Agnès Barthélémy

Yannick Fagot-Revurat

Keywords: Spectroscopie d'électrons, ARPES, Physique des matériaux, Spintronique, Ferroélectrique, Matériaux 2D Ferroelectrics-Transition Metal Dichalcogenide heterostructures studied ............... Electron spectroscopy, ARPES, Condensed matter, Spintronics; Ferroelectrics, 2D materials

Les dichalcogénures de métaux de transition (TMD) sont des matériaux lamellaires. La brique de base est un feuillet de formule générique MX 2 , où M est un élément de transition et X un chalcogène (M = Mo, W…; X = S, Se, Te…) liés entre eux par des liaisons covalentes. Le solide tridimensionnel est ensuite obtenu en empilant ces couches MX 2 , faiblement liées entre elles par des interactions de type van der Waals. Dans les TMD à base d'éléments de transition lourds (Mo ou W), le fort couplage spin-orbite et l'absence de symétrie d'inversion dans la structure cristallographique engendre une inversion de la population de spin entre deux point K adjacents dans la première zone de Brillouin, une propriété fort attractive pour des applications en spintronique si ces populations de spins peuvent être contrôlées. Ce travail de thèse vise à étudier l'influence de la polarisation d'un ferroélectrique sur la structure électronique du TMD. Des films de WSe 2 ont été préparés par épitaxie par jet moléculaire (MBE) puis transférés sur un ferroélectrique, ici du BiFeO 3 . Ce ferroélectrique est lui-même épitaxié sur un substrat de DyScO 3 (110) mais sur des couches intercalaires permettant, d'ajuster la direction de sa polarisation électrique : avec un buffer de SrRuO 3 la polarisation est perpendiculaire à la couche et pointe vers le substrat (configuration DOWN), tandis que sur un buffer de La 0.7 Sr 0.3 MnO 3 , la polarisation pointe dans la direction opposée (configuration UP). Grâce à la photoémission résolue en angle (ARPES), nous avons mis en évidence un décalage en én-

ergie de liaison de la structure électronique avec l'inversion de la polarisation. Pour trois et deux couches de WSe 2 , les décalages sont respectivement de 750 meV et 640 meV. Ces résultats démontrent bien, qu'en jouant sur cette polarisation, il est possible d'influer notablement sur la structure électronique de WSe 2 , pour obtenir deux états qui pourraient être facilement différenciés par la conductivité électrique de l'empilement. Ce type d'hétérostructures se comporte donc comme un transistor à effet de champ ferroélectrique et pourrait être utilisé pour fabriquer des mémoires non-volatiles avec un fort contraste ON/OFF. Le même effet (quoique moins prononcé) est observé pour des flakes micrométriques d'une couche d'épaisseur obtenus par Dépôt Chimique en Phase Vapeur (CVD) du même matériau déposés sur des domaines de polarisation opposées d'un substrat de LiNbO 3 et observés par microscopie de photoélectrons (PEEM). Enfin, sur des couches de WSe 2 épitaxiées par MBE sur du graphène, nous avons pu décrire par ARPES la transition de la structure électronique de 2D à 3D lorsque l'épaisseur de TMD augmente. Aux faibles épaisseurs (entre 1 et 3 couches), seuls quelques états électroniques discrets apparaissent à certaines valeurs de kz (perpendiculaire aux couches) dans l'espace réciproque. Au fur et à mesure que l'épaisseur augmente, ces états se multiplient pour commencer à former une dispersion continue. Ces résultats sont très bien reproduits par des calculs de la fonction spectrale dans un modèle à une étape .

Abstract: Transition Metal Dichalcogenides (TMD) are lamellar materials. The building block is a sheet whose generic formula is MX 2 , where M is a transition metal and X is a chalcogene (M = Mo, W…; X = S, Se, Te…), tied by covalent bondings. The three-dimensional solid is obtained by piling those MX 2 sheets, weakly bonded by van der Walls interactions. In TMD made of heavy transition elements (Mo ou W), the strong spin-orbit coupling and the absence of inversion symmetry in the crystallographic structure leads to an inversion of the spin population in two adjacent K points of the first Brillouin zonr, a very attractive property for spintronics applications, provided these two populations can be controled. This PhD work aims to study the influence of the polarisation of a ferroelectric on the electronic structure of a TMD. WSe 2films were prepared by Molecular Beam Epitaxy (MBE) and then transfered onto a ferroelectric material, BiFeO 3 . This ferroelectric is itself deposited on a DyScO 3 (110) substrate with different buffer layers allowing to adjust the direction of the electrical polarization : with a SrRuO 3 buffer, the polarisation is perpendicular to the layer and points towards the substrate (DOWN configuration), whereas with a La 0.7 Sr 0.3 MnO 3 buffer, the polarisation points in the opposite direction (UP configuration). Thanks to Angle-Resolved Photoelectron Spectroscopy (ARPES), we evidenced a binding energy shift of the electronic structure with the polarisation inversion. For bi-and tri-layers of WSe 2 , these shifts are respectively of 750 and 640 meV. These results clearly demonstrate that, by playing on the polarisation, it is possible to notably modifiy the WSe 2 electronic structure to obtain two states which could be easily differentiated by the electrical conductivity of the stacking. This kind of heterostructure therefore behaves like a ferroelectric field effect transistor and could be used to build non-volatile memories with a strong ON/OFF contrast. The same effect (although less pronounced) is observed on micrometric monolayer flakes of the same material made by Chemical Vapor Deposition (CVD) and deposited on domains with opposit polarisation of a LiNbO 3 substrate and observed by Photoemission Electron Microspcopy (PEEM). Last, on epitaxial WSe 2 layers on graphene obtained by MBE, we were able to describe the 2D to 3D transition of the electronic structure by ARPES with increasing thickness of the TMD. At low thickness (between 1 and 3 layers), only few discrete electronic states appear at given kz (perpendicular to the layers) values in the reciprocal space. While increasing the thickness, these states multiply to start to form a continous dispersion. These results are very well reproduced by calculation of the spectral function in a one-step model.

Ô Seigneur; Soleil et Lune, tous les astres de lumière, sur ton ordre ils furent créés ; c'est toi qui posa pour toujours une loi qui ne passera pas. Pour toujours ta parole se dresse dans les cieux. Étonnantes sont tes oeuvres, toute mon âme le sait. Savoir prodigieux qui me dépasse, hauteur que je ne puis atteindre. À voir ton ciel, ouvrage de tes doigts, la lune et les étoiles que tu fixas ; renaît toujours mon émerveillement.

En physicien, je m'astreins à étudier le monde à la lumière des sciences naturelles. Malgré la Vérité qui nous reste voilée, nos lois et modèles limités ; il semble toujours qu'elle se rapproche mais demeure inaccessible. Si le pouvoir prédictif d'une théorie ne nous informe guère de la validité de ses représentations dans l'absolu, la remarquable adéquation entre expérience et théorie instille en moi la tentation de confondre le modèle et le réel. Ces trois ans et demi de thèse auront été l'occasion de me confronter à des aspects bien étranges de la nature. Était-ce vraiment la volonté de Dieu que les électrons obéissent aux lois quantiques ? Qu'ils parfois s'organisent en exquises symétries ? Je ne le saurai jamais dire, tout ce que je peux ici faire, c'est poser ma gratitude d'avoir pu être le témoin de telles manifestations.

Анастасия викторовна, пить чай с тобой всегда радость. мне очень нравится, когда ты рассказываешь твою историю. твоя перспектива очень интересная. я тоже люблю твой стиль немного мятежный, очень классно. Анастасия владимировна, так энергичная ! Спасибо за уроки русского и время за исправить мою фонологию! Beatriz Barata, discreta mas direta. É sempre um prazer ver-te e falar. Eu sou só um impostor, tu es a vera poliglota. Veramente brilhante! Eu compreendo por que a industria do petróleo te queria! Boa sorte para a fim de tesis. Rebecca Cervacio, a franco-brasileira mais legal! Você fez tudo para o grupo de estudantes do síncrotron. Organizar as saídas, preparar as reuniões, tentando criar um verdadeiro espirito de comunidade entre nós. Toda a coragem (e zen no seu caso!) para a conclusão do doutorado, você é a proxima na lista. Myriam Lachheb, j'espère que tu auras apprécié notre aventure italienne autant que moi (le PV aussi). Je te remercie d'avoir eu la patience d'apprendre à me connaître malgré mon politiquement incorrect. Je peux être insupportable mais tu sais bien le rendre. Je te souhaite le

Introduction

Transition metal dichalcogenides are a class of materials that have been heavily scrutinized by thousands of physicists for the last few years. With their formula MX 2 , where a metal is covalently bonded to two chalcogens (M = Mo, W; X = S, Se, Te), one could wonder how materials with such a simple composition could be the object of such a tremendous interest. And for years, even decades, these materials were absolutely not at the center of the attention. One of the most known variant, MoS 2 , was mostly used as mechanical lubricant. Everything changed with the discoveries done on graphene by A.Geim and K.Novoselov. These works, that were rewarded by a Nobel prize in physics in 2010, definitely established the domain of 2D materials. Among other properties, not only graphene had one of the highest conductivity measured to date, and its band structure was showing peculiar properties such as Dirac like dispersion. The only problem with graphene was and remains its absence of gap. This limited its use in optoelectronics in the visible range, and therefore its application in for e.g. in photovoltaics. This is at this point that TMD emerged as semiconductive alternatives to graphene. Like graphite, TMD in their bulk form are lamellar structures composed of individual X-M-X planes weakly bonded to one another by van der Waals (vdW) interactions. Moreover, they even share the characteristic honeycomb structure. This meant that the use of the methods developed to fabricate and modelize graphene were readily extendable to TMD, in addition with decades of investigations on semiconductors. From this point on, TMD as 2D materials proved themselves to be very promising for photonics and electronics. First of all, they show a transition from an indirect to a direct band gap when reaching the monolayer limit. Secondly, they have a large exciton binding energy. This means that from the control of the thickness, one can tune the optical properties of TMD, such a modularity being interesting for photovoltaic applications. On the other hand, the presence of high Z elements ensured strong spin-orbit coupling that could be used for spintronics. Overall, TMD revealed themselves as an ideal platform for experimentation. As simple materials with rich physics, and owing to their ultra thin character, their properties can be easily modified by proximity effects and external stimuli.

A particularly appealing direction consists in coupling TMD with electrically polarized materials such as ferroelectrics. This would enable a remanent modulation of their electronic properties and pave the way towards non-volatile memory devices such as ferroelectric field effect transistors (FeFET) with giant OFF/ON ratios. This thesis reports our attempts to control the properties of TMD using ferroelectric materials. Our aim was to produce hybrid structures composed of thin films of TMD on ferroelectric substrates and analyze them by the means of photoemission, a technique based on the photoelectric effect. The tool box of photoemission gives us access to the chemical state (X-ray Photoemission Spectroscopy), the properties of the valence band (Angle-resolved Photoemission Spectroscopy) and the local surface state (Photoelectron Emission Microscopy). They are, however, best performed in a synchrotron facility that offers the proper photon source for precision measurements.

This thesis was done at the Synchrotron SOLEIL (Saint Aubin, France), at the CAS-SIOPEE beamline specialised in photoemission spectroscopy. The work was done in close collaboration with the IRIG-SPINTEC lab of the CEA (Grenoble, France), the Unité Mixte de Physique CNRS-Thalès (Palaiseau, France) and the Centre de Nanosciences et Nanotechnologies (Palaiseau, France). We also collaborated with the Laboratoire d'Etude des NanoStructures et Imagerie de Surface (LENSIS), part of the Service de Physique de l'Etat Condensé (SPEC) of the CEA (Saclay, France) for the preparation and optimization of the photoemission microscopy experiments. We performed most of the photoemission spectrocopy measurements at the CASSIOPEE beamline. Additional measurements were done in ANTARES beamline at Synchrotron SOLEIL. Photoemission microscopy was done at the NanoESCA beamline at Elettra Sinctrotrone (Trieste, Italy). This report is organized in five chapters:

• Chapter 1 is an introduction on photoemission. The link of photoemission with the study of strongly correlated electrons systems is established along with the main mathematical tools. Different models of photoemission are presented from the simplest picture (one electron picture), to sophisticated descriptions (many-body picture).

• Chapter 2 introduces photoemission from an experimental point of view. This chapter details the production of synchrotron radiation, the light source used throughout this thesis. The detection system is presented in a general way before introducing the specifics of each photoemission technique that we used (X-Ray Photoemission Spectroscopy, Angle-Resolved Photoemission Spectroscopy, Photoelectron Emission Microscopy).

• Chapter 3 introduces the ferroelectricity with different models (phenomenological and microscopic) before addressing the issue of the charge screening of ferroelectric material surfaces. The ferroelectric materials used in this thesis are then presented in detail: the lithium niobate LiNbO 3 and bismuth ferrite BiFeO 3 . The main characterization techniques for ferroelectric materials are also explained.

• Chapter 4 presents the electronic structure of transition metal dichalcogenides through the example of WSe 2 , the TMD studied in this work and the main models in a first part. After a review of photoemission on WSe 2 , the second part of the chapter is dedicated to the study by photoemission of the band structure of WSe 2 of increasing thickness. With the help of theoretical calculations of the band structure we solve the problem of "missing bands" in the band structure.

• Chapter 5 presents our efforts to fabricate and characterize TMD-Ferroelectric hybrid structures. The chapter begins with a review of the state of the art on such systems. We then present the results of the first system: MBE grown centimeter scale thin-films of WSe 2 flakes transferred onto BiFeO 3 thin-films studied by photoemission spectroscopy. Thereupon, we show the results of our combined microscopic-spectroscopic photoemission study on microscopic WSe 2 transferred on bulk ferroelectric LiNbO 3 .

There are two proposed reading orders:

-Chapter 1, chapter 2, chapter 4, chapter 3 and chapter 5. With this order, the reading experience is separated in two blocks. The first one introduces photoemission and directly hops in the characterization of the band structure of WSe 2 . The second block begins with the explanation of ferroelectricity as a prerequisite to our work on hybrid TMD-Ferroelectric systems. This order is most adapted to readers eager to see the results of the thesis.

-Chapter 1, chapter 2, chapter 3, chapter 4, and chapter 5. With this order, the fundamental aspects of the thesis are presented first (the materials and the techniques) and then only the results. The chapter about TMD smoothly introduces our first measurements with angle-resolved photoemission spectroscopy, begining the presentation of the results. We then show the rest of our findings on the work about hybrid TMD-Ferroelectric systems. This order offers a slower reading experience, adapted to readers beginning on the subject.

-Photoemission and condensed matter systems

. Introduction

The main experimental technique used in this thesis is photoemission. It is based on the photoelectric effect. Photons are used to excite electrons in the solid. Some of these "photoelectrons" can escape in the vacuum where they can be collected by an electron analyzer able to determine their energy and momentum (sometimes even their spin) outside the solid. Thanks to conservation laws, the experimentalist can describe the initial state of these electrons, prior to the excitation by the photon.

A photoemission result can be either extremely simple or incredibly complex to interpret. In simple cases, a one-electron picture is sufficient: only the photoelectron is considered while the rest of the system can be seen as frozen. On the other hand, in correlated systems, one has to take all the electrons into account and describe the evolution of the whole system in the presence of the hole left in the electronic structure. Formulating an accurate description of this phenomenon is a many-body problem which solid states physicists have been facing for decades.

