsubsection 1.1.3, the formation of vacancies is a thermally activated process, apparently very favorable in the case of copper. It has been calculated that formation of vacancies increases the strength of the molecule-metal bonds, leading to a gain in the binding energy of ∼ 2 eV-3 eV [START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF][START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF]. In this case, why vacancies do not form under each fullerene deposited on the surface? Regarding the structure, fullerenes on seven-atom vacancies are positioned in a particular binding site of the substrate: C 60 molecules are placed on top of a missing copper atom of the first layer, i.e. on a hollow site of the second copper layer. With this remark in mind, the question we would like to ask is, whether the commensurate structures proposed above are compatible with the formation of vacancies. Or, in other words, is the coincidence between molecules and metal atoms the appropriate one to allow the formation of vacancies? Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] have proposed that small, local distortions of the molecular lattice can take place. Fullerenes could shift their position in the plane of the surface to find the better coincidence with the substrate underneath. Such displacements may lead to local reductions of the intermolecular distances, possibly in a regime where molecule repel each other. There is hence potentially a competition between binding to the substrate and interaction with neighboring molecules. We will examine this idea, starting from the easiest possible assumption: fullerenes on the commensurate phase can shift their position to be on a top site, corresponding to the position in which seven-atom vacancies can form.

Model with reconstruction and local distortions: role of steric effects on the formation of vacancies

Here, we consider all the commensurate structures we calculated and showed in Figure III.13, even the ones we did not experimentally observe. For each commensurate phase we allow local distortions of the molecular lattice. We plot the coincidence of the fullerene and copper layer when we allow each fullerene to move to the nearest position in which a seven-atom vacancy can form (top site). This calculation is done using a home-developed Python code, in which we start from the positions for a perfect commensurate phase and we allow the shifts mentioned above for each fullerene in the unit cell. Remember that commensurate structures are periodic from one unit cell to the other, thus we assume that local distortions are also periodic. For each phase we compute the distortions for four different initial positions of the commensurate supercell, corresponding to four highly symmetric sites (top, hcp, fcc and bridge) on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Three other commensurate phases with larger supercell are not shown in the figure, but the following discussion also applies for these phases. Copper atoms are the gray circles, and when a vacancy forms we eliminate the seven atoms involved in the substrate reconstruction. Blue dots indicate the position of fullerenes in the commensurate phase without distortions. Black dots are the center of C 60 molecules when distortions are considered. Red hexagons indicate the copper atoms with lower coordination (only 4 neighboring atoms) that form strong bonds with the fullerene sitting on a vacancy [START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. Colored frames around the cartoons indicate phases experimentally observed, black frames are commensurate phases that we did not detect in our samples. Full frames correspond to phases for which we observe the formation of vacancies under each fullerene (or at least for 6 7 of the molecules), meaning that the proposed commensurate structure is in agreement with experimental data. Dotted frames are experimental phases in which vacancies were not present under each C 60 molecule, so the proposed commensurate phases with distortions do not agree with our data.

Results of this calculation are presented in

There is a correlation between the orientation of the phases and the formation of vacancies: for five out of six possible commensurate phases with angles smaller than 19°we observe the formation of vacancies below each molecule of the lattice. For the 10.14 Å-19.1°p hase, 3 4 of the fullerenes sit on vacancies. For larger angles, experimental phases 9.68 Å-22.4°, 9.80 Å-26.3°and 10.33 Å-30°, only half of the molecules sit on vacancies. The fraction of molecules on vacancies overall increases for larger values of the orientation angle. It is also worth to remark that there is no correlation between the molecular lattice parameter and the quantity of vacancies in a given phase. For instance, we find phases with expanded (10.22 Å-0°) and compressed (9.8 Å-9.28°) lattice parameters for which vacancies form under each molecule.

Following ideas proposed by Shi et al. [START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF] we consider the possible role of copper atoms with low coordination number. For molecular phases with orientations < 19°neighboring red hexagons never have a common (low-coordinated) Cu atom. The only exception is the phase 9.68 Å-6.6°, that we did not detect experimentally. When orientation increases beyond 19°, red hexagons start to merge, which implies that a number of Cu atoms should bind simultaneously with two C 60 molecules. Shi et al. proposed (for other metal substrates) that this situation cannot be favorable, which prevents the formation of such a high density of vacancies. Another way of understanding the same idea is assuming that vacancies feel repulsive mutual interactions, which can be seen as a steric effect. This qualitative argument is perfectly consistent with our experimental observations: when copper atoms are shared by two fullerenes, vacancies do not form. Thus, we conclude that orientation of the phase plays a major role in the formation of seven-atom vacancies.

Model with reconstruction, local distortions and limit angle

Taking into account all these informations we define new criteria to build the commensurate phases with substrate reconstruction and local distortions of the molecular lattice. We now define a maximum value for in-plane displacement of fullerenes. This radius is chosen to avoid fullerenes to "jumping" from one high-symmetry binding site of the substrate to another one, so that only shifts from low-symmetry sites to top sites are allowed. Moreover, we add a phenomenological criterion based on our experimental data. For commensurate phases with orientation > 19°, only half of the molecules in the unit cell can form vacancies. We come up with the commensurate phases represented in Figure III.21, where we only show the case of the phases experimentally observed.

Resulting height patterns are in agreement with the experimental ones for seven of the commensurate phases (solid lines of the frames), whereas four of the height patterns do not correspond to the ones observed experimentally (dotted lines of the frames). All the commensurate phases with uniform or periodic height patterns are correctly predicted by our model. For the 10.09 Å-16.4°phase we show two different structures, corresponding to different initial positions of the corners of the supercell (hollow or top sites, respectively). In the first one, six fullerenes over seven sit on vacancies and the height pattern forms a periodic lattice with √ 7 × √ 7 periodicity. In the second one, vacancies are present below each fullerene. Both patterns were observed experimentally for this phase. Same reasoning applies for phases rotated by 5.4°, 9.28°, 10.9°and 14.4°. For these phases we always observe unform height patterns in which vacancies form under every fullerene. However, Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] found evidence of √ 7 × √ 7 periodic height pattern in the 10.04 Å-10.9°phase.

The two phases exhibiting non-ordered height patterns in our samples (26.3°and 22.4°) cannot be predicted in the framework of the coincidence model, because we only consider periodic supercells. However, the 9.80 Å-26.3°phase exhibits short range formation of "high" and "low" lines of fullerenes. Our model results in a phase with alternating lines of reconstructed and non-reconstructed sites. We only need to consider 60°rotations of this pattern and combine them to find a non-periodic height pattern with local order (alternating "high" and "low" rows of fullerenes), like the one in our STM images. For the 9.68 Å-22.4°phase we experimentally found non-ordered height patterns without any short range periodic recognizable pattern. This disorder could correspond to a complex mixture of the two proposed structures with 1 3 or 2 3 of vacancies. These two disordered phases will be analyzed using specific methods usually applied to disordered phases frustrated magnets in chapter IV. The phase 10.33 Å-8.2°exhibits a periodic height pattern in our model, but our experi- mental observations with orientation (8.4 ± 0.8)°revealed a uniform height, so they probably correspond to the phase rotated by 9.3°.

Finally, the 10.33 Å-30°phase is not well reproduced by our model. Experimentally we observe zigzag patterns, in which "high" and "low" zigzagging lines of fullerenes alternate. This type of pattern has a 2 × 1 unit cell, while the supercell of the commensurate phase is a ( √ 3 × √ 3)R30°cell. Using our periodic coincidence model we never find height patterns compatible with the observations. Nevertheless, considering a global shift of part of the unit cell to tile the molecular lattice, we are able to account for our experimental observations for this phase. This case is out of the application of our model but will be explained in more details in the following paragraphs.

Experimental evidence for local and periodic in-plane distortions

In the previous section we assumed the existence of local distortions in the molecular layers, but do we experimentally observe these displacements? Here, we verify the existence of in-plane displacements in STM images (previously shown, Figure III.12), which requires a careful analysis because molecular shifts with respect to the perfect commensurate lattices are very small < 0.5a Cu ≲ 10%a C60 .

In Figure III.22 we show two STM images of commensurate phases with a periodic molecular height contrast. For these images we were able to semi-quantitatively reveal the presence of local in-plane distortions. We also verify that they are consistent with the commensurate models presented before. In both STM images we observe larger gaps (black regions) between some of the neighboring fullerenes, indicating local variations of the nearest neighbor distance. We superpose two types of lattices to these images. First, we build a perfect commensurate lattice without in-plane distortions. In each top left panel of Figure III.22a,b this superposition is represented. Corresponding ball models are shown in the bottom left panels. Second, we proceed the same way for a commensurate lattice with the expected in-plane displacements that would allow the formation of seven-atom vacancies. Superposition of those lattices on STM images are shown in the top right insets for (a) and (b). The second type of superposition better fits the experimental data, in particular larger nearest neighbor distances between fullerenes coincide very well with the intermolecular gaps observed on STM images.

We conclude that our model including lateral displacement of C 60 molecules on the commensurate phases is in agreement with our experimental data. We have demonstrated that local displacements occur inside each unit cell of the superstructures and repeat from one cell to another in the lattice. This behavior has been verified for phases with periodic height contrasts. Looking at our coincidence models we also expect this type of local distortions in the case of uniform and disordered phases. For disordered phases there is not repetition of the same supercell, but several rotations and combination of different supercells could exist. In this case, localizing periodic in-plane displacements turned out to be impossible with the tools we used so far. In Figure III.12, for the phases rotated by 22.4°and 26.3°, we also observe some gaps between neighbor molecules, pointing out that local non-periodic displacements are probably present on these phases too.

Another type of in-plane displacement is also present in our samples. It does not occur inside each unit cell, but global lateral shifts of entire unit cells are observed. This class of displacement gives rise to the apparition of domains separated by walls. This behavior was only observed for the 10.33 Å-30°phase. In Figure III.23 we show an evidence of this type of displacements. In (a) we show a zoom on the zigzag height pattern, in which we see molecular gaps (black lines) parallel to the zigzag pattern (DW marked with red dotted lines). We extracted a height profile perpendicular to these lines (Figure III.23b). We experimentally measure two characteristic distances d 1 and d 2 , where d 1 corresponds to short distances between fullerenes and d 2 to (10 ± 2)% longer distances. To explain both the local displacement of fullerenes and the zigzag height pattern we propose a coincidence model in Figure III.23c. In the bottom part of the illustration fullerenes are sitting on non-distorted positions (blue dots) which naturally corresponds to a formation of vacancies underneath half of the molecules, and to a zigzag height pattern. On the top part we find the same kind of pattern, in which the whole structure is shifted. A longer distance (red arrows), by 8% of the lattice parameter, is expected according to this coincidence model. This displacement can be seen as the formation of a domain wall, and places fullerenes on sites where vacancies can form according to the zigzag pattern.

Distortions of the molecular lattice hence seem to be the rule for all the commensurate phases. Fullerenes on Cu(111) form complex structures, in which in-plane displacements of the molecules, that can be associated with an elastic energy cost, are compensated by the formation of vacancies. In section 4 we address the energetic cost of each one of these mechanisms and deduce the energy of each commensurate phase.

Summary of observed structures

We have analyzed all the observed commensurate phases and described their in-plane and out-of-plane degrees of freedom. We considered the coincidence with the substrate, including local distortions of the molecular lattice; and apparent height of fullerenes, addressing the formation of vacancies and molecular orientation. Our STM measurements, combined with nc-AFM measurements at low temperature, and careful analysis of the possible commensurate relations allow us to propose a structure for each observed phase. In Figure III.24 we represent the most frequently observed phases that have different height patterns. For each phase we indicate the coincidence with the substrate when distortion and reconstruction of the substrates are considered. We also show the orientation of each fullerene on the phase, deduced from high resolution STM images.

For phases with uniform height contrast we found the formation of seven-atom vacancies below each fullerene and most of the molecules are sitting on pentagonal faces, with some exceptions sitting on hexagonal phases. These phases correspond to the case of 0°orientation (Figure III.24 B), for which no local distortions are needed to reconstruct the surface; and the phases with angles between 5°and 17°, in which distortions appear (only one example is shown for orientation 16.4°, Figure III.24 C). The phase rotated by 0°also exists without reconstruction of the substrate (not shown in the figure), as it was discussed in subsubsection 2.1.1. In this case, no vacancies form and fullerenes can sit on different binding sites of the substrate that have very similar binding energies. Most of the fullerenes are sitting on hexagonal faces, but some molecules on pentagonal faces were also observed.

Phases exhibiting periodic height patterns have larger orientation angle, from 14°to 20°. In these phases vacancies form only below a certain ratio of fullerenes ( 67 or 3 4 ) and important lateral distortions are needed for the reconstruction of the substrate. Molecules on vacancies are orientated with a pentagonal face towards the surface. Molecules on non-reconstructed sites are sitting on apex sites on most of the cases, but some molecules sitting on hexagonal faces were also observed. These two types of molecules (apex vs hexagonal faces) differ not only by their orientation, but also by their height. With our current data we are unable to conclude if another type of reconstruction (one-atom vacancies) exist on the samples or if we simply observe a topographic contrast linked to more or less strong C-Cu bonds, but this question is answered in section 3.

Phases showing non-ordered molecular height patterns are cases in which steric effects between vacancies play an important role. Their orientation is > 20°, and forming seven-atom vacancies is only possible underneath half of the fullerenes. These phases can have an even number of molecules per cell (9.98 Å-26.3°phase) or an odd number of molecules per cell (9.68 Å-22.4°phase). In the first case, forming half of vacancies inside the unit cell is possible. In the second case, note that the experimentally-observed formation of vacancies below 50% of the molecules cannot be obtained within a single unit cell, but rather with clusters consisting of an even number of unit cells, each with 1 3 or 2 3 of vacancies. These phases exhibit complex height patterns. For the 9.98 Å-26.3°phase the height pattern repeats but including rotations respecting a three-fold symmetry. On the 9.68 Å-22.4°phase no clear height pattern can be visually detected, but a mixture of the two possible unit cells seems a reasonable explanation. In both cases in-plane distortions are present. Concerning the molecular orientation, molecules sitting on vacancies are also on pentagonal faces, while the other molecules are mainly on apex configuration, with some exceptions observed on hexagonal faces.

Finally, the phase exhibiting a zigzag height pattern (10.33 Å-30°) cannot be well described by the repetition of a unit cell. Indeed this phase is formed by laterally shifted unit cells, separated by domain walls. In this phase we also observe half of the molecules sitting on vacancies. Molecular orientations were on pentagonal faces on vacancy sites and on apex configurations for non-reconstructed ones. It is worth to note that this phase, with the corresponding zigzag height pattern, was not frequently observed and never extends over large domains (typically, 7 nm at most).

Until now we have described commensurate structures of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], considering all the usual degrees of freedom for fullerenes adsorbed on metals. Figure III.24 is a complete description of our proposed structures, based on our experimental observations. A thorough analysis of all the phases has never been published in the literature for this system. In the next sections we address the energy of each of these structures, taking DFT results as inputs for a model in which molecule-substrate and intermolecular interactions are considered. Details about the structure allow us to estimate the energies of each phase, in order to build an energy landscape and answer the question we asked at the beginning of this section: why does the molecular system exhibit a coexistence of ten different structural phases, rather than a single ground state?

DFT study of C 60 on Cu(111): clarification of STM contrasts and molecule-substrate vs. molecule-molecule interactions

In this section we present our DFT calculations for the C 60 on Cu( 111) system (details about the method were presented in section 2 of chapter II). The goal of the section is twofold: 1 -compare the predicted structures with our measurements and with results from the literature; 2 -determine the energy scales of the different interactions present in the system (molecule-substrate, molecule-molecule). In section 4, these energies will be used as input parameters to estimate the energy of each commensurate phase and build the energy landscape of the C 60 on Cu( 111) system.

First, we address the binding energy of the fullerenes depending on the binding site, the molecular orientation and the reconstruction of the substrate. Second, we discuss the apparent height observed in STM images and the orientation of the molecules, using DFT calculations of the STM tip-molecule conductance and DFT simulations of the STM images. Third, we present the calculation for the intermolecular interaction. Fourth, we discuss the charge transfer from the copper substrate to the C 60 molecules and the existence of a long range dipolar interaction between fullerenes.

Binding sites and energies

On the unreconstructed Cu(111) surface

Details about the calculation of the binding energy

Here we discuss the binding energy and structure of fullerenes adsorbed on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] for different configurations, when the substrate is not reconstructed. For each of the computed configurations we chose a binding site and a molecular orientation, and we relaxed the structure to find the minimum of the total energy of the system comprising copper atoms and fullerenes. Once the structure is relaxed, fullerenes were displaced from their equilibrium position along the axis perpendicular to the copper surface (z) and the total energy of the system was calculated at different distances. We consider the energy at "infinite" distance, i.e. z > 15 Å where no interactions between the copper and the molecules are expected, as a reference. Thus, the molecule-Cu interaction energy is E C60-Cu (z) = E total (z < 15 Å) -E total (z = 15 Å). The binding energy is defined as the minimum of this energy vs z. This type of calculation is performed in two different cases. In one case, far-appart fullerenes on the copper surface are considered. In the other, a periodic lattice of fullerenes forming a (4 × 4)R0°phase is considered. The only difference between the two cases is the lateral size of the supercell used for the calculations. A supercell much larger than the interaction distance between C 60 molecules is considered in the first case. In the second case, a small copper cell, corresponding to the (4 × 4)R0°periodicity is used. In the case of farapart fullerenes, we expect the binding energy to be determined only by the copper-fullerene interaction. In the other case, intermolecular interactions via the substrate can appear and modify the binding energy. It is important to note that van der Waals interactions are not taken into account in these calculations, neither in the case of far-apart fullerenes, nor for the (4 × 4)R0°lattices, where they may play an important role. Indeed, van der Waals interactions are included in the Fireball code using perturbation theory, which requires to know the energy of each separate object (molecule, atom) in the system, before including perturbations. When a metal substrate is considered, it is not possible to separate the system in different sub-systems to compute their energy, so a perturbative approach is not suitable.

One of the goals of our calculations was to compare binding energies corresponding to different adsorption configurations, each corresponding to different molecular orientations and binding sites. Some of the configurations have been addressed in the literature [START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF][START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF][START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF], but no systematic study, including all the variety of configurations we observe experimentally, has been published. When binding energies are not computed with the same code, DFT results cannot be easily compared. Thus, we calculate the binding energies for many combinations of molecular orientation and binding sites, using the same code (Fireball, see section 2 of chapter II for details) and the same calculation cell (same number of copper atoms), in order to be able to directly compare the energies.

In Table III

Far-apart fullerenes: molecular orientations

For all the configurations we considered, the structure of the system was relaxed and we find negative binding energies corresponding to stable configurations, Table III.3. All our binding energies are between -4.3 eV and -2.1 eV.

Previous DFT studies of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] mainly considered molecular configurations in which fullerenes have a hexagonal face towards the surface, corresponding to a three-lobe contrast in STM. This studies were in agreement with STM experiments at low temperature, for which it was demonstrated that isolated fullerenes bind with an hexagon carbon ring towards the surface [108]1 . Our own DFT calculations show that other orientations are stable and even have lower energy, which is in agreement with our experimental observations. We found using DFT that fullerenes sitting on the 6-6 bond and in the apex configurations (in some particular binding sites) are more stable than the hexagon configuration previously reported in literature. We emphasize that for all the phases with non-uniform height patterns we observed high molecules (height h 2 in our notation) with a two-lobe contrast, which may correspond to a 6-6 bond or an apex configuration (see Figure III.19). In our analysis of STM images, fullerenes on hexagonal faces (three-lobes contrast and molecular height h 3 ) were much less observed than the case of fullerenes with two-lobe contrast and h 2 height. Looking at the energy values we conclude that the apex orientation is favorable for the fcc binding sites, which are the ones we predicted in our study of the structure (see Figure III.7). Moreover, in subsection 3.2 we show that the experimental contrast observed in STM is well reproduced with simulations based on the DFT-optimized structure, for the apex orientation.

In addition, our DFT calculations predict that the apex configuration has a smaller molecular height than other orientations (on different binding sites), again consistent with our experimental data. The apparent height in STM for molecules exhibiting a two-lobe contrast is lower by 0.76 Å than the ones with hexagonal orientations. DFT predicts molecular heights smaller by 0.3 -0.8 Å for fullerenes with apex orientation on fcc sites, compared to fullerenes sitting on hexagonal faces (top, bridge, fcc and hcp sites considered). This observation is also in agreement with our nc-AFM measurements, for which we observe a topographic height difference between molecules with apex and hexagon orientation. We then conclude that most of the "high" fullerenes in our experimental phases have apex orientation and that the height difference between these molecules (h 2 ) and the ones with hexagonal orientation (h 3 ) is due to different binding sites and not to substrate reconstruction as it has been proposed by Forcieri et al. [START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF].

Far apart fullerenes: binding sites

Concerning the binding sites on the Cu(111) surface, we found that all the high-symmetry sites on Cu(111) offer stable configurations. For hexagon orientations, top sites are the most stable ones, followed by fcc and hcp sites, and finally by bridge sites. Energy differences range from 0.01 eV to 1.2 eV, which is larger than the previous values reported in the literature (0.06-0. eV [START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF]).

Drawing conclusions from these values should be done carefully. First, fullerenes in our samples are not isolated, but surrounded by other molecules. As a consequence, a strong competition between different energy terms takes place: coincidence or binding energy (fullerenes occupy high-symmetry sites), intermolecular interactions (we show in subsection 3.3 that the strength of the interaction is comparable to binding energies) and reconstruction of the substrate (that determines which positions can be occupied by the C 60 molecules) compete. We suggest that, when forming lattices, fullerenes can slightly modify their position with respect to a perfect undistorted lattice to find the most stable binding site and orientation, as long as intermolecular interactions do not become too large. In our coincidence model (section 4) we use the data from DFT calculations to derive a binding energy map in the copper cell, which depends on binding sites. Nevertheless, we do not impose one single kind of possibly occupied binding sites, but consider several possible ones.

(4 × 4)R0°commensurate molecular lattice

We now describe our DFT results for the (4 × 4)R0°commensurate fullerene lattice. We could expect this calculation to be more relevant than the one for far-apart molecules to interpret our results for the uniform phases, because the coincidence with the substrate is precisely the (4 × 4)R0°commensurate phase. Nevertheless, van der Waals interactions are not taken into account in our calculations of the binding energies, while they are potentially strong. On the contrary, studying the (4×4)R0°commensurate lattice is a way of estimating intermolecular interactions via the copper substrate (not van der Waals). Is the binding energy larger or smaller when considering a lattice instead of far-appart, quasi-individual molecules, i.e. is there a stabilizing or de-stabilizing intermolecular interaction due to the substrate?

For this comparison we only compute four different configurations: top, bridge, fcc and hcp sites with fullerenes sitting on hexagonal faces. In the unreconstructed (4 × 4)R0°phase we experimentally observe orientations corresponding to hexagons, so we do not repeat the calculations for the other orientations.

We found that the top site is the most stable one, followed by hollow sites and the bridge site. This energy hierarchy is in agreement with our calculation for far-apart fullerenes. However, it is not in agreement with previous studies, which concluded that the energies increase following the sequence hcp < f cc = top < bridge [START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF] or f cc < hcp [START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. We hence cannot conclude based on these calculations (ours or others from the literature) which binding site is preferred, all the more our experimental data suggest that several binding sites can be occupied (see Figure III.10). Taking into account our DFT results, in agreement with the ones by Larsson et al. [START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF], we propose that hollow and top sites can host fullerenes.

Finally, we can compare the binding energies for molecular lattices and far-apart fullerenes. We find that the energy hierarchy is not exactly the same. Top sites are always the most stable ones, and the bridge sites the less stable, but hcp sites are less favorable than fcc sites for far-apart molecules while these two sites are equally favorable in case of the (4 × 4)R0°p hase. We also observe that binding energies are systematically smaller by 8-24% (i.e. more stable) in the case of far-apart fullerenes. This result suggests that a substrate-mediated intermolecular interaction stabilizes the system. The corresponding interaction could be due to electrostatic interactions or exchange interactions between molecules via the copper substrate.

On reconstructed sites: vacancies in the metal surface

Remarks about the calculation of the binding energy in presence of vacancies

We consider six different types of vacancies, four of which are shown in Figure III.25a. All of them are one-atom deep and have different lateral sizes. The smallest one corresponds to one copper atom missing, a configuration that has been reported for fullerenes on Pt(111) [START_REF] Felici | X-ray-diffraction characterization of Pt(111) surface nanopatterning induced by C 60 adsorption[END_REF][START_REF] Pinardi | Tailoring on-surface chemistry of (hetero) aromatics on transition metal surfaces[END_REF] and on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Li | Surface Geometry of C 60 on Ag(111)[END_REF]. The largest vacancy lacks seven atoms. This type of vacancy is well-known to form below fullerenes in the (4 × 4)R0°phase on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. We also compute the structures for three-atoms vacancies centered on two different sites (fcc and hcp) of the copper substrate surface. Finally, we compute the structure for six-atoms vacancies on fcc and hcp sites and found that these structures are not stable, i.e. no convergence was found for the DFT calculations.

Calculating the binding energies for reconstructed sites is a challenging task. To compare our results for different vacancy sizes with the ones for non-reconstructed sites, a constant number of atoms must be kept in the calculation. To do so, we take the atoms missing in the vacancy and place them on the bottom surface of the copper cell. These Cu atoms then necessarily have a lower coordinance and are thus destabilized, which implies that the structure is less stable than the perfect copper cell without reconstruction. The real binding energy of a vacancy is not simply the energy of the final structure with a fullerene on a vacancy, but the energy cost of creating such a vacancy needs to be considered:

E vac = E bind + E cost .
Estimating this energy cost is precisely what is difficult. To estimate it, we can compare the total energy of a perfect copper cell (without the fullerene molecule) with the one of the copper cell with a vacancy on the top surface and, attached on the bottom side of the Cu slab, the Cu atoms from the vacancy.

Shi et al. [START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF] have considered a rather dense arrangement of the Cu atoms on the bottom surface: they found E 7at-vac cost = 3.3 eV (seven-atom vacancy) and E 1at-vac cost = 1.1 eV (one-atom vacancy). We simply considered a seven-atom, three-atom or one-atom (compact) island on the bottom Cu surface, leading to E 7at-vac cost = 8.9 eV, E 3at-vac cost = 6.6eV and E 1at-vac cost = 2.8 eV. The reason for these strong differences is the different number of Cu atoms with low-coordinance, hence with lower stability than the "usual" surface atoms. This shows how difficult the estimate of the energy cost is, especially with simulations dealing with finite-size systems. In reality, Cu atoms expelled from the vacancy sites diffuse at the surface and form large, one-atom thick Cu islands, with a low proportion of edge atoms. An alternative way to estimate the energy cost and avoid this issue is to place the Cu atoms from the vacancies within the bulk of the Cu slab, as interstitials. This is not the actual situation but it provides a lower-estimate of E 7at-vac cost = 1.9 eV and E 1at-vac cost = 0.7 eV [START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF]. From these considerations we estimate intermediates values E 7at-vac cost = 2.6 eV and E 1at-vac cost = 0.9 eV for the two kinds of vacancies.

We also propose another way of estimating the energy cost, using the energy values of different types of vacancies reported by Shi et al. [START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF] and making an approximation. We now consider that each broken bond between copper atoms equally contributes to the energy cost. For a one-atom vacancy we found one single atom that has 6 broken bonds, and the energy calculated by Shi et al. is E 1at-vac cost ∼ 0.9 eV, so E bond ∼ E cost /N bonds ∼ 0.15 eV. For three-atom vacancies each atom participating in the vacancy has 4 broken bonds, so 12 bonds have been broken in total and E 3at-vac cost ∼ 0.15 × 12 = 1.8 eV. For seven-atom vacancies (6 atoms, each one with 3 broken bonds): E 7at-vac cost =∼ 0.15 × 18 = 2.7 eV, in good agreement with the calculation in Ref. [START_REF] Shi | Adsorbate-induced reconstruction by C 60 on close-packed metal surfaces: Mechanism for different types of reconstruction[END_REF]. These estimations are based on a crude, simple approximation, but they give values consistent with both Shi et al. and our calculations 2 , without performing new, complex, DFT calculations.

Binding energies for vacancies

The energies for the vacancies, calculated with the previous estimates of the cost for vacancy formation, are given in Table III.5. The energy of seven-atom vacancies is between ∼ -5 eV and ∼ -6.5 eV, which is significantly lower than binding energies for fullerenes on non-reconstructed sites (between ∼ -2.5 eV and ∼ -4.3 eV).

The energy of one-atom vacancies is ∼ -3 eV, so they are only favored with respect to some non-reconstructed sites. Hence, we suggest that the substrate will preferentially reconstruct with seven-atom-vacancies, rather than with one-atom-vacancies. This again questions the proposal by Forcieri et al. [START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF], who suggested the formation of one-atomvacancies. Moreover, our DFT calculations show that the molecular height for fullerenes on one-atom vacancies (1.9 Å) should be very similar to the one of molecules sitting on nonreconstructed sites with a hexagon orientation (1.5 Å-1.9 Å), which is not consistent with the height difference of 0.7 Å reported by Forcieri et al. for these molecules. Consistent with our previous section, we conclude that the intermediate height h 2 observed experimentally is due to different binding sites on non-reconstructed copper sites, and not to the formation of one-atom vacancies. 2 Considering our DFT calculation of the energy cost for one-atom vacancy we find: E 1at-vac cost ∼ 2.8 eV, so E bond ∼ 0.47 eV. For three and atom vacancies counting bonds leads to values E 3at-vac cost = 5.6 eV and E 7at-vac cost = 8.5 eV, which are similar to the values obtained by DFT within a 15% error.

Table III.5:

Binding energies and distances between the base of a C 60 molecule and the copper surface for a given molecular configuration, calculated by DFT for far-apart fullerenes on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface. Reconstruction of the surface in the form of one-atomdeep Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] vacancies of different kinds is included. Note that the origin of the height values is the same as in Table III.3, i.e. the position of Cu atoms at the topmost surface layer, far away from the vacancy. The 0 Å height of the seven-atom-vacancy for instance indicates that the fullerene is lower than without a vacancy, in which case the height is > 1 Å. Finally, there is no energy gain for three-atom vacancies neither on a fcc or hcp sites, compared with binding energies for non-reconstructed sites. We suggest that these types of reconstruction do not experimentally occur.

Bind

Orientation of fullerenes on seven-atom vacancies

In literature it has been proposed that fullerenes on seven atom vacancies on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] sit on a hexagonal face, in the (4 × 4)R0°phase [START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. However, in our STM images of periodic and non-ordered phases, C 60 molecules on vacancies exhibit a circular contrast, that has been identified in literature as the contrast for fullerenes with a pentagonal face towards the surface. Our DFT calculations for both configurations, a hexagon and a pentagon towards the surface, indicates that the height differs only by 0.1 Å. We also find that the pentagon orientation has a lower energy than the hexagon one (-6.45 eV vs -5.17 eV). The pentagon orientation is energetically favored. In subsection 3.2 we show a simulation of a STM image of such configuration, which is in agreement with the experimental contrast we measured. Finally, we suggest that in the (4 × 4)R0°phase the hexagon orientation (experimentally observed by Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]) could be favored by intermolecular interactions, which depends on the relative orientations between neighboring molecules [START_REF] Hauptmann | Interactions between two C 60 molecules measured by scanning probe microscopies[END_REF][START_REF] Sweetman | Visualizing the orientational dependence of an intermolecular potential[END_REF][START_REF] Leaf | A Combined Monte Carlo and Hückel Theory Simulation of Orientational Ordering in C 60 Assemblies[END_REF].

Apparent height and molecular orientation

Here we present DFT simulations for the conductance between the STM tip and fullerenes in different configurations. The conductance was computed at the Fermi level, i.e. at zero bias potential. These curves give a qualitative idea of the expected conductance, while a direct comparison would require conductance calculations for different bias potentials. In We observe that for distances of the order of the fullerene diameter (∼ 7 Å) the conductance is large, as expected in this regime where the tip is essentially in contact with the molecules. When the distance increases beyond the contact regime (tip height > 11 Å), the conductance decreases following an exponential law (linear in logarithmic scale). Our STM measurements were performed at constant current, so we can compare curves corresponding to different configurations at a fixed value of the conductance and deduce the expected height differences. The region of conductance corresponding to our measurements is out of the contact regime, at distances > 11 Å.

Fullerenes on seven atom vacancies (independently of their orientation) exhibit the lowest height. The two curves corresponding to C 60 molecules on seven-atom vacancies are ∼ 20 pm lower than the molecules on non-reconstructed sites. This difference has a topographic origin, i.e. fullerenes on seven-atom vacancies are 1.2 -2.0 Å lower than molecules on nonreconstructed sites, according to our calculations shown in Table III.3 and Table III.5.

In Figure III.26b we show a zoom of the conductance curves, in which small height differences between C 60 molecules sitting on non-reconstructed sites are visible. Calculated differences extend in a range of ∼ 0.2 Å, while experimental measurements of the heights exhibited larger differences of ∼ 150pm (we recall that h 2 = 79 ± 22, h 3 = 155 ± 14 and . Thus, the quantitative analysis of our calculation is not in agreement with our measurements, but the order of molecular heights is well reproduced: fullerenes with pentagon orientation and top sites are the highest ones; followed by molecules with hexagonal orientation on top sites; and the lowest non-reconstructed configurations corresponds to fullerenes with apex orientations on fcc sites. Moreover, we conclude that fullerenes sitting on one-atom vacancies would exhibit a molecular height similar to the one of C 60 molecules with hexagonal orientation. Consequently, they cannot correspond to the lower height h 2 we measured experimentally, as it was proposed by Forcieri et al. [START_REF] Forcieri | Origin of C 60 surface reconstruction resolved by atomic force microscopy[END_REF].

In Figure III.26c-e we compare STM contrasts simulated with DFT to experimental images measured for several molecular configurations. Simulated and measured contrasts are in agreement for the three most frequent molecular orientations observed in our samples: fullerenes with apex orientation, with hexagon orientation and with pentagon orientation on seven-atom vacancies.

To sum up, the height differences between fullerenes on seven-atom vacancies and on nonreconstructed sites are consistent with the measured heights. Our calculation do not allow us to quantitatively address the subtle height differences between fullerenes in non-reconstructed sites, but it correctly predicts the qualitative order of molecular heights. Simulations of STM images confirm the correspondence between experimental molecular contrasts and the molecular orientations we proposed for each category of heights (h 1 -h 4 ).

Intermolecular interaction

Interactions between fullerene molecules, which host delocalized electrons, are dominated, at least above distances of a few angstrom, by van der Waals interactions. They must be taken into account in the DFT calculation codes, as can be done in the Fireball code we used [START_REF] Dappe | Intermolecular interaction in density functional theory: Application to carbon nanotubes and fullerenes[END_REF][START_REF] Dappe | Weak Chemical Interaction and van der Waals Forces: A Combined Density Functional and Intermolecular Perturbation Theory-Application to Graphite and Graphitic Systems[END_REF] or following other DFT approaches, for instance time-dependent schemes [START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF]. However always in the absence of a substrate. To analyze van der Waals interactions influence we hence consider C 60 dimers and two-dimensional fullerene lattices in the absence of a substrate.

The variation of the total energy of a two-dimensional fullerene lattice with the intermolecular distance is shown in Figure III.27a. We deduce an equilibirium lattice parameter of 10.05 Å, in good agreement with the one reported for fullerite (fullerene three-dimensional crystal) of ∼ 10 Å [START_REF] Girifalco | Interaction Potential for C 60 Molecules[END_REF][START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF]. We consider this equilibrium distance as the one of our molecular assemblies: all the curves considered in the following have then been shifted to have this equilibrium distance. The minimal energy value is -1.96 eV, which is of the same order of magnitude as the binding energy with the substrate. Thus, competition between two interactions is expected. The repulsion between molecules when the lattice parameter of the layer is compressed below 10.05 Å increases very sharply. For instance, for phases with a compression of -3.7%, corresponding to the 9.68 Å-22.4°phase with non-ordered height pattern, the energy becomes positive, amounting to ∼ 0.4 eV. In the attractive regime (above 10.05 Å), the increase of the energy is slower, and asymptotically saturates to zero.

We now compare this calculation on a two-dimensional layer with the one for two fullerenes forming a dimer, in two different cases (Figure III.27b). First, we consider the intermolecular interaction of fullerenes forming a dimer reported by Pacheco et al. [START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF]. Their calculation used a time dependent DFT (TD-DFT) approach, to treat the van der Waals interactions and reproduced thermodynamic properties of fullerite (3D fullerene crystal) with an accuracy as good as 4%. Moreover, they proposed a parametrized potential that fits their ab initio result. They found a minimal energy of ∼ -0.27 eV and an equilibrium distance of 10.03 Å. We also reproduced the results published by Dappe et al. [START_REF] Dappe | Intermolecular interaction in density functional theory: Application to carbon nanotubes and fullerenes[END_REF][START_REF] Dappe | Weak Chemical Interaction and van der Waals Forces: A Combined Density Functional and Intermolecular Perturbation Theory-Application to Graphite and Graphitic Systems[END_REF] for a C 60 dimer, using DFT combined with perturbation theory and the Fireball code. The equilibrium distance between the closest graphitic surfaces was 2.6 Å and the binding energy for the dimer is -0.38 eV. Both curves, ours and the one by Pacheco et al. [START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF], are very similar at long distances, but substantially differ at short distances.

For the comparison with the two-dimensional case, we multiplied the energy values calculated for a dimer by six, implicitly assuming six nearest neighbor bonds for each molecule in a two-dimensional lattice. The three energy curves vs distance are shown in Figure III.27a. The minimum energy using TD-DFT is 18% smaller than the one for the two-dimensional layer. The one re-calculated with perturbation theory is 17% larger. This comparison illustrates the difficulty in computing the intermolecular energy. Depending on the model and on the size of the simulation, energies can vary by ∼20%. We will thus focus on the order of magnitude of this interaction and consider that the intermolecular interaction in a two-dimensional layer of C 60 is reasonably well accounted for by a sum of two-body energy terms between nearest neighbors:

E 2D ∼ 1 2 ij∈N N E dimer (r ij )
In the following section we take the E dimer reported by Pacheco et al. [START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF], because it is an easy to use parametrized potential, and is in good agreement with the experimental properties of fullerite.

Finally, we wanted to verify a result reported by Reddy et al. [START_REF] Reddy | Two-dimensional van der Waals C 60 molecular crystal[END_REF] using molecular dynamics. In this study a two-dimensional lattice of fullerenes with intermolecular interactions was considered, and the energy variations were analyzed when one of the fullerenes of the lattice was progressively pulled out of the two-dimensional layer. This situation seems relevant to our problem, where part of the molecules are higher than the others. Reddy et al. found a local minimum in the energy when the fullerene molecule is pulled out by 2.5 Å. The reason why is not obvious to us. We repeated this calculation for different lattice parameters, which varies for the different commensurate phases. Results are shown in Figure III.27c. For a molecular lattice at equilibrium distance we find an energy minimum at z = 0, as expected. For compressed molecular lattices, we find a shift in the energy minimum, z = 0.9 Å for ε = -3% and z = 1.2 Å for ε = -3.7%. Buckling of the molecular layer hence appears as a way to relax elastic-like compression effects within the fullerene layers.

Charge transfer and electric dipole

In this section we address the charge transfer from the copper substrate to the C 60 molecules, both in the non-reconstructed and reconstructed configurations. Charge transfers have been measured by photoemission and spectroscopy techniques [START_REF] Tsuei | Photoemission and photoabsorption study of adsorption on Cu(111) surfaces[END_REF][START_REF] Hoogenboom | Charge transfer and doping-dependent hybridization of C 60 on noble metals[END_REF][START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF] and calculated by DFT methods [START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF][START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF]. For fullerenes on non-reconstructed sites a charge transfer of ∼ 1 electron towards the molecule has been reported, for fullerenes on seven-atom vacancies the charge transfer is much larger, of ∼ 3 electrons. Our DFT calculations, considering a basis set of spd orbitals and Lüwdin charges, give a charge transfer of 1.2 electrons from the copper to the molecule on a non-reconstructed site and 1.9 electrons for a fullerene sitting on a seven-atom vacancy.

Such a charge transfer implies the existence of localized dipoles at the molecule/metal interface, which repel each other, altogether representing an additional interaction term between molecules [START_REF] Tsuei | Photoemission and photoabsorption study of adsorption on Cu(111) surfaces[END_REF][START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF][START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF]. The intensity of the dipole strongly depends on the distribution of charges at the interface, which is not directly accessible in experiments. The small work function reduction associated to the presence of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], compared to bare Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], was argued to be compatible with the charge distribution calculated with DFT [START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF], from which a 0.04 eÅ dipole per molecule was estimated. Our own estimate is very different from the previous one, since we find a dipolar moment of 4.8 eÅ for fullerenes on non-reconstructed sites and of 4.4 eÅ for fullerenes on seven-atom vacancies. Our calculation indicates that the charge is more localized in the case in which seven atom vacancies form, leading to a smaller dipole. This result can intuitively be understood by the fact that fullerenes on vacancies are nearer the copper surface than C 60 molecules on non-reconstructed sites, thus charge transfer is more localized at the interface. Our values allow us to explain the non-ordered height patterns discussed in chapter IV, whereas it is not the case if we consider a dipole of 0.04 eÅ.

Further exploring the value of the charge transfer and the dipolar moment constitutes an interesting research path. For example, complementary measurements of the molecular conductance could be compared with calculations of the density of states and the charge transfer could be deduced. It would also be interesting to measure frequency shift maps on non-contact AFM experiments (KPFM technique) to detect charge states of the fullerenes. During this PhD project both types of measurements were performed, but results are not presented in this manuscript. Finally, using X-ray photoelectron spectroscopy (XPS) could allow to better determine the work function in molecular assemblies and give some insights on the charge transfer.

Coincidence model and energy landscape

In the previous sections, we have described the large variety of structural mechanisms playing a role in the organization of fullerenes assemblies on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]: positional coincidence of the molecules and substrate atoms, in-plane local distortions in the fullerene lattices, formation of vacancies in the metal surface and related height patterns of the molecular layer. We also used DFT calculations to prove that the binding energy values and the intermolecular energies are of comparable magnitude, suggesting that a competition between molecule-substrate and molecule-molecule interactions drives the formation of phases. However, some questions regarding phase coexistence have not been answered: Why do we observe ten coexisting phases at room temperature? Do we have a system with many degenerate ground states? Can we describe our experimental observations as a system at thermodynamic equilibrium, or is the system trapped in different metastable states during growth?

To answer these questions, we propose a model that estimates the internal energy of each phase. First, we detail how to build the mode using the information we have obtained with our description of the molecular structures. Second, we present the results for the internal energies of the different phases. Finally, we consider the entropy of each phase, compare the corresponding free energy at non-zero temperature to the internal energy, and discuss the origin of phase coexistence.

Principles of our coincidence model and energy terms

The principle of our coincidence model to compute the internal energy of each commensurate phase is presented in For the case without surface reconstruction, we consider a given commensurate phase and determine the molecular positions (x i , y i , z i ) for each C 60 molecule in the superstructure unit cell. From these positions we compute the coordinates with respect to the copper unit cell (x Cu i , y Cu i ), i.e. the binding site of the fullerenes on the copper surface. From these geometrical parameters we deduce the sum of all C 60 -Cu binding energies, which we call the coincidence energy. We can compute the position vector ⃗ r ij between fullerenes, which allows to compute intermolecular interactions. Details about the binding energy term and the intermolecular energy term are given in the following subsubsection 4.1.1 and subsubsection 4.1.2, respectively.

For phases including vacancies in Cu and local distortions in the fullerene lattice, the geometry is determined in the same way as in subsubsection 2.2.3. We start from the geometry of the non-reconstructed phase and update molecular positions. Fullerenes that are near a top position, which is the preferred site to create seven-atom vacancies, can change their spatial coordinates to occupy the top site and form a vacancy. In plane coordinates (x i , y i ) are modified to allow for in-plane dislocations. Formation of a vacancy implies also the modification of the out-of-plane coordinate z i . This procedure is performed for all the fullerenes in the superstructure unit cell that are near a top site within a critical radius r C . If the orientation of the molecular phase is φ > 20°, we consider that seven-atom vacancies cannot form under all the fullerenes because of steric effects (see subsubsection 2.2.3 for details) and we only allow the presence of vacancies underneath half of the fullerene molecules. When the superstructure unit cell contains an odd number of molecules, we consider two configurations with ((N cell C 60 -1)/2) vacancies and ((N cell C 60 + 1)/2) vacancies, respectively. From the updated coordinates of each molecule (x vac i , y vac i , z vac i ) we deduce the binding sites on copper (x Cu,vac i , y Cu,vac i ) and the position vectors between fullerenes ⃗ r vac ij . From these geometrical parameters we compute the coincidence and intermolecular energies, as explained in subsubsection 4.1.1 and subsubsection 4.1.2, respectively.

To perform this calculation we developed a Python code (resumed in Figure III.28), divided in several blocks, each one allowing the computation of the different energy terms for the non-reconstructed and reconstructed cases. The structural and energy data are then saved for their analysis and representation.

Figure III.28:

Flowchart of the coincidence model used to compute the energy of each commensurate phase of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], both for the phases with no-reconstructed surface and with reconstructed surface. The geometry of each phase is determined, and the internal energy is deduced from the coincidence with copper atoms and the intermolecular distances.

Binding energy with the substrate

To compute the binding energy between fullerenes and the Cu(111) surface we use a twodimensional potential U coinc-2D that only depends on the binding site of a molecule in the copper cell (x Cu i , y Cu i , z Cu i ). For a commensurate phase with N molecules in the supercell the coincidence energy term writes:

E coincidence = 1 N N i U coinc-2D (x Cu i , y Cu i , z Cu i )
The function U coinc-2D is built from DFT results for binding energies of fullerenes on different copper sites (see subsection 3.1). We consider two different cases, the one without surface reconstruction and the one including seven-atom vacancy formation. The potential U coinc-2D is thus differently defined if z i = -0.1 (presence of a vacancy) or if z i = 1.2 Å (apex fcc configuration). For the high-symmetry sites on the copper cell (top, bridge, fcc, hcp sites on the unreconstructed copper and seven-atom vacancy) we consider the lowest binding energy calculated by DFT, i.e. we consider that fullerenes are free to choose the molecular orientation that best fits the binding site. Note that these configurations coincide with the molecular orientations observed in our high resolution STM images. More explicitely:

• Top site on non-reconstructed Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], with 6-6 orientation and binding energy of E top = -3.06 eV.

• Bridge site on non-reconstructed Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], with 6-6 orientation and binding energy E bridge = -3.38 eV.

• Fcc site on non-reconstructed Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], with apex orientation and binding energy E f cc = -4.26 eV.

• For hcp site on non-reconstructed Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], we consider that the site is very similar to the fcc one when a molecule is in its apex configuration (which is the case in our DFT calculations for molecules with a hexagon on the fcc and hcp sites), thus E hcp = -4.26 eV.

• Seven-atom vacancy (centered where a top site of the non-reconstructed Cu(111) would be) with pentagon orientation and binding energy E vac = -6.45 eV.

Between these high-symmetry sites we assume that the energy map is continuous and no significant energy barrier exists. This seems reasonable for two reasons. First, a previous DFT studiy by Wang et al. [START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF] showed that energy barriers between binding sites, when rotation and translation are both allowed, are smaller than 0.3 eV. Moreover, isolated fullerenes freely diffuse and orient in our growth conditions (room temperature) on the surface to attach to the different phases in the proper configuration, thus the corresponding barriers are much smaller than 0.3 eV, comparable with or smaller than thermal energy at room temperature (∼ 26 meV). This indeed represents very small variations, at least 10 times lower than those in binding energies from one high-symmetry site to the other. Thus,

C 60 -C 60 intermolecular interaction

To take into account the intermolecular energy E intermolecular (⃗ r ij ) in commensurate phases we consider two different contributions: the interaction energy associated to chemical repulsion and van der Waals interactions (see subsection 3.3) and the dipolar interaction between charged fullerenes (see subsection 3.4). The intermolecular energy for a commensurate phase with N molecules in the superstructure unit cell writes as:

E intermolecular = 1 2N N i j∈(N N of i) E C60-C60 (⃗ r ij ) + E dip (⃗ r ij )
where N N are the nearest neighbors, E C60-C60 is the energy associated to chemical repulsion and van der Waals interactions and E dipolar is the energy due to electric dipoles at the copperfullerenes interface.

The term E C60-C60 (⃗ r ij ) in the energy can be directly computed from the parametrized curve for a dimer of C 60 molecules reported in the literature [START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF] and presented in Figure III.27b. We consider the distance between molecules ||⃗ r ij || (not simply the in-plane projection of their distance).

The dipolar interaction is simply calculated according to the classic electrodynamics formulation for two parallel dipoles µ 1 , µ 2 both perpendicular to the surface and separated

E dip = µ 1 µ 2 4πϵ 0 r 3 12 cos(θ) -3cos(θ 1 )cos(θ 2 )
θ being the angle formed by the two dipoles µ 1 and µ 2 (zero for molecular dipoles perpendicular to the copper surface). The angles θ 1 and θ 2 are the angles formed by each dipole with the line connecting their centers, and they always have the same value in our calculations. These angles are of 90°if dipoles µ 1 and µ 2 correspond to fullerenes with the same z coordinate; and smaller for fullerenes with different z coordinate (i.e. when one fullerene is sitting on a non-reconstructed site and the other on a vacancy). This equation is applied for each fullerene i in the supercell and its nearest neighbors j.

Note that by considering only nearest neighbors we made a strong hypothesis about the short range character of the interactions. This assumption is reasonable for the intermolecular interaction given the length-scales in the lattice. First nearest neighbors are at distances of the order of 10 Å, so the typical nearest neighbor dimer interaction energy (chemical repulsion + van der Waals) is ∼ -0.3 eV. Second nearest neighbors are at distances of ∼ √ 3 × 10 ∼ 17 Å, for which the interaction energy is two orders of magnitude smaller.

Concerning dipolar interaction, it is by definition long range. To correctly compute this energy term we would need to consider the interaction of each dipole with all the other dipoles in the lattice. Nevertheless, the dipolar term can be (roughly) considered as a constant shift of the internal energy for all the considered commensurate phases. Our model aims at computing the energy of each commensurate phase of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] in order to compare them, determining the ground state and the metastable states. Thus, having a different energy reference is not a problem. In order to prove that dipolar energy is similar for all the considered phases, we compare the energy landscape for all the phases without considering dipolar interactions and including dipolar interaction between nearest neighbors. The dipolar interaction between first neighbors is the most important term, because of the r -3 dependence of the dipolar energy. The second neighbors contribution is ∼ 5 times smaller than for nearest neighbors, ∼ 8 times smaller for third neighbors, etc. This comparison is detailed in the next section and shown in Figure III.31.

Energy of each phase 4.2.1 Energy landscape and metastable states

In In both cases we find similar results, exhibiting a complex energy landscape with many metastable states.

We observe that for the phases without formation of vacancies the internal energy of different phases extend in a range of 1.33 eV (with dipolar interaction) / 1.19 eV (without dipolar interaction). Moreover, the phases with lower energy correspond to the smaller super cells, in which more fullerenes sit on high symmetry sites of the substrate. This is in agreement with the common hypothesis used in literature, which considers that smaller super cells are necessarily more stable [START_REF] Grimmer | Coincidence-Site Lattices and Complete Pattern-Shift Lattices in Cubic Crystals[END_REF][START_REF] Smith | Bollmann's 0-lattice theory; a geometrical approach to interface structure[END_REF]. Nevertheless, we observed vacancy formation in our samples. The phases we need to consider are thus the reconstructed ones, which have indeed lower energy than the non-reconstructed phases in our model.

The ground state of the system is the 10.22 Å-0°phase (uniform height contrast) when it forms vacancies below each molecule. Its energy is lower by 0.60 eV (considering dipolar interaction) / 0.55 eV (without dipolar interaction) with respect to the second phase with lowest energy, and by 2.00 eV/1.90 eV (with / without dipolar interaction) with respect to the highest energy phase observed experimentally. Such an energy difference indicates that this phase should preferentially form. We recall that it is the case when the samples are deposited at high temperature or annealed (see subsubsection 2.1.1), suggesting that supplying thermal energy to the system allows to overcome energy barriers and leads to the conversion of metastable phases to the ground state.

The internal energy of different phases, when reconstruction of the substrate is included, increases with the orientation angle φ, i.e. with the number of fullerenes sitting on sevenatom vacancies. The experimental phases with an angle 0°< φ < 19°correspond to uniform height patterns with all the C 60 molecules sitting on seven-atom vacancies. Their energy is smaller for smaller angles, because the local distortions needed to accommodate the formation of vacancies are less important, thus the cost in intermolecular energy associated to these distortions is also smaller. The experimental phases with larger angles φ > 19°h ave slightly higher energies, their supercell size is smaller and their height patterns can be periodic (10.14 Å-19.1°) or non-ordered (9.80 Å-26.3°).

Height patterns and limits of the periodic model

There are two phases for which our model does not predict the experimentally observed height contrast, the 10.33 Å-30.0°and the 9. For the 10.33 Å-30.0°phase we described the formation of domain walls (Figure III.23), that are not considered in our model, so the energy we show in the energy landscape is the one for a periodic phase with a √ 3 × √ 3 height pattern and 1 3 of fullerenes sitting on vacancies. The experimental phase exhibits 1 2 of fullerenes sitting on vacancies, which necessarily decreases the energy of the phase (E vac < E non-rec ). In addition, the formation of DWs increases the fullerene distances, which should reduce the intermolecular interaction energy cost. For this 10.33 Å-30.0°phase we hence expect a smaller energy than the one shown in the energy landscape.

Concerning the 9.68 Å-22.4°phase, our model predicts a periodic √ 3 × √ 3 height pattern with 1 3 of fullerenes sitting on vacancies. Experimentally we observe a non-ordered height pattern with 1 2 of fullerenes sitting on vacancies. Simply considering a mixture of cells with 1 3 or 2 3 of C 60 molecules on vacancies gives an energy of -3.65 eV (in presence of dipolar interaction, Figure III.31b), which is 1.01 eV higher than the energy we show in our energy landscape. In our model we consider periodic cells, which is obviously not the case in the experimental phase. We think that the energy of the experimental phase might be smaller than the one shown in the energy landscape, because the elastic cost associated to the formation of vacancies and in-plane local distortions may be reduced by locally ordering the molecules to reduce the constraints. Moreover, formation of vacancies below 1 2 of fullerenes instead of 1 3 also decreases the energy of the phase.

Finally, we can comment on the height patterns of the 9.8 Å-9.28°, 10.04 Å-10.9°, 10.18 Å-14.4°and 10.09 Å-16.4°phases. Our model predicts that the lowest energy for each phase corresponds to a uniform height pattern (seven-atom vacancies form below each fullerene). However, √ 7 × √ 7 periodic height patterns (not shown in the energy landscape) are only 0.49 eV, 0.02 eV, 0.02 eV and 0.05 eV higher in energy, respectively. These small energy differences are perfectly consistent with our experimental observations, in which we detect the presence of both uniform and √ 7 × √ 7 height patterns for the 10.18 Å-14.4°and 10.09 Å-16.4°phases. It is also in agreement with the observations published by Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF], who reported a √ 7 × √ 7 height pattern for the 10.04 Å-10.9°phase.

Summary

We conclude that fullerene lattices on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] can adopt a large variety of metastable states, whose energies span a range of ∼ 2.0 eV. The model reproduces correctly the experimental observed phases and their height patterns, except for two non-periodic phases. Phases with important compressive biaxial strain (small circles in Figure III.31) are not experimentally observed, in good agreement with the higher energy calculated within our model. Some of the phases shown in the energy landscape were not experimentally observed. However, this might simply indicate that they readily transform into phases with very close-by orientation angles, but lower energy phases, that the model predicts.

Entropy of each phase

So far we only considered the internal energy of the phases, without taking into account the role of entropy at finite temperature. Nevertheless, we have shown that fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] have multiple degrees of freedom which produces configurations with equivalent energies in our model. Consequently, a configurational entropy is expected for the molecular phases. Here we estimate the Boltzmann entropy, S = k B ln(Ω) where Ω is the number of different possible configurations in a given phase. In particular, we discuss if the entropy for different phases can be extensive, i.e. if it scales with the size of the molecular assemblies. In our energy landscape we show the internal energy per molecule: the total internal energy is extensive and corresponds to the energy per molecule multiplied by the number of molecules in the assembly. If the entropy is also extensive we could find a competition between the internal energy and the entropy term, at non-zero temperature (-T S). If the entropy is not extensive, its weight becomes negligible compared to the internal energy as the size of the molecular assemblies increases.

We recall that fullerenes on the Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface have many degree of freedoms, that lead to numerous different configurations corresponding to the same commensurate phases. The degrees of freedom producing different configurations are:

• The molecular orientation: in some phases we observe coexisting molecular orientations. For example, in the (4 × 4)R0°(uniform) phase we find both fullerenes on hexagon and pentagon faces.

• The azimutal angle (R z ) of the fullerenes: for some fullerenes, such as the ones with apex in periodic phases, different azimutal angles of the molecules are evidenced.

• The binding sites and the translation: for a given phase we have shown that placing the superstructure unit cell on one precise binding site minimizes the energy, but the whole lattice can be translated from one class of binding site to another site of the same class inside the unit cell. Doing so we find molecular lattices with the same coincidence with the substrate but different positions on the copper surface. For example, the uniform phase with Woods' notation (4 × 4)R0°phase can be laterally shifted in 9 equivalent (with equal energy) ways. From the experimental point of view, between shifted domains we find domain walls (see Figure III.9).

• The mixture of different unit cells on phases with non-ordered height patterns, i.e. the 9.68 Å-22.4°and 9.98 Å-26.3°phases. We proposed earlier that the disorder here can be reproduced by a mixture of different unit cells. In the case of the 9.68 Å-22.4°phase we considered two different unit cells with different number of vacancies, which can be rotated respecting the C 3 symmetry of the triangular lattice. The mixture then consists in a tiling problem using 6 unit cells. For a 9.98 Å-26.3°phase we observe unit cells with three different rotations and also with inverse low/high contrasts (timereversal symmetry), such that the molecular phase can here as well be obtained with a tiling based on 6 different unit cells.

Below, we estimate the entropy corresponding to each of these possible sets of configurations.

Molecular orientations are mainly fixed for each phase: in the 10.22 Å-0°phase only some fullerenes appear on pentagon orientations, while most of them are sitting on hexagons. For all the other phases, all "low" molecules exhibit the same orientation. "High" fullerenes show mainly apex orientation and experimentally we observed less than 3% sitting on hexagons. This observation points to an energy difference between these molecular orientations. Here, we compute the upper limit of the entropy, assuming that all "high" molecules are free to choose their orientation and that the internal energy of such phases is equivalent. The entropy term is:

S = k B ln(2 N )
The entropy is extensive and the entropy per site writes

s = S N = k B ln(2)
At room temperature the contribution of this entropy term would be sT RT ∼ 26 meV×0.7 ∼ 18 meV, which is negligible with respect to the energy differences between phases that are at least one order of magnitude larger.

Three different azimutal angles for fullerenes are observed for "high" molecules on all the phases, except for the uniform ones. "High" fullerenes with azimutal degree of freedom constitutes a significant ratio of the molecules in periodic, disordered and stripes phases. This ratio is 1 4 of the molecules in the 2 × 2 periodic phase, 1 7 of the molecules in the √ 7 × √ 7 periodic phase and 1 2 in the stripe and disordered phases. Thus, the maximum entropy value for this contribution is:

S = k B ln(3 1 2 N )
The entropy is also extensive and the entropy per site value at room temperature is:

s = k B 1 2 ln(2) → sT RT ∼ 14
meV which is again much smaller than the energy differences between different commensurate phases.

The entropy corresponding to configurations with different translation vectors inside the Moiré cell is always negligible compared to the energy, because:

S = k B ln(# translations)
is not extensive.

Finally, we consider the mixture of different unit cells. In a tiling problem of space, constraints might exist about how cells can be actually placed. Here we give an upper limit for the entropy considering that to tile a phase of N molecules we can combine all the possible unit cells in a random way. If we consider 6 different unit cells, each one being formed by n molecules, then:

S = k B ln(6 1 n N )
The entropy is extensive and for n = 3 (the case of the 9.68 Å-22.4°phase) we find an entropy per site at room temperature:

s = k B 1 3 ln(6) → sT RT ∼ 16
meV This entropy term is also negligible compared to the energy differences between different commensurate phases.

In conclusion, the orientational and azimutal degrees of freedom of fullerenes in molecular phases lead to an extensive entropy. The same occurs for phases with non-periodic height patterns, for which an additional term in the entropy appears because the phases are a mixture of different unit cells. Nevertheless, even if the entropy is extensive, its value at room temperature is one or two orders of magnitude smaller than the energy differences between commensurate phases. Thus, the entropy term cannot play a dominant role in the stabilization of the phases and no significant modification of our energy landscape due to entropy is expected.

Message to take home: multiple degrees of freedom & phase coexistence on C 60 /Cu(111)

In this chapter we have presented a detailed microscopic study of the C 60 on Cu(111) system. We have described the coexistence of ten different commensurate phases, five of them not being previously reported in the literature, and characterized their structure including molecular orientation, formation of vacancies and coincidence with the substrate. We used DFT calculations to determine interaction energies in the system, considering binding with the substrate, intermolecular interaction and the presence of electric dipoles due to charge transfer at the C 60 /Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] interface. We demonstrated that energy scales for moleculesubstrate and molecule-molecule interactions are of the same order of magnitude. Thus, they are expected to compete in the formation of commensurate phases. Finally, we used the description of the structure and the energies given by DFT calculations as inputs to construct an energy landscape considering different commensurate phases. We concluded that many commensurate phases have similar energies (energy range is ∼ 2 eV, and phases with similar energies can differ by only ∼ 0.1 eV) and we proposed a quantitative picture of the energy landscape for these metastable states.

The quantitative values obtained with our model could be improved including several contributions. On one hand, intermolecular interactions are expected to depend on the relative orientation of molecules [START_REF] Hauptmann | Interactions between two C 60 molecules measured by scanning probe microscopies[END_REF][START_REF] Sweetman | Visualizing the orientational dependence of an intermolecular potential[END_REF][START_REF] Leaf | A Combined Monte Carlo and Hückel Theory Simulation of Orientational Ordering in C 60 Assemblies[END_REF]. On the other hand, the intermolecular and dipolar interactions could be influenced by the molecular neighborhood due to screening effects [START_REF] Torrente | Spectroscopy of C60 single molecules: the role of screening on energy level alignment[END_REF], meaning that one molecule surrounded by neighbor fullerenes on different molecular orientations could feel different interactions. However, our simplified model allows us to tackle the problem using reasonable first order approximations, it gives consistent results when compared with experiments and sets a basis to build more complex models which might refine energy values of each molecular phase.

Moreover, we have considered the role of configurational entropy (in the sense of Boltzmann entropy) and have proposed, using simple arguments, that entropy is extensive for all the phases we observed experimentally. In the particular case of non-periodic phases, we expect a more important entropy term. Despite the extensive character of the entropy, its role, compared to the energy term, is negligible in the stabilization of different metastable states at room temperature. At higher temperature, the weight of the entropy with respect to the energy increases linearly (F = U -T S), and we could think that increasing the temperature would stabilize the phases with larger entropy terms, i.e. those featuring disorder in their tiling. Yet, this is not what is observed experimentally: depositing molecules at high temperature or post-annealing of the samples lead to one single phase, the (4 × 4)R0°phase, with uniform height pattern and formation of vacancies below each fullerene. This phase is the ground state of the system and it is reached when samples are heated to ∼ 340 K, i.e. only ∼ 40 K above room temperature.

In principle, energy barriers exist to the transformation of one metastable state into another state. The probability to cross these energy barriers ∆E follows an Arrhenius law: P (T ) ∝ e -(∆E/k B T ) . The probability of crossing the barrier being exponential, it grows much faster than the value of the -T S term, which would increase by ∼ 10% when the temperature changes from 298 K to 340 K. This semi-quantitative discussion explains why increasing the temperature leads to a vanishing of the phase coexistence and the observation of the ground state.

We suggest that the phase coexistence at room temperature, together with the observation of one single ground state at higher temperatures, can only be explained taking into account the complex energy landscape we reported and the presence of energy barriers between metastable states. Estimating the values of the energy barriers is not possible using our approach at thermodynamic equilibrium. To characterize these energy barriers we would need to address the growth mechanisms of the molecular system. Doing so is a complex problem that may be tackled using kinetic Monte Carlo simulations. We would then need to make guesses about the transition rates and probabilities for different phenomena during growth: diffusion of fullerenes on the Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface, binding of one molecule on a given copper site, attachment of fullerenes at the edges of fullerene islands, creation of vacancies, etc. Seeking to reproduce the domain size and population distribution of the different phases could allow to estimate the energy barriers associated to the different elementary kinetic processes. This type of simulations could be a next step in the characterization of the phase coexistence in the C 60 /Cu(111) system.

Generalization of the coincidence model to other metal surfaces

The methodology developed in this chapter can be generalized to other systems formed by organic molecules deposited on metal substrates. Here we qualitatively discuss the case of fullerenes deposited on other close-packed metal substrates, in particular the metals presented in Table B.2, i.e. Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. On Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], five and four phases coexist, respectively. These phases have different height patterns, including uniform, quasi-periodic (for some phases on Au(111) the height pattern is periodic in average, but many deviations from the perfect period are observed on the fullerenes lattices [START_REF] Schull | Orientationally Ordered (7 x 7) Superstructure of C 60 on Au(111)[END_REF][START_REF] Zhang | The C 60 /Au(111) interface at room temperature: A scanning tunnelling microscopy study[END_REF][START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF][START_REF] Tang | Complex orientational ordering of C 60 molecules on Au(111)[END_REF]) and non-ordered height patterns (which depends on the thermal history of the samples, i.e. deposition temperature and post-annealing). On Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] two phases coexist, one with uniform and the other with periodic height patterns. On Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] three phases with different structure coexist, but all of them show uniform height patterns. Finally, on Al(111) one single commensurate phase have been reported, but it can exhibit uniform or periodic height patterns depending on the way the sample is prepared (thermal treatments).

Below, we briefly describe the geometry of each phase and the relative strength of the molecule-substrate and substrate-substrate interactions. Taking into account these information, we show that complex energy landscapes, including several metastable states, are not specific to Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Moreover, we propose a link between the structure of the phases, their coincidence energies and the formation of different height patterns.

Geometry of commensurate phases

The first step is to generate the commensurate phases for each metal. The geometry of a commensurate phase depends on the lattice parameter of the metal substrate. When changing the metal surface, different molecular lattice parameters (a C60 ), orientations (φ) and coincidences are expected. In Figure III.32 we plot the molecular lattice parameter depending on the orientation, for different commensurate phases of fullerenes on M [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. The colorscale corresponds to the size of the superstructure unit cell. We limit our analysis to reasonable strain values in the molecular lattice (|ε| < 4%) and small superstructure lattice parameters (a S < 30 Å). We observe that commensurate phases have similarities in different metals, as long as the substrate lattice parameters are similar. In these cases, commensurate phases appear for the same orientation and have similar supercells sizes, but their lattice parameters are shifted. It is the case for Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. It is also the case of Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF].

We compare the calculated commensurate phases with the phases that have been reported in the literature. In Figure III.32 blue circles indicate the phases experimentally observed and dotted blue circles correspond to the phases exhibiting non-ordered height patterns. Note that most of the phases reported in literature are the ones with smaller superstructure unit cells (dark color in the figure), which corresponds to unit cells with one single fullerene (black dots) or to unit cells with three or four molecules per cell (dark red and violet dots).

In Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag(111) other commensurate phases have been identified (see Table B.2), considering the orientation of the lattices and the periodicity for height patterns measured by STM. These phases have large superstructure lattice constants (a S > 30 Å and from 9 to 19 molecules per unit cell!), which we do not show in our figure. What are the mechanisms that lead to such large supercells in Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and not in other metals remains unclear.

This type of plots can be build for any heteroepitaxial system. Although, the problem we address for the moment is exclusively geometrical, a realistic description of different systems requires to include other considerations, such as the substrate-fullerene interactions and the formation of vacancies.

Binding energy with the substrate

We have described the geometry of the molecular phases on different M(111) surfaces. To go further, we now consider interactions for each metal. We could perform a detailed analysis of all the possible configurations, including binding sites and molecular orientations, on each substrate, but here the goal is simpler. We aim to give a general picture of the role of the interactions with different metals, without entering into details. Can molecule-substrate interactions drive the physics? Is a competition between molecule-molecule and moleculesubstrate interactions expected? We used our DFT calculations, using the same code (Fireball, see section 2), to estimate the molecule-metal binding strength for different metals. To do so, we consider two configurations for each metal: a fullerene on a top site, without reconstruction of the substrate, and a fullerene sitting on a vacancy. For each metal we choose the type of vacancy that has been reported in the literature (see subsubsection 1.1.3): one-atom for Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and seven-atom for Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Binding energies are calculated in the same way as for Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], which is detailed in subsection 3.1. In Figure III.33 we plot the binding energy of the relaxed configuration and the height between the bottom of the fullerene and the metal surface. In Table III.6 we give the numerical values for each configuration on different metals.

For non-reconstructed sites (full symbols) the order of binding energies for different metals is E P t( 111) < E N i( 111) < E Al( 111) < E Cu( 111) < E Au( 111) < E Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] . Molecular heights for different metals have the same values within a range of ∼ 0.5 Å. Note that binding energies are different by several eV, depending on the substrate.

For the reconstructed sites, we found that fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ni(111) "sink" by ∼ 2 Å below the metal surface, which is a consequence of the formation of seven atom vacancies. The gain in energy when vacancies are created with respect to the non-reconstructed configuration are important for both metals, of ∼ 9 eV in Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and ∼ 5 eV in Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. For the other metals, the height difference between fullerenes on top sites and on vacancies is much smaller, of 0.1 -0.4 Å. Energy gains are also smaller than in the case of the formation of seven-atom vacancies, of 1 eV-3 eV. The values of the binding energies when vacancies form need to be carefully considered because the energy cost for the creation of vacancies is not taken into account. We know that the energy cost should be larger for seven-atom vacancies on Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] than for one-atom vacancies, because more metallic atoms are removed from the surface. Moreover, for different metal surfaces, kinetic hindrance to the formation of vacancies might be much different (Cu and Ag are known to produce, already at room temperature, significant densities of adatoms on the (111) surfaces, which is not the case for Pt and Ni for instance). We should not forget that the discussion here is only grounded on thermodynamics arguments, while some of the systems might be experimentally kinetically "frozen".

We recall that the intermolecular energy for a fullerene on a two-dimensional layer is ∼ -1.5 eV. Comparing this energy with the values of the binding energies, we conclude that ordering of fullerenes on Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] should be dominated by interaction with the substrate. In these cases, we expect fullerenes to strongly bind with the substrate and height patterns to be determined by coincidence with the metal atoms. Indeed, phases observed on these metals exhibit uniform or periodic height patterns that reproduce the coincidence with the substrate underneath. For the other metals presented here, Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], we expect a competition between interaction with the substrate and between molecules. In those systems, coexistence of commensurate phases has been observed and uniform, periodic and disordered height patterns have been reported in the literature. Our semi-quantitative analysis allows us to generalize one of the conclusions for the C 60 /Cu(111) system to other metals: phase coexistence and formation of complex height patterns is related to competition of different interactions in the system. Nevertheless, it is worth to note that the Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface is different from the Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surfaces. Fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] form seven-atom vacancies, whereas they form one-atom vacancies on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. In subsubsection 2.2.3, we have discussed steric effects on commensurate phases with large orientations (φ > 20°) in the case of Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. For these phases, the formation of vacancies below each fullerene is not possible, because vacancies are too large and many copper atoms with low coordinance would be present in the phases. We proposed a phenomenological rule: only half of fullerenes can sit on vacancies for these phases with φ > 20°. In Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], this rule is not longer valid and the formation of one-atom vacancies below each molecule is always possible, meaning that the phases with disordered height patterns (rotated by 30°both in Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]) do not necessarily have half of the fullerenes on "low" positions and the other half in "high" positions. Looking at disordered phases on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], shown in the STM images in Figure III.8, we confirm that the population of high/low molecules is not 50%/50% and there are more low molecules than high ones (detailed analysis of these populations is presented in chapter V).

Intermolecular interactions

Independently of the nature of the metal surface, fullerenes interact between them due to chemical repulsion and van der Waals interactions at short distances (< 10 Å). Depending on the lattice parameter of the molecular lattice this interaction can be in the repulsive or attractive regime. Knowing the value of this intermolecular interaction is straightforward if we know the geometry of the commensurate phases (and if we disregard, in first approximation, the presence of local lattice distortions). Considering graphs in Figure III.32 we can determine which molecular phases are compressed (a C60 < 10.05 Å) and would feel repulsive interactions with their neighbors. Indeed, compressed phases exist in all the metals we consider. For metals in which interaction with the substrate is not too strong, i.e. Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] (see discussion in previous section), compressed phases exhibit disordered height patterns. We propose that observing non-ordered height patterns is linked to the presence of repulsive interactions between fullerenes. Nevertheless, we have shown for Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] that this condition is necessary for the formation of complex non-ordered height patterns, but is not sufficient. For C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], only molecular phases with large orientations (φ > 20°) exhibit some kind of disorder on height patterns. We recall that we found phases with compressed lattice parameter (for instance the phase 9.8 Å-9.28°) and uniform height patterns.

Besides this interaction, fullerenes deposited on metals can be charged, because electrons are transferred from the metal to the molecule. This charge transfer leads to a dipolar interaction between fullerenes and depends on the binding with the metal surface. For C 60 molecules on Au [START_REF] Hunt | Charge transfer and structure in C 60 adsorption on metal surfaces[END_REF][START_REF] Lu | Charge transfer and screening in individual C 60 molecules on metal substrates: A scanning tunneling spectroscopy and theoretical study[END_REF][START_REF] Schulze | Resonant heating and substrate-mediated cooling of a single C60 molecule in a tunnel junction[END_REF], Ag [START_REF] Wertheim | Interfacial reaction of C 60 with silver[END_REF][START_REF] Goldoni | The interaction of C 60 with Ag(100): strong predominantly ionic bonding[END_REF][START_REF] Hsu | Aperiodic incommensurate phase of a C 60 monolayer on Ag(100)[END_REF], Cu [START_REF] Tsuei | Photoemission and photoabsorption study of adsorption on Cu(111) surfaces[END_REF][START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF][START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF] and Ni [START_REF] Hunt | Charge transfer and structure in C 60 adsorption on metal surfaces[END_REF], the formation of ionic bonds with electron donation from the metal to the fullerenes has been reported. Hoogenboom et al. [START_REF] Hoogenboom | Charge transfer and doping-dependent hybridization of C 60 on noble metals[END_REF] measured charge transfers for fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and found values of 1.7, 1.0 and 1.8 electrons, respectively. In Pt [START_REF] Cepek | Chemisorption and fragmentation of on Pt(111) and Ni(110)[END_REF] and Al [START_REF] Maxwell | Electronic and geometric structure of on Al(111) and Al(110)[END_REF], bonds have a covalent character and charge transfer is expected to be small. Thus, for Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] we expect dipolar interactions to exist. For fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], binding with the substrate is weaker than on Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], such that on these metals dipolar interactions would likely play a relatively more important role. However, we cannot estimate the strength of the dipolar interactions without knowing the spatial distribution of charges at the interface, it is thus not possible to determine whether the intensity of this interaction is comparable with other interactions in the system. In Figure III.34 we sum up all the phases that have been reported for different metals, and classify them according to the observed height patterns in STM measurements (different symbols). For each metal we consider the typical binding energy with the substrate that we compute by DFT (y-axis). In the x-axis of Figure III.34a, we plot the orientation of the molecular phases with respect to close-packed rows of the substrate. In the x-axis of Figure III.34b, we plot the lattice parameter of each phase. The color of the symbols corresponds to the strain in the molecular lattice (which is directly linked to the lattice parameter). The orientation of the molecular phases is linked to steric effects: for metals in which seven-atom vacancies form (Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]), only half of the fullerenes can sit on vacancies if φ > 20°. The lattice parameter is linked to the intermolecular interaction (without considering dipolar interactions): for compressed phases, repulsive interaction between fullerenes is expected, which should favor the buckling of the fullerene layer.

Summary: competition of interactions, phase coexistence and complex height patterns

Our diagrams reveals that phase coexistence is more frequently observed for metals in which different interactions compete. For metals in which interaction with the substrate dominates (Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]) one to three commensurate structures are observed, whereas for metals with weaker binding with the substrate (Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]), four to ten different metastable states coexist. Moreover, uniform and periodic height patterns appear on any type of metal, while complex non-ordered height patterns were only observed on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Colored pockets on the diagram gather phases with the same type of height patterns. This representation suggests a correlation between the parameters we analyzed and the type of height patterns. Uniform phases appears for orientations verifying φ ≲ 15°both for compressed phases with strong binding with the substrate and for dilatated phases, independently of the binding energy. Periodic phases correspond to intermediate values of binding energies and orientations, and can be compressed or dilatated. Finally, non-ordered phases are characterized by small binding energies with the substrate, large orientations and compressed lattice parameters. In addition, non-ordered phases do not present the same height patterns when seven-atom vacancies or one-atom vacancies form: for fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] height patterns seem randomly disordered, and vacancies seem to cluster under extended regions of the molecular lattices, which is not the case for Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] (see discussion in subsubsection 1.2.2 and chapter Ve). We suggest that this difference is due to the steric effects existing for seven-atom vacancies, which are not expected for one-atom vacancies.

We confirm that our analysis of the C 60 /Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] system can be generalized to other metal substrates. In addition, this semi-quantitative analysis is useful to guide the analysis of fullerenes on other M(111) samples. Finally, if we want to design new systems in which phase coexistence and disordered height patterns could appear, we know which type of substrates we should select. Fabricating this type of systems could be of great interest to study disorder in molecular samples, in particular correlated disorder associated to geometrical frustration, which we address in the next chapter.

Chapter IV

Non-magnetic C 60 molecular assemblies as highly frustrated "magnets" Contents 1 The Triangular Ising Antiferromagnet (TIAF)

1.1 First nearest neighbor coupling and geometrical frustration In the introduction of this thesis we have discussed some basic notions about order and disorder of matter. Later on we introduced the particular behavior of some frustrated magnetic systems, in which all the interactions in a lattice cannot be simultaneously satisfied. Frustration in these systems prevents ordering, even at very low temperature. In this chapter we study this particular aspect of the C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Only a few on-surface nonmagnetic systems, essentially molecular assemblies [START_REF] Charra | Surface-induced chirality in a self-assembled monolayer of discotic liquid crystal[END_REF][START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF][START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF] and metallic alloys [START_REF] Carpinneli | Charge rearrangement in the Ge x Pb 1-x /Ge(111) interface[END_REF][START_REF] Ottaviano | Short-range order in two-dimensional binary alloys[END_REF][START_REF] Azizi | Frustration and atomic ordering in a monolayer semiconductor alloy[END_REF], have been proposed as realizations of frustrated magnetic-like systems in the last twenty years. This point of view has not be much explored, neither the powerful tools and approaches usually employed to analyze magnetic frustration have been applied to the previous on-surface systems 1 .

In this chapter we extensively revisit our structural surface system, fullerenes on a copper surface, and analyze the molecular height patterns observed in different phases under the point of view of frustration. Despite the absence of magnetism in this system, we show that frustration emerges in our molecular assemblies. This non-magnetic frustration can be compared with the one found in water ice [START_REF] Pauling | The Structure and Entropy of Ice and of Other Crystals with Some Randomness of[END_REF]. For both water ice and fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] the degree of freedom is positional: distance of hydrogen atoms with respect to oxygen atoms in water, and molecular height with respect to copper substrate for fullerenes. In both cases geometry of the lattice, tetrahedral and triangular lattices respectively, makes it impossible to satisfy all the constraints and frustrated states of matter are observed. Our study proposes an original understanding of on-surface systems, using concepts and analysis tools inherited from the (very active) magnetic frustration community. This chapter is also a way to overcome the limits of the analysis developed in chapter III. In this previous chapter we described fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] assemblies from an atomic-scale viewpoint, in which we addressed the rich structure of different molecular phases. We could call this point of view a "surface science" approach or a "physico-chemistry" approach. We realized that the analysis in this framework is complex, involving many degree of freedoms (in-plane position of the molecules, out-of plane height, molecular orientation) and a variety of microscopic mechanisms accounting for the experimental observations (coincidence with the copper atoms, formation of vacancies, lattice distortions). This description was useful to estimate the energy of different molecular phases that coexist at room temperature, thereby to build a phase diagram and better understand why the system shows such astonishing phase coexistence and not a single, low energy ground state. Nevertheless, this way of understanding the molecular system, and in particular the non-ordered height contrasts of some of the phases, showed its limits. Some molecular assemblies show complicated height patterns, neither uniform nor periodically ordered. We were only able to propose a way of forming these patterns by combining different molecular unit cells, but using our coincidence periodic model we cannot account for complex, not ordered, patterns. Describing disorder requires developing a different type of analysis, based on statistical quantities, as we show in this chapter.

To sum up, in this chapter we develop an analogy between C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and an Ising spin system on a triangular lattice, which is usually studied in frustrated magnetism.

The chapter is divided in three parts.

In section 1 we review the literature devoted to the Triangular Ising Antiferromagnet (TIAF) model. First, we introduce the Hamiltonian with only short range coupling between nearest neighbor spins, which is an archetype of a Hamiltonian generating geometrical frustration. We recall the properties of its peculiar ground state, a spin liquid phase with infinite degeneracy, characterized by an extensive entropy [START_REF] Wannier | Antiferromagnetism. The Triangular Ising Net[END_REF], and by so-called algebraic spin-spin correlations at T = 0 [START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF]. The algebraic decay with spin distance is slow (∝ r -α ), meaning that, even if the phase is apparently unable to order, spins can "see" their neighbors and correlations are long range2 . Second, we extend the Hamiltonian to include second and third neighbor couplings, as well as a magnetic external field. Including further couplings in the TIAF model generally results in ground states that are ordered, and not in spin liquid ground states. Eventually, going beyond a review of the literature, we also propose a complete phase diagram at zero temperature, based on previously published results, for the range of couplings we are interested in section 2 and section 3.

In section 2 we address one of the corrugated phases observed in our C 60 assemblies from the point of view of the TIAF model. We establish the ingredients needed to describe the observed experimental disorder in the framework of the TIAF model: triangular lattice, Ising variables and antiferromagnetic-like coupling. We manage to determine an Ising spin Hamiltonian accounting for the experimental observations. To demonstrate that, we compare simulated and experimental correlations in reciprocal and real space. We also discuss whether the experimental phase is at thermodynamic equilibrium, i.e. whether an effective value of the temperature can be defined. Finally, we link the effective couplings found in the TIAF model with actual microscopic interactions in the molecular system. We conclude that the observed height pattern is disordered but correlated, and correlations are those of a spin liquid phase at a temperature close to the critical temperature of transition from the spin liquid phase to an ordered zigzag phase.

The section 3 is based on the same analysis and tools as those introduced in section 2. We apply them to another fullerene phase observed in our samples. In this phase, molecular height patterns seem to form alternating lines of "high" and "low" molecules, suggesting the emergence of an ordered phase. 3 We discuss what could be the corresponding Ising spin Hamiltonian, if the experimental system is at thermodynamic equilibrium, the influence of defects and the link between effective and microscopic couplings. We conclude that our experimental phase is a realization of the TIAF with effective couplings close to an isoenergetic region in which two ordered phases coexist. Indeed, structural differences between the molecular phase considered in this section and the one in the previous section are accounted for by slightly different values of the effective couplings. Despite the small change in the couplings, the corresponding phases are qualitatively different: the first one being a spin liquid with emergent zigzag order 4 and the second one being a coexistence of zigzag and 1. The Triangular Ising Antiferromagnet (TIAF) stripe phases. Section 4 sums up the main messages from previous sections.

In section 5 we generalize our analysis to include other fullerene phases which exhibit, this time, ordered height patterns. First, we show that a unique Hamiltonian with different J 1 , J 2 , J 3 and h ext values is able to predict all the molecular height patterns observed in our experiments. Second, we explain how our analysis can be extended to fullerenes deposited on other close-packed metals. We determine the link between microscopic structures (binding with the substrate, charge transfer, intermolecular interactions and formation of vacancies) and ingredients in the TIAF model.

The Triangular Ising Antiferromagnet (TIAF) 1.First nearest neighbor coupling and geometrical frustration

The magnetic frustrated model on which we base our analysis is the triangular Ising antiferromagnet (TIAF) model. We start our discussion addressing the case with only first neighbor coupling, for which the ground state is a very peculiar disordered phase. We discuss both the theoretical studies concerning the behavior of the ideal model, and the experimental systems that have been proposed as realizations of the TIAF.

The spin Hamiltonian of this model is an Ising Hamiltonian [START_REF] Ising | Beitrag zur Theorie des Ferromagnetismus[END_REF]. It writes:

H = -J 1 ⟨ij⟩ σ i σ j
where σ i,j are Ising spins with ±1 values, sitting on a triangular lattice that represents the crystalline structure of a solid. J 1 < 0 is the antiferromagnetic coupling between nearest neighbors ⟨ij⟩.

It is worth to note that this Ising spin model in presence of an external field is mathematically equivalent to an Ising gas model [START_REF] Mihura | New Type of Multicritical Behavior in a Triangular Lattice Gas Model[END_REF]. For this model the variables σ i are not spins on a crystalline lattice, but particles adsorbed on a solid surface. In this case Ising variables can take values σ i = 0, 1, corresponding to the absence or presence of a particle. Magnetic couplings are here replaced by attractive or repulsive interactions between neighbor particles. The chemical potential, quantifying the number of particles on the system, plays the role of the external field. In this chapter we always refer to an Ising spin model, by analogy with magnetic problems, but it is worth to keep in mind that the model is equivalent to an adsorption problem. Indeed, it could be more intuitive to think of fullerenes as adsorbed particles rather than as spins, but the model is exactly the same in both cases.

The model -theoretical studies

This model is considered as the text-book example of geometrical frustration. Due to the triangular geometry, the low-energy antiparallel spin arrangements cannot be obtained simultaneously on the three bonds of a triangular plaquette. Only two out of three are obtained, see Figure IV.1. In the TIAF model two spins have the same Ising state (for example "up") and the third one the opposite state (then "down"). This is reminiscent of the so-called ice rule: in water ice [START_REF] Pauling | The Structure and Entropy of Ice and of Other Crystals with Some Randomness of[END_REF] two hydrogen atoms in a tetrahedron form short covalent bonds with the oxygen atoms, while the two others form long hydrogen bonds. The distribution of "short" and "long" bonds in the lattice is disordered, but it always respect this local rule, usually expressed as "two in-two out". In the ground state of the TIAF the situation is not strictly the same (two spins up/down and one down/up), but in the following we will call this local rule ice rule. 5 The global low-energy state of the TIAF model is hence obtained in the absence of plaquettes with three "up" or three "down" spins, or, said differently, when the formation of such plaquettes is forbidden.

By imposing this apparently simple rule, frustration gives rise to a very particular state of matter. At zero temperature, the ground state of the TIAF model is a spin liquid phase with residual entropy [START_REF] Houtappel | Order-disorder in hexagonal lattices[END_REF][START_REF] Husimi | The Statistics of Honeycomb and Triangular Lattice. I[END_REF][START_REF] Newell | Crystal Statistics of a Two-Dimensional Triangular Ising Lattice[END_REF][START_REF] Temperley | Statistical mechanics of the two-dimensional assembly[END_REF][START_REF] Wannier | Antiferromagnetism. The Triangular Ising Net[END_REF] and algebraic spin-spin correlations [START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF].

Liquid phases at zero temperature are not a usual behavior in matter, neither for atoms nor for spins. At low temperature, matter usually crystallizes, giving rise to ordered phases, both for positional or magnetic degrees of freedom. For the TIAF, it is not the case: the system remains in a disordered, highly fluctuating and extensively degenerate state at very vocabulary requires to be careful. Indeed, this phase can be called a spin liquid but its correlations are not algebraic like for the ideal TIAF we will describe in the introduction. Algebraic correlations only exist at T = 0.

5 Naming our local rule after the one found in water ice is not an arbitrary choice, but is in fact reminiscent of the geometrical frustration and of the extensively degenerate ground state in both systems. low temperature. In this sense such a state violates the third law of thermodynamics, as formulated by Planck's as: "The entropy of all systems in internal equilibrium is the same at absolute zero, and may be taken to be zero" [START_REF] Wilks | The Third Law of Thermodynamics[END_REF]. Indeed, in the TIAF entropy is extensive and does not tend to a constant value when temperature approaches to zero. Moreover, spin-spin correlations decay with distance in an algebraic way (∝ r -α , with α a positive real number), thus the phase is a critical phase with diverging correlation length, which is not the case for conventional liquids that exhibit exponential decay of the correlations (∝ e -r ). This algebraic, slow decay implies that correlations propagate far away through the lattice, i.e. spins are correlated over several lattice parameters. This might seem surprising for a disordered phase where only short-range couplings are at play. Such peculiar correlations are usually detected in reciprocal space thanks to diffraction techniques (such as neutron scattering [START_REF] Lacroix | Introduction to Frustrated Magnetism[END_REF]) in the form of non-analytical singularities called pinch points occurring at well-defined wavevector values (Figure IV.2). Profiles of the diffracted intensity taken perpendicular to the pinch point's axis show a characteristic non-analytical increasing intensity.