In this chapter, we will start by introducing some notions of the multi-electron problem: how can we write a Hamiltonian describing these systems and how can the Green function formalism help solve it. We will then describe the photoemission process, first in the classical three-step model, and then in the more sophisticated one-step model, used in this thesis to interpret some of our results. Last, we will introduce Angle-Resolved Photoemission Spectroscopy (ARPES), an experimental technique allowing to map the band structure of materials.

. General aspects of the multi-electron problem 1.2.1 . The multi-electronic Hamiltonian

Under the assumption of frozen atomic positions, the most general Hamiltonian for a multi-electronic system writes:

H = N n H 0 (r n ) + 1 2 N n̸ =m U (r n , r m ) (1.1)
Where H 0 is the differential operator corresponding to either the non-relativistic Schrödinger equation:

H 0 = - h2 2m e ∇ 2 + V ext (r) (1.2)
Or the relativistic Dirac equation: (1.4) with the four 4x4 Dirac matrices α and β. V ext and A ext are the potential and vector potential generated by all the nuclei and U (r, r ′ ) the Coulomb interaction between two electrons.

H 0 = -ihcα • D + βmc 2 + V ext (r) (1.3) D =∇ -i e hc A ext

. The field operators: a tool for many body systems

In the context of quantum field theories, field operators allow for a simplified route to describe systems with many particles. The field operator ψ and its conjugate ψ † respectively annihilates and creates an electron at a given position. They are defined as follows:

ψ † (r) |0⟩ = |r⟩ (1.5) ψ(r) |r ′ ⟩ =δ(r -r ′ ) |0⟩ (1.6) ψ(r) ψ(r ′ ) = -ψ(r ′ ) ψ(r) (1.7) ψ(r), ψ † (r ′ ) + =δ(r -r ′ ) (1.8)
Since the field operators for fermions are anticommutative, they encodes the fermionic statistics and in particular the Pauli exclusion principle. Using these, one can easily write multielectronic states by stacking creation operators N i=0 ψ † (r i ) |0⟩ = |r 0 , ..., r N ⟩. These states will describe indistinguishable particles with the correct statistics. It is to be noted that this formalism is not restricted to position states but can be generalised to any complete family of states. The family {|n⟩} will have the associated creation and annihilation operators satisfying similar relations:

ĉ † n |0⟩ = |n⟩ (1.

. Writing operators for many body systems

An operator O 1 acting on a single particle can always be extended into an operator O acting on the Hilbert space of the many body system with the following formulae (1.14) and (1.17). For a continuous variable such as position the formula is:

O = drdr ′ O 1 (r, r ′ ) ψ † (r ′ ) ψ(r) (1.14)
If the operator is diagonal in position representation, it reduces to:

O = drO 1 (r) ψ † (r) ψ(r) (1.15)
For a general family of states {|n⟩} the formula is:

O = mn O mn c † m c n (1.16)
O mn = ⟨m|O 1 |n⟩ (1.17)

. The multielectronic Hamiltonian: restatement

The Hamiltonian in equation (1.1) can be rewritten using the field operator ψ(r) and its conjugate ψ † (r). The multielectronic Hamiltonian becomes [1]:

H = dr ψ † (r)H 0 ψ(r) + 1 2
drdr ′ ψ † (r) ψ † (r ′ )U (r, r ′ ) ψ(r ′ ) ψ(r) (1.18) or again for a family of states {|n⟩}: 19)

H = mn ε mn c † m c n + 1 2 mnop U mnop c † m c † n c o c p (1.
ε mn = drψ * m (r)H 0 ψ n (r) (1.20
)

U mnop = drdr ′ ψ * m (r)ψ * n (r ′ )U (r, r ′ )ψ o (r ′ )ψ p (r) (1.21)
This reformulation is at the basis of all calculation schemes. One can derive increasingly complex theories by making assumptions on the expectation value of ψ † (r ′ ) ψ(r ′ ). Among these are the Hartree-Fock method and other mean-field theories. One can also map the multielectron problem to an equivalent single body problem [2]. This mapping has the attractive property of being exact for all ground state properties and is the basis of all the density functional theory (DFT) approaches. With further approximations, one can derive the Kohn-Sham equations [3]. DFT methods based on the Kohn-Sham theory have had lot of success in predicting the band structure of many materials. Finally, one can tackle the problem from the angle of Green function formalism. While being usually more computationally expensive than DFT, Green function based methods have led to promising results for the computation of the band structure of materials in which electron correlations have a dramatic impact on the band structure [4; 5].

. Green function methods

. Green function formalism for a one-particle Hamiltonian Basic definitions

The formalism of Green functions is of special utility when it comes to solve boundary value problems. For a given differential operator D, there may exist several Green functions G such that: D(x)G(x, x ′ ) = δ(x -x ′ ). In some sense, the Green function is the inverse operator to the differential operator. Knowing the Green function associated with a differential operator will allow to propagate a solution at given point x to another point x ′ . That is why Green functions are also known as propagators. This approach is particularly fruitful when applied to the Schrödinger equation. Instead of solving the standard eigenvalue problem:

[E -H] ϕ(x) = 0 (1.22)
One can try to solve the following equation:

(E -H) G(x, x ′ , E) =δ(x -x ′ ) (1.23)
Or equivalently, in any basis:

(E -H) G(E) =I (1.24)
It is useful to introduce a complete basis |n⟩ in which the Hamiltonian is diagonal such that H |n⟩ = E n |n⟩ with the wavefunctions ψ n (x) = ⟨x|n⟩. In that case we can write:

n,m (E -E n )G nm (E) |n⟩ ⟨m| = n |n⟩ ⟨n| (1.25)
Solving for G we obtain:

G(E) = nm δ nm E -E n |n⟩ ⟨m| (1.26)
As defined in (1.26), the Green function G does not fulfill any particular boundary condition.

It is the choice of the contour of integration that will fix the time ordering of the propagation.

One can equivalently encode this choice of contour by evaluating the function G ± (E) = lim η-→0 +

G(E ± iη). G + and G -are respectively the retarded and the advanced propagators.

They can be written as:

G ± (E) = lim η-→0 + n |n⟩ ⟨n| E -E n ± iη (1.27)
If one knows the state of the system at a given time, the retarded (advanced) propagator predicts (recovers) the state of the system in a future (previous) time. With this in hand, we can calculate the time dependent propagator by performing a Fourier transform of expression (1.27):

G ± (t, t ′ ) = dE 2π e -iE(t-t ′ ) G ± (E) (1.28) = dE 2π e -iE(t-t ′ ) n |n⟩ ⟨n| E -E n ± iη (1.29) = ± i n |n⟩ ⟨n| e -iEn(t-t ′ ) θ(±(t -t ′ )) (1.30)
with θ(t) the Heaviside function.

We are usually interested in the spatial Green function defined as:

G ± (x, x ′ , E) = ⟨x|G ± (E)|x ′ ⟩ (1.31)
Evaluating the expression (1.31) using (1.27) we get:

G ± (x, x ′ , E) = lim η-→0 + ⟨x| nm δ nm E -E n ± iη |n⟩ ⟨m| |x ′ ⟩ (1.32) = lim η-→0 + n ψ n (x)ψ * n (x ′ ) E -E n ± iη (1.33)
In this case the time dependent propagator is given by:

G ± (xt, x ′ t ′ ) = ⟨x|G ± (t, t ′ )|x ′ ⟩ (1.34) = ± i n ψ n (x)ψ * n (x ′ )e -iEn(t-t ′ ) θ(±(t -t ′ )) (1.35)

Green function of a perturbed system

When the Green function of a system is known, it can be used to solve more complex Hamiltonians. Let G 0 be the Green function of the Hamiltonian H 0 , for a Hamiltonian H = H 0 + V the Green function of the full system is given by the following equation:

(E -H 0 -V ) G(E) =I (1.36)
The equation (1.36) can be solved formally:

(E -H 0 ) G(E) -V G(E) =I (1.37) G -1 0 (E)G(E) -V G(E) =I (1.38) G(E) -G 0 (E)V G(E) =G 0 (E) (1.39) G(E) =G 0 (E) + G 0 (E)V G(E) (1.40)
The equation (1.40) is also often written as:

G(E) =G 0 (E) + G 0 (E)V G(E) (1.41) G(E) =G 0 (E) + G 0 (E)T (E)G 0 (E) (1.42)
By introducing T (E) the scattering operator. It can also be solved formally with an infinite series if we recursively inject the expression of G(E) in itself:

G(E) =G 0 (E) + G 0 (E)V G 0 (E) + G 0 (E)V G 0 (E)V G 0 (E) + . . . (1.43) = n [G 0 (E)V ] n G 0 (E) (1.44)

The spectral function A(E)

This Green function G(E) is related to what is called the spectral function A(E) of the system. This quantity is defined as:

A(E) = - 1 π ImG(E) (1.45)
This definition is to be understood as the limit case of:

A ± (E) = ∓ 1 π ImG ± (E) (1.46) = ∓ lim η-→0 + 1 π Im n |n⟩ ⟨n| E -E n ± iη (1.47) = n lim η-→0 + 1 π η (E -E n ) 2 + η 2 (1.48) = n |n⟩ ⟨n| δ(E -E n ) (1.49)
We see that TrA ± (E) = n δ(E -E n ) is the density of states of the system. With the spectral function A(E) we see how an abstract property of the system (its Green function) is related to physical observables. This will prove useful when applied to multi-electron systems.

. Green function for a many body Hamiltonian Green function of a multielectronic system

We now have the tools to write the Green function of a multielectronic system1 . This will help to describe photoemission, where an electron is emitted out of the system after an interaction between light and matter. We follow here the derivation of [6] of the time dependent spatial Green function. It is defined as the overlap integral between a N -particle state where an electron is removed at a time t ′ and position x ′ with another N -particle state where an electron is removed at time t and position x. Such a state can be written using the Heisenberg representation of field operators as defined in equations (1.50) and (1.51).

ψ(x ′ , t ′ ) =e iHt ′ ψ(x ′ )e -iHt ′
(1.50)

ψ † (x, t) =e iHt ψ † (x ′ )e -iHt (1.51)
The Green function then writes:

G(xt, x ′ t ′ ) = θ(t -t ′ ) ⟨N | ψ † (x, t) ψ(x ′ , t ′ ) |N ⟩ (1.52)
We introduce the basis {|N, n⟩}, which diagonalizes the N -electron Hamiltonian as follows:

H |N, n⟩ = E(N, n) |N, n⟩ (1.53)
Since a state with N -electrons cannot be written with states with a different number of electrons, the basis is fully diagonal, i.e. ⟨M, m|N, n⟩ = δ M,N δ m,n . We denote the N -electron ground state |N ⟩ and its energy E(N ). The closure relation is therefore:

I = N n |N, n⟩ ⟨N, n| (1.54)
By inserting (1.54)

= n θ(t -t ′ )e i(E(N )-E(N -1,n))(t-t ′ ) ⟨N | ψ † (x) |N -1, n⟩ ⟨N -1, n| ψ(x ′ ) |N ⟩ (1.57)
This provides access to the excitation energies of the system ε n = E(N ) -E(N -1, n). We can then compute the propagator G(x, x ′ , E) by Fourier transformation of G(xt, x ′ t ′ ).

G(x, x ′ , E) = n ⟨N | ψ † (x) |N -1, n⟩ ⟨N -1, n| ψ(x ′ ) |N ⟩ E -(E(N ) -E(N -1, n)) -iη (1.58)
G(x, x ′ , E) can finally be brought to a form similar to the one-electron Green function definition (1.33) by defining the photoemission amplitudes:

ϕ n (x ′ ) = ⟨N -1, n| ψ(x ′ ) |N ⟩ (1.59) ϕ * n (x) = ⟨N | ψ † (x) |N -1, n⟩ (1.60)
This gives the so called Lehmann representation of the Green function [7]:

G(x, x ′ , E) = n ϕ * n (x)ϕ n (x ′ ) E -ε n -iη (1.61)
In general any Green function

G lm (t, t ′ ) = θ(t -t ′ ) ⟨N | c † l (t)c m (t ′ ) |N ⟩ (1.62)
will have such a representation with appropriate amplitudes ⟨N -1, n| c l |N ⟩ and ⟨N | c † m |N -1, n⟩. This applies in particular in the momentum basis which we will use from now on:

G(k, k ′ , E) = n ϕ * n (k)ϕ n (k ′ ) E -ε n -iη (1.63)
The spectral function A(k,E) Similarly to the definition given in (1.45) for general Hamiltonian, we can summarize the system with the spectral function A(k, k ′ , E). This quantity is defined as:

a k E E F E E F k b Incoherent background Coherent quasiparticle peak ε(k) ε'(k)
A(k, k ′ , E) = 1 π ImG(k, k ′ , E) (1.64) = n ⟨N |ĉ † k ′ |N -1, n⟩ ⟨N -1, n|ĉ k |N ⟩ δ(E -ε n (k)) (1.65) = n ϕ * n (k)ϕ n (k ′ )δ(E -ε n (k)) (1.66)
The most important properties of the system are contained in the diagonal

(k = k ′ ) spectral function A(k, E) = n |ϕ n (k)| 2 δ(E -ε n (k)).
In the case of a system of quasi-particles, the dispersion ε n (k) stays close to the non interacting system dispersion ϵ n (k) up to a renormalisation factor Z and is given a finite lifetime Γ. For a single band, the spectral function writes as follows:

A(k, E) = 1 π ZΓ (E -ε(k)) 2 + Γ 2 + (1 -Z) [Incoherent background] (1.67) ε(k) = ϵ(k) + δϵ (1.68)
The finite lifetime is associated with a Lorentzian broadening of the peak. The renormalization "constant" Z is associated with the apparition of an incoherent background in the spectral function. Since Z < 1, it can be interpreted as a spectral weight giving more or less importance to the coherent part of the spectral function [9; 10]. This idea is usually generalised by introducing the self-energy Σ = Σ ′ +iΣ ′′ . The self-energy is a complex object that encapsulates both:

• renormalization effects (different binding energies/bands changing shape comparatively to the non-interacting case) through its real part Σ ′

• broadening effects through its imaginary part Σ ′′

The spectral function writes consequently:

A (k, E) = 1 π Σ ′′ (k, E) (E -ϵ(k) -Σ ′ (k, E)) 2 + Σ ′′ (k, E) 2 (1.69) Σ (k, E) =Σ ′ (k, E) + iΣ ′′ (k, E) (1.70) ε(k) =ϵ(k) + Σ ′ (k, E) (1.71)
The self-energy Σ is obtained when deriving the equation of motion for the propagator G(kt, k ′ t ′ ) from the multielectronic Hamiltonian. The equation gives the so-called Dyson equation (equation 1.72) which relates the full propagator G and the Green function of the non-interacting problem G 0 in a similar way to the perturbation seen in equation (1.40). The derivation of the Dyson equation can be found in [11].

G = G 0 + G 0 ΣG (1.72)
This essentially shows how powerful the Green function formalism is, since we were able to, albeit formally, reduce a complex strongly interacting multi-electron problem to the computation of a single quantity Σ that summarizes the effect of those interaction on a more readily tractable single electron problem. Not only that, it also opens the way to the computation of response functions. We will show in the next section how this approach helps understanding photoemission as a process and also offers pathways to compute results close to experiments in chapter 5. Let's now describe the photoelectric effect and the experimental technique taking advantage of it: photoemission.

. The photoelectric effect: Historical excerpts

The photoelectric effect is a process involving the emission of electrons when matter interacts with electromagnetic radiations of sufficient energy. The first evidences of the photoelectric effect were gathered in the second half of the 19th century amid the numerous experiments investigating the nature of light and matter and their interactions. One notable evidence came as a byproduct of Hertz studying electric sparks generated by electromagnetic waves. While demonstrating their existence, he also observed that by encasing the spark generator, it was more difficult to generate a spark in the detector (figure 1.2). Investigating further by placing windows of materials with different UV transmittance on the path of the radiation, he identified that UV light had an impact on the production of sparks without being able to explain why [12; 13]. The first proper characterization of the photoelectric effect was done by Lenard in 1899. Working in high vacuum and with several light sources (spark lamp -Funkenlicht, carbon arc lamp -Kohlebogenlampe, zinc arc lamp -Zinkbogenlampe) he generated UV radiations to illuminate a metallic plate. The plate would emit a beam that his experiments showed to have the properties of the so called cathode-rays (Kathodenstrahlen), i.e. electrons. The rays could be deflected with a magnet, had a negative charge and had the appropriate charge-on-mass ratio [14; 15; 16; 17]. By controlling the acceleration voltage, he could tune the kinetic energy of the cathode-rays and therefore determined that:

• The photoemitted current was proportional to the intensity of the light • The kinetic energy was not dependent on the intensity of the UV light • The kinetic energy was dependent on the frequency of the source In 1905, Albert Einstein proposed a theory of quantized light (photons) to explain, among others, the results of Lenard [18]. In his work he introduced the famous E = hω equation [19; 20]. With his theory, an electron would be emitted as soon as the energy of the photon interacting with it is higher than its binding energy, the rest being converted into kinetic energy by energy conservation. In other words: Einstein was thoroughly tested by Millikan. His goal was to determine the value of the h constant. He improved on Lenard's methodology with better vacuum, control of the photon energy with filters and took great care of the quality of the surface of his samples. By testing different metals he provided a first determination of the work function (usually noted Φ) of those metals: the energy necessary to extract an electron of the material with kinetic energy E k = 0 [21]. All these experimental and theoretical results are at the origin of the equation that we will use throughout this work:

E k ≈ hω -E B .
E k = hω -|E B | -|Φ| (1.73)

. The minimal model: one-electron photoionization

Let an electron with a spectrum H 0 |n⟩ = E n |n⟩ interacting with a photon of energy hω by the interaction potential H int . In the approximation of an instantaneous interaction, Fermi's golden rule tells us that the photoemission cross-section (and therefore the photoemitted intensity) is:

I(ω) ∝ n | ⟨f |H int |n⟩ | 2 δ(E f -E n -hω) (1.74)
The electron in the initial state |n⟩ is propelled into the final state |f ⟩ of energy E F that belongs to the continuum of states above the Fermi energy E F as illustrated in figure 1.3. Since the energy of the outgoing electron is directly tied to its binding energy inside the solid, this picture readily allows for interpretation: the photoemission process can be seen as a direct measurement of the photoemission spectrum of the electron. While appealing, the model falls short as it completely ignores the electronic interactions induced by the perturbation of the system. Many intermediate processes can change the final state of the system and induce new or modified energy states in the measured spectrum. The three-step model is an approximation in which the photoemission process is broken down as follows:

E n ℏω a b c E k ℏω -E n E k E k 0 E F E n ℏω
• A photoexcitation step where the electron interacts with the incoming photon and gains energy, transitioning from the initial state to the final state while staying inside the crystal.

• A transport phase where the electron propagates from the bulk of the material to its surface. In this step, inelastic scattering involving other electrons will build up a secondary electrons background.

• An emission step in which the electron jumps into the vacuum.

Such a process is illustrated in figure 1.4. We will explicit each step in the following discussion, taking into account the many body aspects of the problem.

. Step 1 -Photoexcitation

The real system is composed of many electrons. Photoemission is the process that emits an electron from a N -electron ground state |N ⟩ into all the possible excited states of the N -1 electron system |N -1, n, k⟩. These excited states are indexed by the quantum number n, and k is the wavevector of the ejected electron. The conservation of energy gives:

E(N ) + hω = E(N -1, n) + ε k (1.75)
The Fermi's golden rule writes:

I(ω, k) ∝ n |⟨N -1, n, k|H int |N ⟩| 2 δ(ε k + E(N -1, n) -E(N ) -hω) (1.76) ∝ n |⟨N -1, n, k|H int |N ⟩| 2 δ(ε k -ε n -hω) (1.77)
With a general interaction Hamiltonian written as:

H int = ij ⟨i|H 1 int |j⟩ ĉ † i ĉj (1.78) = ij M ij ĉ † i ĉj (1.79)
This formula describes how the one-electron interaction Hamiltonian H 1 int acts on the many body states (see equation (1.17)). While this expression generally holds true, several approximations are made to simplify its interpretation.

Approximations on the final state

One can approximate the final state in many ways. We will follow a progression in the spirit of the work of Hedin [24; 25] taking increasingly complex final states. The one-electron theory described in section 1.5 uses the final state |N -1, n, k⟩ = ĉ † k ĉn |N ⟩ (fully factorized wave function). A less stringent approximation is |N -1, n, k⟩ = ĉ † k |N -1, n⟩, comparatively to the previous one, the state |N -1, n⟩ is not an eigenstate of the Hamiltonian. The quantity ⟨N -1, n, k|H int |N ⟩ then becomes:

⟨N -1, n, k|H int |N ⟩ = ⟨N -1, n|ĉ k ij M ij ĉ † i ĉj |N ⟩ (1.80) = - ij M ij ⟨N -1, n|ĉ † i ĉj ĉk + δ ki ĉj |N ⟩ (1.81) ≈ - j M kj ⟨N -1, n|ĉ j |N ⟩ (1.82)
The last step assumes that the emitted electron does not interact with the rest of the system. It is the so-called sudden approximation. In this case the photoemitted intensity writes:

I(ω, k) ∝ n j M kj ⟨N -1, n|ĉ j |N ⟩ 2 δ(ε k -ε n -hω) (1.83) ∝ n   j M kj ⟨N -1, n|ĉ j |N ⟩   i M * ik ⟨N |ĉ † i |N -1, n⟩ δ(ε k -ε n -hω) (1.84) ∝ ij M kj n ⟨N -1, n|ĉ j |N ⟩ ⟨N |ĉ † i |N -1, n⟩ δ(ε k -ε n -hω)M * ik (1.85) (1.86)
Using (1.66) we get:

I(ω, k) ∝ ij M kj A ji (ε k -hω)M * ik (1.87)
If we keep only the diagonal terms of the spectral function:

I(ω, k) ∝ i |M ki | 2 A ii (ε k -hω) (1.88)
This derivation allows for a simple interpretation of the photoemission current. The photocurrent is the product of the matrix element |M ki | 2 and the spectral function A ij studied in section 1.3.2. The matrix element is non-zero only when the initial and final states as well as the interaction Hamiltonian have common symmetries. The symmetries in question will depend on the details of the Hamiltonian and ultimately, on the geometry of the detection. In the case of core levels |n, l, m⟩, one obtains the well known dipolar selection rules: ∆m = 0, ±1 and ∆l = ±1. Alternatively, with Bloch states, the selection rules translate the crystal symmetries [26; 27]. In addition to these considerations, the matrix element also incorporates the polarisation and photon energy dependence of the photoemission cross-section. The photoionization cross section data in figure 1.5b shows that cross section of valence states such as W 5d is high at low photon energies and very low at high photon energies. It decays dramatically of three orders of magnitude for hω ∈ [100, 200] eV. Because of that, one generally has to carefully select the photon energy when possible.

Approximations on the interaction Hamiltonian

The one-electron interaction Hamiltonian H 1 int can be taken as either relativistic or not. In the non-relativistic case it has the form:

H 1 int = e 2m (A • p + p • A) + e 2 2m A 2 + e m S • (∇ × A) (1.89)
In the relativistic case, it writes [29]:

H 1 int = ecα • A (1.90)
With A being the vector potential.

In the three-step model, the transition is assumed to happen inside the solid so that the variations of the electromagnetic field at the surface of the sample are ignored giving ∇•A = 0. The interaction plane contains the wave vector of the electromagnetic plane wave and the impulsion of the outgoing electron, adaptated from [8]. (b) Total photoionization cross section of a W -atom with partial photoionization cross section of W 5d valence state and W 4f core level. Data taken from [28].

Two photons processes implied by the A 2 term are neglected, further simplifying the expression. Since the source is assumed to be monochromatic as it is the case of most experimental setups, we have:

A(r) = A 0 e iq•r (1.91)
For low photon energies (UV and soft X-rays), the dipolar approximation can be made. The lower the energy, the larger the wavelength (see 1.1: Photon energies and associated wavelengths.

. Step 2 -Transport to the surface

After interacting with the photons, the electrons are transported to the surface of the material. In the way, many of them are scattered away from their original path, loosing energy in inelastic scattering processes and thereby producing a significant proportion of so-called secondary electrons. The distance an electron is expected to cross in between two scattering events is the inelastic mean free path. In the strictest sense, this quantity is material dependent but it has been shown that it was mostly dependent on the energy of the electron, giving the universal inelastic mean free path curve that can be seen in figure 1.6b [30; 31]. Using this value and the Beer-Lambert attenuation law, one can estimate the portion of photoelectrons that successfully comes to the surface. The law is formulated as in (1.92). The collected intensity dI coming from a band of thickness dz at the depth z with emission angle θ and inelastic mean free path λ is:

dI(z) = I 0 exp - z λ sin θ dz (1.92)
The described situation is schematically represented in figure 1.6c. This curve shows that 95% of the signal comes from 3λ inside the material. This demonstrates the surface sensitive nature of photoemission techniques in the range of UV/Soft X-Rays regimes where the probed depth is of tens of angströms. Inelastic Mean Free path for electrons with data points from reference [30], the fitted curve in plain color has the equation λ IM F P = 1430E -2 + 0.54 √ E [31]. (c) Emission geometry and contribution to the photoemitted signal (in %) of the sample as a function of the depth z.

. Step 3 -Emission into the vacuum

After transport, the electron has to leave the surface. The potential felt by the electron is typically lower in the crystal than outside, meaning it has to overcome a potential barrier (cf. figure 1.7.a). For that, energy must be consumed. This energy is related to the work function of the material. The work function is the energy necessary for the electron to be removed from the surface of a material, so far away that it does not experience any potential due to the material anymore but close enough so that it is not under the influence of external potentials yet. At the surface, the electron experiences the atomic potential of the last atoms and also interacts with other electrons. This means that it experiences the full one-electron potential, or in other words, the electrostatic potential at the surface added to the exchange-correlation potential [32]. In an experimental context, the surface is usually modelled as a sharp step with a depth V 0 = µ + Φ S with µ the chemical potential and Φ S the work function (cf. figure 1.7.b). This sharp step forms a boundary that implies the refraction of the electron on the surface (cf. figure 1.7.c). This refraction has important consequences on the measurement of valence state photoelectrons that we will describe in the following section.

k f p k f,|| p || = k f,|| k f,⟂ p ⟂ E z E vac E F μ Φ s a c V 0

Core levels

Valence states

k i k f =k i +G G ℏω ℏω E i E F E f V 0 E B E kin E kin Φ s E vac E 0 π _ a - π _

. Three-step model: application to ARPES

Valence states inside a crystal are modelled as Bloch waves, which are periodic functions of the crystal. Those states (noted |n, k⟩) are labelled with their band index n and crystal momentum k defined up to a vector of the reciprocal lattice. As a result, the electrons organise in what is called the band structure where the energy levels are dependent on the momentum k. The most appropriate tool to study those valence states is angle resolved photoemission spectroscopy. We will see that the interaction of a photon with an electron is a momentum conserving process and therefore, one can take advantage of the angular resolution of the analyzer to relate the outgoing momentum p (see figure 1.8a and 1.7c) and kinetic energy E kin of the electron to its crystal momentum k f (see figure 1.8b and 1.7c) and binding energy. In practice this means that one can effectively map out the dispersion relation E(k). One can do such an analysis using the three-step model presented in 1.6. Let a photon noted γ(hω, q) with energy hω and momentum q interact with an electron in the state |n, k i ⟩ whose initial crystal momentum is k i . In the three-step model, the optical transition occurs inside the crystal, where the conservation of momentum will be allowed up to a reciprocal vector G. The electron is propelled to the state |m, k f ⟩ with final crystal momentum k f . For the transition γ(hω, q) + |n, k i ⟩ -→ |m, k f ⟩ we can write the crystal momentum conservation:

q + k i + G = k f (1.93)
For photon energies in the UV range, the photon momentum q can be neglected. Photoemission of valence states performed at higher photon energy where q becomes larger will need more refined analysis as done in ref [33]. The second important event is the crossing of the surface by the excited electron. Indeed, the surface of the sample is characterised by a discontinuity of the potential which induces electron refraction as discussed earlier. This makes that only the component of the momentum parallel to the surface is conserved as depicted in figure 1.7c. Finally, when the electron is far away from the sample, it can be safely considered as a free electron. Put into equation, this gives:

E f = h2 k 2 f 2m -|E 0 | (1.94) (1.95)
Where we consider that the final state is described by a free electron parabola whose bottom energy is an unknown E 0 (see green parabola in figure 1.8b). On one hand we have:

E kin = E f -|Φ| = h2 k 2 f 2m -|E 0 | -|Φ| (1.96) = h2 k 2 f 2m -V 0 (1.97)
On other hand, with an outgoing free-electron, we can write:

E kin = p 2 2m (1.98)
Which implies:

h2 k 2 f 2m = p 2 2m + V 0 (1.99)
With parallel momentum conservation:

hk f ∥ =p ∥ (1.100)
and perpendicular projection of equation (1.99):

h2 k 2 f ⊥ 2m = p 2 ⊥ 2m + V 0 (1.101)
we can work out the relations connecting the initial and final states provided that we have an expression for p. In an ARPES experiment, as sketched in figure 1.8a, one measures the kinetic energy of the photoelectron (connected to p by equation 1.98) and its emission direction (parametrized by the angles θ and ϕ in figure 1.8a). Thanks to these measurements, p is completely determined:

p = 2m h2 E kin   sin θ cos ϕ sin θ sin ϕ cos θ   (1.102)
Since the final momentum k f is essentially the initial momentum k i up to a reciprocal wave vector, we relax the notation of the momentum to k. This finally gives:

k ∥ = 2m h2 E kin sin θ cos ϕ sin ϕ (1.103) k ⊥ = 2m h2 (E kin cos 2 θ + V 0 ) (1.104)
At the end of the chapter 2, dedicated to the experimental aspects of photoemission, we will see how these formula help us navigating in the reciprocal space and describe any part of the band strucure of a solid by changing either the photon energy or the electron detection angles.