Until now we have only mentioned analytical studies of the short range (only J 1 )TIAF model. These studies discussed thermodynamics quantities, such as the entropy [START_REF] Houtappel | Order-disorder in hexagonal lattices[END_REF][START_REF] Husimi | The Statistics of Honeycomb and Triangular Lattice. I[END_REF][START_REF] Newell | Crystal Statistics of a Two-Dimensional Triangular Ising Lattice[END_REF][START_REF] Temperley | Statistical mechanics of the two-dimensional assembly[END_REF][START_REF] Wannier | Antiferromagnetism. The Triangular Ising Net[END_REF], and the spin-spin correlations [START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF][START_REF] Stephenson | Ising-Model Spin Correlations on the Triangular Lattice. III. Isotropic Antiferromagnetic Lattice[END_REF]. More recent analysis relies on numerical methods, such as MC simulations. The simplest MC simulations are based on local updates, but it has been observed that the TIAF model ground states exhibit topological sectors, which makes local updates inappropriate to explore the whole phase space [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] (see subsection 3.4 in chapter II for a brief discussion about topological sectors, local and global MC updates). This problem has been tackled using different global MC algorithms, for instance by Zhang et al. [START_REF] Zhang | Cluster Monte Carlo dynamics for the antiferromagnetic Ising model on a triangular lattice[END_REF] with a cluster algorithm method. This method updates configurations by flipping clusters of spins, instead of single spins like in local algorithms. The same applies to loop algorithms, which consist on updating spin loops [START_REF] Wang | Generalized Monte Carlo loop algorithm for two-dimensional frustrated Ising models[END_REF]. Well-chosen global updates improves the sampling of the phase space at low temperature, and thus better characterize the ground state. (a,b) Ferromagnetic disks arranged on a triangular lattice, exhibiting out-of-plane magnetization. Each cluster is formed by seven disks, each one identified as an Ising pseudo-spin up (white) or down (black). Adapted from [START_REF] Mengotti | Dipolar energy states in clusters of perpendicular magnetic nanoislands[END_REF]. (c) Triangular lattice of nanomagnets, similar to the ones in (a,b). Black/white contrast reveals magnetization of the disks, that can be compared with up and down spins. A disordered pattern of up/down Ising pseudo-spins is observed. Adapted from [START_REF] Kempinger | Imaging the stochastic microstructure and dynamic development of correlations in perpendicular artificial spin ice[END_REF]. (d) Colloidal particles trapped between two plates. Each dot is one colloidal sphere, contrast corresponds to their position (bright/dark = up/down). The phase looks disordered. Adapted from [START_REF] Han | Geometric frustration in buckled colloidal monolayers[END_REF].

Experimental realizations of the TIAF model

Possible experimental realizations of the TIAF model with only J 1 coupling have been discussed, but in many cases it was difficult to conclude whether the experimental systems presented all the ingredients of the model (two-dimensional triangular lattice, classical bivalued spins and two-body short range antiferromagnetic couplings). For example, if we wanted to look for a purely two-dimensional TIAF in experimental magnetic compounds, we would find much more complex situations, including three-dimensional structures, spins not necessarily confined to an Ising axis, presence of quantum fluctuations, existence of long range interactions, etc. Here we present experimental examples that might realize the TIAF model with short range couplings, despite the fact that for most of them the existence of a spin liquid phase remains an open question 6 . In fact, neither entropy nor spin-spin correlations were characterized in detail, which makes it impossible to conclude about the nature of the observed phases. We divide our review in three parts, each devoted to a specific scientific community. In the first group we sum up examples corresponding to artificial frustrated systems, in which matter is designed ad hoc to study frustrated phases of matter. In the second and third categories we review on-surface systems, namely the surface of metallic alloys or organic molecules deposited on inorganic surfaces.

Artificial frustrated systems

In most cases the community of artificial frustrated systems fabricates arrays of nanomagnets, each of which is considered as an Ising pseudo-spin. Using deposition and litography techniques, the geometry of the lattice can be controlled, and then the magnetic dipolar interactions coming from the stray field generated by nanomagnets can be tuned. The state of each Ising pseudo-spin can be observed in real space, using microscopy techniques, such as magnetic force microscopy and photoelectron emission microscopy [START_REF] Nisoli | Colloquium: Artificial spin ice: Designing and imaging magnetic frustration[END_REF][START_REF] Rougemaille | Cooperative magnetic phenomena in artificial spin systems: spin liquids, Coulomb phase and fragmentation of magnetism-a colloquium[END_REF][START_REF] Skjaervø | Advances in artificial spin ice[END_REF]. In this context, several studies have analyzed the behavior of nanomagnetic disks with perpendicular anisotropy, placed on a triangular lattice. Magnetic imaging of some of these systems are shown in Figure IV.3.

Mengotti et al. [START_REF] Mengotti | Dipolar energy states in clusters of perpendicular magnetic nanoislands[END_REF] fabricated this type of samples, focusing on small clusters formed by two to seven nanomagnets (Figure IV.3a). They tuned the dipolar interaction, observed the resulting configurations for the clusters and compared them with the expected ground state according to dipolar energy calculations. Their measurements allowed to test the experimental protocol to thermalize samples, i.e. to bring the configuration of the nanomagnets to a low-energy state. They found agreement between the calculated and experimental ground states. Due to the small number of magnets in the clusters, no cooperative phenomena, so no spin liquid phase, are expected in such samples.

The same type of samples, this time in the form of lattices of hundreds of nanomagnets (Figure IV.3b), was analyzed by Kempinger et al. [START_REF] Kempinger | Imaging the stochastic microstructure and dynamic development of correlations in perpendicular artificial spin ice[END_REF]. The study focused on the dynamics of the magnetization during hysteresis cycles and the stochasticity of the process, and they did not determine the nature of the ground state of these lattices.

Finally in 2021, Pip et al. [START_REF] Pip | Direct observation of spin correlations in an artificial triangular lattice Ising spin system with grazing-incidence small-angle neutron scattering[END_REF] studied magnetic correlations in the same type of samples. Nevertheless, they measured the lattice in reciprocal space, using neutron scattering, so they did not have access to configurations in real space. They found spin liquid-like correlations that they identified as a realization of the TIAF model with antiferromagnetic dipolar interactions. In this case spin-spin interactions are not short range but infinite range, which modifies the nature of the ground state. At zero temperature an antiferromagnetic striped ground state is expected [START_REF] Smerald | Spin-liquid behaviour and the interplay between Pokrovsky-Talapov and Ising criticality in the distorted, triangular-lattice, dipolar Ising antiferromagnet[END_REF]. Nevertheless, they observed disordered phases corresponding to the TIAF with dipolar interactions at temperatures above the transition to the ground state.

The last example of an artificial system is a non-magnetic, unusual one. Han et al. [START_REF] Han | Geometric frustration in buckled colloidal monolayers[END_REF] used a two-dimensional colloidal system to study the TIAF physics. In their experiment colloidal spheres are trapped between two plates separated by a distance slightly larger than the diameter of the particles (Figure IV.3d). Colloidal particles self-organize, forming a close-packed lattice, and buckle, forming a corrugated layer in which the position of the particles can be considered as an Ising variable. In this case the degree of freedom was not magnetic but positional, and the interactions correspond to those of a hard-sphere model. The resulting phase was not a spin liquid-like phase, but a mixture of zigzagging antiferromagnetic stripes. The authors conclude that in-plane lattice distortions partially relieve frustration and lead to a sub-extensive entropy, instead of the extensive entropy of the spin liquid phase.

On-surface systems: binary alloys

Our second group of examples includes binary alloys on surfaces, in which lattices are [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. On the bottom right region of the image a complex bright (Pb atoms) and dark (Si atoms) pattern is observed. Adapted from [START_REF] Ganz | Submonolayer phases of Pb on Si(111)[END_REF]. (b) Ge 0.37 Pb 0.63 /Ge(111) alloy. A disordered pattern with formation of local antiferromagnetic-like stripes is observed by STM. Adapted from [START_REF] Carpinneli | Charge rearrangement in the Ge x Pb 1-x /Ge(111) interface[END_REF]. (c) STM image of an Sn 0.51 Si 0.49 /Si(111) alloy, exhibiting a disordered contrast with some antiferromagnetic-like stripes. Adapted from [START_REF] Ottaviano | Short-range order in two-dimensional binary alloys[END_REF]. (d) Re 0.5 Nb 0.5 S 2 observed by darkfield scanning transmission electron microscope, exhibiting disordered contrast. Adapted from [START_REF] Azizi | Frustration and atomic ordering in a monolayer semiconductor alloy[END_REF]. close-packed. These systems are non-magnetic, neither the degree of freedom nor the interactions [START_REF] Ortiz-Ambriz | Engineering of frustration in colloidal artificial ices realized on microfeatured grooved lattices[END_REF][START_REF] Lee | Energetics and the ground state quest in an artificial triangular colloidal ice[END_REF]. The Ising variable is the chemical nature of the elements, that can be determined by microscopy techniques. Atoms in these alloys tend to be surrounded by atoms of the other chemical kind, than by atoms of the same type. This interaction is equivalent to an antiferromagnetic coupling. These types of systems cannot be tuned at will, like it was the case for artificial frustrated systems, but "natural" chemical interactions drive the ordering of the phases. However, the ratio of species in the alloy can be controlled, which is equivalent to control the chemical potential, or in the Ising spin Hamiltonian the magnetic field. Some examples are shown in Figure IV.4.

To our knowledge the first proposition of such a system as an example of geometrical frustration was published by Ganz et al. [START_REF] Ganz | Submonolayer phases of Pb on Si(111)[END_REF] who proposed Pb phases on Si [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] as a realization of the TIAF model (Figure IV.4a). However, the only argument for this comparison were the disordered patterns observed by STM, but no detailed statistical analysis of the disorder was proposed.

Carpinelli et al. [START_REF] Carpinneli | Charge rearrangement in the Ge x Pb 1-x /Ge(111) interface[END_REF] studied another binary alloy formed by Ge and Pb atoms (Figure IV.4b). They compared their results with the ones by Ganz and collaborators and started a statistical analysis in which they demonstrated that the observed disorder is not random but correlated. However, they only analyzed correlations between nearest neighbors, which is not sufficient to determine the nature of the observed phase.

Ottaviano et al. [START_REF] Ottaviano | Short-range order in two-dimensional binary alloys[END_REF] analyzed different types of two-dimensional alloys: Sn-Si, Pb-Si and Pb-Ge (Figure IV.4c). They pushed their analysis much further than previous works and compared experimental correlations with calculated ones for perfectly ordered phases and simulated ones using a TIAF model and MC simulations. They demonstrated that the Adapted from [START_REF] Charra | Surface-induced chirality in a self-assembled monolayer of discotic liquid crystal[END_REF]. (b) C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface, each molecule is one dot in the image. The contrast corresponds to the apparent height of the molecules, which was proposed to be a bivalued degree of freedom. Adapted from [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. (c) C 60 molecules on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface, exhibiting two different contrasts linked to the height of the molecules. Adapted from [START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF].

studied binary alloys can be understood in the TIAF framework when the concentration of the two different atomic species is the same.

Azizi et al. [START_REF] Azizi | Frustration and atomic ordering in a monolayer semiconductor alloy[END_REF] studied a semiconductor alloy, which is in fact a monolayer of a twodimensional material: Re 0.5 Nb 0.5 S 2 (Figure IV.4d). Re and Nb atoms sit on a triangular lattice and disregarding the S lattice, they can be considered as a binary alloy. The authors analyzed the probability distribution of having different nearest neighbors, which is linked to atom-atom correlations in real space. They compared the experimental values with MC simulations and demonstrated that the system is correlated, frustrated and compatible with a TIAF description. They also proposed values for the effective couplings, based on DFT calculations. This study demonstrated in a quantitative way that binary alloys are a promising platform to look for frustrated phases of matter.

On-surface systems: organic molecules

Finally we describe the kind of surface system that was studied during this thesis, organic molecules on surfaces. For all the cases presented here molecules self-assemble forming close-packed lattices. Like in the previous examples for binary alloys, these systems are nonmagnetic. Ising degrees of freedom are structural, linked to the position of the molecules. The role of the interactions is difficult to study, because interactions both with the substrate and with neighbor molecules are expected. This type of systems can be observed in real space thanks to scanning tunneling microscopy (STM).

The first example was published by Charra et al. [START_REF] Charra | Surface-induced chirality in a self-assembled monolayer of discotic liquid crystal[END_REF]. They studied the assembly of hexakis-7,10,11-alkyloxytriphenylene, complex organic molecules with four aromatic rings on their center and carbon chains forming external branches, on highly oriented pyrolitic graphite (Figure IV.5a). They observed that these molecules can take two different orientations rotated by 180°, and arrange forming structures similar to the ones observed for a TIAF model. They proposed an Ising spin Hamiltonian with only first neighbor couplings, but the patterns they identify corresponds to ordered phases that arise when long range interactions are considered, as we will see in next section.

Fullerene assemblies on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] and Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF] have also been proposed as realizations of the TIAF model. On both substrates the structural degree of freedom is the height of the molecules with respect to the metal surface. Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] proposed that, for fullerenes on copper (Figure IV.5b), molecular repulsion induces a buckling of the molecules, thus alternating heights like in the case of an antiferromagnetic interaction. They compared local patterns observed in the STM images with a configuration derived from a MC simulation, but did not propose a quantitative statistical analysis. Paβens et al. [START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF] observed similar patterns on C 60 assemblies on Au(111) (Figure IV.5c) and also proposed to interpret them on the framework of a TIAF model. They verified that correlations are present on the system, despite the apparent disorder. Nevertheless, they found a different arrangement of "high" and "low" molecules, which they tried to explain including a coupling that depends on the height of the molecule. This model is not the usual TIAF, thus we cannot predict the nature of the ground state.

Here we have listed all the experimental systems (to our knowledge) that can be observed in real space and are candidates for the realization of the TIAF with short-range interactions. This review allows us to contextualize the work we present in next sections, and to identify missing analysis in previous works: visual comparison of disordered patterns is not suited to decide the nature of a phase, statistical analysis of correlations needs to include next nearest neighbors and beyond, and a combination of analysis in real [START_REF] Ottaviano | Short-range order in two-dimensional binary alloys[END_REF][START_REF] Azizi | Frustration and atomic ordering in a monolayer semiconductor alloy[END_REF] and reciprocal space [START_REF] Pip | Direct observation of spin correlations in an artificial triangular lattice Ising spin system with grazing-incidence small-angle neutron scattering[END_REF] is an insightful strategy to confirm and characterize spin liquid like phases.

Further-distance couplings and ordered ground states

In subsection 1.1 we have only described theoretical and experimental results for the case of TIAF with first neighbor couplings, which is a frustrated model exhibiting an exotic spin liquid ground state at low temperature. However, many other experimental realizations of the TIAF model exist in different physical systems, such as layers of gases adsorbed on solid surfaces, intercalation compounds, quasi 2D magnetic compounds, charge systems, etc. These TIAF examples are usually dominated by long range interactions giving rise to the formation of ordered phases and not to the spin liquid phase. Long range interactions can have different physical origins, depending on the system: dipolar interactions are expected for magnetic moments, Coulomb interactions for charged particles, etc. Now, we address the case of the TIAF model with longer range interactions. In particular we introduce the ingredients we need for our analysis in the following sections: first, second, third neighbor couplings (J 1 , J 2 , J 3 ) and an external field (h ext ). In this case the Ising Hamiltonian writes:

H = -J 1 ⟨ij⟩ σ i σ j -J 2 ⟨⟨ij⟩⟩ σ i σ j -J 3 ⟨⟨⟨ij⟩⟩⟩ σ i σ j -h ext i σ i
First, we describe theoretical results published in the literature, considering couplings up to the third neighbor. Using this type of Hamiltonian with a cut-off for the range of the interactions is a good approximation for many experimental systems. Moreover, tuning the relative values of the coupling strengths and their signs gives rise to a rich phase diagram, in which many different ordered phases have been predicted. We also review studies including an external magnetic field, or in the language of gas Ising Hamiltonians a fixed chemical potential. A literature review of such a rich phase diagram requires to analyze more than 30 publications, in which 50 different ordered ground states have been identified. To simplify this review, we divide our discussion in three parts: TIAF Hamiltonian including J 1 , J 2 couplings and an external field h ext ; J 1 , J 2 and J 3 couplings; and finally, J 1 , J 2 , J 3 and h ext . At the end of the review we briefly sum up the experimental systems in which ordered phases of matter have been identified as realizations of the TIAF with further neighbor couplings.

After the review of the literature, we build the phase diagram at zero temperature that we need for the discussion in section 2 and section 3. This phase diagram can be understood as a compilation and a global analysis of previous results from the literature, for coupling strengths related to our experiment. A similar approach, based on our literature review, is used to propose a phase diagram (for other coupling strengths and external field) in section 5. Phases I-VB are adapted from [START_REF] Metcalf | Ground state spin orderings of the triangular Ising model[END_REF], phase C is adapted from [START_REF] Tanaka | Ground state spin configurations of the triangular Ising net with the nearest and next nearest neighbor interactions[END_REF]. The most studied phases have been named after their magnetic contrast or the periodicity of the cell.

Theoretical studies for the TIAF including first and second neighbor couplings can be grouped depending on three main criteria, whether:

• they address the whole phase diagram or the thermodynamic properties of a given phase or a phase transition;

• they are based on analytical or numerical methods;

• they consider Ising spin models or Ising gas models.

The first phase diagrams at zero temperature for the Ising model with J 1 , J 2 and h ext were proposed in the seventies. First theoretical studies were published by Metcalf [START_REF] Metcalf | Ground state spin orderings of the triangular Ising model[END_REF], who used MC simulations, and Tanaka et al. [START_REF] Tanaka | Ground state spin configurations of the triangular Ising net with the nearest and next nearest neighbor interactions[END_REF], who used analytical methods. Metcalf proposed five different ground states, four of them exhibiting ordered phases and a fifth one which is a coexistence of two different ordered patterns. Tanaka et al. completed this phase diagram, adding a sixth ground state with larger periodicity than the cells considered by Metcalf. In Figure IV.6 we show the phases proposed in this studies, all of them being ordered and having different magnetization depending on the intensity of the external field. Kaburagi et al. studied the Ising gas Hamiltonian, but using analytical methods. They found the same ground states for gases adsorbed on solids, and related the theoretical ground states with some experimental examples. In 1986, Takasaki et al. [START_REF] Takasaki | Magnetic Phase Diagram of the Ising Model on a Triangular Lattice with Antiferromagnetic Nearest-Neighbors and Next-Nearest-Neighbor Interactions[END_REF] completed the study of the phase diagram at finite temperature.

Since the end of the seventies many studies have addressed particular regions of the phase diagram and the nature of phase transitions when temperature is non-zero. The most studied phases are the √ 3 × √ 3 phase and the antiferromagnetic stripe or 2 × 1 phase (see Figure IV.6 for phase nomenclature). The next paragraphs sum up the main results for these phases when temperature and magnetic field vary.

In particular the √ 3 × √ 3 phase has been intensively studied. This phase is the ground state when J 1 is antiferromagnetic and J 2 ferromagnetic and is favored by the presence of an external field. These studies were motivated by the fact that He 4 adsorbed on graphite and magnetic chains of CsCoCl 3 are experimental systems corresponding to the Hamiltonian. Mihura et Landau proposed a first MC study in 1977 [START_REF] Mihura | New Type of Multicritical Behavior in a Triangular Lattice Gas Model[END_REF], in which they studied the effect of density and temperature for the gas model. They showed that multicritical points between solid, liquid and gas phases are present in the phase diagram. Landau continued the MC study of phase transitions in this regime [START_REF] Landau | Critical and multicritical behavior in a triangular-lattice-gas Ising model: Repulsive nearest-neighbor and attractive next-nearest-neighbor coupling[END_REF] and determined the critical exponents for the transition. He also found evidence of critical behavior and topological excitations, pointing out to a unusual behavior that was described in next studies. Fujiki et al. [START_REF] Fujiki | Possibility of the Kosterlitz-Thouless Phase Transition in the Two Dimensional Fully Frustrated Ising Model[END_REF] used MC simulations to describe the presence of topological defects and proposed that the phase transition from the ordered √ 3 × √ 3 phase to a disordered phase at high temperature is a Berezinskii-Kosterlitz-Thouless transition. Qian et al. [START_REF] Qian | Triangular Ising model with nearest-and next-nearestneighbor couplings in a field[END_REF] employed MC and transfer matrix calculations to study the same region of the phase diagram, with better determination of thermodynamic properties. A slightly different Hamiltonian, including antiferromagnetic J 1 with random fluctuations and an external field h ext , has been used to simulate the magnetization plateaux observed in Ca 3 Co 2 Cl 6 , whose ground state at high field is the √ 3 × √ 3 phase [START_REF] Yao | Steplike magnetization of spin chains in a triangular lattice: Monte Carlo simulations[END_REF][START_REF] Yao | Monte Carlo simulation of magnetic behavior of a spin-chain system on a triangular lattice[END_REF]. Nishino et al. [START_REF] Nishino | Termination of the Berezinskii-Kosterlitz-Thouless phase with a new critical universality in spin-crossover systems[END_REF] studied the nature of the Berezinskii-Kosterlitz-Thouless transition in a TIAF when elastic interactions are present; they showed that these interactions are equivalent to the presence of a ferromagnetic coupling.

Many studies have also addressed the region of the phase diagram in which the antiferromagnetic stripe or 2×1 phase is the ground state. This region corresponds to J 1 and J 2 both antiferromagnetic. Glosli et al. [START_REF] Glosli | A Monte Carlo and renormalization group study of the Ising model with nearest and next nearest neighbor interactions on the triangular lattice[END_REF] used MC calculations and renormalization group theory to study the phase transitions from the stripe phase to other phases when temperature and magnetic field are varied. They found three different ground states depending on the value of the field: the stripe phase at low field, the √ 3 × √ 3 phase at intermediate field and the 2 × 2 phase at higher fields. They detected both first and second order transitions, depending on the transition line that is considered. Slotte et al. [START_REF] Slotte | The triangular Ising antiferromagnet with first-and second-neighbour interactions[END_REF] analytically determined the critical temperature for the phase transition from the stripe phase to the disordered one, when temperature is increased. Einevoll et al. [START_REF] Einevoll | The triangular antiferromagnet with infinite nearestneighbour coupling[END_REF] used analytical methods to show that the transition is first order in the J 1 → ∞ limit. Rastelli et al. [START_REF] Rastelli | Monte Carlo simulations on a triangular Ising antiferromagnet with nearest and next-nearest interactions[END_REF] used MC simulations to study the nature of the transition. They confirmed that it is a first order transition, and not a Berezinskii-Kosterlitz-Thouless transition like in the case of ferromagnetic J 2 .

TIAF with J 1 , J 2 , J 3 couplings The phase diagram for the TIAF with J 1 , J 2 , J 3 couplings has been reported in three different papers. The first one was published in 1976 by Tanaka and Uryu [START_REF] Tanaka | Ground State Spin Configurations of the Triangular Ising Net with the First, Second and Third Nearest Neighbor Interactions[END_REF] who used analytical calculations to find the phase diagram at zero temperature. They obtained seven ordered ground states, shown in Figure IV.7, and 4 degenerate states that coexist on the isoenergetic lines between different regions on the phase diagram. Takagi and Mekata [START_REF] Takagi | New Partially Disordered Phases with Commensurate Spin Density Wave in Frustrated Triangular Lattice[END_REF] analyzed the same phase diagram using mean field theory and MC simulations, including the effect of temperature. They considered eight candidates for the ground states at zero temperature and identified five of them as really being the ground states. At finite temperature they observed phases with partially disordered states, i.e. configurations in which some spins are fixed and form ordered patterns, while others fluctuate. They identified these partially disorder phases as intermediate states forming because of a Kosterlitz Thouless transition.

Korshunov [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF] focused on the stripe phase and studied how the phase evolves when temperature is increased. His analysis is based on the formation of domain walls (DW) between stripe domains, in the case J 1 → ∞. In this context DWs energies depend only on J 2 and J 3 couplings, and for the particular ratio J 3 J 2 = 0.5, the formation of DWs does not cost any energy. In this particular case, which will be important in our discussion in section 3, proliferation of DWs is expected. In fact the stripe ground state can be considered as a background in which excitations with no energy cost exist. This mechanism transforms the first order phase transition from a stripe to a disorder phase into a continuous phase transition [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF].

Figure IV.8: (a)

Phase diagram for the TIAF model with J 1 , J 2 and J 3 as published by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF]. Each colored area correspond to one ground state of the TIAF. Black points indicate that a first order transition was found in MC simulations. The region enclosed by red points is a critical region (which becomes a point in the thermodynamic limit) in which stripe and zigzag phases are degenerate. (b-d) MSF of the associated phases.

Finally, in 2016 Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] studied the phase diagram of the Ising Hamiltonian with J 1 , J 2 , J 3 depending on temperature. They considered the topological nature of the phases and developed MC simulations based on a worm algorithm. This algorithm used global updates in which well-chosen chains of spins are updated to probe the whole phase space. This study corresponds to the J 1 → ∞ limit, considers antiferromagnetic couplings and the region of the phase diagram in which J 3 J 2 < 0.75. The phase diagram is reproduced in Figure IV.8 and constitutes the starting point of our study, as we will see later. At low temperature, the phase diagram shows two ordered ground states, called stripe and zigzag phases. For higher temperatures a correlated paramagnet, i.e. a spin liquid has been predicted, in good agreement with the ground state when only J 1 is present in the Hamiltonian.

The region near the J 3 J 2 = 0.5 point at low temperature was also described by the authors7 , who found good agreement with the previous study by Korshunov [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF]. At this point of the phase diagram, the stripe phase and the zigzag phases are degenerate. Indeed, we can imagine the ground state at this point as a stripe phase in which the formation of DW has no energy cost. These DW locally correspond to zigzag patterns, as we will explain in detail in subsubsection 3.5. The magnetic structure factor (MSF) of the spin liquid phase (critical paramagnet in Smerald's denomination, which makes reference to the fact that the temperature is always zero with respect to the couplings, because J 1 → ∞ in their calculation) shows structured intensity and presence of pinch points (Figure IV.8c).

It is worth to note that the work by Smerald et al. demonstrated and characterized the topological nature of the phases in the TIAF model. Besides the exotic properties we have mentioned for the spin liquid phase (extensive entropy, correlated disorder) topological properties play a key role in the TIAF model. For instance, ground states of the TIAF are divided in topological sectors (i.e. regions of the space phase characterized by different topological numbers), which are not accessible by local updates in MC simulations. Flipping single spins is not enough to fully explore the phase diagram, and spin dynamics needs to include collective updates. If we imagine an experimental system, in which spins usually flip independently one from each other, it would never explore the whole phase space of the TIAF.

TIAF with J 1 , J 2 , J 3 couplings and h ext magnetic field Kaburagi and Kanamori [91] extended the study of the phase diagram to the Ising gas model including J 1 , J 2 , J 3 couplings and an external magnetic field. Their analysis used analytical methods and proposed 35 phases as potential ground states of the model, see Figure IV.9. All the ground states are ordered and they correspond to different fixed chemical potentials. The phase diagram proposed in this study is particularly complex, because of the 4-dimensional space formed by couplings and magnetic field. One of the interesting aspects of their study is the comparison of the phase diagram with phases observed experimentally in different physical systems. They identified different ordered phases as realizations of the TIAF with first, second, third neighbor couplings and magnetic field, both for magnetic and adsorption problems.

Experimental realizations of the TIAF with couplings beyond first neighbors and magnetic field

We will not detail each experimental system that can be identified as a phase existing in the TIAF model, but give examples for different classes of physical systems that can be found in the following references.

Figure IV.9:

Schematics of different ordered phases that have been proposed as ground states of the Ising gas Hamiltonian on a triangular lattice with J 1 , J 2 , J 3 and h ext , as published by Kaburagi and Kanamori [START_REF] Kaburagi | Ground State Structure of Triangular Gas Model with up to 3rd Neighbor Interactions[END_REF].

For adsorption of atoms on solid surfaces, usually interpreted in the framework of Ising gas models, many examples exist, among which: He 4 on graphite (mentioned in the case of antiferromagnetic J 1 and ferromagnetic J 2 ) [START_REF] Mihura | New Type of Multicritical Behavior in a Triangular Lattice Gas Model[END_REF][START_REF] Landau | Critical and multicritical behavior in a triangular-lattice-gas Ising model: Repulsive nearest-neighbor and attractive next-nearest-neighbor coupling[END_REF]; Kr and Xe in graphite [START_REF] Villain | Static Properties of Physisorbed Monolayers in their Solid Phases[END_REF]; N on graphite [START_REF] Kjems | Neutron scattering from nitrogen adsorbed on basal-plane-oriented graphite[END_REF]; C or Co on Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Kaburagi | Ground State Structure of Triangular Gas Model with up to 3rd Neighbor Interactions[END_REF]; H, C0 2 , CH 4 on Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Kaburagi | Ground State Structure of Triangular Gas Model with up to 3rd Neighbor Interactions[END_REF]. For compilations of many different examples see Refs. [START_REF] Dash | 8 -Epitaxial Monolayers[END_REF][START_REF] Kaburagi | Ground State Structure of Triangular Gas Model with up to 3rd Neighbor Interactions[END_REF].

Intercalation compounds have also been identified as realization of the TIAF [START_REF] Berlinsky | Theory of Litium Ordering in Li x TiS 2[END_REF][START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF][START_REF] Scholz | Electron Diffraction Investigation of the Ag x TaS 2 System[END_REF][START_REF] Plischke | Intercalate order-disorder transition in stage-2 Ag x TiS 2[END_REF][START_REF] Glosli | A Monte Carlo and renormalization group study of the Ising model with nearest and next nearest neighbor interactions on the triangular lattice[END_REF]. These systems are compounds with chemical formula M x TiS 2 or M x TaS 2 in which the metal atoms (M) are intercalated between layers of TiS 2 or TaS 2 , forming ordered supercells like the ones in the TIAF model.

Concerning magnetic compounds there are two main categories of materials that can be considered as two-dimensional systems. The first ones are magnetic compounds with layered structures, in which magnetic interactions between layers are much smaller than intra-layer couplings. It is the case of FeI 2 [START_REF] Uryû | Ordered spin structure of FeI2[END_REF], CsCoCl 3 [START_REF] Mekata | Magnetic Structure of CsCoCl 3[END_REF][START_REF] Yoshizawa | Neutron Scattering Study of a One-Dimensional Ising-Like Antiferromagnet CsCoCl 3 . I. Instantaneous Correlation[END_REF], CsCoBr 3 , MnBr 2 and FeCl 2 compounds [START_REF] Kaburagi | Ground State Structure of Triangular Gas Model with up to 3rd Neighbor Interactions[END_REF]. The second type includes three-dimensional compounds in which chains of atoms have strong anisotropies, and are thus considered as Ising pseudo-spins coupled between them in a two-dimensional plane. Some examples with structures M 2 X (with M a metal and X carbon or nitrogen) are presented in [START_REF] Kaburagi | Ground State Structure of Triangular Gas Model with up to 3rd Neighbor Interactions[END_REF], a mineral found in nature and called microsommite have also been reported [START_REF] Bonaccorsi | Microsommite: Crystal chemistry, phase transitions, ising model and Monte Carlo simulations[END_REF] and RuBr 3 and MoBr 3 compounds [START_REF] Merlino | Order-disorder transformation in RuBr 3 and MoBr 3 : A two-dimensional Ising model[END_REF] have been identified as relizations of the TIAF.

Our phase diagram at

T = 0 for antiferromagnetic J 1 , J 2 , J 3
Here we focus on the region of the phase diagram we need for the analysis of our experimental molecular lattices, i.e. the case with antiferromagnetic J 1 , J 2 , J 3 couplings. Dressing a full picture of the ground states in this region, using the previous review of the literature, can be a difficult task: many phases have been proposed as ground states, values of the couplings need to be renormalized depending on the type of Hamiltonian (spin or gas model) and no unified discussion exist. The most similar phase diagram to the one we need is the one proposed by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] that we show in Figure IV.8, but where the J 1 → ∞ limit is considered, which is not necessarily the case in our experiments.

Figure IV.10:

Example of the calculation of the energy of an ordered ground state, considering an Ising spin Hamiltonian with first to third neighbor couplings. First, the unit cell is defined. Second, for each spin in the unit cell the energy for each bond with first (1NN), second (2NN) and third neighbors (3NN) is computed. Third, all the terms are summed and normalized by the number of spins and by 1 2 to avoid counting each bond twice. In this case the two spins in the unit cell are equivalent and give energies:

E 1N N = -2J 1 , E 2N N = -2J 2 , E 3N N = 6J 3 ; so total energy is E = 1 2 (-2J 1 -2J 2 + 6J 3 ) = -J 1 -J 2 + 3J 3 .
The energy term corresponding to the field can be computed simply as

E hext = Nup-N down N spins
with N up/down the number of up/down spins in the unit cell and N spins the number of spins inside the unit cell.

In order to build the phase diagram we proceed as follows:

1. We identify all the phases that have been proposed as low energy phases for the TIAF including J 1 , J 2 , J 3 couplings and a magnetic field h ext . To do so we consider 31 publications with more than 50 different phases.

analytically compute isoenergetic lines between phases.

5. Finally, we use MC simulations to confirme the validity of our phase diagram for certain values of the couplings.

In Figure IV.11 we show the result of the above-listed five steps for the phase diagram at zero temperature in the region J 1 , J 2 , J 3 < 0, for a fixed value of J 1 = -1. We also consider that |J 2 |, |J 3 | ≤ |J 1 |. This hypothesis seems natural, since most interactions in matter decay with the distance. We found three ordered ground states, depending on the value of the couplings:

• A stripe phase with (2 × 1) cell, with energy

E s = J 1 + J 2 -3J 3 ;
• a zigzag phase with (4 × 2) cell, with energy E z = J 1 -J 2 + J 3 ;

• and a more complex phase with (8 × 4) cell and energy E 4s = J 2 + J 3 . This phase shows groups of four equal spins and we will call it the 4 spin phase.

We use Monte Carlo simulations to verify that indeed these phases are the ground states in different regions of the phase diagram. The circles in Along the isoenergetic lines between ground states the behavior is complex. For the isoenergetic line J 3 J 1 = 1 4 that separates the stripe and the 4 spin phases, we found the coexistence of these two phases with another ordered ground state that is only present for this particular ratio of couplings. This new phase has a 6 × 6 cell and shows a mixed pattern with stripes and groups of three neighbor spins (3 spin phase) in the same up/down state. In this particular line, the three phases are degenerate. MC simulations along this line shows a mixture of the three phases at low temperature: we can identify local patterns corresponding to the expected phases but the system is unable to bifurcate and choose one of the phases. In addition, the phase at low temperature strongly depends on lattice size because the unit cells for each of the coexisting phases are different ((2 × 1) for stripe, (8 × 4) for 4 spins and (6 × 6) for the stripe/3 spin phase). A more detailed study would be necessary to properly determine the thermodynamic properties and the type of phase transition along this peculiar line, but it is beyond the scope of our work.

Along the line between the stripe and zigzag phase, J 3 J 2 = 1 2 , the behavior has been characterized by Korshunov [101] and by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF], as we described in the review of the literature. For this ratio of the couplings coexistence of stripe and zigzag phase is observed, or in an equivalent description a stripe background in which zigzag-like domain walls form is observed. Indeed, the zigzag phase is a fully packed set of straight domain walls. Korshunov [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF] demonstrated that DWs cost zero energy in this line of the phase diagram. Moreover, Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] proved that for finite lattices the presence of domain walls extent to a small region of the phase diagram around the J 3 J 2 = 1 2 value, giving rise to "an additional phase" that does not exist in the thermodynamic limit. Our MC simulations are in agreement with this study. We found that MC snapshots on this line show signatures of both stripe and zigzag phases. Further details about the behavior of MC simulations in this region are presented in section 3 and Appendix C.

In the isoenergetic line between the zigzag and the 4 spin phase we expect a coexistence between both phases. MC simulations show a mixture of patterns present in both phases without bifurcation.

At the triple point in which the three isoenergetic lines merge ( J 2 J 1 = 0.5, J 3 J 1 = 0.25) we expect the coexistence of 7 ordered phases, following our analysis of the energy of the phases at zero temperature. These phases include zigzag and stripe patterns with different periodicity and groups of 3 and 4 neighbor spins in the same up/down state. In MC simulations we observe a mixture of local patterns that seem to correspond to the expected ones.

Our phase diagram at zero temperature allows us to verify the presence of zigzag and stripe phases even when J 1 does not tend to infinity. These two phases are of particular interest in the description of our experimental results. Moreover, we also found that the coexistence of both phases along the isoenergetic line J 3 J 2 = 0.5 is also present in our phase diagram.

In principle, MC simulations are an excellent tool to address the effect of temperature in the previous phase diagram, determine which phases have lowest energy at finite temperature, develop a detailed thermodynamic analysis, determine the nature of the phase transitions, etc. However, these goals are out of the scope of this thesis. First, because they would be useless for the comparison with our experimental system. Second, because our MC simulations are in agreement with previous studies, thus we combine our simulations with results reported in the literature.

In some cases, curiosity pushed us to verify some particular thermodynamic properties calculated with our simulations, or the agreement between our numerical results and the expected behaviors for some phases. These results are not essential for the discussion in next sections, so we present them in the Appendix C.

Spin liquid phase with emergent zigzag order in C 60 /Cu(111)

We aim to show that the non-ordered height contrast of some assemblies of C 60 molecules on a copper surface, that we described in chapter III, can be compared with arrested configurations of a TIAF (also called microstates in the text). We now focus on the peculiar height pattern exhibited by the 9.68 Å-22.4°phase. In the following we call this phase disordered phase. We want to verify whether the observed disorder is correlated or not, and if it can be understood as a structural equivalent of a spin liquid phase at finite temperature.

We have described the limitations of the previous analysis (chapter III) in molecular assemblies, in particular the fact that our model considered supercells that repeat periodically. Thus, the model is not adapted to describe non-ordered phases. In this chapter we propose a different approach, based on Ising models that capture disordered, correlated phases. We start our discussion with a phenomenological analysis in which we forget about the microscopic properties of the molecular lattice and establish the analogy with the ingredients of a TIAF model. Later on, we analyze the equivalent of spin-spin correlations, both in real and reciprocal space, to decide if the signatures of fullerenes assemblies are comparable to the ones of a spin liquid phase. Finally, we come back to the real molecular lattice and its properties with a two-fold goal: clarifying the microscopic origin of the couplings in the TIAF model and determining the limits of our effective analysis.

With our analysis we are able to demonstrate that the 9.68 Å-22.4°phase is a experimental realization of a frustrated TIAF model. In particular, we show that the non-ordered height patterns are correlated in the same way as in a spin liquid phase at a temperature just above the critical temperature of a transition from the spin liquid phase to an ordered zigzag phase. Exploring the link between molecular structural non-magnetic systems and frustrated magnets was the main goal of this thesis, so this chapter is of particular interest.

Molecular TIAF: ingredients

The TIAF Hamiltonian applies to systems where three specific ingredients are present. First, a triangular two dimensional lattice. Second, Ising variables. Third, an antiferromagnetic coupling between nearest neighbors. We start our analysis by showing that each of these ingredients is indeed present in C 60 assemblies on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF].

Lattice geometry -The copper substrate is a single crystal with FCC structure. The crystal is cut along the (111) crystallographic plane, which exposes a triangular arrangement of copper atoms at the metal surface. Fullerenes are known to form commensurate phases on top of metal surfaces [2,[START_REF] Altman | Determination of the orientation of C 60 adsorbed on Au(111) and Ag(111)[END_REF][START_REF] Sakurai | Scanning Microscopy of Fullerenes Study[END_REF], matching somehow the substrate lattice underneath [START_REF] Shi | Survey of structural and electronic properties of C 60 on close-packed metal surfaces[END_REF]. We then expect C 60 molecules to self-assemble following a triangular lattice. In Figure IV.12a we show a scanning tunneling microscopy (STM) image of a fullerene assembly on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. In the inset of Figure IV.12b a perfect triangular lattice is superposed to the STM image, showing that each C 60 molecule occupies one site on the triangular lattice.

Ising degree of freedom -In our non-magnetic molecular system we do not find the usual Ising degree of freedom (spins or presence of atoms). It is the height of the fullerenes with respect to the copper surface that can be considered as a bi-valued variable in our samples. Accessing to the true corrugation of a C 60 island in STM is not an easy task because both topographic and electronic effects generally contribute to the measured apparent heights. As detailed in chapter III there are two different molecular topographic heights in the disordered phase, corresponding to the presence or absence of vacancies under the C 60 molecules. In Fig- ure IV.12c, we show a molecular height profile measured by STM, thus including topographic and electronic effects. For the molecular phase under analysis, two main heights differing by 85 ± 17 pm were measured in chapter III of subsubsection 2.2.2. This value, that was averaged over many height profiles, is in agreement with the profile shown in Figure IV.12c. The two visible heights in the profile correspond to fullerenes appearing in blue and yellow in our STM images. A third height (light yellow) is also observed for a few C 60 molecules 8 . Following our structural analysis, we know that fullerenes appearing in blue sit on vacancies, while yellow and light yellow colors correspond to molecules sitting on the non-reconstructed copper surface, in different binding configurations (for more details see subsubsection 2.2.2 and subsection 3.2 in chapter III). The average "magnetization" 9 is 2%±5% for the 6,200 molecules we analyzed, over 10 STM images. 3 molecular islands out of 10 have a negative "magnetization", the 7 remaining ones having a positive one.

Another way of measuring heights -We also performed an analysis of the molecular heights based on a color analysis of the STM images, instead of using height profiles. We measured locally the average height on a disk positioned at the center of each C 60 molecule and build the height histogram for 4 different STM images, analyzing the configurations for more than 3,700 molecules. Results are shown in Figure IV.12d. In agreement with the observations in the profile, we find two main peaks, with apparent heights of 94 pm and 172 pm. The lowest peak corresponds to blue C 60 molecules in the STM images, the higher one to yellow ones. A third smaller peak with an apparent height 245 pm also appears (corresponding to light yellow fullerenes in the STM image). The peaks are fitted using gaussian distributions. We obtain standard deviations of 19 pm, 24 pm and 10 pm. These values are in agreement with the height study presented in chapter III, subsubsection 2.2.2. We interpret the dispersion in height values as a consequence of: 1 -the non-uniform contrast associated to each single molecule (fullerenes do not appear as homogeneously colored disks, but show two-and three-lobes structure); 2 -the various orientations of the molecules and the crystallographic sites they occupy on the substrate. The intersections between the distributions are used as a cutoff to define three categories of fullerenes: blue color, height with respect to copper h < 128 pm; yellow, 128 pm< h < 228 pm; and light yellow, h > 228 pm. We find ∼50%, ∼47% and ∼3% of the molecules in these categories. Magnetization 8 An attentive reader will remark that this "very high" molecules are surrounded by a particular environment, i.e. they are almost always surrounded by three "high" and three "low" neighbor fullerenes, with "low" and "high" molecules alternating around the central "very high" molecule. We think that this local patterns are due to the coincidence with the substrate (see Figure III.7) in which three different binding sites exist for the 9.68 Å-22.4°phase. The same "very high" molecules are observed in the 9.98 Å-26.3°phase (described in section 3) but the local pattern appear less often, which is consistent with the different coincidence model proposed for this structural phase. 9 Magnetization is defined as the signed difference in the number of molecules appearing in blue (σ=-1) and (light) yellow (σ=+1) is 0.6% in good agreement with previous estimates.

Antiferromagnetic couplings -Finally, we are interested in the correlation of the molecular heights between nearest neighbor C 60 molecules, and which J 1 coupling could account for these correlations. To address this question we count the number of nearest neighbors with equal (blue/blue or yellow/yellow) and distinct (blue/yellow) contrast for every molecule in ten different C 60 assemblies. We only consider molecules having six nearest neighbors, i.e. not in the borders of the molecular island. In Figure IV.12e a histogram showing the number of nearest neighbors with distinct contrast is shown. If the effective coupling between first neighbor fullerenes (for the moment we neglect longer range couplings) was ferromagnetic we would expect less than 3 nearest neighbors with distinct contrast. If the coupling was antiferromagnetic we would expect more than 3 different neighbors. In the particular case in which the ice rule (no triangle with all-three identical spins at their corners) is perfectly respected, we would find an average of 4 distinct nearest neighbors. In the case of a non interacting system, i.e. a random distribution of molecules, we would observe a symmetric histogram with, as the central value, 3 nearest neighbors of distinct contrast. The mean value of our measured distribution is 3.76, definitely off the value of 3 for the random case and close to the value of 4 for the spin liquid phase. This reveals the presence of an antiferromagnetic-like coupling between nearest molecules.

Evidence of correlated disorder -The mean of the distribution of different nearest neighbors is equivalent to the spin-spin (or molecule-molecule in our case) correlations for nearest neighbors,

C 1 = 1 2N N 1 ij∈N 1 σ i σ j = N 1,= + N 1,̸ = N 1
with N 1,= the average number of neighbors with the same contrast, N 1,̸ = the average number of neighbors with opposite contrast and N 1 the total number of nearest neighbors per molecule. Our mean value of ∼3.76 corresponds to a correlation value of -0.25. The expected value for a spin liquid phase is -1/3 [START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF]. The experimental correlation value is negative and comparable with the value for the spin liquid phase. We conclude that the height disorder observed in our STM images does not correspond to a random disorder but to a correlated one. We will now see whether the observed value, however different from -1/3, is consistent with a spin liquid phase.

Height correlations in reciprocal space: ground state and range of effective couplings

In order to verify if the three ingredients discussed in the previous subsection really lead to the formation of a spin liquid phase, we have computed the MSF (see subsection 3. In the experimental MSF three peaks appear around each corner of the Brillouin zone (white circles in the figure). Such peaks in the MSF are a signature of order in the phase, so an underlying form of order must be present in the C 60 assemblies. This tendency to order is sufficiently weak that it cannot be detected directly in direct space, in the STM images. This observation does not correspond to the usual spin liquid phase, which is disordered. We actually need to include longer range interactions in the Hamiltonian to account for our experimental results. Indeed, the peaks observed in the experimental MSF are located at the positions of the characteristic Bragg peaks of a zigzag phase, as demonstrated by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] (see Figure IV.8) and confirmed by the MSF of our MC simulations shown in Figure IV.13c. The zigzag phase is the ground state of the TIAF model when antiferromagnetic second and third neighbor couplings are included. Specifically, the couplings need to verify the following conditions: J 3 J 2 > 0.5 [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] and J 2 J 1 < 0.5 [START_REF] Tanaka | Ground State Spin Configurations of the Triangular Ising Net with the First, Second and Third Nearest Neighbor Interactions[END_REF][START_REF] Takagi | New Partially Disordered Phases with Commensurate Spin Density Wave in Frustrated Triangular Lattice[END_REF] (see Figure IV.11). Surprisingly10 , C 60 molecules do not only interact via first nearest neighbor couplings, but in one way or another, an interaction with second and third neighbors is present in the molecular system.

We conclude that the molecular phases are neither in a purely disordered nor in an ordered phase. Indeed, perfect zigzag ordering of the height pattern would lead to the presence of Bragg peaks in reciprocal space, while the peaks we observe have lower intensity than that expected for Bragg peaks, and should hence be considered as emergent Bragg peaks 11 . Correlations are present in the system and, even if disorder seems to dominate in real space configurations, C 60 molecules tend to form an ordered zigzag pattern.

To interpret this observation with the help of a model Ising spin Hamiltonian, we compare experimental and MC simulated pairwise spin correlations. This comparison of correlations in real space is presented in the next section.

Pairwise correlations in real space: coupling strengths and thermodynamic equilibrium

Here we discuss which values of the effective couplings (J 1 , J 2 , J 3 ) permit to describe our experimental results using a TIAF Hamiltonian. We want to figure out whether we can reproduce the observed short range antiferromagnetic molecular height correlations (meaning J 1 < 0) and the peaks observed in the MSF pointing to a zigzag kind of order (J 2 < 0 and J 3 < 0 needed). We set J 1 = -1 and we limit the analysis of our phase diagram (see Figure IV.11) to the ratio in which the zigzag phase is the ground state, i.e. J 2 J 1 < 0.5 [START_REF] Tanaka | Ground State Spin Configurations of the Triangular Ising Net with the First, Second and Third Nearest Neighbor Interactions[END_REF][START_REF] Takagi | New Partially Disordered Phases with Commensurate Spin Density Wave in Frustrated Triangular Lattice[END_REF] and J 3 J 2 > 0.5 [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF].

Then, we used MC simulations to study the spin Hamiltonian for different values of the J 2 J 1 and J 3 J 2 coupling ratios. In particular, we computed the spin-spin correlations at low temper-ature and compared the values of the second (C 2 ) and third (C 3 ) pairwise spin correlations to the values we measured. The results are reported in Figure IV.14 where the points are the numerical values (infered from MC simulations) with their associated standard deviation.

The colored shaded regions are the experimental values averaged over ten STM images. We choose to show the minimum value of the second correlation and the maximum value of the third correlation in the low temperature regime (T ≤ J 1 ). Other representations could have been chosen (for instance we could represent the maximum and minimum value of each correlation in the low temperature regime) but the one we chose is especially instructive. Indeed, C 2 and C 3 show a non monotonous behavior at low temperature with a local minimum for C 2 and a local maximum for C 3 (see Figure IV.15). These local minimum/maximum appear to be very sensitive to the variation of the J 2 J 1 and J 3 J 2 ratios, hence they are good indicators of the compatibility between the model and the experiment.

Looking at Figure IV.14 we found a range of values for the J 2 J 1 and J 3 J 2 ratios that accounts for the experimental measurements. Simulated and experimental values are compatible if J 2 is substantially smaller than J 1 (5J 2 < J 1 ) and if 0.55 < J 3 J 2 < 0.60. Thus, we determine a range of parameters in which both the second and third pairwise correlations are captured. The ground state expected for these couplings at zero temperature is a zigzag state, which is consistent with the intensity distribution observed in the MSF. It is worth to note, that this result is not trivial at all. For instance, we could have found non overlapping ranges of the J 2 /J 1 or J 3 /J 2 ratios compatible with both C 2 and C 3 . Both second and third neighbor correlations in our experiment are in agreement with the TIAF model for a well-defined range of couplings. We can wonder if experimental correlations beyond third nearest neighbors are consistent with the TIAF model and the range of couplings we have found. Another remaining question is whether it is possible to define an effective temperature for the experimental data in the framework of the TIAF. To answer these questions, we compare the experimental values to those derived from MC simulations (from first to fifteenth neighbors) depending on temperature.

In Figure IV.15b the agreement between the experiment and the simulated Hamiltonian is shown, for a set of couplings compatible with the previously determined range. We chose

J 2 J 1 = 0.1, J 3
J 2 = 0.57, but agreement for other coupling values is shown in section 2 of Appendix C. First, we show the evolution of the five first correlation values with temperature. Experimental correlation values are not spread all over the T-axis, but they are placed at a single temperature value. In fact, we observe an excellent agreement of the experimental values with the simulated ones at a single value of the temperature, T |J 2 | ∼ 1 (except for C 1 , discussed further in subsection 2.4). This result indicates that our experimental system can be understood as a realization of the TIAF at an effective temperature of T |J 2 | ∼ 1.

In the following paragraph, we concentrate on the comparison of the experiment and the simulations at the values found for the couplings and the effective temperature. We wish to answer the following question: does the system exhibit the same long range correlations as those predicted by the MC simulations?

To produce the data shown in Figure IV.15c, we fix the couplings ( J 2 J 1 = 0.1, J 3 J 2 = 0.57) and the temperature ( T |J 2 | = 1) used in the simulation, and compare the pairwise correlations from the first to the fifteenth neighbor, an even more stringent test for our model. All the experimental points fall within the errorbars for the MC simulated correlations, which confirms that the molecular phase can be analyzed as a microstate of the TIAF model, with fixed values for the couplings and for the temperature. MC correlations correspond to a system at thermodynamic equilibrium and our experimental data are in agreement with this values, for a fixed temperature. This result is absolutely not trivial. We recall that the fullerene on copper samples are grown using molecular beam epitaxy, i.e. a C 60 molecule powder is evaporated and molecules aggregate on the copper surface following a succession of kinetic processes, which is by definition an out-of-equilibrium situation (not necessarily very far from equilibrium though). Nevertheless, our effective analysis demonstrates that the experimental phase can be understood as an arrested configuration at thermodynamic equilibrium of the TIAF model with J 1 , J 2 , J 3 couplings. These results are robust when the J 2 J 1 ratio is modified in the range 0.001 < J 2 J 1 ≤ 0.1, as we show in section 2 of Appendix C.

When looking at the experimental MSF we detect a diffuse background, characteristic of liquid phases, coexisting with emergent Bragg peaks associated to an ordered zigzag state. The same patterns are observed for a MC-simulated MSF, with the couplings and temperature mentioned before (

J 2 J 1 = 0.1, J 3 J 2 = 0.57, T |J 1 | = 0.1), see Figure IV.
15d for comparison 12 . We attribute the coexistence of these two signatures in the MSF to the fact that the effective temperature is close to the critical temperature of the transition from an ordered zigzag ground state to a spin liquid phase. Near the ratio J 3 J 2 = 0.57 Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] predicted a critical temperature T c = J 2 , which is the effective temperature we found for our experimental phase. To confirm this, we compute the specific heat using MC simulations (shown in Appendix C). A peak marking the phase transition is indeed observed slightly below the effective temperature we determined. Another way of discussing the same idea is considering the evolution of pairwise correlations with temperature (Figure IV.15a). Our experimental values are placed in the region in which correlations start to change from the values expected for the spin liquid phase (0.1 < T |J 1 | < 1) to the ones expected for a zigzag phase ( T |J 1 | < 0.1). We conclude that our experimental fullerene phase on Cu( 111) is a spin liquid with emergent zigzag order.

Magnetic-like defects and local lattice distortions

As previously mentioned, all the experimental correlations values from C 2 to C 15 are in agreement with the MC simulations. This is not the case for C 1 , which never fits with the numerical results. In a spin liquid phase or a zigzag phase C 1 = - 1 3 , which means that three spins in the same state never occupy the corners of a triangular plaquette, i.e. the ice rule is obeyed. In our experiments, we found C 1 = -0.24 ± 0.02. We also observe a small intensity at the gamma point (0,0) in the MSF, which corresponds to a ferromagnetic signal. Even if the molecular lattice correlates in an antiferromagnetic fashion the ice rule is not satisfied everywhere. Closer inspection of the STM images shows the presence of ferromagnetic-like defects formed by triangular plaquettes in which 3 molecules are "low" or "high". In particular, we observe many defects with 3 "low" fullerenes, while defects with three "high" molecules are rarely observed. In Figure IV.16 we show some of those defects. Interestingly, the STM contrast reveals an extended black background in the center of the corresponding triangular plaquettes. This indicates that C 60 molecules are shifted from the sites of the triangular lattice, suggesting that the lattice parameter is slightly increased and molecular repulsion is reduced. As we explained in chapter III, measuring the lattice parameters of our molecular assemblies in STM images at room temperature is not an easy task. Indeed, artifacts such as thermal drift during imaging and hysteresis of the piezoelectric actuators moving the STM tip make difficult to achieve the resolution needed to detect local variations of the lattice parameter, that are expectedly less than 5%. Moreover, different corrections were applied for different STM images, implying that we cannot compare lattices parameters measured in different images.

Here, we compare intermolecular distances in one single STM image, for fullerenes next to the position in which we observe magnetic-like defects. To do so, we considered 23 molecular profiles through the defects, see Figure IV.16. Profiles were drawn through six C 60 molecules, the two centered ones corresponding to "low" molecules in the defect. Then the distance between molecules, d 1 to d 5 , were measured. For each profile we have computed the average distance between molecules and compared local distances d i with this average value. Thus, we have 23 average values ⟨d⟩ and the difference for each local distance d i -⟨d⟩ ⟨d⟩ . We find and average distance considering all the profiles of 1 23 23 k ⟨d⟩ = 10.09 Å, with a standard deviation of ±0.02 Å. Recalling that STM images were corrected (see subsubsection 1.2.2), this value is not the real lattice parameter of the fullerene lattice but a reference to compare with local distances. We then computed the average for all the profiles for each local distance, shown in Table IV.1. Local distances vary with respect to the average value within their standard deviation, intermolecular distances are then inside the dispersion of our measurements. The only exception is the distance between fullerenes forming the defect. The local distance between C 60 molecules forming the defect is, on average, 9% larger than the average distance. This data analysis points in the same direction than our initial hypothesis: the presence of defects breaking the ice rule seems to be correlated to local distortions of the molecular lattice. In addition, the standard deviations have significant values, between 6% and 10% of the measured values, which could correspond to local variations of the local lattice parameter. The presence of local distortions in fullerenes' lattices has been reported in the literature (see subsubsection 1.2.3 in chapter III) and demonstrated in our analysis for C 60 phases on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] with periodic height contrasts (see subsection 2.2, also in chapter III). Thus, fullerene's lattices on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] are not rigid but show some degree of elasticity. We also know that a "low" molecule, i.e. sitting on a vacancy, has a lower energy than a "high" molecule, i.e. binding to the unreconstructed copper surface, which explains why the configuration with 3 molecules "low" is more likely. For details about vacancy formation see subsubsection 1.1.3, subsubsection 2.2.2 and subsection 3.1 in chapter III. This elastic property of the molecular layer is not taken into account in the Ising spin Hamiltonian. Nevertheless, the experimental system develops correlations up to the fifteenth neighbor in agreement with the Ising spin model. Surprisingly, the presence of defects does not seem to affect the values of C 2-15 correlations and the only effect we detected was on C 1 . We suggest that the effect of the defects is screened at short range by neighboring molecules. We see in Figure IV.16 that most of the defects, three "low" molecules, are mainly surrounded by "high" molecules. The environment of a defect exhibits a particular structure, which is in agreement with the idea of screening the presence of defects. Despite the elastic character of the lattice and the possibility of relaxing frustration forming magnetic-like defects, the experimental phase is very well described as an arrested configuration of the TIAF model at thermodynamic equilibrium.

Table IV.1:

Difference between the average distance between molecules and the distance for each pair of molecules in the 9.68 Å-22.4°phase. Notation is shown in Figure IV. [START_REF] Boltzmann | Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen[END_REF]. The distance between fullerenes forming the defect (d 3 ) is always larger than the average distance.

d i -⟨d⟩ ⟨d⟩ (%) d 1 d 2 d 3 d 4 d 5 2 ± 6 -8 ± 8 9 ± 8 -3 ± 7 -0.2 ± 10

Microscopic origin of antiferromagnetic couplings

We have demonstrated that couplings from first to third neighbors are necessary to explain our experimental observations. Short range coupling between first nearest neighbors has already been considered in previous studies. Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] discussed the compression of the C 60 layer on the Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] substrate in the case of the disordered phase. The lattice parameter for the molecular lattice is 9.68 Å, i.e. 3.7% smaller than the equilibrium lattice parameter of 10.05 Å (see subsection 2.2). In this regime a strong molecular repulsion associated to electron interactions is expected to overcome the attractive van der Waals interactions between molecules [START_REF] Girifalco | Interaction Potential for C 60 Molecules[END_REF][START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF][START_REF] Dappe | Intermolecular interaction in density functional theory: Application to carbon nanotubes and fullerenes[END_REF]. This molecular interaction explains the tendency of nearest neighbors to organize alternating "high" and "low" fullerenes, which increases the intermolecular distance. In most of the cases, the molecular layer buckles to increase the distance between first nearest neighbors, and in some cases in-plane distortions allow to further decrease molecular repulsion (see subsection 2.4).

Contrary to first neighbor coupling, second and third neighbor couplings have never been considered in the literature. We propose that the origin of these antiferromagnetic couplings is a dipolar interaction between molecules. Indeed, it has been shown that C 60 molecules on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] substrate are charged by electrons provided by the copper substrate [START_REF] Tsuei | Photoemission and photoabsorption study of adsorption on Cu(111) surfaces[END_REF][START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF][START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF]. This charge transfer has been associated with an electric dipole [START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF], which means that C 60 on Cu(111) could repel each other via a dipolar interaction, as it is the case for other organic molecules on metal substrates [START_REF] Fernandez-Torrente | Long-range repulsive interaction between molecules on a metal surface induced by charge transfer[END_REF]. Dipolar interaction is a repulsive long range interaction, and we propose that it is the essence of the antiferromagnetic effective J 2 and J 3 couplings that we invoked in our previous analysis.

Estimates of the strength of microscopic couplings -It is even possible to estimate the order of magnitude of the intermolecular and dipolar energies and to compare them with the J 1,2,3 couplings. To estimate the interaction energy of a C 60 dimer we used the calculation by Pacheco et al. [START_REF] Pacheco | First-principles determination of the dispersion interaction between fullerenes and their intermolecular potential[END_REF] and extract the energy corresponding to two C 60 molecules at the distance of first, second and third neighbors in our lattices. We do so both for distances in "high"-"high" dimers (r 1 =9.68 Å, r 2 =16.76 Å, r 3 =19.35 Å) and "high"-"low" dimers (r' 1 =9.76 Å, r' 2 =16.81 Å, r' 3 =19.39 Å) 13 . The dipolar momentum calculated in our DFT calculations (subsection 3.4 in chapter III) is µ 1 = 4.8eÅ for "high" fullerenes and µ 2 = 4.4eÅ for "low" molecules sitting on vacancies. We compute the usual dipolar interaction:

E dip = µ 1 µ 2 4πϵ 0 r 3 12 cos(φ) -3cos(φ 1 )cos(φ 2 )
where r 12 is the distance between dipoles, φ is the angle formed by the two dipoles µ 1 and µ 2 (zero for molecular dipoles perpendicular to the copper surface) and φ 1 and φ 2 are the angles formed by each dipole with the line connecting their centers.

We define 2J the energy difference between "high"-"low" and "high"-"high" configurations. The energies are reported in Table IV.2. We find a ratio between couplings, J 2 J 1 ∼ 0.08 and J 3 J 2 ∼ 0.62, in quantitative agreement with the analysis we made earlier using a spin Hamiltonian.

The magnetic, purely effective, analysis we developed appears here as a very powerful tool. First, it allows us to deduce the presence of long-range interactions in the molecular lattices, which was an unexpected result at the beginning of this project. Second, the values we found in the effective model are in quantitative agreement with the microscopic interactions in the real molecular system. Both approaches, the one inherited from magnetism and the other from surface science, are based on very different concepts and use different analysis tools, but, at the end of the day, they give the same results for the disordered phase of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF].

Summary

We have shown that the TIAF model, including molecule-molecule couplings with first to third neighbors, reproduces the observed height patterns observed in C 60 molecular assemblies on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], both in real and reciprocal space. We have determined the region of the phase diagram in which the TIAF Hamiltonian is in agreement with the fullerene's configurations, i.e. the possible values of the couplings ( J 2 J 1 > 5 and 0.55 < J 3 J 2 < 0.6). We were also able to define an effective temperature (

T ef f J 2 ∼ 1)
and thus prove that the arrested configuration is at thermodynamic equilibrium. We have demonstrated that the fullerene assemblies are on the verge of a transition between a disordered spin liquid and an ordered zigzag phase. The analysis performed in this section is a key result of the thesis in which we demonstrated, in a detailed and quantitative way, that a non-magnetic molecular system behaves as a highly frustrated magnet.

In addition, analysis of the molecular layer in the framework of an Ising spin Hamiltonian helped us to determine the range of the interactions between molecules and to establish a range of coupling values that account for our experimental observations. Coming back to the microscopic origin of the interactions leads to the same conclusion as with the effective analysis, both about the long range character of interactions and the quantitative value of the couplings.

We also observed the presence of defects breaking the "ice rule". These defects are rapidly screened by surrounding molecules. Although this type of defects is not taken into account in the TIAF model we considered, the observed molecular configurations are otherwise remarkably well described by the spin Ising Hamiltonian. We could wonder how the molecular lattice will change if supplementary defects were introduced in the lattice. For instance, it should be possible to locally modify the state of some fullerenes using the STM tip [START_REF] Cuberes | Manipulation of C 60 molecules on Cu (111) surfaces using a scanning tunneling microscope[END_REF][START_REF] Xu | Construction of a Molecular Switch Based on Two Metastable States of Fullerene on Cu(111)[END_REF] and to observe how the lattice adapts to the presence of excitations.

Even if defects are present in our samples, the experimental system is in a very low temperature regime immediately after deposition, whereas in the case of usual magnetic frustrated artificial systems an experimental protocol is needed (demagnetization or thermal annealing [START_REF] Rougemaille | Cooperative magnetic phenomena in artificial spin systems: spin liquids, Coulomb phase and fragmentation of magnetism-a colloquium[END_REF]) to bring the samples to the low temperature regime. Surprisingly, the effective temperature we found is comparable with the smallest temperature we were able to simulate numerically, meaning that the "thermalization" in the experiment is at least as efficient as the MC one. We could wonder if changing experimental parameters would allow us to bring the system to higher temperatures, corresponding to the purely disorder spin liquid manifold.

Energy degenerate stripe and zigzag phases in C 60 /Cu(111)

This section develops the same type of analysis than the section 2, but applied to a different molecular phase. We consider an effective Ising spin Hamiltonian to analyze the height patterns found in the 9.98 Å-26.3°phase of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], also called Linear wall phase by Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. This molecular phase is characterized by the formation of alternating rows of "high" and "low" molecules. If these rows were perfectly ordered, we would identify the phase as a realization of the stripe phase present in the TIAF with, at least, first and second nearest neighbor couplings (see phase diagram Nevertheless, we observed stripe order only at local scale, with many defects between domains with different stripe orientations. Here, we describe the phase in the framework of the TIAF model and determine the appropriate coupling strengths, that explain the experimental observations. Finally, we will briefly discuss the origin of the couplings and compare them with the ones of the disordered (9.68 Å-22.4°) phase described in the previous section.

Our analysis proves that the experimental 9.98 Å-26.3°phase of C 60 molecules on Cu( 111) is also a realization of the TIAF model. The comparison between experimental and simulated correlations strongly suggest that the observed phase is indeed located in a very specific region of the phase diagram, in which coexistence of two ordered phases (stripe and zigzag) is expected. This region of the phase diagram has been described [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF][START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] as a stripe ordered ground state in which domain wall defects can form with a zero energy cost. We show that our structural molecular non-magnetic system behaves as a frustrated magnet, not only forming phases that are analogous to a spin liquid (section 2), but also realizing theoretical predictions of another interesting region of the phase diagram.

TIAF ingredients

In Figure IV.17a we show two STM images of the 9.98 Å-26.3°phase, formed by ∼ 1, 000 and ∼ 500 fullerenes, respectively. In both regions, local order develops forming rows of alternating "high" and "low" molecules. First, we analyze if the three ingredients of the TIAF model are present in the 9.98 Å-26.3°molecular phase. Results are shown in Figure IV.17. Like in the case of 9.68 Å-22.4°phase: 1 -molecules sit on a triangular lattice; 2 -the molecular heights can be considered as an Ising variable; 3 -nearest neighbor interactions are antiferromagnetic.

As in the other molecular phases, fullerenes match the Cu(111) substrate underneath and form a close-packed triangular lattice, see Figure IV.17c.

To study the height difference between "high" and "low" molecules, height profiles are plotted (Figure IV.17c). For this profile, height differences are in agreement with our previous statistical analysis of molecular heights, which revealed a height difference of 87 ± 14 pm for this phase (subsubsection 2.2.2 in chapter III). We also extracted the height histogram from the STM images (Figure IV.17d), and observed two peaks separated by ∼ 83 pm. The number of "high" and "low" molecules is not perfectly balanced, and we find a "magnetization" Finally, we plotted the histogram for nearest neighbors of opposite contrast. For a perfect stripe phase all the fullerenes would have 4 (2) neighbors of distinct (identical) contrast. In the case of a random phase, the histogram would be symmetric and centered at 3. In Figure IV.17d we show that the histogram is non-symmetric, with a majority of opposite nearest neighbors, indicating antiferromagnetic-like couplings between nearest neighbors. Considering both images we find an average value of the histogram of 3.7, implying that the first neighbor correlation is C 1 = -0.23. The system is not perfectly correlated, which would give an average of 4 and C 1 = - 1 3 , but correlations associated to an antiferromagnetic coupling are clearly present. Even if the average of the histogram is very similar to the one of the disordered phase, the histogram itself is different. Here we observe more C 60 molecules surrounded by 4 opposite neighbors than by 3, which was not the case in Figure IV.12e. Thus, we conclude that this molecular phase presents all the ingredients of a TIAF model for both analyzed images. However the nature of the phases remains unclear. Which couplings can explain the observed stripe patterns? Are these couplings different from the ones of the phase studied in section 2? Can we explain the observed disorder as the effect of temperature on a system at thermodynamic equilibrium? To answer these questions we analyze the correlations of the phase both in reciprocal and real space. Results are detailed in the following sections.

Height correlations in reciprocal space: ground state and range of the effective couplings

The MSF was computed for both STM images and averaged (Figure IV.18a). The experimental MSF does not show Bragg peaks but rather a structured intensity pattern forming an hexagon inside the Brillouin zone (dotted white hexagon). Intensity is also present at the gamma point (center of Brillouin zone) which corresponds to a ferromagnetic signal.

We compared the experimental MSF with the one simulated by MC for a stripe phase, see Figure IV.18b. To make this comparison, we built a spin Hamiltonian with a minimal set of parameters, including only first and second neighbor couplings (J 1,2 < 0). The simulated MSF is characterized by Bragg peaks located in the center of the edges of the Brillouin zone. We do not observe Bragg peaks at these points in the experimental MSF, but intensity is non-zero at this location. In Figure IV.18c, the MSF for a stripe phase simulated at finite temperature is shown. The temperature was chosen here near the critical temperature corresponding to the phase transition from a spin liquid phase to a stripe phase. At this finite temperature, we observe a broadening of the Bragg peaks and the emergence of a diffuse background. Visual comparison between experimental and simulated MSFs reveals that the simulated patterns lacks some of the intensity found in the experiment. It is worth to note that this intensity cannot be explained by the presence of a spin liquid phase (see MSF of this phase in Figure IV.13b).

We conclude that the simplest model, including only J 1 and J 2 , does not capture the features of the experimental MSF. We observe the formation of stripes in real space, so we know that we need to keep J 1 and J 2 antiferromagnetic. Comparing the experimental MSF with the one for a zigzag phase (see Figure IV.13c), we realize that the position of the Bragg peaks for the zigzag phase are also present in the experimental MSF. This intensity could be explained by the presence of an antiferromagnetic J 3 coupling, in particular for a ratio J 3 J 2 ∼ 0.5, which corresponds to the transition between a stripe and a zigzag ground state (a detailed discussion has been presented in subsubsection 1.2.2). In Figure IV.19 we show the MSF from MC simulations for a spin Hamiltonian including third neighbor couplings with J 3 J 2 = 0.5. At low temperature, we observe intensity at the position of Bragg peaks of the stripe phase, together with a hexagon-structured intensity pattern at where intensity is also observed in the experimental MSF. We additionally plot the MSF at finite temperature, for T ∼ |J 2 |, and found a similar MSF, with more intensity on the hexagonal pattern and less at the position of Bragg peaks. For the ratio J 3 J 2 = 0.5 experimental and simulated MSF are comparable, which was not the case when J 3 = 0 (see Figure IV In the following analysis we determine the range of possible couplings reproducing the experiment and compare it with predictions by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] and Korshunov [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF]. Is J 3 J 2 = 0.5 the only point of the phase diagram that reproduces our experimental results? If it was the case, we would be in a region of the phase diagram in which phase coexistence is expected. We also wonder about the various sets of J 1 , J 2 , J 3 that allow to explain our experimental results. Does a link exist between those couplings and the microscopic structure of the phase? We decide to focus on the region of the phase diagram in which J 1 , J 2 and J 3 are antiferromagnetic and consider J 3 J 2 ≤ 0.5 to be sure that the ground state is a stripe phase (in agreement with the stripe height patterns observed in STM images).

Pairwise correlations in real space: coupling strength

In the previous paragraphs we have detected the presence of stripe pattern in our samples and computed the MSF. We concluded that, in order to reproduce these experimental features, we need to consider a TIAF Hamiltonian with antiferromagnetic J 1 , J 2 and J 3 . Here we use correlations in real space to determine which values of the couplings explain our experimental data. In particular, we demonstrate that our experimental phases can only be explained for couplings verifying J 3 J 2 ∼ 0.5, thus near the region of the phase diagram in which stripe and zigzag phases are expected to coexist.

In section 2, dedicated to the analysis of the disordered phase, we based our determination of the compatible couplings on the analysis of C 2 and C 3 correlations. These correlation values exhibited variations with temperature that made them very sensitive probes to determine the coupling strengths. We tried to use a similar approach in the case of the 9.98 Å-26.3°p hase, without clear results. Indeed, C 2 and C 3 correlation values vary smoothly in the region of interest of the phase diagram, hence it is difficult to use them as a criterion to choose a set of couplings.

Here we show another type of analysis based on the comparison of experimental and simulated correlations in real space. This approach is more general than the one presented before, and we think that it could be used for other experimental systems or phases. It is based on the minimization of the residuals of the fit of experimental correlation values to the simulated ones. Recall that in subsection 3.3 of chapter II we defined two types of sum of residuals:

• SSR: the absolute value of the distance between experimental correlation values and simulated ones at one given temperature of the model.

• wSSR: the absolute value of the distance between experimental and simulated correlation values, weighted by the standard deviation of MC correlations at a given temperature. This quantity is defined in such a way that wSSR< 1 implies that, in average, J 2 ratio, for three fixed J 2 J 1 = 0.2, 0.3, 0.4 ratios corresponding to the region in which SSR and wSSR are minimal in (a). all the experimental correlation values are inside the MC errorbars. If wSSR> 1 it is not the case, and the larger is the wSSR value, the worse is the fit. We will consider that a "good" agreement between experiment and simulations occurs when wSSR≤ 1.

To perform the comparison between experiment and simulations we used MC simulations and computed correlation values, from second to fifteenth neighbor, for different sets of couplings. We did not include the first neighbor correlations, which is justified in subsubsection 3.5.1, but we verified that our results are robust when C 1 is included (not shown). Following our analysis of the MSF, we fixed the value of J 1 = -1 and explored the region 10 -3 < J 2 J 1 < 0.5 and 0 < J 3 J 2 < 0.53. For each MC simulation we compared the simulated correlations depending on temperature with the experimental ones, chose the value of temperature that minimizes the residual of the fit (SSR) and save the values of the SSR and of the weighted residual (wSSR). The set of couplings that better captures our experimental data is the one with minimal SSR and wSSR. Note that we have imposed a strong hypothesis about thermodynamic equilibrium, because we compare correlations for fixed values of the temperature in the model. This will be justified a posteriori. At this point we do not impose any condition on the value of J 3 J 2 , so we considered three different cases ( J 3 J 2 = 0, 0.25, 0.5). Looking at the coupling values for which of wSSR≤ 1 we note that only the model with J 2 J 1 ≥ 0.2 and J 3 J 2 = 0.5 reproduces experimental correlations. Moreover, for the three J 3 J 2 ratios we considered, SSR is minimal when J 3 J 2 = 0.5. wSSR is more sensitive to the choice of J 3 J 2 , but it is also minimized in the region J 2 J 1 ≳ 0.2. We conclude that the stripe phase of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] is potentially captured by a TIAF model with

Results of this comparison are presented in

J 2 J 1 ≥ 0.2.
Which set of J 3 couplings reproduces best the experimental data remains now to be determined. In Figure IV.21b, we plotted the evolution of SSR and wSSR when the J 3 J 2 ratio changes. Three cases corresponding to J 2 J 1 = 0.2, 0.3, 0.4 are shown. The three curves exhibit similar behaviors: when J 3 J 2 ∼ 0.5, SSR and wSSR decrease. wSSR is smaller than one when

J 3
J 2 = 0.5, and also in the regions

[ J 2 J 1 = 0.3, 0.4 ≤ J 3 J 2 < 0.53] and [ J 2 J 1 = 0.2, 0.5 ≤ J 3 J 2 < 0.53]. Thus, 0.5 ≤ J 3
J 2 < 0.53 and J 2 J 1 ≥ 0.2 are necessary conditions to find a quantitative agreement between experimental correlations and MC simulations.

The condition on J 3 J 2 has strong physical implications. We have detailed in subsubsection 1.2.2 that the line J 3 J 2 = 0.5 is an isoenergetic line between the stripe and zigzag phases of the TIAF model. Korshunov [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF] first studied this region of the phase diagram and Smerald et al. have described it as "a high-degeneracy point, in which any number of straight, parallel domain walls can be placed into the stripe state (the zigzag state corresponds to a fully packed set of straight domain walls)". In subsubsection 3.5.2 we show the existence of DWs in our samples. We have determined the coupling strengths allowing to capture the experimental correlations. However, to do so, we considered comparison of correlations at a given temperature, i.e. we pre-supposed thermodynamic equilibrium. But is it actually the case? In this section we compare experimental correlations and simulated ones, plotting their values together to verify the agreement. In J 1 = 0.3 and J 3 J 2 = 0.5. We also plotted correlation values computed from the experimental measurements (colored dots). The experimental points are inside the standard deviations of MC curves for a single temperature T ∼ |J 2 |, except the C 1 value. We recall that it was also the case for the disordered phase. This feature is addressed in subsubsection 3.5.1.

Thermodynamic equilibrium

In Figure IV.22b we compare experimental (circles) and simulated (rectangles) correlations from first to fifteenth neighbor, for the fixed temperature T ∼ |J 2 |. All the correlation values agree, except the C 1 that we mentioned before and C 11 for which experimental and simulated standard deviations almost overlap. It is worth to note that experimental correlations were calculated and averaged for two molecular assemblies, while for the disordered phase (which exhibits a better agreement with the model, wSSR∼ 0.4 instead of wSSR∼ 0.6 for the stripe phase) we averaged over 10 molecular regions. Increasing statistics and measuring other STM images of the stripe molecular phase might lead to a better agreement. In conclusion, the 9.98 Å-26.3°phase can also be interpreted as an arrested configuration of the TIAF model at thermodynamic equilibrium. This phase also requires considering longer range antiferromagnetic interactions (J 2 , J 3 ), which are not very different from the ones found for the disordered phase but produces a different phase. In this case, experimental configurations correspond to a degenerate phase in which stripe and zigzag patterns coexist.

In the phase diagram by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] (reported in Figure IV.8), the effective temperature for our experimental configurations. T ∼ |J 2 |, is in fact slightly above the critical temperature of the transition from the spin liquid phase to the stripe/zigzag degenerate phase. Nevertheless, this comparison needs to be considered critically, because we found J 2 J 1 = 0.3, which is not anymore in the J 1 → ∞ limit considered in Ref. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF]. We are not sure if the spin liquid phase exists for this set of couplings, and we will let this question open for future studies (even if a first attempt to answer is presented in Appendix C).

We found these results puzzling: considering the same experimental playground as before, fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], in which only structural parameters slightly change, we found a different phase of matter. Moreover, this phase arises in highly frustrated systems and is expected to host particular DWs, which are excitations with zero energy cost.

Magnetic-like defects

In this section we describe the magnetic-like defects in the molecular lattice. The goal is two-fold. On one side we want to understand why the experimental value for C 1 is never captured by the model. On the other side, we aim to verify the presence of DWs in our experimental images, and compare them with those reported in the literature [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF][START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] for the region of the phase diagram in which stripe and zigzag phases are degenerate.

In the 9.98 Å-26.3°phase most of the observed defects are also formed by "low" fullerenes, see Figure IV.23. Nevertheless, those defects show some differences with respect to those in the disordered phase. In the stripe phase, groups of "low" C 60 molecules are not systematically surrounded by "high" molecules. Moreover, the distortions we mentioned above for the defects are not systematically observed, as can be seen in the insets of Figure IV.23 (occasional absence of a black background at the center of the triangle, indicating the molecules around are not significantly shifted outwards laterally).

To quantify the local distance between molecules, we repeat the same procedure used in subsection 2.4. From profiles like the one shown in Figure IV.23, we extract intermolecular distances d 1-5 (d 3 , distance between the two molecules around the defect). In Table IV.3 we show the average and standard deviation of the difference between the d 1-5 values extracted from each profile and the average lattice parameter ⟨d⟩. The average distances are always smaller than their standard deviations, hence it is not possible to conclude about a systematic lattice distortion around the defects. However, standard deviations are large (5-9%), indicative of important variations of the local lattice parameter. We think that, like we showed for other phases of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], the molecular layer is elastic and is able to accomodate some local defects that are not allowed in the Ising spin model. This elastic property explains the disagreement between the measured and calculated first neighbor correlation C 1 and also the presence of a small ferromagnetic contribution at the gamma point (0,0) in the experimental MSF. Surprisingly, the molecular lattice seems to be able to adapt the local distortion and the formation of defects without breaking the global stripe ordering. For the disordered phase, we proposed that ferromagnetic-like defects were screened because they were surrounded by "high" molecules. Here it is not the case, but the phase can still be described as a realization of the TIAF model.

Table IV.3:

Difference between the average intermolecular distance and the distance for each pair of molecules for regions showing ferromagnetic defects in the 9.98 Å-26.3°phase. Notation is shown in Figure IV.23. All the distances present significant standard deviation, but not clear variation in the average value is measured.

d i -⟨d⟩ ⟨d⟩ (%) d 1 d 2 d 3 d 4 d 5 0.5 ± 9 -2 ± 7 4 ± 5 -0.3 ± 6 -2 ± 5

Magnetic-like domain walls

Analyzing correlations in real and reciprocal space we have determined that our experimental phase can be understood as an arrested configuration of the TIAF model in the region of the phase diagram where J 3 J 2 ∼ 0.5. This region corresponds to a degenerate state between stripe and zigzag phases and formation of DWs is expected. Do we actually observe DWs in the experimental system? Do they correspond to those proposed in the literature? Here we inspect the experimental configurations in real space to detect the presence of DWs and compare them to the ones reported by Korshunov [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF]. J 2 ratios and the effective temperature T ef f that best fit the experimental data for each J 3 J 2 ratio.

Korshunov proposed several types of DWs to explain how the stripe phase can transit from a perfectly ordered ground state with symmetry breaking to a liquid disordered phase. DWs are grain boundaries between different stripe domains, that can be rotated (by 60°or 120°) and shifted. In Figure IV.24 we show the type of DWs that he proposed, including formation of kinks, corners and merging of DWs. For each of these structures it is possible to compute the energy cost in a spin model, compared to a reference that is the ordered stripe ground state. We will not detail this calculation, but it is similar to the one used to compute the energy of ordered ground states (see Figure IV.10a-c). Korshunov found the following energies, per unit length, for the structures shown in the figure 14 : • Single (straight) DW:

E 1DW = -2J 2 + 4J 3 • Kink: E k = -2J 1 + 4J 3 • Double (parallel) DWs: E 2DW = -4J 2 + 8J 3 = 2 × E 1DW • Corner: E c = -2J 2 • 120°intersection: no energy cost in J 1 , J 2 , J 3 so E 120 = 0 • 60°intersection: E 60 = -4J 3 • Double 60°intersection: E 2-60 = -12J 3
It is worth to mention some of the properties of these DWs. First, all the proposed structures have no energy cost in J 1 , except for the kink that is a magnetic defect breaking the ice rule. Indeed, these DWs are proposed in the J 1 → ∞ limit, meaning that even at finite temperature defects with an energy cost in J 1 will not appear in the system. For our experimental phase we found J 2 J 1 ∼ 0.3 and T |J 1 | ∼ 0.3, so we are not in this limit. In principle, we could find some "kink" defects in our phases, because of the effect of temperature. We do observe some kinks in MC snapshots of the configurations. However, we discussed in the previous section that the density of defects in the experiment is larger than that expected in MC simulations (experiments give larger C 1 values than the MC simulations), because of the elastic nature of the molecular lattice. We keep in mind this difference between the experimental and the modeled system, and focus our attention on the other types of defects. All the DWs have an energy cost in J 2 and J 3 , but for the single and double DWs the energy cost vanishes when J 3 J 2 = 0.5, which is precisely the coupling ratio we found for our experiment. Thus, we would expect a proliferation of these types of DWs in our molecular lattices. Moreover, the energy cost of a 120°intersection between three DWs is always zero, suggesting that this structure would also be favored in our samples. Nevertheless, even if at J 3 J 2 ̸ = 0.5 DWs have a certain energy cost, a non zero density of DWs is expected at finite temperature. To compare the expected densities of different DWs, we can compute their Boltzmann weight, which depends on the energy of each DW (e -E/T ). The energies are the ones specified below, and the temperature is chosen to be the effective temperature that best fit our experimental data for a given J 3 J 2 ratio. Note that using this quantity as a direct indicator of the density of defects is equivalent to suppose that the system we are looking at is at thermodynamic equilibrium. J 2 ratio. For each J 3 J 2 the value of J 1 was fixed to -1. We choose T ef f value corresponding to the effective temperature we find when we fit our experimental correlations to the MC simulated ones with a certain J 3 J 2 ratio. These temperatures range from T ∼ 1.6 (when J 3 J 2 = 0) to T ∼ 0.3 when ( J 3 J 2 = 0.5). We verified that the evolution of the curves with respect to the J 3 J 2 ratio were qualitatively the same when we plot them for a fixed temperature, i.e. constant relative density of defects whatever the coupling ratios and only changes in the curves' slopes.

For kinks, corners and intersections we plot the product of their Boltzmann's factor and the one of DWs needed for their formation. Kinks like the one proposed by Korshunov cannot form without the presence of a single DW, the same is true for corners and double DWs and for intersections and single DWs. Looking at Boltzmann's factors for each type of defects we can distinguish two main regions. One, for 0 < J 3 J 2 ≲ 0.25 where single DWs with the three type of intersections (120°, 60°and double 60°) exist. The second one, 0.25 ≲ J 3 J 2 ≤ 0.5, features a proliferation of double and triple DW, and 120°intersection between single DW favored with respect to 60°. In conclusion, we expect the formation of many single, double and triple DWs in our experimental phases, with a preference for 120°intersections over other intersection structures.

In Figure IV.25a,b we present a binary representation of our STM images and highlight some of the DWs, with red lines. We indeed observe the presence of the defects we expected, specially the triple DWs. Performing a quantitative analysis of these images would need to automatize the detection of defects, and we were lacking time for that. However, we can propose a provisional conclusion from this qualitative analysis.

Our analysis of the molecular phase as an arrested configuration of the TIAF is (qualitatively) robust when we switch from an averaged statistical analysis to a local one. We have found a very good agreement of experimental and simulated correlations (which are global, averaged quantities) in real and reciprocal space. We wonder if we would find the same agreement considering two molecular regions separated by one single DW, one with zigzag order and the other with stripes order. The answer to this question does not seem trivial to us. Indeed, making the difference between formation of many defects on a stripe background and a spatial separation of phases is not possible using global averaged quantities. The advantage of our experimental system is that we can look at molecular configurations in real space, and directly know if there is phase separation in real space or a complex phase formed by many defects. We have proposed a path to investigate order in the molecular phase, based on detection of DWs in real space. We demonstrate the presence of defects, in agreement with theoretical expectations for the region J 3 J 2 = 0.5 in the TIAF phase diagram. We confirm that we can indeed understand our experimental phase as a stripe background with DW excitations with zero-energy cost.

Microscopic origin of antiferromagnetic couplings

In subsection 2.5, we discussed the microscopic origin of the antiferromagnetic couplings in the disordered phase. Here, we will apply an analogous reasoning for the stripe phase. In chapter III we detailed the structure of each phase and found that the main difference between disordered and stripe phases is the coincidence with the copper substrate. For the stripe phase, the coincidence lattice (with the substrate) is a 2 × 2 super-cell, whereas for the disordered phase it is a √ 3 × √ 3 super-cell. Moreover, the two types of lattices have different lattice parameters. For the stripe phase, the lattice parameter is 9.98 Å, ∼1.2% smaller than the equilibrium distance between fullerenes on a two-dimensional layer. For the disordered phase compression is larger, ∼ -3.7%. Except for these structural parameters, the interactions at-play in both phases are expected to be the same. The stripe phase being compressed with respect to the equilibrium intermolecular distance, repulsive interactions between fullerenes should be present in the system. Concerning long range interactions, charge transfer from copper to C 60 molecules is expected also in stripe phase because adsorption of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] is similar to the one in the disordered phase. Thus, we expect dipolar interaction to play a role in the ordering of the molecules.