. The one-step model 1.8.1 . The essence of the one-step model

While the three-step model helps understanding photoemission to a reasonable level, the separation between the different stages of the process remains artificial. In particular, difficulties arise when considering phenomena near the surface. As discussed by Caroli [34], the three-step model is not entirely satisfactory from a conceptual point of view, it is hard to provide a clear cut between the steps and it misses some unavoidable scattering. In that matter, the one-step model constitutes an improvement: in this framework the final state is treated fully coherently from the inside of the solid to far away near the detector (see figure 1.9). The development of the one-step model for ARPES owes a lot to the work of Pendry [35]. One of the main features of his work was to rewrite Fermi's golden rule using only Green function formalism. This allows for affordable computation of the spectral function as will be shown later. Pendry's formula derives from equation (1.87) that we rewrite here:

I(ω, k) ∝ ij M kj A ji (ε k -hω)M * ik (1.105) Since M kj = ⟨k|H 1
int |j⟩ with |k⟩ being the final state we can write:

I(ω, k) ∝ ij ⟨k|H 1 int |j⟩ A ji (ε k -hω) ⟨i|H 1 † int |k⟩ (1.106) ∝ ⟨k| H 1 int ij |j⟩ ⟨i| A ji (ε k -hω)H 1 † int |k⟩ (1.107) ∝ ⟨k| H 1 int A(ε k -hω)H 1 † int |k⟩ (1.108) Remembering the relation A(E) = - 1 π ImG(E) (equation (1. 45
)) we obtain:

I(ω, k) ∝ - 1 π Im ⟨k| H 1 int G + (ε k -hω)H 1 † int |k⟩ (1.109)
In addition, Pendry realised that a more accurate description of the final state was needed. He envisioned it as a "time reversed LEED state". In essence, photoemission is a process where an electron comes out of the sample after absorbing a photon. Low energy electron diffraction (LEED) is the process where a low energy electron penetrates the sample and is scattered.

One can see the two processes as being mirror images of each other. This allows to write the final state as:

|k⟩ = G - 2 (ε k -hω) |k ∥ , ε k ⟩ (1.110)
Where G - 2 is the advanced propagator which takes a plane wave state |k ∥ , ε k ⟩ and brings it to scatter with the crystal potential. In comparison, G + is the retarded propagator that encodes the initial state of the crystal. This work opened the way to band structure calculations using multiple scattering techniques, taking advantage of the flexibility of Green function formalism.

E i ψ i (r) ψ f (r) E f

. The SPR-KKR code

In this section, we present the SPR-KKR code, whose name stands for Spin Polarized Relativistic Korringa-Kohn-Rostoker. The last 30 years saw a great effort in the development of a flexible package able to compute spectral functions to high accuracy. The SPR-KKR code aims to calculate the spectral function of general systems. In general, band structure computation codes relying on Bloch-like states break down when the translational symmetry of the crystal is broken. To compute the properties of a system with reduced symmetries (2D systems, systems with disorder, semi-infinite bulk systems...), one has to artificially introduce periodicity so that the code is applicable. This puts severe limits to the reliability of these codes. By using a material specific input, SPR-KKR makes it easier to compute the properties of disordered or confined systems. Instead of relying on the translational symmetry of the crystal, the problem is approached from the point of view of multiple scattering. The original idea of KKR was to divide the space into smaller cells where the scattering problem could be solved. Korringa showed that in a periodic potential, a Bloch wave in a unit cell Ω i was writable as the sum of incoming waves scattered from distant cells and an outgoing wave from the current cell [36; 37]. In this perspective, the aim becomes to write the retarded Green function G + 2 as a function of the single site scattering.

We will overview the method following the discussion of Ebert et al. in reference [38]. The reader should refer to this paper for a more throughout presentation. The first step is to solve the single site scattering problem in the cell n with a potential V (n) (null outside the domain Ω n , spherical or not) such that the full potential of the crystal is V = n V (n) . It is computed performing a partial wave analysis of the potential whose result is the scattering matrix at site n t n l (E). In a relativistic system, this essentially means solving the Kohn-Sham-Dirac equation in the angular momentum basis using the potential V (n) . The next step is to compute the scattering between neighbouring sites. This will give the single site Green function G n (E). Since it is assumed that free propagation (associated with propagator G 0 (E)) occurs in between sites, the Green function G n (E) is given by the Dyson equation:

G n (E) = G 0 (E) + G 0 (E)V n G n (E) (1.111) G 0 (r, r ′ , E) = - e -i √ E|r-r ′ | 4π|r -r ′ | (1.112)
Which can be written in terms of the single site t-matrix as:

G n (E) = G 0 (E) + G 0 (E)t n G 0 (E) (1.113)
In the same way, the full Green function G(E) of the system is obtained:

G(E) = G 0 (E) + G 0 (E)T (E)G 0 (E) (1.114)
Where T (E) is the t-matrix of the system as a whole. It can itself be decomposed as sequences of single site scattering and free propagation events. In the case of layered systems, it is possible to take advantage of the in-plane periodicity to calculate the Green function in the reciprocal space, speeding-up the results. The third dimension is then accounted for by keeping track of scattering events between planes, i.e. multiple transmission and reflection phenomena. This method is illustrated in figure 1.10. In the end, the photoemission current is calculated as the sum of atomic (single site), intralayer, interlayer, surface and incoherent (in a disorder averaged system) contributions [39; 40]:

I = I atom + I intra + I inter + I surf + I inc (1.115)
As suggested, the method is readily expandable to disordered systems. In this case it is called KRR-CPA (Coherent Potential Approximation). The method can be modified into a self-consistent version that brings of the complexity of the multi-electron problem through the self-energy Σ as in KKR-DMFT (Dynamical mean field theory, see [41]) implementations [42; 43].

. Conclusion

In this chapter we reviewed several perspectives on photoemission from the simplest possible model to more sophisticated approaches. These insights will guide the interpretation of photoemission experiments throughout this thesis. In chapter 4, we will use one-step calculations made by our collaborator Ján Minár (University of West Bohemia, Pilsen, Czech Republic). Before that however, we will focus on the experimental aspects of photoemission in the next chapter, explaining how the measurements can be done in a synchrotron radiation facility.

-Experimental Photoemission

. A source: The synchrotron radiation

. Basics of Synchrotron Radiation

Photoemission experiments require a photon source. Historically and to this day, these sources have been UV-lamps or X-ray tubes. Synchrotron radiation is however, when available, the ne plus ultra. Synchrotron radiation is the light emitted by relativistic particles (usually electrons) when their trajectory is bent by a magnetic field. First accelerated in a linear accelerator and then possibly by a small circular accelerator called a booster, these electrons are then injected in a storage ring made of straight sections connected by bending magnets. The latters create a vertical magnetic field which bends the trajectory of the electrons in between two straight sections (Figure 2.1). Because of the radial acceleration due to this magnetic field, electrons radiate an intense light which contains all the wavelengths from infrared to hard X-rays: the synchrotron radiation. The first observation of synchrotron radiation was made in 1947 at the General Electric laboratory in Schenectady (NY, United States of America). According to G. C. Baldwin and D. W. Kerst: "...it was made possible by a trivial design change [the removal of the silver coating of the glass vacuum chamber in which the electrons were circulating] and by a conscious disregard for the rules of radiation safety" [1]. It is indeed an outstanding source:

• It is very intense and focused: the source point (the electron beam) is very small and the emitted light is concentrated in a small cone of aperture 1/γ (where γ is the relativistic Lorentz factor of the electrons). For SOLEIL, γ=5400 and 1/γ=0.19 mrad (≈0.01°).

• The emission of a bending magnet contains all the wavelength from infrared to X-rays, up to a critical energy which increases with the energy of the electrons and the magnetic field generated in the magnet (E c (keV)=0.665BE 2 , with B in tesla and E in GeV).

For SOLEIL (E=2.75 GeV), the critical energy for a bending magnet (1.71 T) is 8.6 keV. At ESRF (E=6 GeV and B=0.86 T), it reaches 20.9 keV. It is often said that synchrotron radiation is a "white" source.

• It is polarized. The light emitted in a bending magnet is linearly polarized in the plane of the ring. Circularly polarized light can also be collected slightly above (or below) the machine plane.

• It a pulsed source: the electrons do not circulate as a continuous current in the ring but are rather grouped together into bunches, generating flashes of light when experiencing the magnetic field inside the magnet. One can use this temporal structure (the flash duration is typically of the order of some ps) for time-resolved experiment.

. Insertion devices

At the begining, only bending magnets were used as light sources, with the characteristics listed above, whereas the straight sections were "useless". Then quickly came the idea to install series of magnets above and below the trajectory of the electrons to create a periodic magnetic field in which the electrons wiggle, creating light at each curve (figure 2.3). Insertion devices are sorted into two categories:

-The electron beam has a maximum deviation angle much larger than 1/γ inside the insertion device. In this case, the insertion device behaves like a series of bending magnets: the emission spectrum is wide, more intense than on a single bending magnet and, since larger magnetic fields can be used, possibly shifted to higher photon energy. This is called a wiggler.

-If the electron beam has a maximum deviation angle of the order of 1/γ, the electron beam only slightly oscillates around the axis of the insertion device and the different source points can interfere. The emission spectrum is then not continous any more but made of an intense line (at an energy position given by the magnetic period of the insertion device and the magnetic field seen by the electrons) and harmonics of this line at (at two times, three times… the photon energy position of the first harmonics). One can put these first harmonics at the desired photon energy, generally by changing the magnetic field. When the insertion device is made of permanent magnets, this is achieved by changing the gap in between the magnets. This is called an undulator.

All these possible light sources available on a synchrotron radiation facility and their characteristics are summurized on figure 2.4. At the early days of synchrotron radiation, only bending magnets were used on so-called first generation machines, which were mainly machines made for high energy physics. The second generation was made of machines designed exclusively for the use of synchrotron radiation. Both bending magnets and straight sections were used. As indicated on figure 2.4, the source having the highest performance in terms of brilliance (i.e. providing the largest photon flux concentrated in the smallest emission angle; the brilliance also takes into account the size and divergence of the electron beam) are the undulators. The third generation machines took advantage of these unique properties of undulators with as numerous as possible straight sections, even partly giving up the use of bending magnets. As generation goes by, the properties of the electron beam was also greatly improved.

These photons produced by the storage ring are sent into beamlines. Each beamline is generally dedicated to one or two experimental techniques. The beamline light source as well as its optics are designed to provide a photon beam adapted to the needs of those experimental techniques.

. The CASSIOPEE Beamline Introduction

The CASSIOPEE beamline is installed on a straight section of the SOLEIL storage ring (Saint-Aubin, France) and is dedicated to photoemission. It is the main beamline that was used during this PhD. In the following, its main elements will be described.

Insertion devices

CASSIOPEE provides photons in the VUV and soft X-ray range, from 8 to 1200 eV. To cover this energy range, the beamline uses two undulators placed one behind the other in the storage ring straight section. The 8-100 eV photon energy range is covered by a HU256 undulator (see figure 2.5). As indicated in the name, it has a 256 mm magnetic period. The magnetic field is created by coils, placed either above and below the electron beam oscillates with large deviation angles : the emission spectrum is similar to a bending magnet, although more intense and possibly shifted in energy. (c) In an undulator, electrons oscillate close to the insertion device axis. The interferences between the different emission points gives a emission spectrum made of a main line (first harmonics) and its harmonics. Remade from [2].

(in this case it creates a vertical magnetic field, the electrons wiggle in the horizontal plane and the emitted photons have linear horizontal polarization), or along the horizontal direction (horizontal magnetic field and vertical polarization of the light). A combination of the different fields can give an helical trajectory to the electron to produce circularly polarized light. The energy position of the first harmonics can be changed by changing the current in the coils.

Optics of the beamline

As schematized in the figure 2.8 the main optical elements of the beamline are:

• An entrance optics made of two mirrors: it is cooled around 90K to absorb the parasitic power emitted by the ring (300 W in the worst case). It also shifts the useful photons to an axis parallel to the undulator axis, so that the very hard x-rays and gamma rays emitted by the machine can be shielded. Last, it focuses the beam on the grating of the monochromator.

• The monochromator is made of a grating and a plane mirror. The grating disperses the different wavelength contained in the white beam. The incident angle on the grating can be changed. The mirror is then used to select an emergent angle and sends the reflected photons on an exit slit, whose aperture determines the photon energy resolution. This two-rotation system allows to have a quite high resolving power on the whole 8-1200 eV photon energy range with only two gratings (400 lines/mm and 1200 lines/mm), using the so-called modified Petersen mode (a constant ratio between the incident and emergent angles). The gratings also focus the beam vertically, thanks to a variable line spacing.

• A focusing optics made of two (spherical and toroidal) mirrors: it focuses the beam at the sample position in the ARPES experiment. The size of the beam varies with photon energy but stays of the order of some tens of micrometers (vertically and horizontally).

• A toroidal deviation mirror can be inserted in the beam after the monochromator to send the beam towards the spin-resolved photoemission experiment. It also focuses the beam down to a size of around 300 µm 2 .

Experimental endstations

CASSIOPEE hosts two experimental setups. An ARPES experiment is equipped with a Scienta R4000 hemispherical analyzer and a cryogenic (temperature from 15 to 400 K) 6-degree of freedom sample holder. The second experiment is a spin-and angle-resolved photoemission experiment, equipped with a MBS A1 analyzer, with a moveable entrance optics able to collect electrons in 30°x30°angular range. Part of the collected electrons are sent to a spin rotator. Depending on its configuration, any spin component can be aligned with one of the two detection axis of a FERRUM spin analyzer made of a magnetized iron oxyde layer [3; 4].

. Measuring electrons: the hemispherical electron analyzer

Once one has a photon source, the photoexcited electrons are collected by an electron energy analyzer whose main role is to determine their kinetic energy. Different strategies can be implemented in order to achieve this goal such as time of flight electron analyzer which will be left out of the discussion [5]. We will instead focus on the "condenser" based detector, for instance the hemispherical deflector analyser (HDA). The HDA is in essence composed of two concentric half spheres of radii R 1 and R 2 which will be brought to the voltages V 1 and V 2 respectively. The electrons inside the apparatus will be deflected according to their initial kinetic energy, their trajectory will be determined by the potential in-between the two semi-spheres and imaged at the exit of the analyzer. The electrons are then collected into a detector, either a channeltron (punctual detector) or more commonly nowadays microchannel plate plates (MCP) which provide spatial resolution and can be imaged with a CCD camera.

. The ideal hemispherical electron analyzer

V 2 V 1 ΔR R 1 R 2 r(θ) r π r 0 v 0 α θ=π/2 E k > E PE E k = E PE E k < E PE R 0 R π Figure 2
.9: Diagram of a HDA. R 0 is the radius where the entry slit is located, r 0 the radius of the incoming electron. R π is the radius where the exit slit is located, r π the radius of the outgoing electron.