In Table IV.4, we estimate the values of these energies and the corresponding effective couplings in the Ising model. We find J 2 J 1 ∼ 0.13 and J 3 J 2 ∼ 0.62. These values are of the same order of magnitude as those found using the magnetic-like analysis, but we do not find J 3 J 2 ∼ 0.5 as expected. Indeed, our calculation of the microscopic interactions does not allow to distinguish the J 3 J 2 ratio of the stripe and disordered phases. Understanding the difference in the microscopic couplings ratios for the disordered and stripe phases is not straightforward. Indeed, lattice parameter being different for both phases both intermolecular and dipolar energy are modified. Intermolecular interaction for the lattice parameter of the stripe phases changes very abruptly because we are close to the equilibrium distance. For the lattice parameter of the phase, 9.98 Å, the intermolecular interaction is repulsive, but very small variations of the distance (∼ 1.5%) would make the interaction attractive. This consideration was irrelevant for the disordered phase, for which a change from repulsive to attractive regime implied larger variations of the distance, by ∼ 4%. Our calculations are good average estimates of the coupling strengths, but we should not overlook that the lattice features local distortions, that can alter the results.

We conclude that the estimated stripe microscopic interactions are of the same order of magnitude as those we found using an effective magnetic approach. For both disordered and stripe phases we have calculated J 3 J 2 ∼ 0.6, i.e. the experimental phases are close to the degeneracy point J 3 J 2 = 0.5 in the TIAF phase diagram. Tiny differences in the structural parameters of the phases caused different height patterns, identified as two different phases of the TIAF at finite temperature.

Summary

Our analysis of the 9.98 Å-26.3°or stripe phase of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] confirms the conclusions of the previous section about the 9.68 Å-22.4°or disordered phase. Fullerenes on copper can be understood as frozen configurations of the frustrated TIAF model with antiferromagnetic J 1 , J 2 and J 3 couplings. Furthermore, the molecular phases are configurations at thermodynamic equilibrium in the spin Hamiltonian language.

Despite the similarities between the two phases, we have proved that small differences in the structure dramatically change the resulting phase. The stripe phase can be captured by the TIAF model with coupling strengths corresponding to the region of the phase diagram in which stripe and zigzag order are degenerate. This phase is in fact a very particular coexistence of two ordered phases, which, to our knowledge, has never been experimentally observed. Moreover, this peculiar phase coexistence is characterized by the zero-energy cost for formation of domain walls. We wonder how the molecular system will react to excitations generated with the STM tip: could we create as much DWs as we want? Is the system able to fluctuate to accomodate new constraints? A first hint is visible in our STM images, in which some fullerenes seem to be bi-colored (blue and yellow at the same time), indicating that they fluctuate during the measurement (Figure IV.17a-c).

Except for the presence of defects breaking the ice rule, which we attribute to an elastic character of the molecular layer, all the other features analyzed in the phase indicate a low effective temperature. We found T ∼ |J 2 |, or T |J 1 | ∼ 0.3, a surprisingly low temperature almost corresponding to the lowest temperature we could reach by MC simulations with local updates. Thermalization in the real experimental samples is almost as good as the one achieved in MC simulations. As we mentioned for the disordered phase, in other experimental two-dimensional systems such as artificial frustrated systems reaching such low temperature is very unusual. We wonder if it would be experimentally possible to increase the effective temperature of the phase. And if it was the case, could we observe a spin liquid-like phase? Note that for the stripe phase we found J 2 J 1 ∼ 0.3, so the first coupling is not much larger than the other ones and we do not know if it is possible to stabilize a spin liquid phase under these conditions.

4 Take home message: non-magnetic molecular system as a frustrated TIAF model

In this chapter we have demonstrated that fullerenes phases on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] can be understood as experimental realizations of the frustrated triangular antiferromagnetic Ising Hamiltonian.

We have identified the experimental phases as configurations of the TIAF, determined the region of the phase diagram which corresponds to the experimental observations and verified that the phases are at thermodynamic equilibrium.

To obtain these results, we have established a set of methods allowing to compare experiment and theory. We used STM microscopy to access molecular configurations in real space and MC simulations to determine physical observables from the TIAF Halmiltonian. We developed analysis tools that can be applied to both experiment and simulations, and which allow us to compare measured and simulated quantities. Most of these tools, correlations in real and reciprocal space magnetic structure factors, are statistical quantities averaged over the lattice. Local analysis was also used to characterize the presence of magnetic-like defects in the molecular lattices. Finally, we were able to apply our microscopic study of the lattice (mainly developed in chapter III) to understand the origin of effective couplings in the Ising spin Hamiltonian: van der Waals and chemical repulsion at short range and dipolar interactions at longer distances. Indeed, we have shown that surface systems, and in particular C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], are very rich experimental systems to study frustration. Further studies on the manipulation of the samples, for example modifying experimental conditions during evaporation or switching molecules from one state to another using the STM tip, are promising research paths.

Moreover, this type of samples stands at the interface of several communities in physics. They are usually envisaged as complex systems formed by many atoms, with many degrees of freedom (positional, rotational), and studied following approaches that are typical of surface science communities including physicists and chemists. But they can also be described by effective models, within the framework of statistical physics, in which the studied system is reduced to an on-lattice problem with only an Ising degree of freedom. Our samples exhibit some of the advantages of artificial frustrated systems: they are two-dimensional, some fabrication parameters can be tuned and we can observe configurations in real space to resolve the state of each individual element on the lattice. At the same time, the system orders following "natural" non-chosen interactions, like many magnetic three-dimensional compounds or like water ice, which are studied in condensed matter. In this sense we see fullerenes on metals, together with other surface systems that have been suggested as frustrated phases of matter (see subsection 1.1 for details), as fantastic experimental platforms to study frustration and build bridges between different communities.

Generalization of the effective analysis to other phases of fullerenes on metal surfaces

At the microscopic level, simply varying the nature of the metal surface offers the opportunity to address different regimes molecule-metal interactions and different values of the substrate lattice constant. The variety of available densed-packed metal substrates hence altogether represents a very rich playground to explore different kinds of order/disorder phenomena. In particular, our analysis based on the TIAF model might be generalized to other fullerenes phases.

In chapter III we have shown the coexistence of ten different phases of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], with six different height patterns. We may wonder whether they could all be described by a TIAF model. In subsection 5.1 we address this question and propose some basis to further develop our analysis. We propose that the six height patterns observed in fullerene assemblies on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] can be interpreted as experimental realizations of a single TIAF Hamiltonian, with different interactions depending on the structure of each molecular phase.

In the introduction to this chapter, and also in chapter III, we mentioned that other C 60 on close-packed metal surfaces present similar height patterns. For example, fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], etc. have been reported to form commensurate phases in which vacancies can form. In subsection 5.2 we shortly review some of these phases and propose how they can be understood in the framework of spin Hamiltonians: Do they correspond to ground states of a TIAF model? Which effective couplings need to be included? Are effective couplings linked to the microscopic interactions? We found that almost all the experimental phases can be understood as realizations of a TIAF. Besides, in most of the cases a semi-quantitative link can be proposed between, on one hand, the structure of a molecular phase, so its molecule-molecule and molecule-substrate interactions, and on the other hand the couplings in a TIAF.

Triangular Ising model under an external magnetic field: ordered C 60 /Cu(111) phases

We start by analyzing other commensurate phases formed by C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. We need to first make a short reminder about each of the phases and their specific height pattern, that were described in chapter III and are shown in Figure IV.26a. We can (artificially) sort out the observed phases in three groups.

The first group includes the phases analyzed in the previous sections. These phases exhibit non-ordered height patterns (Figure IV.26a,b), but we have shown that this apparent "disorder" is in fact correlated and that a short range order exists. For these phases, the ingredient that dominates the Hamiltonian is the effective antiferromagnetic J 1 coupling, but further couplings J 2 and J 3 need to be taken into account. The microscopic origin of the J 1 coupling is the intermolecular interaction. Concerning J 2 and J 3 couplings, they were interpreted as the effect of dipolar interactions, caused by the electronic charge transfer from the copper surface to C 60 molecules.

The second group consists of the 10.33 Å-30°phase, for which an ordered zigzag height pattern is observed. For this phase, we will not propose any analogy with the TIAF model, because we have shown that fullerenes do not form an isotropic triangular lattice but a globally distorted one (see Figure III.23 in chapter III). This phase is formed by a periodic repetition of structural DWs. To study this phase as a realization of an Ising spin Hamiltonian we would need to include a triangular lattice with periodic distortions, related to the presence of these DWs, or to mimic the effect of distortions including site dependent coupling strengths. This is clearly interesting, but goes beyond this PhD work.

The third group regroups all molecular phases exhibiting uniform or periodic height patterns. Below, we propose how we could analyze these phases using a TIAF model. ). The 2 × 2 periodic phase (10.14 Å-19.1°in the notation used for the structure) has the same height pattern than the ground state of an Ising spin Hamiltonian which includes antiferromagnetic J 1 , J 2 and an external field h ext [START_REF] Metcalf | Ground state spin orderings of the triangular Ising model[END_REF][START_REF] Kaburagi | Ordered structure of adatoms in the extended range lattice gas model[END_REF][START_REF] Tanaka | Ground state spin configurations of the triangular Ising net with the nearest and next nearest neighbor interactions[END_REF][START_REF] Glosli | A Monte Carlo and renormalization group study of the Ising model with nearest and next nearest neighbor interactions on the triangular lattice[END_REF][START_REF] Takasaki | Magnetic Phase Diagram of the Ising Model on a Triangular Lattice with Antiferromagnetic Nearest-Neighbors and Next-Nearest-Neighbor Interactions[END_REF]. The √ 7 × √ 7 periodic phases (10.04 Å-10.9°, 10.09 Å-16.4°or 10.14 Å-14.4°in the notation used in chapter III) exhibit the same periodicity as the ground state of an Ising spin Hamiltonian with antiferromagnetic J 1 , J 2 , J 3 couplings and an external field h ext [START_REF] Kaburagi | Ground State Structure of Triangular Gas Model with up to 3rd Neighbor Interactions[END_REF].

For the disordered phases, we have been able to relate, in a semi-quantitative way, the values of the effective magnetic couplings to interaction potentials of well-defined microscopic origin. An open question is whether this is also the case for the ordered phases.

We propose that for these uniform and periodic phases the dominant term in the Hamiltonian is not the intermolecular or dipolar repulsion, but rather the molecule-substrate interaction via the formation of vacancies. Formation of vacancies would be a one body term in the Hamiltonian. The energy difference between a fullerene "low" (sitting on the nonreconstructed copper surface) and "high" (in a vacancy) depends only on the state of the C 60 molecule, without considering the state of neighboring molecules. Following this reasoning we can consider the formation of vacancies as the effect of an external field h ext , whose value can be linked to the energy difference between a fullerene sitting on a non-reconstructed site of the copper or on a vacancy: 2h ext = E vac -E norec . From DFT calculations presented in subsection 3.1 we can estimate the order of magnitude of the external field (per molecule): h ext ∼ 1 eV, which is 10 or 100 times larger than the energy scale corresponding to J 1 , J 2 and J 3 ! Note that this order of magnitude is a priori the same for disordered phases. This may seem surprising since in our previous analysis of these phases we did not take into account an external magnetic field.

Nevertheless, we need to recall a discussion we made in subsubsection 2.2.3 of chapter III. There we concluded that the density of vacancies in a certain molecular phase depends on the orientation of the phase. If the fullerene lattice is rotated by φ > 20°only half of the fullerenes can form a seven-atom vacancy on the copper surface. The argument was that, when the orientation of a molecular phase increases, a larger density of vacancies would imply that some copper atoms bind simultaneously to two fullerenes, which is not favored. This constraint on the formation of vacancies can be understood as a fixed chemical potential, µ(φ), and for our phases we would have different values of the chemical potential (or external field, remember the equivalence between spin and gas Ising models) depending on the number of vacancies that can be formed in the phase:

µ ∝ N vac N molecules
or in the spin language,

h ext = N vac -N no vac N molecules × E vac -E no rec 2
where N vac is the number of vacancies that can form in the unit cell of a given phase, N molecules the number of fullerenes in the unit cell and N no vac = N molecules -N vac . For the case of disordered and stripe phases the number of vacancies is 1 2 of the molecules (due to steric effects). The value of the field for these phases is hence zero, despite the fact that binding energy for fullerenes on vacancies is lower than the one of fullerenes on non-reconstructed sites. Thus, there is no contradiction between the previous analysis and including a magnetic external field in the study of other molecular phases.

In addition to this magnetic field, we expect antiferromagnetic couplings coming from intermolecular and dipolar interactions. We are able to estimate them in the same way as for the disordered and the stripe phases, and we know how to translate these energies into J 1 , J 2 and J 3 couplings. We recall that our calculation gives values for energies per bond and each fullerene has 6 nearest neighbors, so to compare the intermolecular and dipolar energies (J couplings) with the energy of formation of a vacancy (linked to h ext ) a factor 6 needs to be taken into account. Uniform, 2 × 2 and √ 7 × √ 7 periodic phases coupling ratios are always around the same values J 2 J 1 ∼ 0.16 and J 3 J 2 ∼ 0.62. The precise values are reported in Table IV.5.

A detailed phase diagram of the Ising spin Hamiltonian exploring all the phase space when J 1 , J 2 , J 3 and h ext change, and including the effect of temperature is out of the scope of this thesis. However, we can plot the phase diagram at zero temperature for coupling strengths close those corresponding to the microscopic interactions at play in the system. In IV.5: Coupling strengths for different molecular phases. Magnetic couplings J 1 , J 2 , J 3 are calculated taking into account intermolecular and dipolar interactions for C 60 dimers, which only depend on the geometry of each phase (as described in subsection 2.5). The external magnetic field is computed as

h ext = Nvac-Nno vac N molecules 1 6 (E vac -E norec ) =
Nvac-Nno vac N molecules ×0.18 eV, where 1 6 is a normalization factor used to compare the intermolecular energy of a dimer with the energy per molecule associated to E vac and E norec . consider the energies E(J 1 , J 2 , J 3 , h ext ) of all the GSs that have been reported in the literature and identify the phase with lowest energy for each set J 1 , J 2 , J 3 , h ext of parameters.

Phase

Following our previous analysis we would expect to find our experimental phases in the region 0.45 ≲ J 3 J 2 ≲ 0.65, which is indeed the case. Moreover, we expect the stripe and zigzag phases to appear for zero field, in good agreement with the phase diagram. The uniform (ferromagnetic-like) phase appears for an external magnetic field h ext > 8.5 and we estimate that the microscopic energy related to the vacancy formation is ten times larger than the intermolecular energy (Which we fixed to J 1 = -1), so the value for the field would be h f erro ext ∼ 10. If we assume that the field intensity is proportional to the number of vacancies that can be created for the periodic phases we obtain for the 2 × 2 periodic phase h 2×2 ext = 1 2 × 10 ∼ 5. Note that for the line given by this value of the field (h ext ∼ 5) the 2 × 2 phase is the ground state if J 3 J 2 < 0.5. In the case of the √ 7 × √ 7 phase the magnetic external field would be h

√ 7× √ 7 ext = 5
7 × 10 ∼ 7. Looking at the phase diagram we observe that this phase is indeed the ground state for h ext ∼ 7 and J 3 J 2 < 0.25

We did not have enough time to explore the whole phase diagram using MC simulations, but we verified that we were able to stabilize these periodic phases in our MC simulations. One example for each phase is shown in Figure IV. 26. We conclude that all the observed height patterns correspond to phases existing in the framework of the TIAF model with antiferromagnetic J 1 , J 2 , J 3 couplings and an external magnetic field h ext . We established a correspondence between microscopic interactions and effective parameters in the model, that we sum up in Figure IV.28. In addition, we proposed a semi-quantitative analysis, in which we consider the microscopic values expected for the interactions, translate them in the language of an Ising spin Hamiltonian and verify the correspondence between experimental phases and the GS of the model. Our analysis of the C 60 /Cu(111) system using a TIAF model is robust and allows to explain all the height patterns observed experimentally. Interestingly, the phase diagram in Figure IV.27 includes 12 different ground states. We wonder if it would be possible to find other phases formed by fullerenes on close-packed metal surfaces that realize some of those phases. We hope that future experimental studies will explore the theoretical phase diagram, using molecular lattices of fullerenes on close-packed metal surfaces.

Ising spin Hamiltonian applied to other C 60 /M(111) systems

In the previous sections we have investigated the equivalence between height patterns in C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and phases predicted for a triangular Ising frustrated spin Hamiltonian. Considering different metal surfaces comes with several important changes in the microscopic structure, thus changes in the effective magnetic couplings appearing in the Hamiltonian are expected. For instance, changing the metal modifies the binding energy of fullerenes with the substrate, the type of vacancies that can form and the energy cost to reconstruct the surface. It also modifes the charge transfers, thus the dipolar interactions. Each metal has a given lattice parameter, which implies that commensurate phases of C 60 molecules are different, in particular the distance between fullerenes in the molecular layer, a parameter on which intermolecular interactions depend very sensitively.

Here, we discuss to which extent our analysis could be extended to other close-packed metal surfaces. First, we describe which structural characteristics and microscopic interactions are expected in different metals, and how they could be considered in the frame of an effective Ising spin model. Second, we recall the different phases of fullerenes that have been previously observed by various research groups on selected close-packed metal surfaces, and compare them with the phases predicted by the TIAF model. This section can be understood as an attempt to generalize our previous analysis to other metal surfaces.

We consider different structural phases of fullerenes deposited on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. For all of them, fullerenes form close-packed lattices and can take different heights, depending on the formation of vacancies on the metal substrate. In all cases at least two ingredients of the triangular Ising model are present: the triangular lattice and bi-valued (structural, non-magnetic) variables. But, which kind of effective couplings could we expect? Do we have exotic liquid-like phases? Can we find a link between microscopic couplings and observed height patterns?

To answer these questions we discuss, in a qualitative manner, the differences between metal surfaces and how they translate into the effective Ising spin Hamiltonian. In the analysis of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] we discussed the role of three structural characteristics and the associated interactions: intermolecular lattice parameter, associated to intermolecular interactions; charge transfer from the metal to the molecules, which determines the dipolar interactions; and formation of vacancies, which depends on the geometry of the phase and on the binding with the substrate. We can now discuss these same properties for different metal surfaces, using the analogy illustrated in Figure IV.28.

• Intermolecular distances and molecular repulsion: lattice parameters of different phases of fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Pt(111) range from 10.22 Å (+1.7% strain) to 9.60 Å (-4.5% strain). Compressed molecular phases (a C60 < 10.05 Å) appear for all the considered metals, with some strongly compressed phases thus with expectedly strongly repulsive interactions, that can be interpreted as an antiferromagnetic J 1 in the TIAF model.

• Charge transfer and dipolar interaction: C 60 molecules form ionic bonds with Au [START_REF] Hunt | Charge transfer and structure in C 60 adsorption on metal surfaces[END_REF][START_REF] Lu | Charge transfer and screening in individual C 60 molecules on metal substrates: A scanning tunneling spectroscopy and theoretical study[END_REF][START_REF] Schulze | Resonant heating and substrate-mediated cooling of a single C60 molecule in a tunnel junction[END_REF], Ag [START_REF] Wertheim | Interfacial reaction of C 60 with silver[END_REF][START_REF] Goldoni | The interaction of C 60 with Ag(100): strong predominantly ionic bonding[END_REF][START_REF] Hsu | Aperiodic incommensurate phase of a C 60 monolayer on Ag(100)[END_REF], Cu [START_REF] Tsuei | Photoemission and photoabsorption study of adsorption on Cu(111) surfaces[END_REF][START_REF] Wang | Rotation, translation, charge transfer, and electronic structure of C 60 on Cu(111) surface[END_REF][START_REF] Larsson | Orientation of individual C 60 molecules adsorbed on Cu(111): Low-temperature scanning tunneling microscopy and density functional calculations[END_REF][START_REF] Pai | Optimal Electron Doping of a C 60 Monolayer on Cu(111) via Interface Reconstruction[END_REF] and Ni [START_REF] Hunt | Charge transfer and structure in C 60 adsorption on metal surfaces[END_REF]; they bind in a covalent way with Pt [START_REF] Cepek | Chemisorption and fragmentation of on Pt(111) and Ni(110)[END_REF] and Al [START_REF] Maxwell | Electronic and geometric structure of on Al(111) and Al(110)[END_REF], i.e. with not much charge transfer. Charge transfer for fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Cu(111) of 1.7, 1.0 and 1.8 electrons, respectively, have been measured [START_REF] Hoogenboom | Charge transfer and doping-dependent hybridization of C 60 on noble metals[END_REF]. Thus, for Au, Ag, Cu and Ni dipolar interactions may play a role, or in the Ising spin language, long range antiferromagnetic couplings might be present.

• Vacancy size and steric effects: vacancies are formed by 7 atoms in the case of Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], and for all the other metals they have been reported to consist of 1 atom (except for the case of Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], in which both types of vacancies have been proposed [START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF][START_REF] Tang | Complex orientational ordering of C 60 molecules on Au(111)[END_REF][START_REF] Tang | Probing the buried C 60 /Au(111) interface with atoms[END_REF]). Changes in the size of the vacancies could be a key parameter, because if vacancies are small they are not expected to "interact" with one another, and steric effects would not be present in the system. We think that steric effects should be important in Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. For all the other metals, even if the orientation of the commensurate phases is large (φ > 20°), vacancies could form below each fullerene. We recall that the number of vacancies that can be created in a phase is translated into the amplitude of an external magnetic field, so for phases in which one-atom vacancies form and have an important binding energy, strong effective external field is always expected, which was not the case for seven-atom vacancies.

• Strength of the molecule-metal binding: in section 6 of chapter III we present binding energies for fullerenes attached to different metal surfaces. These binding energies were calculated by DFT for configurations that can be compared from one metal surface to another (same number of atoms, same code used for the calculations). We do not pretend that the absolute values of the energies calculated by DFT are perfectly those in the experimental systems, but we can compare relative values on different metals. We found that for a C 60 molecule on a top site, the binding energies for different metals are ordered as follows E P t( 111) < E N i( 111) < E Al( 111) < E Cu( 111) < E Au( 111) < E Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] . Even if these energies can change depending on the orientation of the fullerenes, the adsorption site and the reconstruction mechanism, they give a first order approximation for the strength of metal-C 60 interaction. When the binding energy with the substrate is very large we expect the effective magnetic field to be important with respect to intermolecular and dipolar interactions. On the contrary, if the binding energy is small we expect intermolecular repulsion to play a more important role. In previous sections we observed an influence of intermolecular interaction for fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], so we also expect some influence in the case of Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], which have weaker binding energies with the substrate according to our DFT calculations.

In Table B.2, we sum up the different C 60 /M(111) phases that have been reported in the literature. We also show some examples of imaging on these phases in Figure IV. [START_REF] Charra | Surface-induced chirality in a self-assembled monolayer of discotic liquid crystal[END_REF]. In these images we observe three types of height patterns that were reported in the literature: phases with uniform height, with periodic or quasi-periodic (the height pattern is periodic in average, but many deviations from the perfect period are observed on the fullerenes lattices) heights and "disordered" patterns, in which visually determining a periodic order is not possible. Now, we will look at different commensurate phases on each metal surface and propose how to interpret these observations in a spin Ising Hamiltonian. Adapted from [START_REF]Observation of a Metastable Honeycomb Arrangement of C 60 on Ni(111) with (7 × 7) Periodicity: Tailoring an Interface for Organic Spintronics[END_REF]. Pt(111): high resolution image of a uniform phase, in which coexistence of several molecular orientations is observed. Adapted from [START_REF] Liu | Molecular orientations and interfacial structure of C 60 on Pt(111)[END_REF].

Looking at Table B.2, we note that Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] only hosts uniform phases. In this case, the interaction with the substrate seems to dominate. The binding energy for non-reconstructed sites is of ∼ -7 eV according to our DFT calculations, indeed much larger than on copper ∼ -3 eV (Figure III.33 and Table III.6 in chapter III). Being small compared to this binding energy, repulsive C 60 -C 60 interactions are not expected to play a decisive role on Pt [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] (E intermolecular ∼ -1.8 eV at most), even if strain in the molecular phases is between -0.7% and -4.5%. The translation to an Ising model will imply a strong external field driving the physics and producing a ferromagnetic phase.

Fullerenes on Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Al(111) also form uniform phases, that can become periodic if samples are annealed. The thermal energy brought by this process likely allows substrate reconstruction. Probably, an important energy barrier needs to be overcame to form the periodic-reconstructed height pattern. On both substrates the height pattern is a √ 3 × √ 3 periodic lattice (see Figure IV.29), which is one ground state of the Ising spin Hamiltonian when antiferromagnetic J 1 coupling and external field h ext are included.

The molecular phase in Al [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] is compressed (ε = -1.4%), which could explain the antiferromagnetic J 1 (the presence of a repulsive interaction was discussed in Ref. [START_REF] Maxwell | Electronic and geometric structure of on Al(111) and Al(110)[END_REF]). Binding with the substrate is important (∼ -7 eV for fullerenes on vacancies and ∼ -4 eV for molecules on non-reconstructed sites), so we expect to find an external field in the Hamiltonian. Further couplings J 2 and J 3 do not play a major role, because of small charge transfers (covalent binding). This way, we found agreement between the ingredients needed in the Hamiltonian (antiferromagnetic J 1 and h ext ) and the experimental √ 3 × √ 3 periodic height pattern.

For the periodic phase on Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] seven-atom vacancies are the preferred reconstruction structure, which implies that creating vacancies everywhere is not possible because the phase is rotated by 30°. Thus, an effective external field could be relevant but its value cannot be too large. This phase on Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] is not compressed, so we do not expect repulsive intermolecular interactions. However, fullerenes on Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] form ionic bonds so charge transfer and dipolar interactions are expected. Dipolar interactions would correspond to antiferromagnetic J 1 , J 2 and J 3 . The periodic √ 3 × √ 3 phase exists as well as the GS of the TIAF with these couplings, as long as an external field is taken into account. We conclude that the antiferromagnetic J 1 coupling needed for the formation of a √ 3 × √ 3 height pattern could be produced by dipolar interaction, but we are not sure of the possible origin of the effective magnetic field. Further insights on the structure of this phase would be needed to decide if the analogy with the TIAF model is consistent. Another commensurate phase, with uniform height pattern, forms for C 60 molecules on Ni [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. This phase is aligned with the Ni(111) lattice (φ = 0°) and it is the equivalent of the (4 × 4)R0°uniform phase observed for fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. In all the phases (except when samples are annealed) most of the molecules sit on non-reconstructed sites, indicating that formation of vacancies might have a substantial energy cost. The uniform phase is characterized by a large lattice parameter with strain +3%, so no repulsion between fullerenes is expected. Dipolar interactions could be present in the system because charge transfer has been measured for fullerenes on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Based on the experimental observations, we propose that interaction with the substrate dominates over dipolar interactions. Careful inspection revealed that, while the molecular height pattern of this phase is essentially uniform, a molecular rotational pattern exists, with well-defined (and different) orientation of each molecule with respect to that of its neighbors [START_REF] Tang | Complex orientational ordering of C 60 molecules on Au(111)[END_REF]. Quasi-periodic phases exhibit complicated height patterns, involving the formation of vacancies and rotational order of the fullerenes inside large supercells (including 9 or even 49 fullerenes per cell) [START_REF] Schull | Orientationally Ordered (7 x 7) Superstructure of C 60 on Au(111)[END_REF]. The periodicity of such phases is related to the coincidence with the substrate. These phases result from a complex balance between substrate-molecule and molecule-molecule (a C60 < 10.05 Å for all of them) interactions. Moreover, the molecular rotational degree of freedom is not considered in the Ising model. A more advanced analysis would be necessary to decide if an analogy between the experiment and the Ising model is possible. Finally, a "disordered" phase is observed for a lattice parameter of 9.98 Å (-0.7% strain). We suggest that this disorder is a balance between intermolecular repulsion, dipolar interactions and interaction with the substrate. In an Ising spin Hamiltonian we would have a competition between antiferromagnetic J 1 , J 2 and J 3 and external field h ext . To determine the type of disorder observed in this phase (random or correlated) a detailed analysis would be needed. We address this phase in chapter V, in which we quantitatively investigate the analogy with the TIAF model, and conclude that J 1 antiferromagnetic and an external field are sufficient to capture the observed (weak) correlations.

Fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] form mainly uniform phases and a disordered phase when the molecular lattice is compressed by -0.4%. Pussi et al. [START_REF] Pussi | Elucidating the dynamical equilibrium of C 60 molecules on Ag(111)[END_REF] have estimated the energy barrier to form a vacancy and found a value of 0.84 eV. This large energy barrier could explain why most of the fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] do not form vacancies, except when samples are annealed (in particular for the disordered phase). Uniform phases have a lattice parameter that is larger or close to the equilibrium intermolecular distance, which could explain why repulsive interaction effects do not seem to be present. Thus, the interaction with the substrate dominates, i.e. h ext drives the physics in the Ising model. A detailed analysis of the "disordered" phase is presented in chapter V, where we include the effect of antiferromagnetic couplings and a field and successfully explain the weak correlations observed in STM images.

In conclusion, mapping between microscopic interactions and effective couplings in an Ising spin Hamiltonian is a promising way to understand the formation of different height patterns in C 60 assemblies on a variety of close-packed metal surfaces, and not only on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. In particular, analyzing disordered phases using model Hamiltonians is a powerful manner to understand the role of the substrate, intermolecular and dipolar interactions.

Changing the perspective, fullerenes on close-packed metal surfaces can be considered as model systems simulating the exciting behavior of triangular Ising spin Hamiltonians. For some particular effective couplings, related to peculiar lattice compression situations, fullerenes on metals behave analogous to frustrated exotic phases. Model systems could be designed to:

• address particular values of the couplings by choosing different metal substrates; • change the interaction between fullerenes and their substrate, for example by intercalating atoms of different species; • or modifying molecular lattice parameters and intermolecular interactions, for instance in heteroepitaxial stacks where the lattice parameter of the metal substrate could be adjusted.

In this chapter we intend to show that the system we have mentioned in previous chapters, fullerenes on a Cu(111) surface, is not the only on-surface system behaving as a highly frustrated magnet. We already generalized part of our analysis to other phases formed by C 60 molecules on different close-packed metal surfaces in section 5 of chapter IV, but they were mainly ordered phases.

Here, we address phases exhibiting structural disorder. We consider six different types of non-magnetic on-surface samples for which microscopy images showing disordered configurations have been published in the literature. For each image exhibiting disorder we perform the same analysis we used for our own scanning tunneling microscopy (STM) images in chapter IV: we binarize the images, thereby determine the effective Ising state of each atom or molecule, and compute the magnetic structure factor and the pairwise correlations in real space. Thanks to this analysis, we prove that all these disordered phases can be statistically described as microstates of frustrated triangular Ising antiferromagnet (TIAF) Hamiltonians at thermodynamic equilibrium, with different interaction ranges and an effective external magnetic field. At the same time, we illustrate how our statistical analysis constitutes a powerful approach to study disordered configurations observed in real space.

We analyze two different types of on-surface systems: atomic binary alloys and fullerenes on close-packed metal surfaces. All the systems we treat have been proposed as realizations of the TIAF model with first neighbor interaction only. Spin liquid phases are thus expected. Our approach allows us to decide whether the observed phases are actually equivalent to spin liquid phases. Moreover, we are able to propose a more accurate model for each of these realizations, and to give an overview of the numerous frustrated phases that can be encountered in non-magnetic, structural, on-surface samples.

In section 1 we briefly sum up some of the phases that have been studied as ground states of the TIAF model. In particular, we present the magnetic structure factor (MSF) of such phases derived from Monte Carlo (MC) simulations, which are compared to experimental MSF in the next sections. This summary is a reminder of what is presented in section 1 of chapter IV.

Our methodology, i.e. how the TIAF Hamiltonian is chosen for each experimental example, is described in section 2. Section 3 presents the analysis of three different binary alloys that have been reported in the literature as being spin liquid-like phases. In these cases the Ising structural degree of freedom is the chemical nature of the atoms. We confirm that these phases are disordered but correlated, and two of them can be interpreted as realizations of spin liquid phases at high temperature, i.e. the experimental phases are not in a spin liquid ground state but somewhere in between the spin liquid phase at low temperature and the paramagnetic state at high temperature. We also show that varying the alloy composition in the samples is a way to tune the value of an external magnetic field in the spin Hamiltonian.

In section 4 we perform a similar analysis for fullerenes on close-packed metals (M(111)). We analyze the cases of C 60 molecules deposited on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], in which the 1. Brief summary of some frustrated phases in the TIAF model structural Ising degree of freedom is the molecular height due to vacancy formation, like in the case of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. We show that in these two cases a small antiferromagnetic nearest neighbor coupling is present in the system, but the dominant term in the Hamiltonian is an external magnetic field. Based on the results published in the literature, we propose a way to control the effective external field during sample fabrication. We also revisit the case of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], and verify that the analysis presented in section 2 and section 3 of chapter IV is robust. We confirm that the C 60 /Cu(111) system is an example of a TIAF Hamiltonian with first, second and third neighbor couplings.

At the end of both section 3 and section 4 we:

1. sum up the conclusions of our analysis; 2. propose a link between the couplings and external field in the effective Hamiltonian and the real microscopic interactions in the experimental systems; 3. and suggest how experimental parameters can be modified to explore the phase space of the TIAF model.

Brief summary of some frustrated phases in the TIAF model

A detailed review of different ground states of the TIAf model, including J 1 , J 2 , J 3 and h ext has been presented in section 1 of chapter IV. Here we only recall some of the reported ground states, because they are those we will compare to the literature data.

Antiferromagnetic J 1 : spin liquid phase

The exotic properties of the spin liquid phase, such as the non-zero entropy at zero temperature, the algebraic correlations and the existence of topological sectors, have been described in detail in section 1 of chapter IV. Here we only recall that this phase is the ground state of the TIAF model when J 1 is antiferromagnetic and no other couplings or an external field are present. In fact, the presence of neighbor couplings at larger distance or of an external magnetic field (with a strength comparable to the one of the first neighbor coupling) would lead to the formation of ordered ground states. Smerald et al. showed [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] that the spin liquid phase exists at finite temperature in the J 1 → ∞ limit, even if other couplings are present. In our analysis, this condition can be rewritten as J 1 ≫ J 2 , J 3 , h ext . In this case the TIAF model is in a paramagnetic uncorrelated state at high temperature, in a spin liquid phase at intermediate temperature and in an ordered ground state at low temperature (which depends on the relative strengths of J 2 , J 3 , h ext ). In Figure V.1 we show the MSF obtained from MC simulations for a spin liquid phase, which we already discussed in chapter IV.

Further antiferromagnetic couplings: stripe and zigzag phases

The phase diagram for the case in which J 2 and J 3 are antiferromagnetic has also been presented in section 1 of chapter IV. Here we focus on the stripe and zigzag phases, which exist if J 2 J 1 < 1 2 and J 3 J 1 < 1 4 . The stripe phase is the ground state in the region J 3 J 2 < 1 2 , the zigzag phase in the case J 3 J 2 > 1 2 . At J 3 J 2 = 1 2 the ground state is a coexistence of stripe and zigzag phases, in which domain walls corresponding to local zigzag patterns form in a stripe background. In Figure V.1 the MSFs are shown for sets of couplings corresponding to these three phases. Note that we chose the same couplings as those discussed in section 2 and section 3 of chapter IV.

External magnetic field:

√ 3 × √ 3

and ferromagnetic phases

In presence of an external magnetic field (and no further couplings are considered) the ground state of the TIAF model is no longer a spin liquid (h ext = 0) but an ordered phase. If h ext < 6J 1 the ground state is the √ 3 × √ 3 phase, while if h ext > 6J 1 the ground state is simply a ferromagnetic phase [START_REF] Metcalf | Ground state spin orderings of the triangular Ising model[END_REF]. It has been predicted that the transition from a disordered state at high temperature to an ordered √ 3 × √ 3 phase at low temperature follows a Kosterlitz-Thouless scenario when J 1 is antiferromagnetic and J 2 is ferromagnetic [START_REF] Fujiki | Possibility of the Kosterlitz-Thouless Phase Transition in the Two Dimensional Fully Frustrated Ising Model[END_REF]. The nature of the transition in a system with antiferromagnetic J 1 and an external field is under debate [START_REF] Qian | Triangular Ising model with nearest-and next-nearestneighbor couplings in a field[END_REF][START_REF] Kwak | Phase transitions of 2D triangular antiferromagnetic Ising model in a uniform magnetic field near h=0 and T=0[END_REF]. In Figure V.1 the MSFs are shown for sets of couplings 2. How to choose a TIAF Hamiltonian corresponding to the √ 3 × √ 3 and the ferromagnetic phases. For both MSFs a peak appears at the gamma point (0,0), this intensity is a ferromagnetic signal which develops in the presence of an external magnetic field, and is not visible for the phases mentioned above.

How to choose a TIAF Hamiltonian

In the next sections we determine a TIAF Hamiltonian for each of the experimental observations reported in the literature. To do so, we perform the following steps:

• We convert the experimental image into a perfect triangular lattice in which Ising variables account for the chemical nature/molecular height of each atom/molecule in the system.

• We compute the associated MSF and we symmetrize it taking into account the C 3 symmetry of the triangular lattice, and we compare it with the MC simulated MSFs presented in Figure V.1. This symmetrization is a choice to easily compare the experimental configurations (which are not necessary symmetric) with the MC MSFs which are averaged over many configurations, thus symmetric (except if the system is at low temperature and it has bifurcate to a ground state with broken symmetry, such as the perfectly ordered stripe or zigzag phases).

• Depending on the location of the intensity in reciprocal space, we chose a TIAF Hamiltonian for which the ground state is one of the phases presented in Figure V.1. In all the cases we fix J 1 = -1 and decide if J 2 , J 3 and h ext need to be considered.

• Once the set of non-zero (J 2 , J 3 , h ext ) values is chosen we compute the real space pairwise correlations on the basis of MC simulations. If the spin liquid phase (i.e. only J 1 present) is not the ground state, we vary the strength of the couplings or the applied magnetic field to explore the desired region of the phase diagram. For example, for a system showing an MSF similar to the one of the √ 3 × √ 3 ground state, we perform MC simulations for J 1 = -1 and h ext = 0.3, 1, 2, 3, 4, 5, 6. In the case of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] we do not repeat the analysis developed in chapter IV, but consider the same set of couplings we found in section 2 and section 3 of chapter IV.

• We compare the experimental and simulated real space pairwise correlations as a function of temperature and determine if we can define an effective temperature. When several sets of parameters are in agreement with the experimental data we plot the case with smaller weighted residuals (wSSR). 1

1 We recall that the wSSR was defined in subsection 3.3 of chapter II as: wSSR=

1 n n k C M C k (T )-C exp k 2 ∆C M C k (T ) 2 ,
where C k indicates the correlation values for the k th neighbor and ∆C M C k are the errorbars of the simulated correlations. Using this definition, wSSR≤1 implies that, in average, all the experimental pairwise correlations are inside the errorbars of the MC simulated values. In consequence we deduce a good agreement between the experimental data and the model.

The goal of this procedure is not to find the TIAF Hamiltonians that best fit the experimental data, but to show that the experimental configurations can be explained in the framework of a TIAF model. In this sense this chapter is different from chapter IV, in which we look for the range of J 1 , J 2 , J 3 values that account for our measurements. Here, for each experimental system we propose a possible TIAF model, with minimal ingredients and a reasonably good agreement with the measurements (wSSR< 1 in all cases).

For some of the experimental images we find high effective temperatures, one or two orders of magnitude larger than the couplings in the system. In these cases the samples are weakly correlated and determining an unique Hamiltonian is not possible. For instance, if we find a system with J 1 = -1 and T ∼ 10|J 1 |, J 2 , J 3 , h ext << 1 could be present in the experimental system, but we would not be able to detect them using our analysis, because they would express (i.e. have an influence in the correlation values) at much lower effective temperatures.

Binary alloys as a TIAF with a tunable external magnetic field

We now analyze two types of metallic binary alloys formed by Sn and Si, and Ge and Pb and a semiconductor alloy formed by Re and Nb atoms in a Re 0.5 Nb 0.5 S 2 monolayer. In all cases, the chemical nature of the atoms is assimilated to an Ising variable and the atoms of interest form a triangular lattice. In these systems, atoms of same chemical nature are expected to experience a repulsive interaction, whereas atoms of different chemical nature experience attractive interactions. These chemical interactions could be understood as the equivalent of antiferromagnetic interactions in an Ising system. In the following we do not consider the nature of the atoms or of the microscopic interactions (which could be Coulombic interactions between individual atoms holding a different partial charge or steric-like effects, related to the different sizes of the different atoms), but simply analyze the images as triangular lattices antiferromagnetically coupled hosting Ising variables.

From an uncorrelated paramagnet to a spin liquid phase

Ottaviano et al. [START_REF] Ottaviano | Short-range order in two-dimensional binary alloys[END_REF] claimed that Sn 0.5 Si 0.5 and Pb 0.5 Ge 0.5 alloys "are very good practical realizations of a frustrated antiferromagnetic 2D Ising system on a triangular lattice". In The comparison needs to be considered with caution, as we are only taking into account one STM image (∼ 2, 000 atoms), while the MC simulations average the correlations over thousands of configurations, each one constituted by 400 spins. We observe that all the experimental correlations from the first to the fifteenth neighbor are captured by this Hamiltonian at an effective temperature T ∼ 3J 1 . We conclude that the observed alloy is indeed 3. Binary alloys as a TIAF with a tunable external magnetic field an experimental realization of the short range TIAF model at thermodynamic equilibrium. Nevertheless, the observed phase is not a pure spin liquid ground state, but is between a paramagnetic phase at high temperature and a spin liquid phase. We perform the same analysis for the image published by Azizi et al. [START_REF] Azizi | Frustration and atomic ordering in a monolayer semiconductor alloy[END_REF] for the Re 0.5 Nb 0.5 S 2 monolayer, in which individual Re and Nb atoms are resolved, while in the S atoms are invisible in the detection mode chosen by the authors (Figure V.3). The MSF of the experimental image (∼ 1, 000 atoms) is very similar to the one of the Sn 0.5 Si 0.5 alloy, but the intensity values are higher. We again considered the TIAF model with J 1 = -1 and found that experimental correlations are consistent with the simulated ones at T ∼ 2J 1 . The experimental image is, again, an arrested configuration of the TIAF model at thermodynamic equilibrium. The observed phase is a low temperature paramagnetic phase approaching the spin liquid phase. 3 and ferromagnetic phases. The diffuse intensity is located at the same positions as those for a spin liquid or a √ 3 × √ 3 phase. We then compare experimental correlations with the simulated ones for a TIAF model with J 1 = -1 and an external field h ext = 0, 1, 2, 3. In Figure V.4c,d we show the comparison for which we found the best agreement (J 1 = -1, h ext = 2). The effective temperature can be defined for all the correlations and gives T ∼ 2.5|J 1 |. The experimental phase is thus a realization of the short range TIAF model under an applied magnetic external field. The observed configuration is a low temperature paramagnetic phase close to the transition to an ordered √ 3 × √ 3 phase.

Discussion and prospects

We have shown that several atomic binary alloys can be described as arrested configurations of a TIAF Hamiltonian. In particular we found that two ingredients can dominate: antiferromagnetic coupling between nearest neighbors or an external effective magnetic field. The antiferromagnetic interaction in the effective Hamiltonian can be understood as a shortrange repulsive interaction between atoms of the same chemical element. The nature of this interaction might depend on the chemical elements and probably on the substrate onto which the alloy is deposited, especially if interactions can be screened by the substrate. A precise determination of these interactions is beyond the scope of our review. The external magnetic field can be experimentally tuned by varying the density of each element in the alloy, as done in another study by Ottaviano et al. [START_REF] Ottaviano | Mechanism of the short range ordering in a 2D binary alloy[END_REF] who used a thermal post-annealing treatment of samples to control the atomic density. We have shown that for equal densities of the two elements, the antiferromagnetic term dominates, whereas a magnetic field can describe the observed arrangements when the proportion of the two atoms is unbalanced. The possibility of experimentally controlling the field term in the Hamiltonian opens the door to future studies. It might be possible to visualize in real space the evolution of a spin liquid phase under an external field. In particular, it might be possible to observe the nature of defects and the mechanisms at play close to the transition from a correlated disordered phase to an ordered √ 3 × √ 3 phase.