The equation of motion of an electron inside the ideal hemispherical analyzer are derived from the classical mechanics of a charge in a central potential V (r) defined using the notations of figure 2.9:

∆R =R 2 -R 1 (2.1) ∆V =V 1 -V 2 (2.2) V (r) = ∆V ∆R R 1 R 2 1 r - 1 R 1 + V 1 (2.3)
For compacity we will write

V (r) = k r + C with k = ∆V ∆R R 1 R 2 and
C is a constant. This gives the two equations of motion in polar coordinates [6]:

   r -r θ2 = - qk mr 2 mr θ + 2mr ṙ θ = 0 (2.4)
The solutions of the equations of motion in this potential are well known to be elliptical trajectories. In particular there is one circular trajectory for an electron traveling on the average radius R 0 = 1 2 (R 1 + R 2 ) at θ = 0 with kinetic energy E 0 = 1 2 mv 2 0 . This trajectory is given by the condition obtained from (2.4) when r 0 = R 0 and mr θ = mv 0 :

mv 2 0 R 0 = qk R 2 0 (2.5)
This defines the pass-energy E p :

E p = 1 2 mv 2 0 = 1 2 q r 0 k (2.6) = 1 2 q R 0 ∆V ∆R R 1 R 2 (2.7)
The HDA works as an energy filter which select electrons of kinetic energy equal to the pass energy E p . In this situation, an electron with kinetic energy E k at input radius r 0 (not necessarily equal to R 0 ) and entrance angle α will be mapped to the exit radius r π :

r π = -r 0 + 2R 0 1 + (1 -E k Ep cos 2 α) (2.8)
Hence, electrons of different kinetic energies are spatially sorted along one dimension of a 2D detector placed at the exit of the HDA. This formula can be generalised to more exotic analyzer configurations taking in account several design parameters. Analyzers can be paracentric (ξ = Rπ R 0 ̸ = 1), which will change the whole potential inside the apparatus and therefore the position of the principal trajectory. This can be necessary since the potential near the entrance slits will deviate from the ideal radial 1/r potential. The presence of a potential step at the entry of the analyzer can induce the refraction of the incoming electrons, changing the entrance angle α into a related α * [7; 8; 9]. Within the ideal HDA framework, the sensitivity (i.e. the way two different kinetic energies are spatially separated by the analyzer) is quasi-linear as shown in figure 2.10. For electrons with kinetic energies close to the pass energy we can write:

r π ≈ -r 0 + 2R 0 E p E k (2.9)
HDA are most of the time used at a given pass energy. An entrance optics first slows down (or speeds up) the electrons to bring their KE around this selected pass energy. Formula (2.9) illustrates how one can use the pass-energy to trade-off between the resolution of the HDA and the measured intensity at high pass energy: for an identical MCP the number of measured electrons will be higher (i.e. higher count rates) but the sensitivity ∆r π /∆E k will be lower. The relation (2.8) also shows that the exit radius has a dependence on the starting angle which impacts the energy resolution of the instrument. Electrons with slightly different entrance angles can end up being measured at the same exit radius. This is known as the α 2 aberration or the non-isochromaticity of the analyzer that ultimately limits the energy resolution of the HDA.

-0.1 -5

E k ∈ [0.95E p , 1.05E p ]
where the relation between r π and E k is close to linear.

. Effect of the entrance slits

In modern HDA, electrons enter the analyzer not through a hole (as depicted in figure 2.9) but through a slit perpendicular to the plane of figure 2.9. The goal is first to collect more electrons taking advantage of the large 2D detector at the exit of the analyzer. Electrons are collected on a angular range corresponding to the length of the slit as shown of figure 2.11. The shape of the entrance slit has a direct impact on the resolution of the instrument. One generally has the choice between straight or curved slits. For a curved slit of the same radius R 0 as the analyzer (referred to as radial slits from now on), whatever is the entrance angle β of the electron on the slit (β is the angle defined in figures 2.11 and 2.12), it has the same radius r 0 . According to equation (2.8), its image formed on the detector will be at the radius r π (E k ) (figure 2.12a). For a straight slit, r 0 depends on β (figure 2 ). This therefore produces a direct imaging of the band dispersion along a direction parallel to the analyzer slit. Strictly speaking, with a curved slit, the dispersion is not really measured along a well-defined crystallographic direction. This penalizes the angular resolved measurements. With straight slits however, we see improved reciprocal space resolution as only electrons coming from the same direction can enter the slit, as illustrated in figure 2.11. Nowadays, all the analyzers dedicated to angle resolved photoemission spectroscopy are equipped with straight slits. Even though it implies the curvature problem that we mentioned before, the issue is usually corrected by the acquisition software.

. Electrical calibration of the analyzer

In addition to the energy filtering aspects of the analyzer, one has to take great care to have a well defined electrical reference so that the measured kinetic energies make sense. First of all, the sample and the spectrometer should be at equilibrium, meaning that their respective Fermi levels are aligned. Secondly the electrons entering the analyzer loose some energy. The lost energy is called the work-function of the spectrometer noted here Φ Spec . The measured kinetic energy is then lower than the true kinetic energy. Knowledge of the work-function of the analyzer allows for an absolute measure of the kinetic energy but one is usually more interested in binding energies. This requires the knowledge of the Fermi level energy E F . In metallic samples, knowing the kinetic energy where the Fermi level lies is not an issue and only requires to measure the last occupied levels. The binding energies can then be found easily without knowing the spectrometer work-function Φ Spec . In semiconductor samples, the Fermi level is in the band gap. Knowing the Fermi-level requires the semiconducting sample to be at equilibrium with a metallic bond on which the measurement will be done. The calculation of the binding energy is then straightforward:

E B = E M eas kin -E M eas kin (E F ) (2.10)
If one knows the work function of the analyzer then a value of the Fermi level can be calculated:

E M eas kin (E F ) = hω -Φ Spec (2.

. Studying core electrons with X-Ray Photoemission Spectroscopy (XPS)

The first use of photoemission spectroscopy was to perform chemical analysis. The historical name of the technique was ESCA: Electron Spectroscopy for Chemical Analysis. This domain was pioneered by Kai Siegbahn who not only improved the detection methods and instrumentation, but also proved the utility of the technique expanding its use outside of research laboratories [11]. The chemical analysis is usually performed with photons in the energy range of soft X-rays (100-1500 eV). With these energies, one can start to measure the core levels of atoms. Since two different elements have different binding energies for their core levels, their photoemission spectrum is in turn specific. The set of binding energies for the electrons of two different atoms being different, one can use XPS to identify an unknown material. Usually, even a subset of the spectral lines is enough to distinguish two elements. However, elements in a material interact and bond with each other. The spectral lines can therefore change as compared to isolated atoms, even more so, they can be modified by the photoemission process itself. We will quickly review the zoology of effects on spectral lines in the following sections.

Orbital like states

Delocalised states

. Shifts of spectral lines Chemical shifts

Core levels have one key property: the associated wave functions lies near the atom nucleus. Owing to their localised character, photoemitted electrons act as probes of the chemical environment: in solids as in molecules, electrons are surrounded by potentials that can modify their measured binding energy. This sensitivity of the binding energy obtained by photoemission on the chemical environment is what is called a chemical shift. The chemical shift of a given core level of an atom indexed by i can be approximated by the following formula [11]:

∆ϵ i = kq i + j̸ =i q j R ij (2.
12)

The origin of this formula is two-fold. First of all, the energy of the core level is changed by the charge q i on the atom at position i: this is the Coulomb interaction of the electron with the other electrons on the same atom. The change in binding energy is proportional to this charge, hence the constant k. The second part expresses the change in potential due to the neighbouring atoms. R ij is the distance between nuclei i and j and q j the charge of its electrons. This simplified picture assumes that the charges of the other electrons are lumped at the nucleus positions.

Electrostatic shifts

Another important source of shifts are charge distributions on the sample. Insulating samples can be subject of charging when under the beam. Because of the poor conductivity of the material, the ejected electrons are not replaced quickly enough leading to local accumulation of positive charges. If the system reaches equilibrium, most photoemission peaks will be shifted by a finite amount. In extreme cases of charging, the peaks also are broadened or even distorted. In the context of this thesis, we take advantage of the polar surfaces of ferroelectric materials to induce electrostatic shifts of the photoemission spectrum of the two-dimensional material WSe 2 . This specific question will be the object of the last chapter of this thesis.

. Spin-orbit coupling

Spin-orbit coupling is a relativistic effect that couples the spin angular momentum of an electron to the orbital angular momentum of the atom. Its origin can be traced back to the Dirac formulation of the electron spectrum in an atom. It gives rise to the so called fine structure of the atom. By virtue of spin-orbit coupling, all orbitals with a non null orbital angular momentum L (i.e. p,d and f orbitals) will show two spectral lines. The separation ∆ s-o between the two lines is dictated by the spin-orbit Hamiltonian:

H s-o = f (r)L • S (2.13) f (r) ∝ 1 r ∂U (r) ∂r (2.14)
where U (r) is the interaction potential between the electron and the nucleus. Eigenstates of H s-o are eigenstates of the total angular momentum J = L + S. In this case there are two possible states with:

j + = l + 1/2 (2.15) j -= l -1/2 (2.16)
The electrons in the g l = 2(2l + 1) degenerated states (if one ignores spin-orbit coupling) are distributed into the states j ± with the degeneracies g j ± where g j ± = 2j ± + 1. The intensity ratio I + Iof the spectral lines is generally dictated by the degeneracy ratio g + g -. Because of these well defined properties (fixed intensity ratio, known ∆ s-o ) spin-orbit splitted lines are especially useful to identify species when the spectral lines are shifted.

Shell

Total angular momentum j

Ratio of degeneracy

g = 2j + 1 Total number of electrons 2(2l + 1) s 1/2 - 2 p 3/2,

. Exploring the reciprocal space with Angle Resolved Photoemission Spectroscopy (ARPES)

. Practical ARPES measurement

As we saw in 1.7, the measured kinetic energy and emission angles of the photoelectron are related to its momentum inside the crystal by the relations:

k ∥ = 2m h2 E kin sin β cos φ sin φ (2.17) k ⊥ = 2m h2 (E kin cos 2 β + V 0 ) (2.18)
With β the angle along the entrance slit as defined in figures 2.11 and 2.12. In practice, the value 2m h2 is approximated to be 0.512

Å -1 •eV -1/2 .
In ARPES settings using a 3-rotation manipulator, one can explore the Brillouin zone at will. Using the angles φ, θ and τ defined in figure 2.17, one can move the portion of the Brillouin relatively to the slit (placed along k x ). The equations in (2.18) can be reformulated as:

k x = 2m h2 E kin sin(β -τ ) (2.19) k y = 2m h2 E kin sin θ (2.20) k ⊥ = 2m h2 (E kin cos 2 (β -τ ) + V 0 ) (2.21)
In general, the vectors of the reciprocal space of the sample are arbitrarily projected onto k x and k y . One can use the angle φ to rotate the sample and get a crystallographic axis to coincide with k x . Since the angular opening is finite, the explorable portion of the Brillouin zone is limited. With a standard opening ±15°, the maximum measurable size of the Brillouin zone will be 0.265 × hν[eV](Å -1 ). For a photon energy of 50 eV, that represents 1.81 Å -1 . Depending on the sample, the measured spectrum can show one or several Brillouin zones. There is therefore a compromise between the k-space resolution and the size of the measurable reciprocal space. In addition, the higher the photon energy, the lower the crosssection for most valence band states. This adds another trade-off between intensity (cross section) and the explored k-space range. Some ARPES settings probe the different angles by deflecting the electrons with the entrance optics of the analyzer. Conversely, another way involves physically moving the entrance optics as is the case for the MBS analyzer (used for spin-resolved experiments at CASSIOPEE). These configurations have the advantage of keeping the projection of the beam on the sample exactly the same.

The matter is more complex when one wants to measure the out-of-plane component k ⊥ .

Using the free-electron model, we can see from equation (2.21) that changing the photonenergy hν allows to map out k ⊥ efficiently. While theoretically accessible, the knowledge on the value of k ⊥ is limited by several factors. First of all, even considering the equation as exact, the value V 0 is never known a priori and is generally a fitting parameter between theory and experiments. Secondly, from Heisenberg uncertainty principle, the incertitude on k ⊥ is dependent on the probed length λ inside the sample. The probed length λ is itself closely related to the inelastic mean free path. We showed earlier that it was mostly dependent on the electron kinetic energy thus scaling with the photon energy. In fine, the quantum number k ⊥ is only approximately conserved [12]. This is to show that there are two kinds of limitations on the knowledge of k ⊥ :

• One coming from the model of the final state

• One arising from a fundamental indetermination: the approximate conservation of k ⊥ in the photoemission process

In practice, this makes surface states particularly attractive in ARPES. Because of their spatial confinement in the z direction, the associated k z (k ⊥ ) is not defined. This means that the bands associated with the surface states is independent on k z and therefore will not disperse when the photon energy hν is changed. This said, dramatic change of their intensity can be observed [13].

. Practical ARPES analysis

When measuring standard ARPES data with 2D detectors one obtains a image with axes being the detection angle/kinetic energy. Such a image can be easily converted to binding energy and wave-vector provided that the Fermi-Energy E k (E F ) is known. As discussed previously, this is straightforward on metallic samples but requires more care for semi-conductors. The relations between the couples (angle, kinetic energy) and (wave-vector, binding energy) implies that the size of the reciprocal space measured on the detector grows with the photonenergy used. This makes the choice of the photon energy critical to access some parts of the reciprocal space. Some materials like WSe 2 or graphene have their band structure spanning a wide area on the reciprocal space (more than 1 Å -1 of distance between high symmetry points) which require high photon energy (> 90 eV) to capture in one take (see figure 2 Oppositely, the features of the band structure one may want to see can be very small. This is the case of the 2D electrons gas in KTaO 3 which occupies a small 0.2 Å -1 range. Measuring it required to use photons with energies as low as 30 eV as can be seen in the following work [14; 15]. This photon energy change, combined with the cross-section and possibly k z dispersion in 3D samples implies some care. More prosaically, the relations used also deform the measurement grid (see figure 2.19) making it visibly non uniform on the span of few eV. While slightly detrimental to the resolution in the reciprocal space, this last fact is only a minor inconvenience. We usually want to measure the energy dispersion along one or several high symmetry directions of the sample. To do so, we take advantage of the angle θ on the manipulator (see figure 2.17) to access the k y direction. This way, we get a 3D data block with coordinates (k x , k y , E B ) roughly describing the whole band structure at a given k z . This 3D data block can then be cut in any direction contained in the k x -k y plane (giving the dispersion in this direction) or at any binding energy E B (giving an isoenergy surface. A cut at E F gives the Fermi-surface). An example of is given in figure 2.20

-1 0 1 -4 -3 -2 -1 0 k || ( Å-1 ) E -E f (eV) hν = 27 eV -1 0 1 k || ( Å-1 ) hν = 85 eV

. Imaging with Photoelectron emission microscopy (PEEM)

PhotoElectron Emission Microscopy (PEEM) is a photoemission based technique that is complementary to photoemission spectroscopy. In comparison to photoemision spectroscopy, this technique conserves the position of incoming electrons, producing images of the surface at the micrometer scale. In addition to standard microscopy, momentum microscopy is also possible. These two imaging modes are respectively called x-PEEM (real space) and k-PEEM (reciprocal space). While there are several kinds of photoemission microscopes, we will only present here the NanoESCA endstation with its characteristic double HDA setup in the broadest terms. The reader interested by an extremely detailed description of the instrument should read the reference [16]. The instrument is composed of a PEEM column, an energy filter and detection systems as illustrated in figure 2.21a. The PEEM column starts with an immersion lens with an extractor usually operated between 12-24 kV, which collects the photoemitted electrons (see figure 2.21b). The extractor voltage roughly translates into the field of view (FoV) of the final image since electrons with high take-off angles end up inside the column where they are deviated by several electrostatic lenses. At this stage, one can choose between the two imaging modes, projecting either the back focal plane (k-PEEM) or the image plane (x-PEEM) onto the slit of the first HDA. When operated in x-PEEM, its is possible to place a contrast aperture in the back focal plane to increase the spatial resolution. Operated in k-PEEM, one can restrict the area from where the photoelectrons are collected using an iris aperture located in the image plane. Inside the first HDA, the incoming electrons are filtered depending on their kinetic energy. Once filtered, the transfer lens gets them into the second HDA stage. The idea of the double analyzer is to improve the quality of the measurement: by having the electrons to do a full rotation, the spherical aberration is corrected and the electrons are refocused as they were at the entry of the first HDA. The final step is to go through the projection optics before arriving on the imaging detector. The detailed trajectories are schematized in figure 2.22.