It is worth to keep in mind that the growth of the analyzed alloys is a kinetic process, but the resulting phases can be well described (even more than fullerene phases presented in chapter IV, we did not observe quenched defects here) in the framework of a TIAF Hamiltonian at thermodynamic equilibrium. The effective temperature of the observed phases is low enough to find clear signatures of the spin liquid or the √ 3 × √ 3 phases, both in the MSF and in real space pairwise correlations. Despite the fact that the alloys are clearly correlated, they are never in the ground state. We wonder if this behavior is intrinsic to the model or due to the strength of the interactions in the experimental system. By "intrinsic" we mean that the dynamics of the model, which is in principle a local dynamics similar to single spin flips in our MC simulations, freezes preventing to reach the ground state. In our MC simulations based on the Metropolis algorithm we found that temperatures of T ∼ 0.1|J 1 | could be reached (see section 1 in Appendix C) and the acceptance rate was still large (r ∼ 0.28 ). Even if our simulations cannot explore the whole phase space which is divided in topological sectors [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF], the spin liquid phase continues to fluctuate at low temperatures and no clear freezing of the dynamics is observed. The experimental systems are at much higher effective temperatures, T ∼ |J 1 |. Why are the effective temperatures not lower? We cannot come up with a definitive conclusion, but we propose a hypothesis. The interaction between atoms in such systems could be too weak. Ottaviano et al. give an estimation of J 1 for alloys of Pb-Ge and Sn-Si, of 30 me V and 10 me V respectively. In a naive way we could propose that these values are of the order of thermal energy at room temperature (E RT ∼ 26 me V) and thus the system is at an effective temperature respecting 0.9 ≲ T |J 1 | ≲ 2.6. For the case of Re 0.5 Nb 0.5 S 2 studied by Azizi et al. [START_REF] Azizi | Frustration and atomic ordering in a monolayer semiconductor alloy[END_REF], they proposed (based on DFT calculations) that the nearest neighbor interactions is ∼ 0.15 eV (1740 K), thus much larger than thermal energy at room temperature. However, the Re 0.5 Nb 0.5 S 2 crystals were grown at T ∼ 1220 K, which could explain why we found an effective temperature T ∼ |J 1 |.

Fullerenes on metal surfaces as a TIAF model with further neighbor couplings and an external field

In this section we address the disordered phases observed in fullerenes on close-packed metal surfaces. These phases have been described in section 5 of chapter IV. Disordered height patterns have been reported in Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. We show that all the disordered phases can be interpreted in the framework of a TIAF model, but the relative strength of the couplings and external field strongly depends on the metal surface (as expected) and on the growth conditions. In Figure V.5 we present an STM image of fullerenes on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. The image shows ∼ 1, 000 molecules, 47% being "high" and 53% being "low". Gardener et al. [START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF] describe the observed disorder as being "random". In Figure V.5b we show the MSF of the experimental image, which looks indeed like the one of an uncorrelated paramagnet. Nevertheless, when we compute the pairwise correlations in real space we note that C 1 is small and negative. The comparison between experimental and simulated correlations for a TIAF model with antiferromagnetic J 1 are consistent at T ∼ 26J 1 , which explain why the MSF does not exhibit any clear structure. In fact, the experimental phase can be understood as a realization of the TIAF at high temperature, which shows a tiny antiferromagnetic signal. but the experimental phase is at high temperature in a region in which the paramagnetic state exhibits small ferromagnetic correlations.

From a spin liquid to an ordered zigzag phase

We decide to revisit the case of fullerenes deposited on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface published by Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] who claimed that the STM images of the "disordered maze" presents the same type of disorder as that obtained with MC simulation of the TIAF model with only antiferromagnetic J 1 . They even show a MC snapshot of a low energy state and compare it with the experimental case. Here we perform our analysis on their STM image and their MC snapshot, and show that they in fact correspond to different states of matter. Our analysis also allows us to confirm that our previous analysis of the disordered phase of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] is robust.

The STM image in Figure V.8a shows ∼ 2, 000 C 60 molecules, 55% of them are "high" and 45% are "low". The MSF is very similar to the one we obtain for our samples (section 2 in chapter IV). We observe the emergence of peaks around the corners of the Brillouin zone, together with a diffuse but structured intensity around the edges of the Brillouin zone. Comparing the experimental correlations with the ones for a TIAF with J 1 = -1; J 2 = -0.1, J 3 = -0.057 leads to a very good agreement except for the C 1 value. These results confirm our previous conclusion: the disordered phase of C 60 on Cu( 111) is an arrested configuration of a (J 1 , J 2 , J 3 ) TIAF model in the transition from a spin liquid phase to a zigzag ordered phase. The phase is at thermodynamic equilibrium, except for the presence of quenched defects formed by triangular plaquettes with three molecules in a "low" state. The MC snapshot presented in Figure V.8a (bottom) is apparently similar to the experimental configuration (top). However, by computing the MSF and the pairwise correlations obvious differences are observed. The intensity is not located at the same points in the reciprocal space and the correlations for further neighbors are clearly different. As expected, comparing the MC snapshot reported in the literature with our simulations for a TIAF model with J 1 = -1 leads to a good agreement of the correlations in real space. From this we confirm that the MC configuration reported by Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF] indeed is well described with a J 1 -only model at a certain effective temperature, but that it actually produces correlators and MSFs that are both very different from experimental values. This example clearly proves the power of our analysis based on the MSF and the pairwise correlations. These two tools make it possible to distinguish between disordered states of matter and to determine the range of the couplings present in the system.

Coexistence of zigzag and stripe ordered phases

∼ 0.4|J 1 | ∼ J 2 .
The agreement between experimental and simulated correlations at this effective temperature gives residuals SSR= 0.01 and wSSR= 0.17 (without considering the C 1 correlation, which is clearly at a different effective temperature).

Finally we also analyze the "Linear wall" height pattern exhibited by fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] (stripe in our notation) and measured by Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. We consider the STM image shown in Figure V.9, with ∼ 1, 000 molecules, 49% "low" and 51% "high". The MSF is similar to the one obtained for our samples (section 3 in chapter IV) and comparable to the one shown in Figure V.1 for the stripes-zigzag phase, which corresponds to the TIAF model at the isoenergetic point between the stripe and zigzag phases. However, it exhibits a lower intensity in the regions corresponding to the zigzag Bragg peaks than in the ones of the stripe Bragg peaks. This intensity suggests that we could be not exactly at the point J 3 J 2 = 0.5. In fact, we analyze the pairwise correlations and found that the best agreement (considering always J 1 = -1; J 2 = -0.3 and varying 0.3 < J 3 ≤ 0.5) is for J 3 J 2 ∼ 0.42. At this temperature the number of domain walls is expected to be smaller than in the case J 3 J 2 = 0.5, which is indeed what we observe in the experimental image. The reason why the J 3 J 2 ratio is slightly different in the sample prepared by Pai et al. and in our own samples remains an open, intriguing question. Nevertheless, the analysis is roughly the same in both cases: the so-called stripe phase is an experimental realization of the TIAF model, near the degeneracy point between two ordered phases (the stripe and zigzag phases) at thermodynamic equilibrium. The discrepancy observed in our samples between experimental and simulated values for the first correlations is also found for the reported image by Pai et al., the same type of local defects formed by three "low" molecules explain this difference.

Discussion and prospects

Our review of the disordered phases formed by fullerenes on different close-packed metal surfaces demonstrates that all the considered STM images can be understood as arrested configurations of a TIAF Hamiltonian. We are also able to determine the ingredients needed to reproduce the experimental observations. Surprisingly, nearest neighbor interactions are in general not enough to explain the observed height patterns. In fact, depending on the metal surface, other couplings or an external field need to be considered.

Fullerenes on Au(111)

C 60 molecules deposited on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] exhibit weak correlations (compared to the ones in binary alloys or in fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]). In the two STM images we analyze we found that the disordered height patterns can be described as TIAF realizations at high temperature (T ∼ 10|J 1 |). In both cases an antiferromagnetic J 1 was considered in the Hamiltonian, which we interpret as repulsive intermolecular interactions due to compression of the fullerene layers. For this experimental phase the strain in the molecular layer is ∼ -0.7% (see section 5 in chapter IV and Table B.2 in Appendix B for details about the structure of the phase). If we only consider intermolecular interactions, without dipolar contribution, and we compare the configuration with and without a vacancy (supposing a height difference ∼ 0.1Å, see Table III.6) we find J 1 ∼ -0.01 meV, which is much smaller than what we found for different phases of C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] for which -45 meV< J 1 < -25 meV including dipolar interaction (subsection 2.5 and subsection 3.6 in chapter IV). Nevertheless, a charge transfer has been measured for fullerenes on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], which could significantly increase the J 1 strength.

For one of the STM images, an external field needs to be included, which is surprising because the two STM images corresponds (according to the publications) to the same molecular commensurate phase. The difference between the two phases is the post-annealing treatment of the samples. Gardener et al. [START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF] reported an increasing density of vacancies (thus of "low" molecules) when the samples are annealed at higher temperatures. In fact, formation of vacancies seems to have an important energy cost, which preludes the formation of vacancies at room temperature. We propose that in this context the C 60 /Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] system could be used to study the evolution of a spin liquid phase (high annealing temperatures to find 1 2 of "low" molecules) under an external field (low annealing temperatures to choose the field values). Here the field is not only related to the binding energy of vacancies (which in any case is less favorable than for fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]), but it is modulated by the energy barrier to create such vacancies, which depends on the thermal treatment of the samples.

Unfortunately, a major problem might arise in this hypothetical study, as the experimental phases exhibit weak correlations, i.e. are at high effective temperatures in which disorder resembles the one of a paramagnetic phase. To find phases with stronger correlations it would be necessary to determine why the system cannot reach lower effective temperatures. In our MC simulations we did not find freezing of the dynamics neither for the TIAF Hamiltonian with J 1 = -1 or for J 1 = -1, h ext ̸ = 0 for temperatures 100 times smaller that the experimental ones, we then deduce that reaching lower effective temperatures is not limited by the model dynamics. The weak correlations in the experimental images could be due to small values of the repulsive intermolecular interaction, but further analysis is needed to confirm this hypothesis. Another hypothesis concerns the creation of gold ad-atoms at the interface between the metal and the C 60 molecules. Copper atoms are known to diffuse on the surface at room temperature, whereas it it not the case for gold atoms [START_REF] Stoltze | Simulation of surface defects[END_REF]. We suggest that in a metal in which adatoms easily form, vacancies can rearrange and low energy states are easily reached. It could be interesting to experimentally increase the mobility of the gold atoms (for example annealing the samples during longer times) to let the molecular phase explore the phase space and verify if correlated states corresponding to lower temperatures form.

Concluding about the origin of high effective temperatures is not easy in the case of fullerenes on Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. In fact, many mechanisms are probably playing a role: smaller strength of J 1 couplings than in Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]; formation of one-atom vacancies instead of sevenatom vacancies, which leads to smaller height differences between "high" and "low" fullerenes, and for which steric effects are not expected; and different mobility of metal atoms on the surface.

Fullerenes on Ag(111)

The disordered phase formed by fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] substrate is similar to the case of Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]: the corresponding TIAF model includes an antiferromagnetic J 1 and an external field; and the disorder is weakly correlated.

The disordered phase formed by C 60 molecules on Ag( 111) is compressed by only -0.4%, so antiferromagnetic coupling is probably small compared to the one in Au(111) (strain ∼ -0.7%) and Cu(111) (strain ∼ -0.7% and ∼ -3.7%). In addition, formation of vacancies only changes the molecular height by ∼ 0.4Å (according to our DFT calculations), so buckling relaxes strain in a less important way than in Cu(111) (remember the height difference between "low" and "high" molecules of ∼ 1.3Å).

The external field we found in the effective model is ten times larger than the antiferromagnetic coupling, thus a ferromagnetic ground state is expected. However, the binding energy to vacancies in Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] is much less favorable than in the case of Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], which means that the higher value of the effective field cannot simply be related to the binding energy of vacancies. In fact, STM images show more high fullerenes than low ones, indicating that thermal energy was not sufficient to create more vacancies. Pussi et al. addressed a thermodynamic study of the formation of vacancies in C 60 /Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], which strongly suggests that modifying the growth conditions and the post-annealing treatment of the samples could allow to control the number of vacancies, i.e. the value of the field. Moreover, they study the switching from "low" to "high" molecules when the samples are annealed. We propose that post-annealing treatments could modify the ability of the silver surface to reconstruct and thus the effective temperature of the samples. If it was the case, the disordered height patterns could exhibit stronger correlations and potentially approach a spin liquid phase. Nevertheless, if the J 1 value is too small, the system might remain in a high effective temperature.

Fullerenes on Cu(111)

Disordered phases of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] are substantially different than the phases reported for C 60 /Au [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and C 60 /Ag(111) systems. Correlations in our samples, and in those of Pai et al. [START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF], correspond to the ones of a TIAF at low temperature, near the transition from disordered states to ordered ground states. Moreover, the effective temperature corresponds to the lower temperatures we were able to reach in our MC simulations, when the single flip dynamics is frozen and the acceptance rates are very small (r ∼ 0.01). We suggest that this low effective temperature is due to a combination of factors. First, the coupling strengths we obtain for fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] are of the order of thermal energy at room temperature, -45 me V< J 1 < -25 me V. Second, the formation of vacancies in Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] implies important height differences between "high" and "low" molecules, which allows to relax strain by buckling of the molecular layer. Third, steric effects between seven-atom vacancies are expected, which translates in a repulsive interaction between "low" fullerenes. Fourth, copper adatoms are known to readily form at room temperature, which seems favorable for an extensive reconstruction of the surface and could explain why fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] are able to reach low energy states.

C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] are not only at low effective temperature but they constitute a particularly rich playground to study frustrated systems, because further neighbor couplings (J 2 , J 3 ) are present in the system. Taking into account these results new research paths appear in the case of fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. Decreasing the growth temperature may change the density of vacancies (external field term) and reduce the mobility of copper atoms (thus higher effective temperature would be expected). As mentioned in the previous chapters, locally modifying the fullerene's height could be a way to observe how the system adapts to the presence of defects.

Conclusion Two approaches, one experimental system

In this manuscript we have extensively described and analyzed ordered and disordered phases in two-dimensional molecular assemblies, mainly in the specific case of C 60 molecules deposited on a Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surface. These phases were described down to the atomic scale, under a microscopic point of view, considering our samples in the framework of on-surface models. A second approach shed a radically different light on our experimental system: a statistical analysis of molecular phases, in which we showed how the physics of order and disorder in non-magnetic molecular lattices can be explored with the help of magnetic-like Ising Hamiltonians. More precisely, we described the structural frustration of our molecular system and related it to some exotic phases known in highly frustrated magnets. Applying both approaches to the same experimental system, and proposing bridges between them constitutes the principal originality of our work. In this way, we demonstrated that fullerenes on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] are not only a very rich on-surface system (including multiple microscopic interactions, various mechanisms to relax the energy of the molecular layer, and showing phase coexistence) but also they constitute a model system to emulate model spin Hamiltonians. In both cases our investigation could rely on real space observations, which is particularly insightful for the analysis of frustrated configurations.

High-resolution microscopy point of view

The viewpoint we developed in chapter III combined a detailed analysis of microscopy images (both STM and nc-AFM) with ab initio calculations, to resolve the structure, down to the atomic scale, of ten coexisting molecular phase. We were able to elucidate, for the first time to our knowledge, the nature of the orientation of the fullerenes on the copper surface, of their binding sites, of the copper vacancies and, related to that, the existence of two well-defined molecular heights, and the presence of local distortions within the molecular lattice. Fullerene phases on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] show intriguing height patterns, corresponding to the formation of vacancies below the molecules. The height patterns are periodic for some of the phases, but for other phases no clear order is observed and instead correlated disorder develops. Using geometrical considerations about the formation of vacancies, which basically translates in a repulsive-like interaction between vacancies, we manage to account for all the periodic height patterns. We also proposed a mechanism to explain the formation of correlated disorder patterns. The observed disorder might be understood as a mixture of different molecular unit cells (including 3 or 4 fullerenes) that respect some local rules, i.e. vacancies cannot form everywhere because of repulsion between each other, and positioning of the fullerenes is favored on certain high-symmetry crystallographic sites of the substrate (favorable coincidences). A complete fully-consistent description of the structure of the ten coexisting phases was lacking in the literature.

Besides the description of the phases, we proposed a model to semi-quantitatively explain phase coexistence at room temperature. Our model is based on energy competition associated to two different contributions: the tendency that the fullerenes have to adopt the above-mentioned coincidences with the copper substrate (driven by a specific moleculesubstrate binding energy landscape) and the intermolecular energy (i.e. molecule-molecule short range van der Waals and electrostatic interactions, and long range dipolar interactions due to charge transfer). We demonstrated that the combination of various degrees of freedom (molecular orientation, binding sites, vacancy formation) together with competing energies lead to a complex energy diagram in which the ground state and various metastable states correspond to the different phases observed in the experiments. We estimated the potential influence of configurational entropy and concluded that it is marginal with respect to internal energy considerations. Despite the existence of metastable states, we found energy differences of the order of 1 me V between coexisting phases, which may indicate that the molecular assemblies are trapped in local energy minima separated by large energy barriers, despite the energy difference between those states. Future studies addressing the kinetic aspect of the problem could improve our understanding of phase coexistence and the possible interconversion of one phase into another.

The low energy states predicted by our model are consistent with our experimental observations of molecular phases. On one hand, we found commensurate phases, some of them exhibiting short superperiodicities, while others are large superperiodicity phases including local, periodic lattice distortions. These phases have periodic height patterns. On the other hand, the nature of the phases exhibiting non-ordered height patterns was not unambiguously resolved. Seeing these phases as tilings of a triangular lattice with different unit cells accounts for their disorder, but we wonder what would happen in between adjacent tilings. Disorder in the height patterns may be described by considering the presence of height defects within an otherwise regular arrangement of heights within commensurate phases. In this sense, following the picture introduced by Frenkel and Kontorova, the defects actually represent solitons, i.e. singularities in the fullerene-on-Cu coincidence field. Moreover, our statistical analysis showed that these phases are frustrated, the height pattern of one of them can be compared to a spin phase with algebraic height-height (analogous to spin-spin) spatial correlations. We ask the question whether such a kind of disorder may not actually correspond, in the paradigm of structural phases, to what is called a 2D floating phase, i.e. a 2D liquid. A more detailed description of the phase in terms of commensurability and defect formation could be an interesting research path.

We did not restrict our structural analysis to the case of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], but we applied it to the case of other close-packed metals. We reviewed fullerene phases previously reported in the literature and discussed their geometry, the molecule-molecule vs. molecule-substrate interactions and the formation of different ordered and disordered height patterns. Even if we did not enter into the the level of details we reached for C 60 /Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], we proposed a synthetic view of different systems, which allowed us to define the necessary ingredients to find phase coexistence and non-ordered phases. In this way, our analysis could guide future research by looking for on-surface fullerene on metal systems in which phase coexistence and disordered height patterns are present. At the same time, our comparison shows that our model is general enough to be applied to different molecular systems including commensurate phases, surface reconstruction and local distortions.

Geometrical frustration framework

Our second approach, based on a statistical analysis of the spatial correlations, focused on the question of geometrically frustrated phases in C 60 on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] samples. We established a quantitative connection between the molecular non-magnetic phases with a non-ordered height pattern and a triangular Ising antiferromagnet (TIAF) model, which is the archetype of geometrical frustration (usually considered for spins). This comparison required to implement a number of tools to compute the quantities of interest in the experimental images: the magnetic structures factors and the pairwise correlations. To determine the behaviors (nature of the phases and correlations) predicted by the frustrated model, we employed MC simulations with a single spin flip dynamics. Combining these tools we were able to demonstrate that the two phases exhibiting non-ordered height patterns in the C 60 /Cu(111) system are experimental realizations of the TIAF model in which an antiferromagnetic first neighbor interaction dominates. Using our structural analysis, we showed that this antiferromagnetic effective coupling is caused by intermolecular repulsion at play when the molecular layer is compressed with respect to its equilibrium lattice constant. Moreover, we discussed how the steric effects experienced by seven-atom vacancies are a key ingredient to find such frustrated phases.

Our effective analysis clearly indicated that short range interactions were not sufficient to describe the non-ordered heigth patterns. Although first neighbor coupling dominates, we needed to include second and third neighbor couplings to account for the experimental correlations. In this way, our effective model informed us about the length of intermolecular interactions in the microscopic fullerene assemblies on copper. This finding was surprising for us, because a priori we only expected intermolecular interactions due to van der Waals and chemical repulsion, which are large at distances ≲ 15Å. It was, in fact, our attempts to account for the observation within this effective model that lead us to consider dipolar interactions in the molecular assemblies.

We proved that one of the non-ordered phases, the one we called disordered phase, corresponds to arrested configurations of a TIAF state close to the transition temperature from a spin liquid phase to an ordered zigzag phase. We could define a temperature for our observations, meaning that the observed phase is a realization of the TIAF at thermodynamic equilibrium. Quantitatively demonstrating that our experimental non-magnetic molecular system behaves like a critical spin liquid is a key result of our work.

Regarding the other non-ordered phase, which we called stripe phase, we found that a slightly different structure (compared to the disordered phase) leads to a completely different state of matter in the effective model. In fact, we concluded that this molecular phase is (dis)ordered in the way predicted by a TIAF model in a region of the phase diagram where two ordered (stripe and zigzag) phases coexist. This coexistence is not a trivial one with some patches of stripe and zigzag orders spatially separated, but a phase in which domain walls (zigzag-like order) can proliferate in a stripe background with no energy cost.

Another unexpected result was that the two previous phases could be described by a Hamiltonian at low temperature. The phases exhibit correlations characteristic of a thermodynamic equilibrium state, and this state is at the lowest temperature we could reach in our MC simulations. For both disordered and stripe phases, the dynamics of our MC simulations experiences a critical slowdown at temperatures slightly smaller than T ∼ J 2 , and our molecular phases are at an effective temperature T ∼ J 2 . The molecular phases are very well thermalized, in principle the best a system with a local dynamics could be. This result is absolutely non-trivial and it is worth to compare our fullerene assemblies with other two-dimensional frustrated systems. For example, if we consider frustrated artificial systems formed by arrays of nanomagnets the lowest temperatures reached are usually much larger, even after applying dedicated protocols to thermalize the samples [START_REF] Rougemaille | Cooperative magnetic phenomena in artificial spin systems: spin liquids, Coulomb phase and fragmentation of magnetism-a colloquium[END_REF].

We also addressed the ordered height patterns observed for commensurate phases of C 60 molecules on Cu [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] in the framework of a TIAF model. We performed a semi-quantitative analysis which showed that all the observed height patterns correspond to a ground state of the TIAF model, including first, second and third neighbor couplings and an external magnetic field. We have already mentioned the origin of the couplings, and we proposed that the magnetic field is related to the tendency of forming vacancies below the molecules. The strength of the external field in different molecular phases is thus related to the binding energy of fullerenes in vacancies and to the steric effects between seven-atom vacancies.

We extend this analysis to the case of fullerenes on other close-packed metals studied in the literature, and concluded that all the experimental height patterns we considered could correspond to ground states of the TIAF with the above cited couplings and an external magnetic field. Further analysis could address the origin of the microscopic interactions in each phase, and the parallel with effective couplings. We already presented some elements for the comparison, including the binding energy of the fullerenes with each metal, the strain in the molecular layers, the formation of vacancies and the charge transfer between the molecules and the metal.

Finally, we systematically applied our analysis of correlations (MSF and pairwise correlations) to different experimental images reported in the literature for C 60 on close-packed metal surfaces and for atomic binary alloys, in which disordered phases are observed. Comparing the experimental correlations with MC simulations of the TIAF model allowed us to determine a model Hamiltonian with the same correlations as those we extract from the experimental images. We thus proposed that our analysis is a powerful tool to characterize disorder in on-surface systems, which indeed behave as structural frustrated phases.

On-surface systems appear as rich platforms to study frustration. In these systems the state of single atoms and molecules can be directly observed in real space, and the properties of the samples could be tuned. For example, we proposed that fullerenes assemblies could be locally modified using the tip of STM, and how the molecules react to this modifications could be observed. Another exciting research path would be to change the nature of the molecule, for example C 70 molecules might be used instead of C 60 molecules. In this case the reactivity with the substrate is expected to be weaker and thus the charge transfer and dipolar interactions. We wonder if the range of interactions could be reduced in this way, leading to the formation of a purely spin liquid like phase2 . We also suggested that modifying the metal surface on which the molecules are deposited can change the nature of the observed phases. In some cases the growth conditions and the annealing treatment could also modify the molecular arrangements. For the case of alloys, the population of each atom can directly be controlled, thus the effective field in the magnetic model. In principle this quantity can be tuned in a continuous way, and it might be experimentally feasible to study the evolution of a TIAF under an external field. current depends on the electronic structure of the measured surface). • Binding sites: different high-symmetry positions on a crystalline substrate, in which a molecule can bind. • Chaotic phase: applied to on-surface systems, i.e. particles adsorbed on crystalline surfaces. It makes reference to an out-of-equilibrium phase, in which the position of the particles adsorbed on the surface correspond to local minima of the energy. • Commensurate phase: state of matter in which two coexisting lattices exhibit a common periodicity. For example, modulated spins on a chain or atoms adsorbed on a crystal. • (Magnetic) couplings: in this manuscript, it makes reference to the exchange-like couplings J 1 , J 2 , J 3 between sites in a spin Ising Hamiltonian. • Critical paramagnet: disordered but correlated phase of matter exhibiting algebraic (critical) correlations. In the TIAF model it only exists at T = 0, or equivalently in the limit J 1 → ∞. • (Magnetic or structural) degree of freedom: in this manuscript magnetic degrees of freedom corresponds to Ising spins, while structural degrees of freedom can be positional, rotational or chemical. • Exponential correlations: pairwise correlations exhibiting and exponential decay with the distance, following the mathematical expression e -ξr , with ξ a real positive number. This type of correlations are characteristic of structural (usual) liquids. • Freezing of the Monte Carlo dynamics: in a Monte Carlo simulation it makes reference to the slowdown of the dynamics, when the probability of accepting new updates decreases and reaching lower energy states takes longer at each step. • Frustration: impossibility of a physical system formed by interacting elements to simultaneously minimize all the interactions each element experiences from the other constituents. • Hysteresis of the piezoelectrics: behavior observed on the piezoelectric actuators of STM tip, in which the position vs. time evolution depends on the previous position of the piezoelectric. This hysteresis translates into a non-constant value of the STM tip velocity during scan cycles. • Ice rule: in this manuscript (by analogy with the arrangement of hydrogen atoms in water ice) it refers to the constraint for a low energy state of the TIAF in which three spins on a triangular plaquette never are in the same up/down state. The low energy state for each triangular plaquette corresponds to two spins in the same up/down state and the third one in the other down/up state. • Incommensurate phase: by opposition to a commensurate phase, state of matter in which two coexisting lattices do not exhibit a common periodicity (even at infinite distance). For two-dimensional systems in the absence of interaction between the two lattices, this definition is more a thought experiment than a real description of physical phases. In fact, in a two dimensional atomic or molecular layer long range order cannot exist [START_REF] Mermin | Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models[END_REF][START_REF] Mermin | Absence of ordering in certain classical systems[END_REF][START_REF] Hohenberg | Existence of long-range order in one and two dimensions[END_REF], thus a layer with a fixed lattice parameter independent of the one of the surface is not expected. Instead, "floating" regions form (the lattice parameter of the adsorbed layer can be fixed at short range) and are mixed with commensurate regions or separated by local distortions. Additionally, for non-vanishing interaction between the two lattices, ordered incommensurate phases can exist that are stabilized by periodic arrays of distortions, so-called static distortion waves.

state is disordered (but correlated) and has an extensive non-zero entropy at zero temperature. • Thermal drift: imaging artifact especially relevant at room temperature, in which the STM sample drifts laterally during the raster scan of a surface region on the sample. • Triangular Ising antiferromagnet: Ising Hamiltonian applied to a triangular lattice, with general form H = -J 1 ⟨ij⟩ σ i σ j -J 2 ⟨⟨ij⟩⟩ σ i σ j -J 3 ⟨⟨⟨ij⟩⟩⟩ σ i σ j + ... + h ext i σ i , where J 1 < 0. • Topological sectors: in this manuscript it makes reference to different states of matter showing the same symmetry properties and correlations, but that can be characterized by different topological invariants. • Quenched disorder or defects: state of matter which does not correspond to a low energy state, due to a rapid cooling which prevents the constituents in the system from reaching thermal equilibrium. In this manuscript, we use the terms quenched disorder and quenched defects to call local defects that are characteristic of phases at high temperatures but appear in low-energy states.

MC results for the TIAF model with antiferromagnetic J : spin liquid ground state

We verify to which extent our MC simulations based on Metropolis algorithm are able to reproduce the main characteristics of the TIAF model with first nearest neighbor couplings.

Local updates such as the one in Metropolis algorithm cannot properly describe the spin liquid phase corresponding to the ground state [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF]. In fact, single spin flips are not suited to explore the whole phase space at low temperature, which is divided in topological sectors. The Metropolis algorithm only samples a certain region of the phase space, and despite the fact that the dynamics does not freeze, the system is trapped in a restricted number of configurations. To solve this problem some cluster and worm algorithms applied to the TIAF have been developed [START_REF] Zhang | Cluster Monte Carlo dynamics for the antiferromagnetic Ising model on a triangular lattice[END_REF][START_REF] Wang | Generalized Monte Carlo loop algorithm for two-dimensional frustrated Ising models[END_REF][START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF].

Even if Metropolis algorithm is not adapted to describe the Ising Hamiltonian, it better reflects the physical behavior of our experimental system. We expect C 60 molecules to selfassemble during sample growth, then to be able to locally change molecular heights before freezing in the observed configurations. Global, cooperative rearrangement of the molecular lattice seem to us improbable.

In Figure C.1 we show the specific heat, entropy and acceptance rate (r, defined in subsection 3.4 of chapter II) depending on temperature for the TIAF with J 1 = -1. We found a smooth peak for the specific heat and an entropy plateau. The entropy per site at low temperature is predicted to be 0.3383R [START_REF] Wannier | Antiferromagnetism. The Triangular Ising Net[END_REF]. For a 20 × 20 = 400 spins we found a value of 0.3281R, i.e. a difference of 3%. For a biggest lattice with 100 × 100 = 10000 spins the value is 0.3309, so the difference with respect to the expected value is slightly smaller (2.2%). The acceptance rate decreases significantly at temperature T ∼ J 1 and reach a constant value ∼ 0.28, which indicates that the ground state does not experiment any freezing of the dynamics but continues to fluctuate between different degenerate microstates. In Figure C.2 we show pairwise correlations from first to fifteenth neighbor in real space for the TIAF with J 1 = -1 and a lattice of 400 spins with periodic boundary conditions. Variation of each correlation is monotonous with temperature. We can compare values at low temperature with the ones published by Stephenson [194] that decrease at T = 0 in an algebraic way following the law C.1. We observe a strong dependence of pairwise correlation values on the size of the simulated lattice. For smaller lattices correlations decrease faster than the algebraic law, in qualitative agreement with the presence of a cutoff (like it has been found for the correlations in the Kagomé antiferromagnet [START_REF] Garanin | Classical spin liquid: Exact solution for the infinitecomponent antiferromagnetic model on the kagomé lattice[END_REF]). Taking a lattice of 10 4 spins with periodic boundary conditions we find a very good agreement with the analytical values (Table C.1). For correlations from first to eighteenth neighbor, i.e. correlations within 3 lattice parameters, the difference between On the profile through q y = 2 √ 3 π we observe a constant intensity around the position of pinch points (q x = 0, dotted blue lines) for each temperature. Around q x = ± 2 3 π a structured intensity is present and it decreases with temperature to tend to a uniform, constant value at high temperature (uncorrelated paramagnetic phase).

We conclude that, despite the fact that Metropolis algorithm is unable to probe all the topological sectors in the manifold ground state, we are able to simulate the quantities of interest of the spin liquid phase within a reasonable error. For lattices of 400 spins entropy is accurate within 3% and correlation values to the eight neighbor within 10%. For bigger lattices with 10 4 spins entropy is accurate within 2% and correlation values to twentieth neighbor within 2%. Nevertheless, the pinch points are always significantly broadened, even for large lattices, indicating that we do not correctly reproduce algebraic decay of the correlations at long distances. J 1 = 0.1 and J 3 J 2 = 0.57. We observe the transition from a spin liquid phase (T > 0.1) to an ordered zigzag ground state (T < 0.1).

In Figure C.7 we show the evolution of the magnetic structure factor of the TIAF model with J 2 J 1 = 0.1 and J 3 J 2 = 0.57 when temperature increases. Around T∼ 0.1 we observe the transition from a zigzag ground state (well-defined peaks) to a spin liquid phase (diffuse intensity).

In the stripe-zigzag region

Here we present the same type of information than in the previous section, but for MC simulations of the TIAF with J 2 J 1 = 0.3 and J 3 J 2 = 0.5. These values correspond to those we found to best capture the stripe phase observed in our C 60 on Cu(111) assemblies (section 3 in chapter IV). We recall that the J 3 J 2 = 0.5 ratio corresponds to a isoenergetic line in the phase diagram of the TIAF, in which zigzag and stripe phases coexist [START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF][START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF]. In Figure C.8 we show the thermodynamic properties of the TIAF model for these coupling values. We observe a peak in the specific heat at T J 1 ∼ 0.3, in agreement with the first order transition reported by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF]. Nevertheless, in our simulations we are not in the J 1 → ∞ limit, which is the case in Ref. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF]. This difference could have consequences in the nature of the transition, which we do not address in our study. The entropy calculated with our MC simulations does not exhibit a plateau corresponding to the spin liquid phase, which suggest that no spin liquid phase form at intermediate temperature.

The acceptance rate indicates a slowdown of the simulation dynamics just below temperature of the specific heat peak, T J 1 ∼ 0. J 2 = 0.15. Our MSF at low temperature can be compared with the ones reported by Smerald et al. [START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF] for J 1 → ∞ and J 3 J 2 = 0.492. The intensity in our MSF at low temperature (T < 0.3) is located at the same positions than the intensity for the reported MSF (J 1 → ∞, J 3 J 2 = 0.492) at T ∼ 1. At higher temperatures, our MSF shows a different intensity distribution, with a diffuse intensity inside the first Brillouin zone. This intensity does not correspond to the one of a spin liquid phase (intensity is expected in the corners of the Brillouin zone). At higher temperatures (not shown) we found the MSF of a paramagnetic phase. We interpret the MSF at T ∼ 0.6 as a transition between the paramagnetic state and the stripe-zigzag phase coexistence.

Size effects on the magnetic structure factor

In this section we briefly describe the effect of the lattice's size considered in the MC simulations on the magnetic structure factor. We illustrate this effect for the particular case of the spin liquid phase in its transition to the ordered zigzag ground state. In particular, we show how the size of the simulated lattice modifies the shape of the emergent peaks on the MSF. This analysis aims to explain the elongation of the emergent peaks observed in the experimental MSF for the "disordered" molecular phase (Figure IV.13a).

In Figure C.11 we show the MSF calculated for three different lattices sizes formed by 400, 1600 and 6400 spins. We chose the same coupling values and similar temperatures which correspond to the transition from the spin liquid phase to the ordered zigzag phase. We observe that the main difference between this simulations appears in the shape of the emergent peaks. For the smallest lattice, the peaks are clearly elongated, while for the largest lattice the peaks appear as circular spots. The small lattices best reproduce the experimental MSF (see Figure IV.13a), in which elongated peaks appear in the reciprocal space. This situation corresponds to the number of fullerenes in the molecular regions measured by STM, which are always between 500 and 1200 molecules.

We also verify that the elongation of the peaks in the experimental MSF does not correspond to a poor statistics, i.e. a small number of measurements (N meas in the MC language, see subsection 3.4 in chapter II). To do so we perform the same MC simulations as usual but consider only a reduced number of configurations to compute the MSF. The result is presented in Figure C.12. The effect of the number of measurements is visible in all the ex- amples shown in the figure: for a small number of measurements the emergent peaks exhibit an internal structure and their positions can be slightly shifted with respect to the expected position for a large number of measurements. In fact, low statistics implies strong deviations from the expected MSF, which are non statistically significant. Surprisingly, despite the fact that we only average ten experimental configurations we did not find this type of effects in the experimental MSF shown in Figure IV.13a.

This type of analysis might be extended to the case of the "stripes" phase, for which we have a low statistics (only two images) and molecular lattices formed by ∼ 1, 000 and ∼ 500 fullerenes. Moreover, for this phase we did not observe the presence of emergent peaks but a structured intensity forming lines. We could wonder how this type of MSF is influenced by lattice size and statistics. However, these simulations are out of the scope of this short appendix.

Figure I. 1 :

 1 Figure I.1: (a) Wooden balls used by John Dalton to explain his atomic model (1810-1842). (b) Reproduction of drawing of atomic formulae by John Dalton. (c) Commercial ball models currently used for high school education. (d) Bernoulli's draw representing gas particles inside a cylinder subjected to an external pressure, published in his book Hydrodynamica (1738) [13]. For (a) and (b): copyright © The Board of Trustees of the Science Museum. Science Museum Group Collection

Figure

  Figure I.1a, b). He was probably the first one to use balls models connected by sticks to represent molecules, which is a representation currently used in every science museum and high-school class (Figure I.1c)! But chemistry was not the only branch of science in which the concept of atoms emerged.

Figure I. 3 :

 3 Figure I.3:Frenkel Kontorova model for atoms (balls connected by strings) adsorbed on a crystal (sinusoidal periodic potential). Commensurate phases with small order ("C", a abs = a subs ) and high order ("HC", 5a abs = 4a subs ). Incommensurate phases, a floating phase with the lattice parameter of the adsorbed atoms corresponding to the natural lattice parameter ("F", a abs = a 0 abs ) and the formation of solitons between commensurate regions (S). Chaotic phase in which adsorbed atoms are randomly distributed but always pinned on minima of the substrate potential. Figure adapted from[4] (for commensurate and incommensurate phases) and[START_REF] Bak | Commensurate phases, incommensurate phases and the devil's staircase[END_REF] (for chaotic phases).

Figure I. 4 :

 4 Figure I.4: Example of incommensurate structure with formation of domain walls, for Xe on graphite. (a) Scanning tunneling image of Xe on graphite (200 × 200nm 2 ). Dark regions correspond to commensurate regions, the white honeycomb structure is a lattice of incommensurate domain walls separating the commensurate regions. Bright spots are contaminations on the sample. (b) Shorter range scanning tunneling microscopy image (16 × 16nm 2 ) in which each Xe atom is visible as a small dot. The commensurate regions here appear bright, while dark regions forming hexagons are incommensurate domain walls (the contrast is the opposite to the one in (a)). (c) Schematics representing the positions of Xe atoms (darker circles) and carbon atoms in graphite (smaller, lighter circles). Three commensurate structures (A, B, C) exist and they are separated by incommensurate domain walls. All the images are adapted from [66].

Figure I. 5 :

 5 Figure I.5: Left: Examples of two-dimensional lattices, each node of the lattice represents an atomic or molecular position in a crystalline solid. Right: Examples of internal Ising degrees of freedom corresponding to three different physical systems: magnetic compounds, adsorption of gases on solid surfaces and formation of binary alloys. Each site in the lattices in the left panel is occupied by one Ising variable.

I. 6 :

 6 Figure I.6: Different examples of frustration in two-dimensional Ising systems. Ising variables are represented by black/white balls. Yellow circles indicate a site in which neither black, nor white, Ising variables can satisfy all the interactions present in the lattice. Ferromagnetic (FM) and antiferromagnetic (AF) interactions are indicated by dotted red/full blue lines, respectively. Left: two examples of frustration due to competing interactions. Right: archetype of frustration due to geometry in a triangular plaquette. Two lattices in two dimensions can be build combining triangles: the triangular lattice (triangular plaquettes sharing their edges) and the Kagomé lattice (triangular plaquettes sharing their corners). They are called triangular Ising antiferromagnet (TIAF) and Kagomé Ising antiferromagnet (KIAF), respectively.

Figure I. 7 :

 7 Figure I.7: Examples of periodically ordered phases according to the Frenkel-Kontorova model (blue balls on sinusoidal substrate) and the spin one-dimensional models (red/black arrows). Note that in these example Ising spins, which have a fixed value (σ = ±1) cannot form highly ordered commensurate phases like the one shown for the Frenkel-Kontorova chain in the top right part of the cartoon.

Figure II. 1 :

 1 Figure II.1: Details about the experimental set-up used in Grenoble. (a) Scheme of the UHV system including the growth chamber and the STM chamber that were used during this thesis. (b) Corresponding photograph of the experimental hall. (c) Photograph of the inside of the growth chamber picturing the different elements used to prepare C 60 on Cu(111) samples.

Figure II. 3 :

 3 Figure II.3: Steps to correct the piezoelectric hysteresis in our STM images. (a) Example of the forward and backward scan of an STM image. (b) Subtraction of the backward STM image to the forward one (left panel) and illustration of the hysteresis effect on the scan speed of the STM tip in the x-axis (right panel). (c) Evidence of the shift between forward and backward profiles on the STM image in (a) (top panel). Resulting profiles after correcting the shift (bottom panel). (d) Parabolic correction used to correct hysteresis on the STM image in (a) (top panel). Final profiles after correction (bottom panel).

Figure II. 4 :

 4 Figure II.4: Correction of the thermal drift in our STM images. Left panel: STM tip at room temperature derive from its initial position to a final one. Right panel: To correct this effect we use the C 60 lattice as reference and use global distortions of the STM images to find a perfect molecular triangular lattice.

Figure II. 5 :

 5 Figure II.5: Picture illustrating the working principle of the iterative loop in DFT, from the initial guess of the electronic structure to the computation of the physical quantities of interest.

Figure II. 7 :

 7 Figure II.7: Value of pairwise correlations with its nearest neighbors (C 1 ) for an Ising variable (blue circle with black edge) in different configurations.

Figure II. 8 :

 8 Figure II.8: Three examples of MSF calculation for different phases on the triangular lattice. Dotted hexagons delimit the first Brillouin zone.From left to right: ordered phase characterized by Bragg peaks with intensity diverging with the number of sites in the lattice, localized at specific positions in the q-space; disordered paramagnetic phase showing a very small intensity almost equally distributed on the q-space; disordered correlated phase exhibiting structured diffuse scattering (dark violet regions interconnecting the main peaks), while the main peaks' intensity remains finite (unlike Bragg peaks).

  pared and used as a measurement of the effective temperature of an experimental image. Two examples of this comparison are shown in Figure II.9.

Figure II. 9 :

 9 Figure II.9: Two examples of the comparison between experimental correlations (colored dots) and calculated correlations (colored curves). Left panel shows an example in which effective temperature can never be defined: the experimental correlation for the fifth neighbor (C exp 5 ), indicated by the yellow circle, is negative while the computed correlations (C M C

5

 5

Figure II. 10 :

 10 Figure II.10: Scheme summing up the main steps of a Monte Carlo simulation based on the Metropolis algorithm. The three main steps repeated for each value of the temperature are indicated by colored rectangles. Grey rectangle describes the loop that is repeated in each one of the dotted colored rectangles.

Figure II. 10 .

 10

Figure II. 11 :

 11 Figure II.11:Conceptual picture to explain the disconnection of different topological sectors in the ground state. The phase space includes the ground state that can be reached using local updates like within the Metropolis algorithm. The ground state is divided in different disconnected topological sectors, represented as colored circles, to explore both of them global updates are needed.

Figure III. 1 :

 1 Figure III.1: A few relevant inter-atomic distances in a C 60 molecule, and its symmetry axes, modified from [186].

Figure III. 2 :

 2 Figure III.2: STM images of fullerene molecules on different M(111) surfaces, together with cartoons of the molecular orientations. Three-dimensional schematics for the orientations corresponding to each molecular contrast are shown in the first rows. The STM contrast (second rows) roughly corresponds to the electronic density of states located in the pentagonal faces of the molecule. (a) Simulated STM contrast for C 60 molecules on Cd(0001) [223]. (b) Measured STM contrast for C 60 molecules on Au(111) [179]. (c) Experimental STM contrast for C 60 molecules on Au(111) [28].

Figure III. 3 :

 3 Figure III.3: (a) Schematics representing the close-packed surface of a M(111) with four different high-symmetry sites: top (C 3 symmetry), hcp (C 3 symmetry), fcc (C 3 symmetry) and bridge (C 2 symmetry). For each site the symmetry axis are indicated by colored lines. White/black/grey dots represent three layers of metal atoms. (b) One of the configurations with coincident symmetry axis for the fullerene (hexagonal face towards the surface) and the metal site (top site). (c) Another example of configuration with coincident symmetry axis for the fullerene (6-6 bond facing the surface) and the metal site (bridge).