. Main imaging modes in PEEM

There are several modes of measurements available when using the NanoESCA endstation.

• The direct imaging mode where all the emitted photoelectrons are collected to form an image. The majority of electrons are secondary electrons and the detected signal has the form I = f (x, y).

• The spectroscopy mode. After the first HDA stage, is located a channeltron which measures the intensity for a given kinetic energy. The signal is of the form I = f (E k ).

• The energy filtered imaging mode: instead of measuring a flat (energy integrated) image, one can add as a third axis the kinetic energy of the electrons. In the k-PEEM mode, the measured signal has the form

I = f (E k , k x ,
E k =4.85 eV E k =4.55 eV E k =4.25 eV E k =3.95 eV E k =3.65 eV E k =3.35 eV

. Origin of contrast in PEEM

The figure 2.23a shows an area imaged at different kinetic energies. The triangle in the picture is a multilayer flake of WSe 2 on a lithium niobate substrate (a system studied in detail in chapter 5). The lithium niobate has flat domains separated by straight thick lines with high rugosity and proeminent peaks in the topography. By changing the kinetic energy, some areas darken or lighten: the straight edges between domains go from completely dark to extremely luminous, the domains light up differently at E k = 3.35 eV and the WSe 2 triangle stays darker than the rest. The technique can even resolve local thickness in some conditions: the shade of gray of the flake is not the same everywhere as evidenced at E k = 3.95 eV in figure 2.23 or 2.24. This gives a collection of images that widely differ, highlighting features invisible at some energies. This leads to the question of the origin of contrast in PEEM. In this example, we see that a contrast can be observed depending on the chemical species involved, the topography of the sample and the local surface charge. In PEEM the intensity is proportional to the number of emitted electrons. This number is therefore affected by the local electric fields that either helps the electron depletion or counteracts against it. The effect of different topographic features on the measured photocurrent has been modelled in references [18; 19]. An example can be seen in figure 2.25.

E k =3.95 eV

-Ferroelectricity

. Generalities on Ferroelectrics

Among the many dielectric materials, the subclass of ferroelectrics is of special interest. Ferroelectrics are at the same time:

• piezoelectric: a stress can induce a internal electric polarization and conversely

• pyroelectric: a change in temperature can change the electric polarization of the material While these properties offer room for engineering, they are not exclusive to ferroelectrics. What makes a ferroelectric special is its capacity to retain a permanent electrical polarization, and have this polarization switched between two or more states at will when subjected to an appropriate electric field. This makes them a particularly versatile class of materials, since one can play on ferroelectricity with a large number of parameters (strain, voltage, temperature). A characteristic of ferroelectric materials is their Curie temperature T c . When reaching this temperature, the material experiences a phase transition. The high temperature phase (T > T c ) is paraelectric, i.e., the electric polarization of the whole crystal is zero, whereas the ferroelectric phase only exists at low temperatures (T < T c ). The highly symmetric phase (paraelectric) is replaced by a phase of lower symmetry (ferroelectric). This phase transition is accompanied by a symmetry breaking into two or more configurations of equivalent energy. There are several kinds of paraelectric-ferroelectric transitions, which are usually classified into either displacive transition or order-disorder transitions.

In displacive transitions, the unit cell of the paraelectric phase has zero polarization. Starting from the paraelectric phase, a change of structural phase occurs when crossing the critical temperature and locks bound charges in a state where electrical polarization is not null: the ferroelectric state. Different polarization states can be observed, corresponding to different distortions of the crystallographic structure of equivalent total energy. The point group of the new structure is often a subgroup of the higher symmetric point group, however, it never keeps the inversion symmetry. This absence of centrosymmetry is an absolute necessity for any polarization to exist within the crystal. This is the case of most perovskites ferroelectrics such as BaTiO 3 , Pb 1-x Zr x TiO 3 , BiFeO 3 . An example of a displacive transition is given in figure 3.1. We see that the cubic BaTiO 3 can have four equivalent polarization states in its tetragonal: one for each <100> axes.

In order-disorder transitions, the unit cell is polarized in both paraelectric and ferroelectric phases. In the paraelectric phase, all the local polarization vectors are oriented in a random direction due to thermal agitation, averaging down to a null net polarization. The ferroelectric phase sees all the polarization vectors oriented in the same direction, adding up to a global polarized state. Strained SrTiO 3 is an example of order-disorder ferroelectric [1]. In this material, the paraelectric phase crystalline structure shows random off-centered Ti atoms giving a little polarization vector to each unit cell that average to zero (see figure 3 Ferroelectricity is often understood in terms of phonons. The locking of the ions composing the lattice at fixed position can be seen as an oscillation mode of frequency zero (static). Since phonons correspond to vibration modes of the ionic lattice, the softening of a transverse optical phonon mode (i.e. the diminution of its frequency) when close to the transition is a signature of ferroelectricity. This is the essence of the model developed by Cochran for BaTiO 3 [2]. In a classical phonon framework, he calculated the displacement of an Ti 4+ ion in the O-Ti-O chain subjected to competing forces: a restoring force R that tends to bring it back to the equilibrium position and long ranges forces (Coulomb forces) A generated by the other dipoles that tend to destabilize the structure and favour ferroelectricity. He derived the relation between the frequency of a mode and several crystalline parameters which, in connection with the Lyddane-Sachs-Teller equation (equation (3.1)), related the described phonon modes frequencies to the dielectric response function. Indeed, Lyddane-Sachs-Teller equation tells us that the static dielectric function ε(0) diverges when a transverse optical (TO) phonon mode softens and that longitudinal optical (LO) phonons do not play any role in ferroelectricity. This way, Cochran could highlight one of the distinctive aspect of ferroelectricity: the discontinuities (hysteresis) seen in the dielectric function of the material at the phase transitions [3] (see figure 3.2).

ε(ω) ε(∞) = N j ω 2 j (LO) -ω 2 ω 2 j (T O) -ω 2 (3.1)

. Ubiquitous hysteresis

As shown before, ferroelectric crystals show hysteresis in their dielectric function as measured by [3]. This comes from the peculiar dependence of the ferroelectric polarization on the electric field, which also describes a loop as pictured in figure 3.3. In an initial state with zero electric field, the ferroelectric will have one defined state of polarization called the remanent polarization P r . Starting with P = -|P r |, when the electric field gets higher, the polarization increases until it reaches a maximum called the saturation value P s . While the electric field is increasing, the electric polarization momentarily cancels. The electric field necessary to nullify the polarization is called the coercive field E c . When E > E c the system switches, i.e. it changes its stable polarization state. The physical quantities that somehow depend on the polarization will also exhibit similar hysteresis. For example, for the dielectric function, from

Maxwell equation D = P + ε 0 E = εE we can write: ε -ε 0 = ∂P ∂E
. This is the reason why ferroelectricity are often said to have memory: it is possible to store a bit of information into the crystal structure and read it as a property of the material.

. Ferroelectricity models

. Landau type models of Ferroelectricity

Ferroelectric behaviour is usually modelled using a Landau type expression of the freeenergy density of the crystal. The model supposes that the free energy of the system can be written as a polynomial expansion of the order parameter near the critical temperature T c . In the case of ferroelectrics, the order parameter can be chosen to be the electric polarization vector P : it has a value of zero in the paraelectric phase and varies continuously in the ferroelectric phase. It is possible to use other order parameters, like the electric induction vector D = P + ε 0 E or the amplitude of phonon mode associated with the transition. The coupling between the electric field and the polarization is often included to describe the biased free energy density:

F (P, E, T ) = α 2 (T -T c )P 2 + β 4 P 4 -E • P (3.2)
The minimal theory seen in equation (3.2), up to order four in the order parameter, corresponds exactly to the Landau theory of phase transitions [4]. The theory is asymptotically valid at the critical temperature for any order parameter in the case of second order phase transitions. It successfully pictures the behaviour of a ferroelectric: an hysteresis in the P-E loop can be derived from the expression of the free energy and the switching mechanism can be explained as seen in the figure 3.4c. Landau-Ginzburg-Devonshire models are phenomenological approaches inspired by the construction of Landau. In the case of ferroelectrics, the development is usually extended to the sixth order. It writes as follows:

F (P, E, T ) = α 2 (T -T c )P 2 + β 4 P 4 + γ 6 P 6 -E • P (3.3)
This additional order allows to describe systems showing first order transitions in which the order parameter is discontinuous at the transition. In these models, the order parameter is the polarization density vector P. It enters the expression of the free energy by its components P x , P y and P z . By keeping the product terms with appropriate symmetry, one can model anisotropic ferroelectricity in crystals. The free energy is written with terms respecting the symmetry of the paraelectric phase. Including the coupling to the strain, the theory is powerful enough to predict the distortions of the lattice. This method was successfully applied to BaTiO 3 by Devonshire in 1949, predicting the correct order phase transitions (cubic -→ tetragonal -→ orthorhombic -→ rhomboedral when temperature drops) [5]. It was later applied to the lead based ferroelectric crystal PbTiO 3 . The expansion limited to the order six is sufficient to model a large number of ferroelectric materials. A systematic derivation of the free-energy density for different symmetry groups can be found in the literature [6]. It cannot, however, be used to predict ferroelectricity in monoclinic systems. The development has to be pursued up to the order eight at least to see these effects as shown by Vanderbilt and Cohen [7]. This will be of importance when discussing the ferroelectricity in BiFeO 3 which displays monoclinic phases.

. Modern theory of polarization: deriving a microscopic model of ferroelectricity

The modern theory of polarization is an attempt to describe ferroelectricity from a microscopic perspective. In the previous section, we used a loose terminology when discussing the polarization. Polarization is easily defined when simple objects are studied. In a collection of punctual charges q i at positions r i , the polarization can be defined as the charge barycentre:

p = i q i r i (3.4)
If one wants to discuss polarization density P in a volume Ω in the case of continuous charge distribution ρ , it would be natural to adopt the following definition:

P = 1 Ω Ω d 3 rρ(r)r (3.5)
While the definition seems clear at first glance, it turns out it cannot be easily used in the case of crystals. To calculate a finite polarization density, one has to choose a unit cell. Wherever this unit cell is located in the crystal should not matter since the crystal as a whole can be reconstructed with its translations vector and symmetry operations. This gives rise to a problem in the definition as P should be invariant under translations, however we see that: Equation (3.7) shows that, owing to the periodicity of the charge density, the polarization density is ill defined. It is clear that it cannot be equivalent to equation (3.5). The first part of the modern theory of ferroelectricity is to acknowledge that the polarization density is not an actual physical property but rather a multivalued object. The work of Resta [8] and Vanderbilt [9] in the 1990s was to redefine the polarization through the formalism of Berry phases. The idea is to separate the ionic contribution P ion (considered as point charges) from the electronic contribution P el (treated by quantum mechanics). The change in polarization is thought of as an adiabatic process following the path of the Hamiltonian H(λ(t)) where λ is a parameter that slowly and continuously varies with time. For example, one can decide that H(λ = 0) = H para and H(λ = 1) = H f erro . From that, we calculate the current J induced by the evolution of the Hamiltonian and relate it to the change in polarization with charge conservation, i.e.:

P = 1 Ω Ω d 3 rρ(r + R)(r + R) (3.6) = 1 Ω Ω d 3 rρ(r)r + R Ω Ω d 3 rρ(r) (3.7)
∇ • d dt P -J = 0 (3.8)
Meaning in bulk:

J = d dt P = dλ dt d dλ P (3.9)
from which the difference of polarization ∆P = P (1) -P (0) between the states λ = 0 and λ = 1 can be calculated by integrating the current J:

∆P = dtJ = dλ d dλ P(λ) (3.10)
This leads to the formulation of the polarization due to the electrons P el in terms of the Berry phase calculated with Bloch waves of occupied states:

P el = occ n i -e (2π) 3 BZ d 3 ke -ik•R ⟨u nk | ∂ ∂k |u nk ⟩ (3.11)
This can be better understood using the Wannier basis. Bloch waves u nk (r) are delocalised and they are dual with a set of localised functions w n (r -R) called the Wannier functions (see figure 3.6). These states are written:

w n (r -R) = Ω (2π) 3 BZ d 3 ke -ik•(R-r) u nk (r) (3.12)
Wannier functions |R,n> Bloch waves |k,n> w 0 (r) They are indexed by the translation vector R and the band index n. Inside the unit cell, we can ignore the translation vector R. Using this basis, the polarization gets elegantly rewritten as [9; 12]:

w 1 (r) w 2 (r) ψ k0 (r) ψ k1 (r) ψ k2 (r) cos(k 0 x) cos(k 1 x) cos(k 2 x)
P el = -e Ω occ n Ω d 3 rw * n (r)rw n (r) (3.13) = -e Ω occ n rn (3.14) (3.15)
Taking P ion as created by the nucleus charges e Ω i Z i r i leads to the full formula [10]:

P = P ion + P el (3.16) = e Ω i Z i r i - occ n rn (3.17)
In this restatement, we recover the initial idea of the charge barycentre. In the simplest models of ferroelectricity, the electronic charges are distributed somewhat arbitrarily between the atomic centers taking into account their electronegativity. The Berry-phase theory of polarization allows however to go beyond this static picture. It becomes possible to quantify the effect of a small ionic displacement on the polarization. In addition, it can support the inclusion of boundary conditions in the statement of the problem. This leads to the definition of the dynamical charge tensor Z * . This tensor encompasses the change in polarization in the direction i by a displacement d k,j of an atom k in the direction j [13; 14]. It is defined as:

Z * k,ij = Ω ∂P i ∂d k,j (3.18)
The tensor roughly expresses how a particular ion in the structure carries the electronic charge.

The computed values of effective charge can be very different from the idealized ionic pictures [15]. As discussed thoroughly by Ghosez, Michenaud and Gonze, the method has been successful into predicting values of polarization for ABO 3 type ferroelectrics such as BaTiO 3 and PbTiO 3 [13]. These methods were later applied to the study of BiFeO 3 predicting values of the polarization in agreement with experimental results [15].

In conclusion, the modern theory of polarization has the main advantage of being accurate to calculate experimentally accessible quantities (dielectric function, susceptibility, currents...) by staying true to quantum theory. Purely ab-initio approaches remain, however, computationally expensive in comparison with Landau type models that capture already the main ingredients of a successful theory of ferroelectricity.

. Surface charges and screening

. Depolarizing field

In the previous sections, we did not discuss one of the most important aspects of ferroelectrics, namely, the boundary conditions. The theory of polarization developed in the previous section was true for bulk ferroelectricity, but the finite size of the samples plays a major role in any application using thin-film ferroelectrics. From standard electrostatics, it is known that the polarization P induces a charge density σ (µC • cm -2 ) at the boundaries of a sample in the following form:

σ = P • n (3.19)
As represented in figure 3.7, this implies opposite charges bound at the top and bottom surfaces of the sample. These charges create in turn in the slab an electric field counteracting the polarization. This field is called the depolarizing field with expression:

E depol = - P εε 0 (3.20)
Where ε is the dielectric constant of the material. In these conditions, the ferroelectric cannot • open-circuit conditions: the charges are not screened, the depolarizing field is maximal.