Figure III. 5 :

 5 Figure III.5: Schematics of different vacancy islands in the metal surface (metal atoms represented with white/gray/black dots), where the molecules (represented by blue circles) can fit in more or less deeply, on side views (top row) and top views (bottom row). (a) Non-reconstructed metal surface. (b) Reconstructed surface with one-atom vacancy. (c) Reconstructed surface with seven-atom vacancy. Colored circles with red edge in (a, b, c) highlight metal atoms that can form strong bonds with the fullerene.

  and the commensurability condition is simply written a S = ia M = ma C 60 (see two examples in Figure III.6a, b). The twodimensional geometry allows a much broader variety of commensurabilities, which is a major source of complexity in the system. An example is shown in Figure III.6c.

Figure III. 6 :

 6 Figure III.6: (a, b) Commensurate lattices in one dimension (one lattice made of grey balls, the other of blue circles). In (a) the blue lattice coincides with the superlattice, which is not the case in (b). (c) Two-dimensional counterpart to (a). One unit cell of the blue lattice is shown.

  surfaces, shown in Figure III.8a,b. In the next section, we show that different commensurate phases exhibit different molecular contrasts in STM images. For example, looking at fullerenes assemblies on Cu(111) (Figure III.7) or on Au(111) and Ag(111) (Figure III.8) we can distinguish different phases by their contrast, without looking for the φ angle of each phase.

Figure III. 7 : 3 )

 73 Figure III.7: Commensurate phases of C 60 molecules on Cu[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] measured by STM, reported by Pai et al.[START_REF] Pai | Structural relaxation of adlayers in the presence of adsorbate-induced reconstruction: C 60 /Cu(111)[END_REF]. Each fullerene appears as one red/orange/white dot. The distance between molecules is ∼ 1nm in all the images, which correspond to different scales. Brightness corresponds to the apparent height of the molecules. (a) (1 × 1)R0°s uperstructure, showing a uniform height pattern (molecules form domains, in which all molecules either sit on top of seven-atom vacancies, or sit on the unreconstructed surface). (b) (2 × 2)R0°superstructure, with a periodic height pattern revealing that3 4 of the fullerenes sit on vacancies. (c) ( √ 7 × √ 7)R19.11°superstructure, with a periodic height pattern revealing that approximately6 7 of fullerenes sit on vacancies. (d) Another kind of (2×2)R0°superstructure, whose height pattern is non-ordered but reveals parallel lines of vacancies. This pattern is not tiled with the superstructure unit cell. Approximately one half of fullerenes sit on vacancies. (e) ( √ 3× √ 3)R30°superstructure, whose height pattern is non-ordered. Like for (d), the height pattern cannot be tiled with the superstructure unit cell and half of the fullerenes sit on vacancies. A more detailed description of the phases, using other notations is given in TableB.2, of Appendix B.

Figure III. 8 :

 8 Figure III.8: Coexistence of different commensurate phases of C 60 molecules on M(111). Black/white lines indicate direction of dense molecular rows, showing orientational ordering for different phases. Note that for each phase different height patterns, including uniform, periodic and non-ordered patterns, are observed. (a) STM image of fullerenes on Ag(111), reproduced from [72]. (b) STM image of fullerenes on Au(111), reproduced from [58]. (c) STM image of fullerenes on Cu(111), reproduced from [233].

  Figure III.7 and Figure III.8 show STM images revealing the vacancy lattices for fullerene lattices on different metal surfaces. Ordered and non-ordered height patterns are obviously observed (for example in Figure III.7b-e), while some lattices show a uniform height (such as the one in Figure III.7a). Why order or disorder develops, sometimes in coexistence on the same surface, remains an open question. It was suggested that intermolecular repulsions are responsible for the observed disorder

Figure III. 9 :

 9 Figure III.9: STM images of C 60 assemblies on Cu(111) deposited while keeping the substrate at high temperature. The small balls are individual molecules, the Cu(111) surface appears dark (lowest apparent height). Height profiles have been extracted along the dotted lines shown in the images. Yellow arrows indicate the position of domain walls. (a, b, c) The substrate was at ∼338 K during molecule evaporation, tunneling bias and current were -1 V, 0.8 nA. (b) inset: zoom of the molecular region marked with a white square, revealing distinct appearances of the molecules. (d) The substrate was at ∼363 K during molecule evaporation, tunneling bias and current were 0.5 V, 0.8 nA.

Figure III. 10 :

 10 Figure III.10: (a) STM image of molecular domains, one with molecules sitting on the unreconstructed Cu(111) surface (bright dots), the other with molecules sitting in vacancies (darker dots). (b) Perfect lattices superposed to the STM image to determine the relative lateral shifts between the grids. (c) Positions of copper atoms on the surface (white circles) and relative position of fullerenes within vacancies (blue) and on the unreconstructed surface (green). The image sums up the analysis of five different STM images. For each of them, the position of atoms within vacancies is taken as reference.

Figure III. 11 :

 11 Figure III.11: Two STM images showing phase coexistence on the same Cu(111) surface, when fullerenes are deposited at room temperature. Different phases show different orientation angles and height patterns.

Figure III. 13 :

 13 Figure III.13:Possible commensurate phases of fullerenes on Cu[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] depending on their orientation and supercell lattice parameter, considering strains < 4% and superstructure lattice constants < 30 Å. Black squares correspond to calculated phases, not observed experimentally. Colored circles correspond to calculated phases matching the experimental phases. Color indicates the strain in the molecular lattice. The two colored squares correspond to the two possible commensurate phases that could explain the orientation we observe at 8.4 ± 0.8°. Curves correspond to the relation between orientation and superstructure lattice parameter for incommensurate structures, for which the fullerenes layer can rotate continuously on the copper surface.

Figure III. 14 :

 14 Figure III.14: Schematic representation of each commensurate structure proposed in TableIII.2, putting in evidence the local positional coincidences between the molecules (represented with large circles) and the substrate atoms (gray circles). When molecules are black, the binding site do not correspond to a highly symmetric site.

Figure III. 15 :

 15 Figure III.15: (a,f) Details of same STM image showing phase coexistence. (b,g) Fourier transform of the STM images shown in (a,f).(c-e) and (h,i) STM images filtered for each component in the Fourier transform. (j) Radial scans across the peaks on Fourier transform marked with full and dotted circles in (b, g).

Figure III. 16 :

 16 Figure III.16: Same type of analysis than the one shown in Figure III.15. (a, f, l) Cropped regions of the STM image. (b, g, m) Fourier transform of the STM regions. (c-e); (h-k) and (n,o) STM images filtered for each component of the Fourier transform. (p) Radial scans crossing peaks on the Fourier transform, for each one of the components (marked with full and dotted circles in b,g,m).

Figure III. 17 :

 17 Figure III.17: Study of the apparent height exhibited by fullerenes on different commensurate phases. Numerical values are presented in Appendix B. (a) Statistical values for the apparent heights on STM measured for many different height profiles on different molecular regions, classified according to the orientation of the molecular phase. Heights can be divided in four categories, h 1 to h 4 . h 1 was taken as reference. Colorbar indicates the number of measurements used to compute the average and standard deviation of each height value. (b-f) High resolution STM images measured at low temperature, different heights h 1 to h 4 are indicated on the images. Correlation between the observed orbitals and the value of apparent heights are observed. Height patterns were identified and compared with the ones in phases with different orientations, colored frames indicate the correspondence with phases on graph (a).

Figure III. 18 :

 18 Figure III.18: Zoom on fullerene assemblies on Cu(111) in which different molecular heights are observed. For each region, STM in constant current mode (first column), current in constant height mode (second column) and the nc-AFM frequency shift also in constant height mode (third column) were measured at low temperature. Current and frequency images were simultaneously measured. Apparent heights h 1 -h 4 are indicated on some of the molecules on each STM image. (A-D) Correspond to different regions, in which local height patterns were similar to the ones observed at room temperature. Picture frames are colored according to the colors used for each phase shown in the previous figures.

Figure III. 19 :

 19 Figure III.19:Summary of the different molecular heights observed experimentally. We detect four different apparent heights in our STM images, that are a combination of three different molecular orientations and interface reconstruction. Orientations are top views of fullerenes on the surface. Interface schematics for molecules with height h 2 indicate that the height difference is partially due to topography, but we were not able to tell if the substrate is reconstructed or not.

  Figure III.20. The figure shows 16 commensurate phases in which seven-atom vacancies are supposed to form under each molecule.

Figure III. 20 :

 20 Figure III.20: Coincidence of fullerene molecules with copper atoms when local distortions of the commensurate structure and formation of seven-atom vacancies below each molecule are taken into account. Copper atoms are represented by gray circles; binding position of the molecules on the non distorted commensurate phase is marked by blue dots; distorted binding sites are marked in black. We plot the molecules inside the superstructure cell and their nearest neighbors. Copper atoms missing indicate the position of vacancies. Red hexagons have their corners on copper atoms with low coordinance number. These atoms form strong bonds with fullerenes. Colored frames indicate phases detected experimentally. Full frames correspond to phases in which every molecule sits on a vacancy. Dotted frames correspond to phase in which vacancies were not present under each molecule.

  10.22Å-0°10.09Å-5.4°1 0.09Å-16.4°1 0.14Å-19.1°9.68Å-22.4°9.68Å-22.4°9.98Å-26.3°1 0.33Å-30°10.33Å-30°1 0.33Å-8.2°9.80Å-9.3°1 0.04Å-10.9°10.18Å-14.4°10.09Å-16.4°F igure III.21: Coincidence model for commensurate phases observed experimentally.Formation of seven-atom vacancies and local distortions are considered. Copper atoms are sketched with gray balls; molecular positions on the non-distorted lattice are indicated by blue dots; black dots are molecular positions when the lattice is distorted. Fullerenes are represented by blue and yellow circles, corresponding to molecules sitting on vacancies or on non-reconstructed sites, respectively. The solid-line frames around each cartoon indicate agreement of the experimental and calculated height patterns, while the dotted frames are more complex cases in which periodic repetition of the unit cell cannot reproduce the experimental height patterns.

Figure III. 22 :

 22 Figure III.22: First row shows STM images at room temperature for two phases with periodic height contrasts: (a) 10.14 Å-19.1°and (b) 10.09 Å-16.4°phases. On top of each image we superpose the expected positions of fullerenes in the case of a commensurate phase without (left column) and with (right column) molecular in-plane displacements. Corresponding models are shown in the second row, with non-shifted and shifted molecules appearing as yellow and blue disks respectively.

Figure III. 23 :

 23 Figure III.23: Evidence of a shift of the molecular cells and formation of domain walls. (a) STM image of the 10.33 Å-30°phase. Domain walls corresponding to larger intermolecular distances are marked by red dotted lines. (b) Height profile perpendicular to the domain walls corresponding to the white dotted line in (a). (c) Coincidence model for the experimental observation, including the formation of domain walls and the reconstruction of the substrate. Gray balls are copper atoms; blue dots positions of fullerenes without considering in-plane displacement; black dots molecular positions considering global shift; blue and yellow circles are molecules on reconstructed and non-reconstructed sites. The red dotted lines mark the position of the DW; the red arrows are the longer d 2 distance.

Figure III. 24 :

 24 Figure III.24: Summary of the commensurate structures for the most frequently observed experimental phases, including all the degrees of freedom we analyze: coincidence with the substrate and nature of binding sites; local in-plane displacements; formation of seven-atom vacancies and molecular orientation. (A) Color legend for binding sites on the Cu(111) surface, top sites are on the corners of the copper unit cell, bridge sites are medium size circles, hollow sites small circles. (B-F) Complete model with molecular displacement, formation of seven-atom vacancies and orientation of each fullerene. Only the bottom part of fullerenes are shown. (C) Phase appears on both configurations shown in the cartoons. (E) Phase is a mixture of the cell shown here, rotated by 0°, 60°and 120°. (F) Phase is a mixture of the two cells shown in the schematics, combined in a disordered non trivial way.

  4 we show the binding energy for different configurations. The corresponding molecular height z of the molecule is also shown, considering z = 0 for copper atoms at the topmost surface layer, away from the molecule. In Figure III.25b,d we plot the curves interaction energies vs the molecular height (E C60-Cu (z)).

  Figure III.26a we show the conductance curve.

Figure III. 25 :

 25 Figure III.25: Binding energy of fullerenes on Cu(111) computed by DFT for different molecular configurations. (a) Cartoon indicating binding sites (colored), structure of vacancies (black) and molecular orientations. (b) Binding energy with respect to the molecular height z for different configurations in the case of a single fullerene on the Cu(111) surface. (c) Same calculation for the case of a (4 × 4)R0°commensurate phase. (d) Binding energy with respect to the molecular height z for fullerenes sitting on different types of vacancies.

Figure III. 26 :

 26 Figure III.26: (a) Conductance between the STM tip and fullerenes on a copper surface, calculated by DFT for different molecular configurations. (b) Detail of the conductance curves in (a). c-e, from left to right: Scheme, DFT simulations of STM images and experimental STM image of different molecular configurations. (c) Apex orientation on a top site. (d) Hexagon orientation on a top site. (e) Pentagon orientation on a seven atom vacancy.

Figure III. 27 :

 27 Figure III.27: Intermolecular interaction between C 60 molecules, computed by DFT and including van der Waals interactions. (a) Intermolecular energy between fullerenes in a periodic two-dimensional lattice (colored dots and black curve). The energy was normalized by the number of molecules and represented with respect to the molecular lattice parameter a C60 . Color scale corresponds to the strain, computed with respect to an equilibrium distance of 10.05 Å. Red and blue curves are the data shown in (b), multiplied by 6 to account for the interaction between 6 nearest neighbors. (b) Intermolecular interaction between two fullerenes separated by a distance r C60 , computed using time dependent DFT [149] or a perturbation theory and DFT (based on [34] calculation). (c) Evolution of the intermolecular energy of a two-dimensional lattice of fullerenes when the central molecule in the inset of (b) is displaced along the z-axis (height coordinate z 60 ). Different curves correspond to molecular layers with different lattice parameters, from the equilibrium one to a compression of 3.7%.

Figure III. 28 .

 28 The commensurate phases we consider are shown in FigureIII.[START_REF] Bernoulli | Hydrodynamica: sive de viribus et motibus fluidorum commentarii (Sumptibus Johannis Reinholdi Dulseckeri[END_REF]. All these phases have supercell lattice parameters a S < 30 Å and biaxial strain ε < 4%.

  we use a cubic interpolation to compute the two-dimensional potential U coinc-2D , shown in Figure III.29.

Figure III. 29 :

 29 Figure III.29: Two-dimensional potential U coinc-2D that gives the coincidence energy (or binding energy) for fullerenes on different sites in the Cu(111) unit cell. Each binding site has a characteristic energy calculated by DFT and between sites a cubic interpolation is used to deduce the energy value. This map gives the values of the function U coinc-2D that depends on the molecular coordinates with respect to copper (x Cu i , y Cu i , z Cu i ). It is used as input to compute the energy of each commensurate phase. Two cases are considered, the first one corresponding to fullerenes on a non-reconstructed surface (z i ∼ 1.2 Å. The second one corresponds to fullerenes forming seven-atom vacancies on the top sites (z i = -0.1 Å).

Figure III. 30 :

 30 Figure III.30: Schematic representation of electric dipoles due to charge transfer from the copper surface (white, gray, black balls) to the fullerenes (blue balls). The notations for the dipoles, the distance and the angles between them are shown in the left sketch.

  Figure III.31 we show our results for the calculation of the internal energy of each commensurate phase, not taking into account (Figure III.31a) and taking into account (Figure III.31b) the dipolar interaction.

  68 Å-22.4°, indicated by yellow arrows in Figure III.31.

Figure III. 31 :

 31 Figure III.31: Energy landscape considering the internal energy of each commensurate phase, in two different cases: 1 -when neither in-plane distortions nor reconstruction of the copper surface are considered (light symbols); 2 -when local in-plane distortions and formation of vacancies are taken into account (dark symbols). The color of the circles indicates the size of the molecular supercell, their size being related to the strain in the molecular layer. Star symbols highlight the phases that have been experimentally observed. The two experimental phases signaled by yellow arrows are non-periodic, so their energy is expected to be lower than the one predicted by our model (as explained in the main text). (a) Only coincidence energy with the substrate and intermolecular interaction due to van der Waals and chemical repulsion are considered. (b) Dipolar interaction between molecules is included in the calculation.

Figure III. 32 :

 32 Figure III.32: Structural parameters for commensurate phases of fullerenes on different M(111) surfaces with different lattice parameters. Blue circles indicate the phases that have been reported in the literature, dotted symbols indicate that height patterns in these phases are not ordered. We only represent phases satisfying |ε| < 4% and a S < 30 Å.

Figure III. 33 :

 33 Figure III.33: Binding energies for fullerenes on different M(111) surfaces, calculated by DFT. Height corresponds to the distance between the bottom of a fullerene and the metal surface. Full symbols indicate non-reconstructed configurations, with C 60 molecules sitting on a top site with a hexagon facing down. Empty symbols correspond to reconstructed configurations in which fullerenes form a vacancy on the metal surface. For Cu[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] and Ni[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF], seven-atom vacancies were considered, and one-atom vacancies for other metals.

First, our review

  of the literature shows that phase coexistence exists for several M[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] substrates. Second, we revisited the structure of different commensurate phases and the typical energy scales for molecule-substrate and molecule-molecule interactions.

Figure III. 34 :

 34 Figure III.34: Commensurate phases of fullerenes on different M(111) surfaces, classified according to the observed height patterns in STM. Different symbols correspond to different types of height patterns. The color of the symbol indicates the strain in the molecular layers. Colored pockets on the diagrams mark regions where phases with same type of height patterns are found. The two-colored pockets correspond to the presence of phases with different height patterns in a certain region of the diagram. (a) Competition between the binding energy with the substrate (metal substrate corresponding to each energy is indicated on the right of the figure) and steric effects hindering the formation of vacancies. (b) Competition between the binding energy and the lattice parameter, i.e. intermolecular interaction on each phase. The vertical dotted line corresponds to the value of the equilibrium lattice parameter for a C 60 layer.

Figure IV. 1 :

 1 Figure IV.1: Schematics of the geometrical frustration on the triangular Ising antiferromagnet. Colored dots correspond to bivalued variables, i.e. Ising variables. Six degenerate configuration with energy E = J 1 exist. All of them have one frustrated (red) and two satisified (green) bonds per triangle.

Figure IV. 2 :

 2 Figure IV.2: Schematic representation of a pinch point in two-dimensions. From left to right: pinch point intensity in the k-space (in yellow); profile parallel to the pinch point showing constant intensity; profile perpendicular to the pinch point showing nonanalytical intensity.

Figure IV. 3 :

 3 Figure IV.3: Examples of frustrated artificial systems that could be realizations of the TIAF model with J 1 coupling. (a,b) Ferromagnetic disks arranged on a triangular lattice, exhibiting out-of-plane magnetization. Each cluster is formed by seven disks, each one identified as an Ising pseudo-spin up (white) or down (black). Adapted from [126]. (c) Triangular lattice of nanomagnets, similar to the ones in (a,b). Black/white contrast reveals magnetization of the disks, that can be compared with up and down spins. A disordered pattern of up/down Ising pseudo-spins is observed. Adapted from [95]. (d) Colloidal particles trapped between two plates. Each dot is one colloidal sphere, contrast corresponds to their position (bright/dark = up/down). The phase looks disordered. Adapted from [70].

Figure IV. 4 :

 4 Figure IV.4: Microscope images of binary alloys proposed as experimental realizations of the TIAF model with J 1 coupling. In all the cases the contrast corresponds to atoms of different chemical elements. (a) STM image of Pb on Si[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]. On the bottom right region of the image a complex bright (Pb atoms) and dark (Si atoms) pattern is observed. Adapted from[START_REF] Ganz | Submonolayer phases of Pb on Si(111)[END_REF]. (b) Ge 0.37 Pb 0.63 /Ge(111) alloy. A disordered pattern with formation of local antiferromagnetic-like stripes is observed by STM. Adapted from[START_REF] Carpinneli | Charge rearrangement in the Ge x Pb 1-x /Ge(111) interface[END_REF]. (c) STM image of an Sn 0.51 Si 0.49 /Si(111) alloy, exhibiting a disordered contrast with some antiferromagnetic-like stripes. Adapted from[START_REF] Ottaviano | Short-range order in two-dimensional binary alloys[END_REF]. (d) Re 0.5 Nb 0.5 S 2 observed by darkfield scanning transmission electron microscope, exhibiting disordered contrast. Adapted from[START_REF] Azizi | Frustration and atomic ordering in a monolayer semiconductor alloy[END_REF].

Figure IV. 5 :

 5 Figure IV.5: STM images of organic molecules on inorganic surfaces exhibiting disordered patterns. (a) Hexakis-11-alkyloxytriphenylene on highly oriented pyrolitic graphite, each molecule is one of the structures with three branches at 120°. The inset shows the two possible orientations for the molecules, that are identified as an Ising degree of freedom. Adapted from [29]. (b) C 60 molecules on Cu(111) surface, each molecule is one dot in the image. The contrast corresponds to the apparent height of the molecules, which was proposed to be a bivalued degree of freedom. Adapted from [150]. (c) C 60 molecules on Au(111) surface, exhibiting two different contrasts linked to the height of the molecules. Adapted from [155].

1. 2 . 1 Figure IV. 6 :

 216 Figure IV.6: Schematics of different ground states for the Ising spin Hamiltonian on a triangular lattice, including J 1 , J 2 and h ext . All the phases exist in the regime J 1 < 0, J 2 < 0 for different ratios of the couplings and different intensities of the magnetic field.Phases I-VB are adapted from[START_REF] Metcalf | Ground state spin orderings of the triangular Ising model[END_REF], phase C is adapted from[START_REF] Tanaka | Ground state spin configurations of the triangular Ising net with the nearest and next nearest neighbor interactions[END_REF]. The most studied phases have been named after their magnetic contrast or the periodicity of the cell.

Figure IV. 7 :

 7 Figure IV.7: Schematics of different phases for the Ising spin Hamiltonian on a triangular lattice, including J 1 , J 2 and J 3 , as proposed by Tanaka and Uryu [202]. Black frame pictures indicate ground states, the other phases correspond to degenerate phases existing in the isoenergetic lines between different regions on the phase diagram. Both ferromagnetic and antiferromagnetic couplings are considered. Phases of interest for our discussion were named after their magnetic contrast or cell periodicity.

2 .

 2 Magnetic order was also characterized by magnetic structure factors (Figure IV.8b-d). Ordered ground states exhibit Bragg peaks in k-space (Figure IV.8b,d).

  Figure IV.11 correspond to values of the couplings for which MC simulations were performed and are fully consistent with the phase diagram at zero temperature. For some values in the zigzag region (open circles in Figure IV.11) MC simulations were unable to bifurcate and find an ordered ground state, and we observe a coexistence of different zigzag domains at low temperature. This behavior corresponds to a dynamical freezing, mentioned earlier when describing the difference between local and global updates in MC simulations. It illustrates one limit of our Metropolis algorithm.

Figure IV. 11 :

 11 Figure IV.11: Central panel: Phase diagram for J 1 = -1 and antiferromagnetic J 2 and J 3 at zero temperature. Colored regions correspond to three ordered ground states, dotted lines to isoenergetic lines between phases. Full colored circles indicate points in which we verified the ground state using Monte Carlo simulations. Empty dotted circles indicate values of the couplings for which the MC simulations were unable to reach the expected ground states because of dynamical freezing. Insets: MC snapshots at low temperature for different coupling values. Insets with a full edge show the three ordered ground states found in the phase diagram. Insets with dotted edges are snapshots on coexistence regions at the values of isoenergetic lines.

Figure IV. 12 :

 12 Figure IV.12: (a) STM image of a typical C 60 island on a Cu(111) surface for the 9.68 Å-22.4°phase. Fullerenes appear as bright yellow/dark blue dots depending on their apparent height. Black background is the copper surface. The island is formed by ∼ 1, 000 molecules. Scale bar is 10 nm. (b) Detail of the image (a) showing the close-packed structure of the molecular lattice. A triangular lattice has been superposed to the STM image. (c) Intensity scan along the dashed line shown in (a). Two molecular heights are observed for the fullerenes. (d) Height histogram measured for 4 different STM images, showing two main heights (in agreement with profile in (c)) and a third much less frequent height. (e) Histogram of nearest neighbor molecules having different STM contrast (in black), measured on 10 STM images. If the height of the fullerenes was randomly ordered we would find the gray histogram. The insets shown in the top of the histogram are examples of configurations with zero to six distinct nearest neighbors.

  2 for details) averaged over 10 different STM images (Figure IV.13a).

Figure IV. 13 :

 13 Figure IV.13: (a) Magnetic structure factor for the experimental molecular lattice, averaged over 10 STM images. (b) Magnetic structure factor for the TIAF with antiferromagnetic J 1 at low temperature, computed by MC simulations. (c) Magnetic structure factor for the TIAF with J 1 , J 2 and J 3 antiferromagnetic at low temperature in the zigzag region, computed by MC simulations. The inset shows the order corresponding to this zigzag phase. Colorbar is the same for the three MSF.

Figure IV. 14 : 3 J 2 ratio (fixed J 2 J 1

 143221 Figure IV.14: Comparison between experimental and simulated values for the second, C 2 , and third, C 3 , correlations depending on (a) J 2 J 1 ratio (fixed J 3 J 2 = 0.57); (b) J 3 J 2 ratio (fixed J 2 J 1 = 0.1). Dashed black lines and colored regions indicate the experimental measured value averaged over 10 STM images with its standard deviation. Colored dots are MC values for the correlations, min(C 2 (T ≤ J 1 )) and max(C 3 (T ≤ J 1 )), at low temperature with their standard deviation.

Figure IV. 15 :

 15 Figure IV.15: (a) Notation for first to fifteenth neighbors in the triangular lattice. (b) First to fifth correlation values depending on temperature calculated by MC (colored curves, model parameters J 1 =-1, J 2 =-0.1, J 3 =-0.057) compared to experimental values (colored dots). (c) Comparison of simulated (colored rectangles) and measured (dots) correlations from first to fifteenth neighbor when an effective temperature T ∼ 0.1 is defined. (d) MSF calculated by MC for the model with J 1 =-1, J 2 =-0.1, J 3 =-0.057 at T ∼ 0.1, to compare with Figure IV.13a.

Figure IV. 16 :

 16 Figure IV.16: Detail on the defects breaking the ice rule, i.e. three "low" C 60 molecules on a triangular plaquette, in the 9.68 Å-22.4°molecular phase. Left bottom panel: STM image of the phase. White lines indicate the molecular profiles through defects. Top insets: Zoom-in on defects in the phase, we systematically observe a black region at the center of a triangular plaquette formed by three "low" fullerenes. Right bottom panel: Height profile through one defect, the two central molecules corresponding to two fullerenes forming the defect. Distance notation corresponds to the statistical analysis of inter-molecular distances presented in Table IV.1.

  Figure IV.11 and Figure IV.8).

Figure IV. 17 :

 17 Figure IV.17: (a, b) STM images of a typical C 60 island on a Cu(111) surface for the 9.98 Å-26.3°phase or stripe phase. Molecules appear as bright yellow/dark blue dots depending on their apparent height. Scale bars are 5 nm. (c) Zoom in STM image (b) showing the closed-packed nature of the molecular arrangement. The triangular lattice on top of which the molecules sit is highlighted in white. (d) Intensity scan along the dashed line shown in (a). Two molecular heights are observed for the molecules. (e) Height histogram measured for the two STM images, showing two preferred heights (in agreement with profile in (d)) and a third, minority component. (f) Histogram of nearest neighbor molecules having distinct STM contrast (in black), measured on 2 STM images. The histogram expected for a randomly disordered arrangement of molecular heights is shown in gray for comparison.

Figure IV. 18 :

 18 Figure IV.18: (a) Experimental MSF averaged for the two STM images presented in Figure IV.17. (b) MSF for a stripe phase at low temperature calculated by MC simulations. (c) MSF for a stripe phase at finite temperature, computed by MC simulations for a model including antiferromagnetic J 1 and J 2 .

Figure IV. 19 :

 19 Figure IV.19: (a) Experimental MSF averaged for the two STM images presented in Figure IV.17. (b) MSF for a stripe phase at low temperature calculated by MC simulations, including antiferromagnetic J 3 coupling ( J 3 J 2 = 0.5). (c) MSF for a stripe phase at finite temperature, for the same parameters as in (b).

  .18) or positive (see Figure IV.20).

Figure IV. 20 :

 20 Figure IV.20: (a) Experimental MSF averaged for the two STM images presented in Figure IV.17. (b) MSF for a stripe phase at low temperature calculated by MC simulations, including ferromagnetic J 3 coupling ( J 3 J 2 = -0.5). (c) MSF for a stripe phase at finite temperature, for the same parameters than (b).

Figure IV. 21 :

 21 Figure IV.21: Residual (SSR, full circles) and weighted residual (wSSR, empty circles) of the fit of experimental correlation values to simulated correlations, for different couplings in the TIAF Hamiltonian. Dotted lines connect points for wSSR and are a guide to the eye. (a) Evolution of SSR and wSSR with the J 2J 1 ratio, for three fixed J 3 J 2 = 0, 0.25, 0.5 ratios. (b) Evolution of SSR and wSSR with the J 3 J 2 ratio, for three fixed J 2 J 1 = 0.2, 0.3, 0.4 ratios corresponding to the region in which SSR and wSSR are minimal in (a).

Figure IV. 21 . 2 J 1

 2121 At first we varied the value of J and plot the SSR and wSSR for each MC simulation (Figure IV.21a).

Figure IV. 22 :

 22 Figure IV.22: Comparison of experimental and simulated correlations for the TIAF with J 1 = -1, J 2 = -0.3, J 3 = -0.15, in both real and reciprocal space. (a) First to fifth correlation values depending on temperature calculated by MC (colored curves) compared to experimental values (colored dots). (b) Comparison of simulated (colored rectangles) and measured (colored dots) correlations from first to fifteenth neighbor when an effective temperature T ∼ 0.3 is defined. (c) MSF calculated from MC simulations for the model with J 1 = -1, J 2 = -0.3, J 3 = -0.15 at T ∼ 0.3, to compare with Figure IV.18a.

  Figure IV.22 we show this comparison. In Figure IV.22a the temperature variations of the simulated five first correlations are represented by colored curves. These curves were calculated for values of the couplings J 2

Finally

  , in Figure IV.22c we show the simulated MSF for J 1 = -1, J 2 = -0.3, J 3 = -0.15 and T ∼ 0.3, for comparison with the experimental one shown in Figure IV.18a. Intensity is structured in a very similar way in both MSFs, except the intensity at the gamma point in the experimental case. Note that the color-scale is the same in both graphs.

Figure IV. 24 :

 24 Figure IV.24: (a-c) Different types of DWs in the stripe phases, proposed by Korshunov [101]. (d) Boltzmann's factors for different types of DWs, considering different J 3J 2 ratios and the effective temperature T ef f that best fit the experimental data for each J 3 J 2 ratio.

Figure IV. 25 :

 25 Figure IV.25: (a, b) Reconstruction of the STM images shown in Figure IV.17 for the 9.68 Å-22.4°phase. Each molecule is considered as an Ising variable and the image is binarized to easily visualize the stripe pattern. Red lines indicate the presence of domain walls between stripe domains with different orientations. (c) Schematics showing the three most frequent domain wall types in the experimental images.

  In Figure IV.24d we plot the evolution of the Boltzmann factor for each structure proposed by Korshunov and also for the triple DW (shown in Figure IV.25), as a function of the J 3

Figure IV. 26 :

 26 Figure IV.26: First row: STM images of different C 60 on Cu(111) phases with different height patterns. Second row: Comparison with snapshot configurations from Monte Carlo simulations obtained for different model parameters. (a) J 1 = -1, J 2 = -0.1, J 3 = -0.057 at T = 0.1; (b) J 1 = -1, J 2 = -0.3, J 3 = -0.15 at T = 0.3; (c) J 1 = -1, J 2 = -0.16, J 3 = -0.10, h ext = -10 in the ground state; (d) J 1 = -1, J 2 = -0.16, J 3 = -0.07, h ext = -5 in the ground state; (e) J 1 = -1, J 2 = -0.16, J 3 = -0.10, h ext = -7 in the ground state.

Figure IV. 27 :

 27 Figure IV.27: Phase diagram at zero temperature of the TIAF with J 1 = -1, J 2 = -0.16 and varying J 3 and h ext . Each inset represents an ordered ground state found in the phase diagram. The colors of the cartoon frames correspond to the colors in the phase diagram. The GSs corresponding to height patterns observed in C 60 on Cu(111) system have frames with full line and their names are shown. A Python code in which the energies of all the ordered phases we considered are defined is presented in Appendix D.

  Figure IV.27 the phase diagram for J 1 = -1, J 2 = -0.16, when J 3 and h ext vary, is shown. This phase diagram was build following the same procedure we used for Figure IV.11: we Table

Figure IV. 28 :

 28 Figure IV.28: Schematic summary of the link between actual microscopic interactions in the experimental systems, C 60 assemblies on M(111) surfaces, and the effective magnetic couplings on the TIAF model, as they are proposed in subsection 5.1.

Figure IV. 29 :

 29 Figure IV.29: STM images of commensurate phases of C 60 molecules on different closepacked metal surfaces. Fullerenes correspond to each dot shown in the images (of different sizes). Ag(111): coexistence of three different phases, characterized by their orientation and their height pattern. On the top right of the image the height pattern is disordered, it is uniform on the top left and quasi-periodic on the bottom of the image. Image adapted from [72]. Al(111): Periodic phase of fullerenes with √ 3 × √ 3 periodicity. Low molecules appear as much smaller dark dots in the image (yellow circle indicates one of these "low" fullerenes). Adapted from [121]. Au(111): Coexistence of three different phases with different orientations and disordered (bottom left), quasi-periodic with 7 × 7 periodicity (bottom right) and uniform (top) height patterns. Adapted from [203]. Ni(111): coexistence of a uniform phase (top left) and a periodic phase with √ 3× √ 3 periodicity (bottom right).Adapted from[START_REF]Observation of a Metastable Honeycomb Arrangement of C 60 on Ni(111) with (7 × 7) Periodicity: Tailoring an Interface for Organic Spintronics[END_REF]. Pt(111): high resolution image of a uniform phase, in which coexistence of several molecular orientations is observed. Adapted from[START_REF] Liu | Molecular orientations and interfacial structure of C 60 on Pt(111)[END_REF].

C

  60 molecules on Au(111) form five different phases, including uniform, quasi-periodic (Figure IV.29) and disordered height patterns. Coexistence of phases indicates that different mechanisms could play a role in the stabilization of the phases, like in the case of fullerenes on Cu

Figure V. 1 :

 1 Figure V.1: MSF calculated with MC simulations for the TIAF model with different coupling and external magnetic field values. The name of each ground state is indicated and the associated real space configurations are shown next to each name. In all the cases the MSF are calculated at finite temperatures: Spin liquid: T ∼ J 1 ; Stripes: T ∼ |J 2 |; Zigzag: T ∼ |J 2 |; Stripes-zigzag: T ∼ |J 2 |; √ 3 × √ 3: T ∼ |J 1 |; Ferromagnetic: T ∼ 4h ext .

  Figure V.2a we show one STM image of a Sn 0.51 Si 0.49 /Si(111) sample. Clearly, the phase exhibits disorder in real space. The MSF of the experimental image (Figure V.2b) shows a diffuse but structured intensity, located at the same positions as those for one of the spin liquid phase (Figure V.1a). Thus, we consider an Ising spin Hamiltonian with J 1 = -1 and compute the correlations from the first to the fifteenth neighbors using MC simulations. The comparison between experimental and simulated correlations is reported in Figure V.2c,d.

Figure V. 2 :

 2 Figure V.2: (a) STM image of a Sn 0.51 Si 0.49 /Si(111) alloy. Adapted from [148]. (b) MSF of the STM image. (c,d) Comparison of the experimental atom-atom correlations in (a) (colored dots) with the MC simulated correlations for a short range TIAF Hamiltonian (J 1 = -1) (colored lines and rectangles). In (c) simulated correlations are plotted as a function of temperature. In (d) the temperature is fixed to T ∼ 3|J 1 |. The agreement between experimental and simulated correlations at this effective temperature gives residuals SSR= 0.006 and wSSR= 0.58.

Figure V. 3 :

 3 Figure V.3: (a) Re 0.5 Nb 0.5 S 2 observed by dark-field scanning transmission electron microscopy. Adapted from [7]. (b) MSF of the STM image. (c,d) Comparison of the experimental atom-atom correlations in (a) (colored dots) with the MC simulated correlations for a short range TIAF Hamiltonian (J 1 = -1) (colored lines and rectangles). In (c) simulated correlations are plotted as a function of temperature. In (d) the temperature is fixed to T ∼ 2|J 1 |. The agreement between experimental and simulated correlations at this effective temperature gives residuals SSR= 0.008 and wSSR= 0.60.

Figure V. 4 :

 4 Figure V.4: (a) STM image of a Ge 0.37 Pb 0.63 /Ge(111) alloy. Adapted from [24]. (b) MSF of the STM image. (c,d) Comparison of the experimental atom-atom correlations in (a) (colored dots) with the MC simulated correlations for a short range TIAF Hamiltonian under an applied field (J 1 = -1, h ext = 2) (colored lines and rectangles). In (c) simulated correlations are plotted as a function of temperature. In (d) the temperature is fixed to T ∼ 2.5|J 1 | ∼ h ext . The agreement between experimental and simulated correlations at this effective temperature gives residuals SSR= 0.006 and wSSR= 0.56.

4. 1

 1 Figure V.5: (a) STM image of fullerenes on a Au(111) substrate after annealing at 573K< T < 673K. Adapted from [58]. (b) MSF of the STM image. (c,d) Comparison of the experimental pairwise correlations in (a) (colored dots) with the MC simulated correlations for a short range TIAF Hamiltonian (J 1 = -1) (colored lines and rectangles). In (c) simulated correlations are plotted as a function of temperature. In (d) the temperature is fixed to T ∼ 26|J 1 |. The agreement between experimental and simulated correlations at this effective temperature gives residuals SSR= 0.003 and wSSR= 0.50.

4. 2 From a paramagnet to an ordered √ 3 × √ 3 phase

 233 We now address another phase of C 60 on Au[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] [START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF]. Like in the example shown in Figure V.5 the observed height pattern exhibits disorder. However, the fraction of "high" and "low" molecules is different, which depends on the annealing conditions after sample growth[START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF][START_REF] Paßens | Enhanced fullerene-Au(111) coupling in (2 √ 3 × 2 √ 3)R30°superstructures with intermolecular interactions[END_REF]. In the STM image we analyze, shown in Figure V.6a, we count ∼ 1, 000 fullerenes, 58% of them are "low" and 42% are "high". The associated MSF (Figure V.6b) exhibits a uniform background in which a peak at the gamma point is observed. The intensity located at the gamma point could correspond to the correlations of a ferromagnetic phase or a √ 3 × √ 3 phase. Thus we consider a TIAF Hamiltonian with antiferromagnetic coupling J 1 = -1 and different values of the external field h ext = 1, 2, 3, 4, 5. In each case we compare the

Figure V. 7 :

 7 Figure V.7: (a) STM image of fullerenes on a Ag(111) substrate after annealing at T = 673K for several minutes. Adapted from [163]. (b) MSF for two STM images published in [163], the one in (a) and another one (not shown). (c,d ) Comparison of the experimental pairwise correlations in (a) (colored dots) with the MC simulated correlations for a TIAF Hamiltonian with J 1 = -1; J 2 = 0, J 3 = 0 and h ext = 10 (colored lines and rectangles). In (c) simulated correlations are plotted as a function of temperature. In (d) the temperature is fixed to T ∼ 29|J 1 | ∼ 3h ext . The agreement between experimental and simulated correlations at this effective temperature gives residuals SSR= 0.007 and wSSR= 0.48.

Figure V. 8 :

 8 Figure V.8: (a-top row) STM image of fullerenes on a Cu(111) substrate; (a-bottom row) and MC snapshot of a simulation considering the TIAF with antiferromagnetic J 1 only. Adapted from [150]. (b) MSF of the corresponding images in (a). (c,d) Comparison of the images pairwise correlations in (a) (colored dots) with the MC simulated correlations for a TIAF Hamiltonian (colored lines and rectangles) with (top row) J 1 = -1; J 2 = -0.1, J 3 = -0.057 and (bottom row) J 1 = -1. In (c) simulated correlations are plotted as a function of temperature. In (d) the temperature is fixed to (top row) T ∼ 0.1|J 1 | ∼ J 2 and (bottom row) T ∼ |J 1 |. Best agreement between the spatial correlations extracted from the images and the corresponding values derived from (our own) sets of MC-configurations as function of temperature gives residuals (top row) SSR= 0.008 and wSSR= 0.43 (without considering the C 1 ); and (bottom row) SSR= 0.01 and wSSR= 0.60.

Figure V. 9 :

 9 Figure V.9: (a) STM image of fullerenes on a Cu(111) substrate. Adapted from [150]. (b) MSF of the STM image. (c,d) Comparison of the experimental molecular heightheight correlations in (a) (colored dots) with the MC simulated correlations for a TIAF Hamiltonian with J 1 = -1; J 2 = -0.3, J 3 = -0.1275 (colored lines and rectangles), which are consistent with the analysis of our samples (section 3 in chapter IV). In (c) simulated correlations are plotted as a function of temperature. In (d) the temperature is fixed toT ∼ 0.4|J 1 | ∼ J 2 .The agreement between experimental and simulated correlations at this effective temperature gives residuals SSR= 0.01 and wSSR= 0.17 (without considering the C 1 correlation, which is clearly at a different effective temperature).

  [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] including vacancies, local distortions and molecular orientations . . . . . . III.[START_REF] Castelnovo | Magnetic monopoles in spin ice[END_REF] DFT results for binding energies of fullerenes on Cu[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] . . . . . . . . . . III.26 Conductance and STM contrast calculated by DFT for C 60 molecules on Cu(111) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.27 Intermolecular interaction between C 60 molecules, computed by DFT and including van der Waals interactions . . . . . . . . . . . . . . . . . . . . . . III.28 Flowchart of the coincidence model used to compute the energy of each commensurate phase of C 60 on Cu(111) . . . . . . . . . . . . . . . . . . . . III.29 Two-dimensional potential giving the binding energy for fullerenes on different sites in the Cu(111) unit cell . . . . . . . . . . . . . . . . . . . . . III.30 Schematic representation of electric dipoles due to charge transfer from the copper surface to the fullerenes . . . . . . . . . . . . . . . . . . . . . . . . III.31 Energy landscape considering the internal energy of each commensurate phase formed by C 60 molecules on Cu(111) . . . . . . . . . . . . . . . . . . III.32 Geometry of commensurate phases of fullerenes on different close-packed metal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.33 Binding energies for fullerenes on different close-packed metal surfaces, calculated by DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.34 Height patterns, molecular lattices and orientations for different commensurate phases of fullerenes on close-packed metal surfaces . . . . . IV.1 Geometrical frustration in triangular plaquettes . . . . . . . . . . . . . . . IV.2 Schematic for two-dimensional pinch points . . . . . . . . . . . . . . . . . . IV.3 Experimental realizations of the TIAF model reported in artificial frustrated systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.4 Experimental realizations of the TIAF model reported in atomic binary alloys142 IV.5 Experimental realizations of the TIAF model reported in assemblies of organic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.6 Ground states of the TIAF model including J 1 , J 2 and h ext . . . . . . . . . IV.7 Ground states of the TIAF model including J 1 , J 2 and J 3 . . . . . . . . . . IV.8 Phase diagram of the TIAF model including antiferromagnetic J 1 , J 2 and J 3 , in the J 1 → ∞ limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.9 Ground states of the TIAF model including J 1 , J 2 , J 3 and h ext . . . . . . . IV.10 Method to compute the energy of an ordered ground state of the TIAF . . IV.11 Phase diagram for J 1 = -1 and antiferromagnetic J 2 and J 3 , at zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.12 Ingredients of the TIAF model in a fullerenes assembly on Cu(111), for the disordered phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.13 Experimental MSF for C 60 on Cu(111) disordered phase compared to simulated MSF for spin liquid and zigzag phases . . . . . . . . . . . . . . . IV.14 Second and third correlation values for experimental C 60 /Cu(111) disordered phase compared to simulated values for the TIAF model . . . . IV.15 Comparison of pairwise correlations and MSF, for experimental C 60 /Cu(111) disordered phase and simulated values for the TIAF model at fixed temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.16 Analysis of quenched defects in the C 60 /Cu(111) disordered phase . . . . . IV.17Ingredients of the TIAF model in a fullerene assembly on Cu(111), for the stripe phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.18 Experimental MSF for C 60 on Cu(111) stripe phase compared to simulated MSF for stripe phase with J 3 = 0 . . . . . . . . . . . . . . . . . . . . . . . IV.19 Experimental MSF for C 60 on Cu(111) stripe phase compared to simulated MSF for stripe phase with J 3 < 0 . . . . . . . . . . . . . . . . . . . . . . . IV.20 Experimental MSF for C 60 on Cu(111) stripe phase compared to simulated MSF for stripe phase with with J 3 > 0 . . . . . . . . . . . . . . . . . . . . IV.21 Pairwise correlations of the C 60 on Cu(111) stripe phase compared to those of the TIAF stripe phase with different coupling strengths . . . . . . . . . IV.22 Comparison of pairwise correlations and MSF for experimental C 60 /Cu(111) stripe phase and simulated values for the TIAF model at fixed temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.23 Analysis of quenched defects in the C 60 /Cu(111) stripe phase . . . . . . . . IV.24 Domain walls in the stripe phase, reported in the literature . . . . . . . . . IV.25 Domain walls in the molecular stripe phase . . . . . . . . . . . . . . . . . . IV.26 STM images of ordered phases of C 60 molecules on Cu(111) compared to ground states of the TIAF model . . . . . . . . . . . . . . . . . . . . . . . IV.27 Phase diagram of the TIAF model with antiferromagnetic J 1 , J 2 , J 3 and an external field, at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.28 Microscopic interactions in the C 60 /Cu(111) samples compared to effective couplings in the TIAF model . . . . . . . . . . . . . . . . . . . . . . . . . . IV.29 Height patterns on fullerenes on M(111) to compare with TIAF ground states191 V.1 Magnetic structure factors of different phases simulated by MC for the TIAF model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  • Wood's notation: notation used in commensurate phases to express the relation between the periodicity in two lattices A and B, each with unit vectors (⃗ a 1 , ⃗ a 2 ) and ( ⃗ b 1 , ⃗ b 2 ). In two dimensions, for lattices with ||⃗ a 1 || = ||⃗ a 2 || and || ⃗ b 1 || = || ⃗ b 2 ||, it can be written as (p × p)Rφ°, where p = a b is the ratio between lattices parameters of each lattice and φ = ∠(⃗ a, ⃗ b) the relative orientation between the lattices.