• short-circuit conditions: the charges are perfectly screened, the depolarizing field is zero.

The presence of a high depolarizing field is detrimental to ferroelectricity, as it destabilizes the polarization. This is readily seen in the Landau model

(3.22) if E = - |L| εε 0 P with L a
proportionality factor, then:

F (P, E, T ) = α 2 (T -T c )P 2 + β 4 P 4 -E • P (3.21) = α 2 (T -T c ) - |L| εε 0 P 2 + β 4 P 4 (3.22)
In the worst cases, it can lead to polarization cancellation or reversal. In general, the depolarization field is dependent on the geometry of the sample [16; 17] and on the quality of the contacting electrode which will provide a screening of the charges.

. Screening mechanisms

There are several possible mechanisms for a ferroelectric system to adapt to the boundary charges. They are of two sorts: intrinsic screening (from defects or domain formation) and extrinsic screening (from adsorbates). The three options are schematized on figure 3.8.

-Domain formation (see figure 3.8a) is the second way of reducing the depolarization field. The crystal will arrange into different domains with compensating polarization states. The crystal balances the domain wall energy with the depolarization energy to stabilize [16; 18]. This process determines the angle between domain walls as well as their sizes.

-Defects come in crystals as dopants or vacancies that contribute to excess of charge carriers. These free carriers are available to screen the bound charges as shown in figure 3.8b. A common source of defects are oxygen vacancies. They can be induced by vacuum annealing, freeing electrons to screen the charges. The change of conductivity helps both stabilizing ferroelectricity (screening) and acting against it (reduction of the overall polarization, as seen in MgO doped LiNbO 3 . [19]). This will be of importance when investigating ferroelectrics with photoemission.

-If atoms or molecules get adsorbed on the surface, it can lead to screening of the charges. However, they can also be a source of polarization reversal [20; 21]. This implies that great care should be taken in the interface chemistry when dealing with ferroelectrics.

. Screening with electrodes: the effective screening length λ ef f

The production of ferroelectric materials as thin films offers the opportunity to minimize the depolarization field by the integration of metallic electrodes. They provide the necessary charges to compensate those creating the depolarizing field (see figure 3.9). The common procedure to make thin ferroelectric films is to have them epitaxially grown on the electrode to ensure the best contact and minimal distance between the screening charges and the ferroelectric surface. However, even in best conditions, the screening is never perfect and occurs over a length λ ef f . This distance is the effective screening length. The simplest model of electrostatic screening in metals is the Thomas-Fermi model [23]. In this model, the chemical potential µ is assumed to vary only with the local value of the electron density n(x) and the local potential φ(x) so that:

µ ≈ h2 2m (3π 2 n(x)) 3 2 -eφ(x) ≈ h2 2m (3π 2 n 0 ) 3 2
(3.23) A demonstration found in Kittel's introduction to solid state physics shows that it implies the following form of the dielectric function:

V Q P E depol
ε(k) = 1 + k 2 s k 2 (3.24) k 2 s = 6πe 2 n 0 E F (3.25)
Where k s is the Thomas-Fermi screening wave-vector. By the Poisson equation, we also have the relation between the total (external + response) potential and density φ(k), ρ(k) with the external potential and density φ ext (k), ρ ext (k):

φ ext (k) φ(k) = ρ ext (k) ρ(k) = ε(k) (3.26)
For a linear potential drop φ ext (x) = -σ ε x (potential due to the surface charges +σ), the screened potential with grounding at x -→ ∞ (φ(∞) = 0) solves into:

φ(x) = σ εk s e -ks|x| (3.27) ρ(x) = -σk s e -ks|x| (3.28)
Perfect screening is reached in the limit k s -→ ∞, absence of screening when k s -→ 0. In a slab of thickness d, this finite length induces a voltage drop across the sample resulting into a non zero field [24]:

E depol ≈ -2 λ ef f dε 0 P (3.29)
The effect of the thickness on the depolarizing field, predicted as early as in the 1970s [25] was demonstrated using Berry-phase calculation of BaTiO 3 /SrRuO 3 interfaces. They could conclude that ferroelectricity could not appear under a critical thickness of 6 unit cells and the magnitude of the polarization would be greatly reduced compared to bulk values [26]. In general, all ordering related properties (including the Curie temperature) change when going from bulk to thin film. The Thomas-Fermi model, while illustrative and tractable, is incomplete. The screening length depends on other factors than just the electron density in the contacting electrodes. There is an interplay between the screening mechanisms presented earlier (domain walls, vacancies) and the purely electrostatic considerations presented above as was shown on BiFeO 3 /La x Sr 1-x MnO 3 interfaces. Numerous works showed that the screening was dependent on accumulation of oxygen vacancies and/or adatoms which in turn depended on the polarization and surface termination of the electrodes [27; 28].

. Ferroelectric perovskites

Ferroelectric perovskites have among the highest reported values of spontaneous polarization and Curie temperatures (see table 3.1). This, in addition with the ample knowledge accumulated on perovskites in the last decades, assured their large utilization. Even if novel 2D-ferroelectrics recently emerged, with exceptionnally high stability for few layer sized films, perovskites remain the ideal playground for applications. In this section, will be presented first the generalities on perovskites before diving into the details of BiFeO 3 and LiNbO 3 which were used in this thesis.

Type

Formula P s (µC 19.9 [37] No -(Theory)

. Generalities on perovskites

Every or so physical property can be found in transition metal oxides. They can be insulators, superconductors, ferromagnets, ferroelectric crystals... The phases are numerous and are usually variables with pressure, temperature or application of fields of all kinds. Small perturbations can dramatically change properties and lead to a complex and rich phase diagram. These structures are based on the bonding of d-orbitals, the cation can show multiple oxidization states and crystal field of different symmetries will lift the degeneracy of the d-orbitals according to the crystallography. The discussion becomes endless when taking into account the effect of vacancies or doping into the crystal. The family of perovskites is a fine example of transition metal oxide family. Perovskites are materials with general formula ABO 3 in which A is an alkali metal (Li, Na, K...) or an alkaline earth metal (Be, Mg, Ca, Sr, Ba...), B is a transition metal and O oxygen. They form a ionic lattice with oxidization degrees of the form:

A +/2+ B 4+/5+ O 2-
. The resulting lattice is called the perovskite lattice which, in its simplest configuration, is cubic, as shown in figure 3.10a. The structure can be seen as a cube with each corners supporting a cation A and containing a BO 6 octahedron. Taking a closer look at the structure in the < 001 > directions, the lattice is actually composed of alternating layers of oppositely charged (or neutral) "A-oxide" AO and "B-oxide" BO 2 (figure 3.10b). In most ferroelectric perovskites the planes are charged.

A 2+ O 2- B 4+ AO 0/1- AO 0/1- BO 2 0/1+
[001]

[100]

[010] The rich physics of perovskites comes from the possible rearrangements of the structure. Most distortions of the structure fall into the three following categories [39]:

• Ferroelectric distortions: off-centered B cation inside the octahedron • Antiferrodistortive distortions: opposite rotation of the octahedra • Jahn-Teller distorsions: deformation of the octahedron Goldschmidt's tolerance factor [40] (equation (3.30)) is used as a crude estimator1 of the stability of ABO 3 compounds into perovskite structure at room temperature. The model assumes a rigid sphere model to assess the stability, relating the ionic radii r A , r B and r O of the species A, B and O.

t = r A + r O √ 2(r B + r O ) (3.30)
When the ratio t is equal to 1, the material is a perfect cubic perovskite (e.g. SrTiO 3 ). t > 1 implies that the B cation is too small as compared to A and free to go off center (as in BaTiO 3 when polarized), whereas t < 1 implies that the B cation is too big as compared to A and tends to predict rotations of octahedra (as in CaTiO 3 or BiFeO 3 ). We easily see from the tolerance factor of LiNbO 3 (t = 0.80) and BiFeO 3 (t = 0.87) that both are far from the ideal perovskite. These ferroelectrics, at the basis of this thesis, indeed show peculiar properties as will be presented in the sections 3.4.2 and 3.4.3.

. Lithium Niobate LiNbO 3

Because of its use in non linear optics, Lithium Niobate -LiNbO 3 (LNO) is among the most studied ferroelectrics on par with BaTiO 3 . Since the second half of the last century, its omnipresence in photonics has led to number of studies on minute aspects of its production, like the impact of doping or defects on its properties . With its T c above 1300K and high spontaneous polarization, it is also a suitable ferroelectric. Here we will review a few of its properties, mainly in regards to ferroelectricity.

Cristallography of LNO

While having the formula ABO 3 , LNO is often not considered as a perovskite since its high temperature paraelectric phase is not cubic. We calculated LNO's tolerance factor to be t = 0.80, which indicates a possible severe deformation of the lattice. In fact, it roughly crystallizes like BiFeO 3 since its paraelectric high temperature phase has space group R 3c while its ferroelectric is phase R3c. The R3c group can be described with rhombohedral or hexagonal axes systems. In the case of LNO, both descriptions are helpful. The lattice parameters are however very different from the ideal cubic perovskite. Because of this heavily distorted unit cell it is often described as a pseudo-ilmenite type crystal [43]. In the [0001] hex ([111]

Growth of LNO

Industrial standard single crystal LNO is generally grown by Czochralski method, i.e., slowly pulled out from the melting mixture of different powders: Li 2 CO 3 , Nb 2 O 5 and possibly MgO in doped versions [START_REF] Chen | Domain switching characteristics of the near stoichiometric LiNbO 3 doped with MgO[END_REF]. While Czochralski growth allows for good quality single crystals, the process is known to introduce punctual defects. There are several type of commercialised LNO depending on the initial mix, i.e., the η =[Li]/ [Nb] ratio that affect the stoichiometry of the final crystal. Congruent LNO (CLN) is the most common type since because of the eutectic point in the phase diagram (see figure 3.13a), it has ratio η = 48.4/51.6 ≈ 0.938. Near stoichiometric LNO (SLN) has a ratio η = 0.990 but has to be obtained from more complicated techniques [START_REF] Sanna | LiNbO 3 surfaces from a microscopic perspective[END_REF]. The amount of defects is ultimately dependent on the exact growth parameters (pulling rate, rotation speed...) [START_REF] Chen | State of the art in crystallization of LiNbO 3 and their applications[END_REF] but generally, industrial LNO tends to be Li deficient. These vacancies tend to be replaced by Nb antisites (Nb cations in place of Li cations) as illustrated in figure 3.13b. The equation reaction is written as [START_REF] Smyth | Defects and transport in LiNbO 3[END_REF]:

6Li Li + 3O O + N b N b -→ 3Li 2 O + 4V ′ + N b •••• Li (3.

Ferroelectric properties of LNO

Number of studies have been able to relate the [Li]/ [Nb] ratio (or LiO 2 concentration) and/or the effect of MgO doping on the ferroelectric properties of LNO. As a general trend, the spontaneous polarization P s and the Curie temperature T c are maximal in CLN and decrease linearly with increasing MgO concentration or when going to a more stoichiometric crystal. The impact of the point defects is dramatic on the values of the coercive field which goes from 22kV • mm in CLN to merely 4kV • mm in SLN (see figures 3.14a and b). This is also true for the value of the internal field. All those properties are closely related to how point defects are compensated. Chen et al. explained that MgO doping introduces Mg 2+ cations that replaces Li deficiencies instead of Nb anti-sites [START_REF] Chen | Domain switching characteristics of the near stoichiometric LiNbO 3 doped with MgO[END_REF]. This overall increases the conductivity of the compound leading to the reduction of its ferroelectric properties. Those considerations will be of importance for the surface preparation of LNO as measurements using photoemissions spectroscopy require some conductivity to alleviate charge effects.

Domain ordering of LNO

In the previous section we overlooked a fact: the P-E loop of LNO is biased (not centered on zero) [50; 32]. This implies that LNO is unipolar, i.e., it does not develop a domain structure naturally but instead has its polarization pointing in a unique direction. In addition, the coercive field of LNO is so high that reversing the polarization using a direct field (poling) the crystal used to be impossible. For a long time, it was considered as a "frozen ferroelectric" [START_REF] Shur | Micro-and nano-domain engineering in lithium niobate[END_REF]. However, nowadays, periodically poled lithium niobate for quasi-phase-matching or second-harmonic-generation is routinely produced in the industry, although this requires specific methods. Indeed, the efforts of Myers et al. showed that it was possible to pole LNO on a large scale using standards techniques for microfabrication. They found a way of obtaining the desired domain structure. They deposited an insulating layer (photoresist) first and overlay a conductive layer (Al in their case), or the reverse. The structure was then completely immersed in a electrolyte bath (LiCl water solution) and a voltage was applied for at least 50ms to switch permanently the domains that were electrically connected to the bath [START_REF] Myers | Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO 3[END_REF]. In chapter 5, we will use this domain structure to study the influence of the polarization direction on the band structure of WSe 2 flakes.

. Bismuth Ferrite BiFeO 3

In the short list of multiferroics, bismuth ferrite -(BiFeO 3 , in short BFO) -is the most studied. This material is the only one showing both magnetic ordering (antiferromagnetism) and ferroelectricity at room temperature, making it the ideal compound to study the interaction of multiple coexisting ferroic orders. The existence of couplings between these orders is a guarantee of complex and interesting physics as well as a rich phase diagram. Owing to the progress in pulsed lased deposition (PLD) techniques, BFO is nowadays mostly studied in its thin film variant rather than its bulk form, since exceptional values of the spontaneous polarization as large as 100 µC • cm -2 were measured. In addition, BFO has large Néel (T N ≈ 640K) and Curie temperatures (T c ≈ 1100K) which make it an ideal candidate for applications.

Crystallography of bulk and thin-film BFO

At very high temperatures (>1206K), bulk bismuth ferrite adopts a cubic structure. In the perfect perovskite picture, BFO is a ionic lattice of Bi 3+ and Fe 3+ with Fe atoms involved in the FeO 6 octahedron. Looking at its value of tolerance factor t ≈ 0.88, the structure is expected to adopt several distortions at room temperature. Indeed, under 1198K, and therefore at ambiant temperature, BFO crystallizes in the rhombohedral R3c (C 6 3v in Schönflies notation) space group [START_REF] Palai | β phase and γ-β metal-insulator transition in multiferroic BiFeO 3[END_REF]. Interestingly, the paraelectric phase has space group R 3c (D 6 3d ) instead of cubic [START_REF] Selbach | The ferroic phase transitions of BiFeO 3[END_REF]. This rhombohedral phase is better understood when looked from the point of view of the pseudo-cubic (pc) crystal system. The cell is doubled to contain two pseudo-cubic cells and shows several distortions. The first one is a tilt of the two FeO 6 octahedra in different directions due to antiferromagnetic ordering. This anti-phase rotation of the FeO 6 octaedra is along the [111] pc ([0001] h ) axis with an angle between 11 and 14° [ 55; 56]. The second distortion is the relative displacement of the Bi and Fe sublattices. The later is considered to be the consequence of the "stereochemical activity" of the Bi atom at the origin of the ferroelectricity in BFO [START_REF] Seshadri | Visualizing the role of bi 6s "lone pairs" in the off-center distortion in ferromagnetic BiMnO 3[END_REF]. The Bi valence configuration is 5d 10 6s 2 6p 3 but in the structure, the ionic character of Bi empties the p orbital while the 6s 2 constitute a lone pair of electrons. According to Ravindran et al., the lone pair is not completely unbounded and partial hybridization occurs with the O-2s and O-2p orbitals. This influences largely the stereochemistry of the Bi atom, i.e. the asymmetric character of the hybridization triggers both the rotation of the octahedra and the lattice displacement from which the ferroelectricity emerges [15]. BFO as a thin-film has a far richer phase diagram. When constrained, it can accomodate three new monoclinic configurations M A , M B and M C from the notation of [7] with signature patterns in X-ray diffraction [59; 60; 61]. The monoclinic phases M A and M B appear when the film experiences respectively moderate compressive or tensile strain. The M C phase appears in more extreme cases of compressive strain. When going from high compressive strain to zero, the M C and M A phases act as bridge phases between heavily distorted tetragonal BFO and rhombohedral BFO. M B phase is the bridge phase between rhomboedral and orthorhombic phases in tensile strain configurations. M C phase has a highly tetragonal character while M A and M B have a rhombohedral character. For this reason they are often called T-like and R-like phases. A summarized view of all these possible transitions can be seen in the figure 3.17. While structurally similar, the M A and M B phases show differences in their ferroelectric properties. The polarization in the M A phase has a more pronounced out-of-plane character (P x = P y < P z ) whereas the other phase has a more in-plane character (P x = P y > P z ). In both cases the polarization stays in the (1-10) pc plane as illustrated in the figure 3.18 [START_REF] Chen | Low symmetry monoclinic mc phase in epitaxial BiFeO 3 thin films on LaSrAlO 4 substrates[END_REF]. Figure 3.17: Effect of the strain on the lattice of BFO. Taken from reference [30].