4 CFigure C. 1 :

 41 Figure C.1: Thermodynamic quantities of the TIAF model with J 1 = -1 computed using Monte Carlo simulations with Metropolis algorithm. From left to right: Specific heat (C), entropy per site (s) and acceptance rate (r).

14 Figure C. 2 :

 142 Figure C.2: Pairwise spatial correlations depending on temperature, from first to fifteenth neighbor for the TIAF model with J 1 = -1. First panel shows the notation and colors used for each neighbor. Curves show MC simulation results for different neighbor pairwise correlations.

C(r) ∝ 1 r

 1 -1/2 so in logarithmic scale: log(C(r)) ∝ -0.5log(r) The comparison with MC simulations is shown in Figure C.3 and Table

Figure C. 4 :

 4 Figure C.4: Magnetic structure factor of the TIAF model with J 1 = -1 computed using Monte Carlo simulations with Metropolis algorithm at different temperatures. First row: Intensity in linear scale. Second row: Same MSF with intensity in logarithmic scale. Third row: Profiles through the red line corresponding to q x = 0. Position of pinch points are indicated by the dotted blue lines. Fourth row: Profiles through the dotted red line corresponding to q y = 2 √ 3 π. Position of pinch points are indicated by the dotted blue lines. Dotted black lines indicates the positions corresponding to high intensity at q x = ± 2 3 π.

Figure C. 5 :

 5 Figure C.5: Evolution of thermodynamic quantities, specific heat, entropy and acceptance rate, depending on temperature. Each column correspond to a different J 2J 1 ratio, in all the cases J 3 J 2 = 0.57.

8 Figure C. 6 : 2 J 1

 8621 Figure C.6: Evolution of spin pairwise correlations depending on temperature, for (a) J 2 J 1 = 0.1 and J 3 J 2 = 0.57; (b) J 2 J 1 = 0.001 and J 3 J 2 = 0.57. Colored rectangles/curves indicate MC results and colored dots experimental values for the phase 9.68Å-22.4°.

Figure C. 7 :

 7 Figure C.7: Evolution of magnetic structure factor of the TIAF model depending on temperature, model parameters were J 2J 1 = 0.1 and J 3 J 2 = 0.57. We observe the transition from a spin liquid phase (T > 0.1) to an ordered zigzag ground state (T < 0.1).

Figure C. 8 :

 8 Figure C.8: Evolution of thermodynamic quantities, specific heat, entropy and acceptance rate, depending on temperature for the TIAF model with J 1 = -1, J 2 = -0.3, J 3 = -0.15.

3 .

 3 Like in the case of the zigzag ground state (Figure C.5) we cannot sample the phase space corresponding to the ground state at low temperature, because of the freezing of the dynamics. In Figure C.9 we show the MC correlations for the region of coexistence of stripe and zigzag phases. We plot both the MC correlations as a function of temperature, and the comparison with experimental values. Correlation values are zero at high temperature, in agreement with an uncorrelated paramagnetic phase. At lower temperatures they are, in general, non monotonous. The agreement of the experimental and simulated correlations is good, as discussed in section 3 in chapter IV.

14 Figure C. 9 :

 149 Figure C.9: Evolution of spin pairwise correlations depending on temperature, for J 1 = -1, J 2 = -0.3, J 3 = -0.15. Colored rectangles/curves indicate MC results and colored dots experimental values for the phase 9.80Å-26.3°.

Figure C. 10 :

 10 Figure C.10: Evolution of magnetic structure factor of the TIAF model depending on temperature, model parameters were J 1 = -1, J 2 = -0.3, J 3 = -0.15. We observe the evolution from a paramagnetic (T > 0.3) state to a coexistence of stripe and zigzag orders (T < 0.3).

Figure C. 11 :

 11 Figure C.11: MSF of the TIAF model depending on the lattice size, model parameters are J 1 = -1, J 2 = -0.1, J 3 = -0.057 and T ∼ 0.1. The number of configurations considered is N meas = 10 4 in all the cases. (a) Lattice size 20×20 sites, averaged over three different MC simulations to restore the three-fold rotation symmetry of the system. (b) Lattice size 40×40 sites, one single MC simulation symmetrized to restore the three-fold rotation symmetry. (c) Lattice size 80×80 sites, one single MC simulation symmetrized to restore the three-fold rotation symmetry.

Figure C. 12 :

 12 Figure C.12: MSF of the TIAF model depending on the number of MC configurations (N meas ) considered to compute the MSF, model parameters were J 1 = -1, J 2 = -0.1, J 3 = -0.057 and T ∼ 0.1. All the cases correspond to lattices with 20×20 sites. (a) MSF averaged for N meas = 5. (b) MSF averaged for N meas = 10. (c) MSF averaged for N meas = 50.(d) MSF averaged for N meas = 100. (e) MSF averaged for N meas = 500.(f) MSF averaged for N meas = 1000.

  e = -j1 + j3 return e def NRGZigzag2(j1, j2, j3, h): e = j1 -1/3 * j2 -1/3 * j3 return e def NRGZigzag3(j1, j2, j3, h): e = j1 -j3 return e def NRGZigzag4spins(j1, j2, j3, h): e = j2 + j3 return e def NRGSemiDoubZigzag(j1, j2, j3, h): e = -1/3 * j1 -1/3 * j2 + j3 + 1/3 * h return e #==Blote 1982============================================ # No new phases? Motsly interested in the type # of transition and critical exponents #==Takagi 1995=========================================== # They found "partially ordered phases" at finite T with # some spins having random behavior # GS are a-stripes, b-sqrt3, c&d-sqrt3 partially disordered, e-zigzag, # f-5 cell zigzag & diamonds, g-5 cell zigzag & diamonds, # h-6 cell zigzag with diamonds, i-7 cell zigzag with diamonds, # j-zigzag with diamonds partially disordered, # k-zigzag with groups of 4 equal spins, # l-9 cell with zigzag & diamonds , # m-9 cell with zigzag & diamonds partially disordered def NRGSqrt3PD(j1, j2, j3, h): e = j1 -2 * j2 + j3 -0 * h return e def NRGSqrt3PDprime(j1, j2, j3, h): e = j1 -3/2 * j2 + j3 -0 * h return e def NRG5ZigzagDiam(j1, j2, j3, h): e = 1/5 * j1 + 1/5 * j2 + j3 -1/5 * h return e def NRG5ZigzagDiamPD(j1, j2, j3, h): , mistake in j1 e = 1/5 * j1 + 2/5 * j2 + j3 -0 * h return e def NRG6ZigzagDiam(j1, j2, j3, h): #verfied e = j1 -1/3 * j2 -1/3 * j3 -0 * h return e def NRG7ZigzagDiam(j1, j2, j3, h): e = 3/7 * j1 + 3/7 * j2 + 3/7 * j3 -1/7 * h return e def NRG7ZigzagDiamPD(j1, j2, j3, h): , factor 2 everywhere e = 3/7 * j1 + 3/7 * j2 + 3/7 * j3 -0 * h return e def NRG9ZigzagDiam(j1, j2, j3, h): , error in the paper e = 5/9 * j1 -5/9 * j2 + 8/9 * j3 -1/9 * h return e def NRG9ZigzagDiamPD(j1, j2, j3, h): e = 5/9 * j1 -2/9 * j2 + j3 -0 * h return e #==Smerald 2016========================================= # Liquid, stripes and zigzag -> Nothing new #==J1, J2, J3, H #==Kaburagi 1978======================================== # 0-ferromagnetic, 1-sqrt7xsqrt7, 2-sqrt7xsqrt3, # 3-sqrt7x2sqrt3, 4-4x4 or 2sqrt3, 5-2x2, # 6-another sqrt7xsqrt7 with M=(2-6)/8, # 7-another 2sqrt3 with M=9-3/12, # 8-another sqrt7xsqrt7 with M=(2-6)/8, # 9-5x5 with M=(7-18)/25, # 10-another sqrt7xsqrt7 with M=(2-5)/7, # 11-20 spins/cell and trapezoids # 12-sqrt3, 13-Metcalf V/semi double stripes, # 14-Metcalf V, # 15-6 spins/cell with double hexagons, # 16-11 spins/cell kind of diamonds, # 17-12 spins/cell and double hexagons, # 18-12 spins/cell parts of zigzag, # 19-12 spins/cell diamonds, # 20-19 spins/cell groups of 6 spins, # 21-16 spins/cell double hexagons non sym, # 22-5 spins/cell double hexagons sym # 23-phase Takagi g-5 cell zigzag & diamonds, # 24-9 spins/cell stripes+double hexagons # 25-15 spins/cell trapezoids, # 26-10 spins/cell diamonds, # 27-25 spins/cell stripes and trapezoids, # 28-9 spins/cell diamonds # 29-9 spins/cell diamonds, # 30-stripes, 31-double stripes, 32-zigzag, # 33-9spins/cell is zigzag with 4spins, 34-double zigzag def NRGKab1(j1, j2, j3, h): e = -9/7 * j1 -9/7 * j2 -9/7 * j3 -5/7 * h return e def NRGKab2(j1, j2, j3, h): e = -3/5 * j1 -7/5 * j2 -3/5 * j3 -3/5 * h return e def NRGKab3(j1, j2, j3, h): e = -j1 -3/5 * j2 -3/5 * j3 -3/5 * h return e def NRGKab4(j1, j2, j3, h): e = -j2 -j3 -1/2 * h return e def NRGKab6(j1, j2, j3, h): e = -1/2 * j1 -1/2 * j2 -1/2 * h return e def NRGKab7(j1, j2, j3, h): e = -j1 -1/2 * h return e def NRGKab8(j1, j2, j3, h): e = -1/2 * j1 -j3 -1/2 * h return e def NRGKab9(j1, j2, j3, h): e = -3/25 * j1 + 9/25 * j2 -39/25 * j3 -11/25 * h return e def NRGKab10(j1, j2, j3, h): e = -1/7 * j1 -1/7 * j2 -1/7 * j3 -3/7 * h return e def NRGKab11(j1, j2, j3, h): e = -3/5 * j1 + 1/5 * j2 + 3/5 * j3 -2/5 * h return e def NRGKab15(j1, j2, j3, h): e = 1/3 * j1 + 1/3 * j2 -5/3 * j3 -1/3 * h return e def NRGKab16(j1, j2, j3, h): e = -2/11 * j1 + 3/11 * j2 + 12/11 * j3 -3/11 * h return e def NRGKab17(j1, j2, j3, h): e = 1/3 * j1 -2/3 * j2 + 1/3 * j3 -1/3 * h return e def NRGKab18(j1, j2, j3, h): e = 1/3 * j3 -1/3 * h return e def NRGKab19(j1, j2, j3, h): e = -2/3 * j1 + 2/3 * j2 + 1/3 * j3 -1/3 * h return e def NRGKab20(j1, j2, j3, h): e = -22/19 * j1 + 3/19 * j2 + 15/19 * j3 -5/19 * h return e def NRGKab21(j1, j2, j3, h): e = 3/4 * j1 -3/4 * j2 -1/4 * h return e def NRGKab22(j1, j2, j3, h): e = j1 -7/5 * j2 + 1/5 * j3 -1/5 * h return e def NRGKab24(j1, j2, j3, h): e = 6/10 * j1 + 6/10 * j2 -14/10 * j3 -2/10 * h return e def NRGKab25(j1, j2, j3, h): e = 1/5 * j1 + j2 -3/5 * j3 -1/5 * h return e def NRGKab26(j1, j2, j3, h): e = -1/5 * j1 + 3/5 * j2 + j3 -1/5 * h return e def NRGKab27(j1, j2, j3, h): e = 13/25 * j1 + j2 -27/25 * j3 -3/25 * h return e def NRGKab28(j1, j2, j3, h): e = -1/3 * j1 + j2 + j3 -1/9 * h return e def NRGKab29(j1, j2, j3, h): e = 1/9 * j1 + j2 + 1/9 * j3 -1/9 * h return e

1 Observing molecules in real space with (sub)-molecular resolution

1.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Scanning probe microscopy . . . . . . . . . . . . . . . . . . . . 1.2.1 Scanning Tunneling Microscope (STM) . . . . . . . . . . 1.2.2 Correction of distortions in STM images and data analysis 1.2.3 Non contact atomic force microscopy (nc-AFM) . . . . . 2 Structural properties and interactions in molecular assemblies: DFT calculations 2.0.1 Schrödinger equation, Born-Oppenheimer approximation and Hohenberg-Kohn theorems . . . . . 2.0.2 Choosing the exchange-correlation term: LDA and GGA 2.0.3 Linear combination of atomic orbitals (LCAO) and basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.0.4 Self-consistent solution and iterative minimization of the energy . . . . . . . . . . . . . . . . . . . . . . . . . . 2.0.5 Our calculations: DFT and Intermolecular Perturbation Theory using Fireball code . . . . . . . . . . . . . . . . . 3 Order and disorder in molecular assemblies: statistical analysis 3.1 Pairwise correlations in real space . . . . . . . . . . . . . . . . 3.2 "Magnetic" structure factor . . . . . . . . . . . . . . . . . . . . 3.3 Definition of an effective temperature . . . . . . . . . . . . . . 3.4 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . 3.4.1 Basic principle . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 MC parameters . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Model parameters . . . . . . . . . . . . . . . . . . . . . . 3.4.4 MC output: thermodynamic quantities . . . . . . . . . . 3.4.5 MC outputs: correlations . . . . . . . . . . . . . . . . . .

III An atomic-scale viewpoint of C molecules self-assembly on Cu(111) Contents 1 Structural properties of C 60 molecules on metal surfaces: brief review of the literature

  1.1 Single C 60 molecules on metal surfaces . . . . . . . . . . . . .

	1.1.1 Molecular orientation and appearance in STM images . .
	1.1.2 Binding sites of fullerenes on M(111) surfaces . . . . . .
	1.1.3 Surface reconstruction and molecular height . . . . . . .
	1.2 Fullerenes assemblies on close-packed metal surfaces . . . . .
	1.2.1 Commensurate phases of C 60 on M(111) . . . . . . . . .
	1.2.2 Corrugation and height patterns in fullerenes assemblies
	1.2.3 Incommensurate, aperiodic phases and local distortions
	of C 60 on metal surfaces . . . . . . . . . . . . . . . . . .

2 Experimental observations on C 60 on Cu(111)

	2.1 Binding of molecules in the (4x4)R0°"uniform" phase . . . .
	2.1.1 Overview of the molecular deposits performed at high
	temperature . . . . . . . . . . . . . . . . . . . . . . . . .
	2.1.2 New insights on the binding sites with STM . . . . . . .
	2.2 Phase coexistence at room temperature . . . . . . . . . . . .
	2.2.1 In-plane structure: rotational ordering, strain and
	coincidence with the substrate . . . . . . . . . . . . . . .
	2.2.2 Out-of-plane structure: molecular heights and orientations
	2.2.3 Relation between in-plane and out-of-plane structures:
	formation of vacancies, lattice distortions and steric effects
	2.2.4 Summary of observed structures . . . . . . . . . . . . . .

3 DFT study of C 60 on Cu(111) 3

  .1 Binding sites and energies . . . . . . . . . . . . . . . . . . . .

	3.1.1 On the unreconstructed Cu(111) surface . . . . . . . . .
	3.1.2 On reconstructed sites: vacancies in the metal surface . .

  .3 and TableIII.

Table III . 3 :

 III3 Binding energy and distance between the base of a C 60 molecule and the copper surface for a given molecular configuration (binding site and molecular orientation), calculated by DFT for far-apart fullerenes on the Cu(111) surface.

	Binding site	Molecular orientation	Molecular height (Å)	Binding energy (eV)
	top	apex	1.4	-2.81
	fcc	apex	1.2	-4.26
	bridge	6-6	1.6	-3.38
	top	6-6	1.5	-3.06
	top	hexagon	1.9	-2.86
	bridge	hexagon	1.9	-2.21
	fcc	hexagon	1.6	-2.54
	hcp	hexagon	1.5	-2.41
	fcc	hexagon R 30 z	1.8	-2.34
	hcp	hexagon R 30 z	1.8	-2.35
	top	pentagon	1.7	-2.35

Table III .

 III

	Binding site	Molecular orientation	Molecular height (Å)	Binding energy (eV)
	top	hexagon	1.9	-3.55
	bridge	hexagon	1.9	-2.69
	fcc	hexagon	1.7	-2.75
	hcp	hexagon	1.7	-2.76

4:

Binding energy and distance between the base of a C 60 molecule and the copper surface for a given molecular configuration (binding site and molecular orientation), calculated by DFT for fullerenes on a periodic lattice with (4 × 4)R0°commensurability with Cu

[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]

.

Table III . 6 :

 III6 Binding energy of fullerenes on different close-packed metals and height between the base of a C 60 molecule and the metal surface, calculated by DFT for single fullerenes.

	Metal	Binding site	Molecular orientation	Molecular height (Å)	Binding energy (eV)
	Pt(111)	top	hexagon	1.7	-7.05
	Pt(111)	top 1-atom vac hexagon	1.6	-7.99
	Ni(111)	top	hexagon	1.8	-6.33
	Ni(111)	top 7-atom vac hexagon	-0.1	-15.39
	Al(111)	top	hexagon	1.7	-4.29
	Al(111)	top 1-atom vac hexagon	1.6	-7.22
	Cu(111)	top	hexagon	1.9	-2.86
	Cu(111)	top 7-atom vac hexagon	0.0	-7.78
	Au(111)	top	hexagon	1.9	-1.93
	Au(111)	top 1-atom vac hexagon	2.0	-4.64
	Ag(111)	top	hexagon	2.4	-1.28
	Ag(111)	top 1-atom vac hexagon	2.0	-2.88

2 Spin liquid phase with emergent zigzag order in C 60 /Cu(111) 2

  Height correlations in reciprocal space: ground state and range of effective couplings . . . . . . . . . . . . . . . . . . . . 2.3 Pairwise correlations in real space: coupling strengths and thermodynamic equilibrium . . . . . . . . . . . . . . . . . . . 2.4 Magnetic-like defects and local lattice distortions . . . . . . . 2.5 Microscopic origin of antiferromagnetic couplings . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	3 Energy degenerate stripe and zigzag phases in
	C 60 /Cu(111)
	3.1 TIAF ingredients . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1 The model -theoretical studies . . . . . . . . . . . . . . 1.1.2 Experimental realizations of the TIAF model . . . . . . . 1.2 Further-distance couplings and ordered ground states . . . . 1.2.1 Literature review . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Our phase diagram at T = 0 for antiferromagnetic J 1 , J 2 , J 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Molecular TIAF: ingredients . . . . . . . . . . . . . . . . . . . 2.2 3.2 Height correlations in reciprocal space: ground state and range of the effective couplings . . . . . . . . . . . . . . . . . . 3.3 Pairwise correlations in real space: coupling strength . . . . 3.4 Thermodynamic equilibrium . . . . . . . . . . . . . . . . . . .

Table IV . 2 :

 IV2 Estimate of the interaction and dipolar energy between C 60 molecules and equivalence with the effective couplings in the Ising spin Hamiltonian.

		E C 60 -C 60 (eV)	E dipolar (eV)	J (eV)
		High-high High-low High-high High-low
	1NN	-0.201	-0.236	0.366	0.309	-0.0456
	2NN	-0.00515	-0.00505	0.0705	0.0629	-0.00375
	3NN	-0.00196	-0.00193	0.0458	0.0411	-0.00232

Table IV . 4 :

 IV4 Estimate of the interaction and dipolar energy between C 60 molecules in the 9.98 Å-26.3°phase, and equivalence with the effective couplings in the Ising spin Hamiltonian.

		E C 60 -C 60 (eV)	E dipolar (eV)	J (eV)
		High-high High-low High-high High-low
	1NN	-0.2661	-0.2664	0.3338	0.2834	-0.0253
	2NN	-0.00417	-0.00409	0.0642	0.0574	-0.0034
	3NN	-0.00158	-0.00160	0.0417	0.0375	-0.0021

  III.12 Orientation and height patterns of different commensurate phases formed by C 60 molecules on Cu(111) . . . . . . . . . . . . . . . . . . . . . . . . . . III.13 Orientations and supercell sizes of different commensurate phases formed by C 60 molecules on Cu(111) . . . . . . . . . . . . . . . . . . . . . . . . . . III.14 Commensurate structures of C 60 molecules on Cu(111) and binding sites . III.15 Analysis of molecular lattice parameters of different C 60 /Cu(111) commensurate phases in reciprocal space (1) . . . . . . . . . . . . . . . . . III.16 Analysis of molecular lattice parameters of different C 60 /Cu(111) commensurate phases in reciprocal space (2) . . . . . . . . . . . . . . . . . III.17 Statistical analysis of fullerene apparent heights in our STM images . . . . III.18 High resolution STM images showing fullerene apparent heights and orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.19 Conclusion about fullerene heights and orientations at room temperature . III.20 Steric effects due to vacancy formation . . . . . . . . . . . . . . . . . . . . III.21 Commensurate structures of C 60 molecules on Cu(111) and binding sites, including vacancy formation . . . . . . . . . . . . . . . . . . . . . . . . . . III.22 Evidence of local distortions on fullerene lattices on Cu(111) . . . . . . . . III.23 Evidence of formation of domain walls on fullerene lattices on Cu(111) . . . III.24 Summary of commensurate phases formed by C 60 molecules on Cu

2 Summary of commensurate C 60 phases on close-packed metalsTable B . 2 :

 B2 Summary of commensurate phases for fullerenes on M[START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF] surfaces reported in literature. Wood's notations (p × p)Rφ and (p ′ × p ′ )Rφ ′ are resumed to (p)Rφ and (p ′ )Rφ ′ respectively, the strain being always biaxial in the considered lattices. Molecular lattice parameter for each phase was computed from the Wood's notation, considering following lattice parameters for the metals: a Ag(111) = 2.890Å, a Au = 2.861Å, a Au = 2.880Å, a Cu = 2.556Å, a N i = 2.492Å, a P t = 2.770Å. The color of the molecular lattice parameter indicates if the phase is compressed (a C60 < 10.05Å, in red) or expanded (a C60 > 10.05Å, in light blue) with respect to a 2D fullerene layer. When a star accompanies Woods' notation in the table, the commensurate phase is our interpretation taken into account data in the literature. When no star is shown, we simply reproduce the guess proposed in the literature.

			C 60 lattice	Super-	a C60	Vacan-	Height	Technique
	Metal		(p × p)Rφ	structure	(Å)	cies	pattern	& Ref.
	(111) Ag	( √	(p ′ × p ′ )Rφ ′ 13)R13.9 • * (1)R0 • *	10.42 1-atom Uniform	STM [174],
								STM [72]
	Ag	(	√	301 5 )R11.52 • * (5)R0 • *	10.03 1-atom Uniform	STM [221]
	Ag	(	√	589 7 )R14.5 • * (7)R0 • *	10.02 1-atom Uniform	STM [174],
								STM [72]
	Ag	(2 √	3)R30 •	(1)R0 •	10.01 1-atom Disorder ("in-	LEED	[1,
								termediate"	113,	163],
								T annealing)	STM [1, 2,
								or	uniform	221,	163],
								(higher	T	DFT	[113,
								annealing)	163]
	Al	(2 √	3)R30 •	(1)R0 • (( √ 3)R30 •	9.91 1-atom Uniform (pe-riodic after	XAS & STM [121], PES &
							when	an-	annealing)	LEED [121,
							nealed)	120], STM &
								STS [87], Ab
								initio [193]
	Au	( 7 2 )R0 •	(2)R0 •	10.08 1-atom Uniform	LEED	[2]
								STM [1, 170,
								69, 203]
	Au	(

√

109

3 )R35.5 • * (3)R0 • * 10.02 1-atom Quasiperiodic STM

[START_REF] Gardener | Scanning tunneling microscopy studies of C 60 monolayers on Au(111)[END_REF]

Table C . 1 :

 C1 Comparison of the published values of spin pairwise correlations at zero temperature[START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF] for the TIAF with only J 1 coupling and our Monte Carlo results for different sizes of the lattice at low temperature.

		N → ∞	400	400		MC 6400	6400		10000	10000
		Step.1964	MC	M C-Step Step	(%)	MC	M C-Step Step	(%)	MC	M C-Step Step	(%)
	C 1	-0.33333	-0.33333	0.0%		-0.33237	-0.3%	-0.33333	0.0%
	C 3	-0.19285	-0.18202	-5.6%	-0.18607	-3.5%	-0.19243	-0.2%
	C 5 +0.35852 +0.33175	-7.5%	+0.34268	-4.4%	+0.35739	-0.3%
	C 8	-0.16561	-0.15409	-6.9%	-0.15671	-5.4%	-0.16489	-0.4%
	C 11 -0.13362	-0.09074	-32.1%	-0.11926	-10.8%	-0.13241	-0.9%
	C 15 +0.25682 +0.18112	-29.5%	+0.22682	-11.7%	+0.25381	-1.2%
	C 20 -0.12320	-0.10413	-15.5%	-0.10781	-12.5%	-0.12128	-1.6%

Ces comportements exotiques ne sont pas cantonnés à des systèmes magnétiques. Historiquement, le premier exemple fut d'ailleurs donné par la glace d'eau, où les atomes d'oxygène forment un cristal alors que les atomes d'hydrogène peuvent se lier aux oxygènes de deux manières distinctes. Il en résulte un très grand nombre de configurations possibles, chacune avec une distribution différente des atomes d'hydrogène. En d'autres termes, l'état fondamental d'une telle phase est massivement dégénéré et son entropie est non nulle à 0 K. Dans ce manuscrit, nous étudions un système non magnétique bi-dimensionnel et démontrons qu'il se comporte comme un réseau magnétique frustré. Notre système d'étude est constitué de molécules de fullerène (C 60 ) déposées en assemblées sur une surface atomiquement plane de cuivre. Dans certaines conditions, ces assemblées moléculaires présentent un désordre qui prend la forme d'une corrugation très particulière. Partant de ces observations, nous élaborons une analogie entre un degré de liberté structural des fullerènes (leur hauteur dans leurs assemblées corruguées) et une variable Ising. Au moyen d'une analyse statistique empruntée à l'étude du magnétisme frustré, nous parvenons à décrire quantitativement nos assemblées de molécules avec des Hamiltoniens de spins Ising frustrés. Notre méthodologie s'avère adaptée pour explorer une variété d'autres systèmes sur surfaces, par exemple d'autres assemblées de molécules et des alliages binaires bi-dimensionnels, du point de vue de la frustration magnétique (et ce, malgré l'absence de magnétisme dans ces systèmes). En outre, nous dévoilons les détails atomiques de la liaison C 60 / cuivre. En particulier, nous posons les bases conceptuelles d'un cadre de description des arrangements moléculaires ordonnés et désordonnés qui, étonnamment, coexistent. Nous combinons pour cela des expériences par microscopie à balayage de haute résolution et des calculs ab initio, et ainsi résolvons la structure des différentes phases moléculaires, en caractérisant l'ensemble des degrés de liberté (hauteur, orientation, position latérale de chaque molécule). Nous introduisons un modèle traitant les interactions en compétition dans le système, qui nous permet de prédire les structures observées et leur stabilité relative. Avec ce modèle, nous déduisons un paysage en énergie riche, comprenant plusieurs phases métastables, et l'appliquons à d'autres interfaces C 60 /métal.

This historical review is based on several sources, including both specialized literature and popular science sources[135, 9,

[START_REF] Brush | The Royal Society's first rejection of the kinetic theory of gases (1821), John Herapath versus Humphry Davy[END_REF]107,[START_REF] Fisher | Statistical theory of liquids[END_REF][START_REF] Croxton | Liquid state physics -A statistical mechanical introduction[END_REF][START_REF] Egelstaff | An introduction to the liquid state[END_REF].

We refer here to the possibility of detect and visualize atoms and molecules in real space, thanks to scanning probe microscopy techniques.

Note that for a 2D adsorbed monolayer this situation is completely theoretical and can never be realized, because long range order is forbidden by the Hohenberg-Mermin-Wagner theorem in these conditions[START_REF] Mermin | Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models[END_REF][START_REF] Mermin | Absence of ordering in certain classical systems[END_REF][START_REF] Hohenberg | Existence of long-range order in one and two dimensions[END_REF].

Like specified in the previous note, long range ordered floating phases with a constant lattice parameter

However, high resolved images of fullerenes using nc-AFM can reveal the internal structure of orbitals, and even of the carbon skeleton of the molecules. Thus, the contrast within each molecule is related to details of the electronic density of the molecule, and not only to its height.

In this manuscript we use the terms "arrested configuration" and "microstate" to refer to a frozen, not fluctuating, configuration of a phase in which we know the state of each degree of freedom present in a lattice.

Why we did not find this configuration as being the lowest energy one in our DFT calculations for isolated fullerenes remains an intriguing question for us, which would require a deeper analysis of the different DFT methods used to address the calculation of binding energies. However, our results reproduce the experimental observations, as explained in the next lines.

Despite the fact that frustration has also been studied in structural non-magnetic systems[START_REF] Sadoc | Geometrical frustration[END_REF], and that the first experimental example of frustration is water ice[START_REF] Pauling | The Structure and Entropy of Ice and of Other Crystals with Some Randomness of[END_REF], which is not magnetic.

Correlations are in fact critical, i.e. the correlation length diverges, as we explain later in the main text.

Many times in this chapter we call the phase studied in section 2 "disordered" phase, and the addressed in section 3 "stripe" phase. These adjectives are a short nomenclature which only accounts for the visual impression given by the height patterns in each phase.

During all this chapter we will speak about a spin liquid in its transition to an ordered state, but this

Even more the existence of algebraic correlations, which are only expected at T = 0.

The J3 J2 = 0.5 point is actually a point in the phase diagram in the thermodynamic limit N → ∞. For finite lattices the properties of this region extend in a small region J3 J2 = 0.5 ± ε[START_REF] Smerald | Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets[END_REF]

As explained in chapter III intermolecular interactions between fullerenes due to van der Waals and chemical repulsion are present at distances < 15 Å, but the distances between second and third neighbor C 60 molecules in the experimental lattices are ∼ 17 Å and ∼ 20 Å.

Here we only address the position and intensity of the peaks, further analysis on their shape is presented in section 3 in Appendix C.

The position of the emergent peaks is the same and the values of the intensity are comparable for both the experimental and simulated MSFs. Nevertheless, a difference in the shape of the peaks is observed: experimental peaks are elongated while simulated ones are circular. This difference is due to lattice size effects detailed in section 3 in Appendix C.

The presence of local distortions is neglected in this calculation, we refer here to the intermolecular distances in the case of a perfect non-distorted triangular lattice.

We use our own convention for the sign of couplings in the Hamiltonian, in which J < 0 correspond to an antiferromagnetic coupling. Korshunov[START_REF] Korshunov | Nature of phase transitions in the striped phase of a triangularlattice Ising antiferromagnet[END_REF] used the opposite convention.

We thank V. Renard for his smart question during the PhD defence, which allow us to propose this new research path.

MC results for the TIAF model with antiferromagnetic J 1 , J 2 , J 3 : ordered ground states

2. We compute the energy of each ordered phase at zero temperature. The energy can be easily calculated from the spin configuration inside the unit cell and the Ising spin Hamiltonian. We need to count the number of neighbor spins with equal and opposite states and sum to obtain a function E(J 1 , J 2 , J 3 , h ext ). A demonstration for the stripe phase is shown in Figure IV.10. A Python code in which the energies of each phase are defined as a function of J 1 , J 2 , J 3 and h ext is presented in Appendix D.

3. We explore the desired region of the phase diagram and identify the phase with lowest energy for each set of couplings.

4. We plot the results to visualize the mean field phase diagram at zero temperature and

The following section changes the point of view from which we analyze the experimental phases. In the previous discussion we used global quantities: correlations are, by definition, averaged over the whole lattice. Here, we do not consider anymore global quantities but local ones, tacking advantage of the fact that we are able to image configurations in real space.

Magnetic-like defects and local lattice distortion

We start by considering magnetic-like defects formed by three spins sitting on a triangular plaquette and exhibiting the same up/down state, i.e. triangular plaquettes breaking the ice rule. The presence of such defects directly modifies the value of C 1 , which is equal to -1/3 when the phase is perfectly correlated at short range and the ice rule is obeyed. We recall that we found C 1 = -0.24, which was always larger than the C 1 value found in MC simulations at low temperature. The defects we observe were also discussed in the disordered phase (subsection 2.4). For this phase, we found that clusters of three spins on the same "low" state are most of the times surrounded by "high" molecules, meaning that defects have a well-defined structure. We were also able to detect local distortions of the molecular lattice at the position of magnetic-like defects, i.e. the three "low" fullerenes were further away than if positioned on the sites of a perfect close-packed lattice. phase, but the experimental system is at an effective temperature well above the transition temperature. In this case, in which the antiferromagentic coupling and the external field are of the same order of magnitude, we are not sure about the nature of the phase at intermediate temperature. In fact, we do not know if the spin liquid phase could exist in a small region of the phase diagram, or if the phase transits from an uncorrelated paramagnet to a √ 3 × √ 3 phase without going through a spin liquid phase. The high value of the temperature T ∼ 12|J 1 | ∼ 4h ext explains why the experimental MSF does not exhibit peaks at the corners of the Brillouin zone, as expected for a √ 3 × √ 3 phase.

From an uncorrelated paramagnet to a ferromagnetic order

Disordered height patterns have also been observed for fullerenes on Ag [START_REF] Leonelli | Light-scattering study of intercalate orderdisorder transition in Ag 0.33 TiS 2[END_REF]

List of Tables

Glossary

• Algebraic correlations: pairwise correlations exhibiting and algebraic decay with the distance, following the mathematical expression r -α , with α a real positive number. These correlations are characteristic of critical states of matter, in which no order develops and correlation length diverges despite the weak strength of correlations. • Apparent height: height measured by STM, which is the result of both topography (distance between the STM tip and the surface) and electronic effects (the measured Local updates are simple to implement, but dynamics is usually slower. Global updates are chosen to optimize the evolution of the simulation, to avoid freezing of the dynamics and to explore the whole phase space of a given model. • Magnetic structure factor: Fourier transform of pairwise correlations, given by the equation:

• Metropolis algorithm: Monte Carlo method based on single spin flips, in which the probability to accept a given update directly depends on the Boltzmann factor associated to its energy. • Molecular azimutal rotation (or orientation): for a molecule on a surface, it is the angular degree of freedom around the axis perpendicular to the surface (usually z-axis). • Molecular orientation: in this manuscript it corresponds to the way in which a C 60 molecule binds to a surface, i.e. which part of the molecule is in contact with the surface and which is exposed to the STM tip. • (Magnetic) monopole: by analogy with electric charges, magnetic monopoles are "particles" appearing for a non-zero divergence of a magnetic field. In fact, magnetic monopoles are not actually charged particles, but they are emerging particles revealing particular magnetic correlations [START_REF] Castelnovo | Magnetic monopoles in spin ice[END_REF][START_REF] Henley | The "Coulomb phase" in frustrated systems[END_REF]. • Pairwise correlations: in a spin system they are defined as the spin-spin (spin represented by σ) correlations at a given distance:

In general, they can be defined for any degree of freedom of particles at a given distance. • Paramagnet or paramagnetic phase: in this manuscript state of matter in which the magnetic degrees of freedom are disordered in an uncorrelated (also called random) way. • (Surface) reconstruction: for on-surface systems it consists on a re-arrangement of the atoms on a surface caused by the presence of adsorbed atoms or molecules. It usually implies mass transfer. In this manuscript surface reconstruction consists in the formation of one-atom deep vacancies on close-packed metal surfaces, caused by the presence of fullerenes. • Pinch point: characteristic feature observed in the magnetic structure factor of systems with algebraic correlations, consisting on a non-analytical divergence of the intensity at some particular points of the q-space. Pinch points are experimental and theoretical signatures of critical liquids. • Single spin flip: local MC update, in which one spin is modified at each MC step.

• Spin liquid: by analogy with structural liquids, magnetic phase of matter in which pairwise (spin-spin) correlations have an exponential (or algebraic, in the case of critical liquids) decay. • Spin ice: by analogy with water ice, magnetic phase of matter in which the ground [START_REF] Liu | Molecular orientations and interfacial structure of C 60 on Pt(111)[END_REF][START_REF] Shi | C 60 on the Pt(111) surface: Structural tuning of electronic properties[END_REF], STM [START_REF] Otero | STM study of C 60 overlayers on Pt(111) surfaces[END_REF][START_REF] Liu | Molecular orientations and interfacial structure of C 60 on Pt(111)[END_REF], DFT [START_REF] Shi | C 60 on the Pt(111) surface: Structural tuning of electronic properties[END_REF]

Appendix B: Details of fullerene phases on closepacked metals 1 Apparent molecular heights of fullerenes on Cu(111)

Appendix C: MC results using Metropolis algorithm

In this appendix we present some of the results of our MC simulations based on Metropolis algorithm. MC simulations are a key tool of the comparison between our experimental molecular system and the TIAF model, as we showed in chapter IV and chapter V. The goal of our MC simulations is to build bridges between our experimental results on molecular assemblies and the TIAF model. Consequently, a detailed study of the thermodynamics of the Hamiltonian, the nature of phase transitions, the description of a complete phase diagram, etc. is beyond the goals of our work. We only address values of the couplings and evolution of thermodynamic quantities when they are useful to understand the C 60 on Cu(111) experimental system.

We present the results of our simulations in two sections. We address separately the case in with only first nearest neighbor coupling (J 1 ) is present in the Hamiltonian and the one in which first, second (J 2 ) and third (J 3 ) neighbor couplings are taken into account.

When only first neighbor coupling is considered, the ground state is a spin liquid phase, i.e. a exotic phase infinitely degenerate characterized by extensive entropy [START_REF] Wannier | Antiferromagnetism. The Triangular Ising Net[END_REF] and, in the limit T = 0, algebraic correlations [START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF]. The aim of studying this phase with MC is two-fold: 1 -we want to determine the limits of our Metropolis MC simulations because we know that the spin liquid phase cannot be well described using local updates; 2 -at the beginning of the project we made the hypothesis that the experimental system was a realization of the TIAF with first nearest neighbor couplings, so we needed to study the spin liquid phase using MC even if finally further couplings were necessary to explain our experimental observations. Here we also verify agreement of the thermodynamic quantities (entropy) and spatial correlations computed with our simulations with those reported in previous studies.

In the case in which longer range couplings are included in the TIAF model, the ground states are ordered phases, exhibiting different ordered patterns depending on the ratio of the couplings. Moreover, for these phases we found a significant slowdown of the simulation dynamics, which was not the case for the spin liquid phase. These qualitative differences (ordered ground states and freezing of the dynamics) between two models makes appropriate a separate discussion. our simulations and the analytical calculation is smaller than 1%. For neighbors within 5 lattices parameters difference is smaller than 1.5%.

We are not able to quantitatively describe the correlations at long range and low temperature if the size of the lattice is too small. However, our approach describes well disorder at short range (even for small lattices) and is accurate within 1% for lattices of 100 × 100 spins. [START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF]. Dotted black line corresponds to the linear fit of algebraic correlations (values reported in Ref. [START_REF] Stephenson | Ising-model spin correlations on the triangular lattice[END_REF]) in logarithmic scale, the expected coefficient is -0.5 to compare with the found values -0.61 and -0.58. Colored empty dots correspond to correlations computed using Monte Carlo simulations with Metropolis algorithm. Size of MC simulations varies from 10 2 (10×10 lattice) to 10 4 (100×100 lattice).

Finally, in first and second rows of Figure C.4 we show the evolution of the magnetic structure factor when varying temperature, for a lattice of 100 × 100 spins. We would expect the apparition of pinch points at the edges of the Brillouin zone (white hexagons in Figure C.4). Those patterns are hardly visible if the intensity is plotted in linear scale (first row). In this type of plot the intensity is much larger in the corners of the Brillouin zone. Better visualization is obtained using a logarithmic scale (second row). In the two last rows of Figure C.4 profiles of the intensity following the lines q x = 0 and q y = 2 √ 3 π are shown. On the profile through q x = 0, i.e. perpendicular to the pinch point, we observe the existence of broadened pinch points at q y = √ (3) 2 , indicated by dotted blue lines. The peaks are broad, meaning that correlations decrease faster than the algebraic dependence, even for a lattice of 10 4 spins. Pinch points amplitude is constant once the MC simulation reach the ground state, T≲ 1. For temperatures below the conversion into the liquid ground state, the pinch points are broader and their amplitude is larger. Background intensity is also larger and distributes uniformly when temperature increases, like in the case of a paramagnetic phase.

MC results for the TIAF model with antiferromag-

netic J 1 , J 2 , J 3 : ordered ground states

In the spin liquid-zigzag region

Here, we present results of our Monte Carlo simulations for the J 3 J 2 region corresponding to a zigzag ground state. In particular we take appropriate values to compare the model with our experimental results, i.e. 0.001 ≤ J 2 J 1 ≤ 0.1 and J 3 J 2 ∼ 0.57 (for details see section 2). We plot thermodynamic quantities, pairwise spin correlations in real space and magnetic structure factor.

In Figure C.5 we show thermodynamic quantities for our model taking different J 2 J 1 ratios. When J 1 = 1000J 2 we observe two jumps in the specific heat, the first one at T J 1 ∼ 0.01 corresponding to the melting of the zigzag phase to the liquid one. The second one at T J 1 ∼ 1, where the spin liquid phase becomes a paramagnetic phase. A clear entropy plateau appears in the temperature regime corresponding to the liquid phase 0.01 ≲ T ≲ 1. At low temperature, our simulations did not reach the zero entropy of the ground state because freezing of the dynamics. Same qualitative behavior was observed for J 1 = 100J 2 . For J 1 = 10J 2 both peaks in the specific heat start to merge together and the entropy plateau corresponding to the liquid phase is not longer well defined. For even lower values of the We also show the acceptance rate in Figure C.5. For all the coupling ratios, a important slowdown of the simulation dynamics is observed at temperatures near the transition from the spin liquid phase to the zigzag phase. For low temperatures, the MC simulations stopped and we could not explore the ground state. Note that at intermediate temperatures, for which the system is in the spin liquid phase, no significant slowdown of the dynamics is observed. Correlations are compared with experimental values. We find a very good agreement between Monte Carlo simulations and experimental values, for both ratios J 2 J 1 = 0.1 and

J 1 = 0.001. In both cases a single effective temperature that matches experimental values can be defined. When J 2 J 1 = 0.1 effective temperature is T ef f ∼ 0.1 and when J 2 J 1 = 0.001 the better agreement is found for T ef f ∼ 0.001, so in both cases Comments (in gray) indicate the equivalence between the phase reported in the literature name used in our code. References are cited as Name Year, details for each one can be found in the bibliography of this manuscript.