Bi 3+ O 2-

Strain engineering of thin-films

Clearly, the choice of a substrate is of primary importance when growing BiFeO 3 . In this work, the BFO grown and provided by our collaborators, the Unité Mixte de Physique CNRS-Thales, was primarily TiO 2 -terminated SrTiO 3 (001) (STO) and DyScO 3 (110) (DSO). The figure 3.19 shows substrates of different pseudocubic lattice parameters compared to those of the materials often deposited as thin films. In both cases, there is compressive strain leading to the growth of BFO in an M A phase (R-type). DSO substrate is close to the optimal substrate minimizing the misfit strain. The impact of the strain is minimal on the Curie temperature (see figure 3.20) and the BFO can stay fully unrelaxed when the strain is between -2.6% and +1.2% to thicknesses of 60-70 nm and even higher for lowest values of strain like on DSO with thickness up to 300 nm [30]. In those samples, there was always an intermediate epitaxial layer of metallic electrode either La x Mn 1-x SrO 3 (LSMO) or SrRuO 3 (SRO) of circa 5 nm thick to provide the necessary screening. The impact of the electrode on the strain is small as can be seen in the extensive review of Burns et al. [START_REF] Burns | The experimentalist's guide to the cycloid, or noncollinear antiferromagnetism in epitaxial BiFeO 3[END_REF]. In the case of SRO electrodes, the strain tends to be reduced [START_REF] Fischer | Imaging and tailoring electric and antiferromagnetic textures in multiferroic thin films of BiFeO 3[END_REF]. From the previous discussion, we got that the polarization of the BFO in the M A phase is more or less identical to the polarization of the rhombohedral bulk BFO, nearly along the main diagonal of the cell. Looking at the perfect cubic BFO in figure 3.21a, one can see that there are eight equivalent <111> directions. Each of them is a possible direction around which the crystal can stabilize. This implies a domain structure of the BFO: there are in fact four different ferroelastic variants (figure 3.21b) and three kinds of domain walls [66]. This means that even in samples with uniform out-of-plane polarization, there is still an in-plane domain structuration. Miscuts along carefully selected directions of the substrate can eliminate a fraction or all of the possible variants as represented in figure 3.21c. This patterning of the polarization is experimentally accessible by piezoresponse force microscopy, a technique that will be presented later in this chapter.

Interface control of the polarization

The choice of the substrate miscut is already a powerful engineering tool to select the desired polarization variants. However, it does not allow to choose the direction of the outof-plane component of the polarization. In a series of papers, it has been demonstrated that it was possible to fully determine the final polarization state of the BFO using a few unit cells thick appropriate buffer layer. As told in the general description of perovskites, BFO can be described as a stacking of (BiO) + and (FeO 2 ) -charged planes. The boundary conditions are fixed by conditions at the interface, i.e., the termination of the electrode.

When using perovskite electrodes such as LSMO or SRO, epitaxial growth of BFO on top guaranties that the AO-BO 2 sequence is respected as illustrated in figure 3.22. LSMO is a perovskite whose charged plane decomposition along <001> directions is (La,SrO) +δ - Similar conclusions were drawn from other experiments using other substrates and theoretical arguments [68; 69; 28]. It was also determined that with the bottom electrode grounded, the potential inside the BFO increases with each unit cell up to the surface. This potential create a slight bias of the hysteresis measurable in the P-E loop of the BFO, hinting at a preferred polarization state. In the end, controlling the polarization state boils down to choosing the termination of the initial substrate. MnO 2 terminated LSMO will grow on TiO 2 terminated STO [START_REF] Yu | Oxide interfaces: pathways to novel phenomena[END_REF] which will lead to an UP polarization state of the deposited BFO film. The same kind of explanation can be invoked for SRO electrodes. Indeed, SRO is a metallic perovskite whose planes along <001> are alternatively (SrO) 0 -(RuO 2 ) 0 planes. In principle it can be either RuO 2 terminated or SrO terminated but the RuO 2 termination was shown to be volatile [67; 71]. As a consequence, the termination is always SrO. The preservation of the AO-BO 2 sequence constrains the BFO to start its growth with FeO - 2 layer, leading to a valence mismatch of -1. This implies a negative polarization in the long run (figure 3.23). The most important consequence is that this technique is the route to samples with uniform out-of-plane polarization, alleviating at once the difficulty of working with multiple domains especially with techniques like photoemission. This will help us in the fabrication and characterization of our hybrid structure WSe 2 /BFO in chapter 5.

. Measuring ferroelectricity 3.5.1 . Piezoresponse force microscopy

Piezoresponse Force Microscopy (PFM) is a declension of Atomic Force Microscopy (AFM) used to analyse ferroelectric materials. It can analyse the domain ordering and prove the switchable character of ferroelectrics. We will first describe standard AFM before explaining the peculiarities of the PFM mode.

Atomic force microscopy

AFM is a type of scanning probe microscopy that allows the measurement of the topography of a sample with a lateral resolution of few nm and vertical resolution of the order of a few Å. The principle of an AFM is schematically represented in the figure 3.24. A nanometer sized tip with a small curvature (5 to 50 nm) is mounted on a cantilever. The tip points towards the sample while the cantilever has a reflective coating on its opposite side. To measure the topography of a sample, a laser beam is sent on the cantilever reflective area. When the tip is close to the sample, it experiences forces that bend the cantilever, which in turn deviate the laser beam. This deviation is detected by a four sectional photodiodes system and can be related to the forces exerted on the cantilever whose spring constant is known. A 3D-imaging of the surface can finally be done by scanning the surface in a x-y grid. This is achieved by having the sample mounted on piezoelectric steppers that control its coordinates in the three directions. This helps achieving the lateral resolution of the technique, but limits the measurable area to a few hundreds of micrometers.

U (r) = E 0 (r -12 -r -6 ) (3.33)
The attractive term (∝ -r -6 ) grossly corresponds to the van der Waals forces (among others) that dominate at moderate distances (1 to 100 nm). The repulsive term (∝ r -12 ) is a phenomenological approximation of the "Pauli repulsion" [72; 73] The existence of these two kinds of forces allows for two modes of mapping, either in contact mode using the repulsive forces or in non contact mode in the attractive regime. There are also dynamical modes with intermittent contact with the sample. These modes help reducing the pressure exerted by the tip on the sample when the forces at play in contact mode are much stronger.

The sample topology can be quite rough in some samples. It is required to adjust the tipto-sample distance if one does not want to deteriorate the tip and the sample. To achieve this, the measurement of the force is fed into a feedback loop so that the tip-to-sample distance is adjusted. The measurement can be done by keeping contant the tip-to-sample distance: this is the so-called constant height mode. The aim can be to keep the force constant (constant force mode) to avoid extreme forces on the sample. The latter remains the most used mode for our application of the technique.

Piezoresponse Force Microscopy

Piezoresponse force microscopy is based on the piezoelectric effect. As defined in the very beginning of the chapter, this effect relates the stress in a material with the electric field. The direct piezoelectric effect is the induction of a polarization by stress while the converse piezoelectric effect is the induction of stress by an electric field. The latter is exploited in PFM measurements. Since the effect is linear in the electric field it can be written as follows:

σ ij = d kij E k (3.34)
Where σ ij is the stress tensor, d kij the piezoelectric tensor and E k the electric field. In a ferroelectric materials it can be shown using a thermodynamical approach that the piezoelectric tensor is related to the polarization. In the following, we will sketch the demonstration found in [5; 74]. Other demonstrations can be found in the literature [START_REF] Damjanovic | Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics[END_REF]. The core idea is to take into account the electrostrictive effect. This effect occurs when an electric field is applied to a dipole. The dipole will tend to stretch under the effect of the field, the positive charges being pulled in a direction, the negative charges in the other (see figure 3.26). This effect exists in all the materials, especially if they are composed of different elements (impliying dipoles).

Adding up this effect to the size of the crystal, we see that there is a global strain due to the application of the field. The electrostriction is usually quadratic in the field [START_REF] Newnham | Electrostriction: Nonlinear electromechanical coupling in solid dielectrics[END_REF] and writes:

x ij = Q ijkl P k P l (3.35) x ij = M ijkl E k E l (3.36)
With x ij the strain tensor, P k the polarization and Q ijkl the electrostriction tensor. If written as a function of the electric field, M ijkl is related to Q ijkl through the electrical susceptibility χ ij (P i = ε 0 χ ij E j ). We will take the free energy density of Landau theory that we presented in section 3.2.1 taking in account electrostriction and the stress-strain energy density of a linear material

1 2 σ ij x ij = 1 2 C ijkl x ij x kl .
For simplicity, we drop all the indexes:

F = α 2 (T -T c )P 2 + β 4 P 4 + 1 2 Cx 2 + xQP 2 (3.37)
The strain σ is given by:

σ = ∂F ∂x = Cx + QP 2 (3.38)
From this relation, we can find the piezoelectric conversion factor d:

d = ∂σ ∂E = ∂σ ∂P ∂P ∂E (3.39) = 2QP χ (3.40)
In a ferroelectric there is an intrinsic polarization. Therefore, with zero field, we have P = P 0 . This finally gives:

d = 2χQP 0 (3.41)
The relation (3.41) allows the mapping of the ferroelectric domains with the piezoresponse. This is close to the relations used in the literature d 33 = 2ε 33 QP [77; 78] where the probed piezoresponse is the d 33 component of the piezoelectric matrix (matrix notation of the tensor component d 333 , corresponding to the z response in the z direction for a z field). In practice however, the piezoresponse amplitude is low and has to be mesured through a lock-in amplifier.

To do so, an AC-voltage is applied between the down electrode of the ferroelectric and the tip.

The amplitude of oscillation can be increased but it has to remain below the coercive field of the material. The strain response x to the electric field E(t) = E AC cos(ωt) will be:

x = ± 2ε 33 Q|P | C (E AC cos(ωt)) + M (E AC cos(ωt)) 2 (3.42)
With the lock-in amplification on the first harmonic ω, one detects:

∆Z ∝ x(ωt) = ± 2ε 33 Q|P | C E AC cos(ωt) (3.43) ∆Z = A cos(ωt + ϕ) (3.44)
In the case of a down domain, the crystal oscillates in phase with the voltage. In the case of an up domain, the oscillation is 180°out of phase as pictured in figure 3.26. If the amplitude is usually difficult to interpret beyond qualitative appreciation, the phase shows the domain configuration of the crystal. It is possible to extend the measurement to the in-plane component of the polarization vector to obtain a vectorial mapping of the domain ordering [START_REF] Kalinin | Vector piezoresponse force microscopy[END_REF]. If one applies a DC component in addition, it becomes possible to switch the domains under the tip. This is a route to local patterning of the ferroelectric material and it can also provide hysteresis curves called a V-d 33 loops. Knowing the electrostrictive coefficient and the local thickness, it is possible to retrieve the P-E loop and have a full characterization of the sample.

. Photoelectron emission microscopy

Less used than PFM, photoemission in PEEM mode is also a way of characterizing ferroelectrics. With a simple model, it is possible to relate the polarization with image contrast using PEEM. Let an electron coming from a domain P + (P -). As discussed before the surface of the ferroelectric is not perfectly screened and, on a length of d + (d -), the outgoing electron is slowed down by an electric field. Because of that, electrons coming from different domains should have different work functions. As discussed in the previous chapter, differences in work functions are measurable with energy filtered PEEM, by measuring the secondary electron edge at very low kinetic energy (see figures 2.14 and 2.25). Rault et al. have shown that out-of-plane polarisation relates to the work function difference: whether from a qualitative perspective (see reference [START_REF] Rault | Polarization sensitive surface band structure of doped BaTiO 3 (001)[END_REF]) or even a quantitative point of view (reference [START_REF] Rault | Thickness-dependent polarization of strained BiFeO 3 films with constant tetragonality[END_REF]). The figure 3.28 shows the excellent agreement between PFM imaging and measurements done with a PEEM. At a given kinetic energy, the contrast between two oppositely polarized domains can be very high (to be seen in figure 3.28b). On a practical experiment, this is a first good indicator of a sizable polarization difference. On the quantitative side, the work function map (3.28c) derived from energy dependent PEEM images completes the demonstration of this relation. To some extent, even in-plane ferroelectricity is accessible with PEEM measurements thanks to in-plane field at domain boundary which deviates electrons [START_REF] Rault | Reversible switching of in-plane polarized ferroelectric domains in BaTiO 3 (001) with very low energy electrons[END_REF]. This link is summarized in equation (3.45) where the difference in work function is shown proportional to the spontaneous polarization.

∆Φ F , ∆E ∝ e ε 0 P + d + -P -d -≈ 2 e ε 0 P s d (3.45)
To this idealised view, one has to add the inevitable variations due to local topography and/or chemistry that limit partially the quantitative treatment of PEEM data to measure ferroelectric physical quantities but it is nonetheless an extremely powerful tool. We will make extensive use of this versatility of the PEEM to study the hybrid system WSe 2 /LNO in chapter 5.

P

-Electronic structure of transition metal dichalcogenides

. Introduction

The study of transition metal dichalcogenides (TMD) has been at the forefront of research in 2D materials for about ten years. The interest in this family of materials emerged when searching alternatives to graphene, the first thoroughly studied 2D material. Discovered about 15 years ago, graphene became a trendy material among condensed matter physicists owing to its peculiar properties. Among them, let's cite exceptionally high conductivity, Dirac like dispersion relations, novel fabrication methods... This led to the 2010 Nobel prize, shared by Geim and Novoselov. This research effort put into graphene improved greatly the knowledge on growth of 2D materials which spawned the domain as it is nowadays. While showing many interesting properties, graphene is a semi-metal, meaning that it is a metal with zero density of states at the Fermi level, but no gap altogether. This last fact prevents its extensive use in optoelectronics in the visible range. TMD in most forms, are semi-conductors. They are lamellar materials. As we will show below in more details, the building block is a sheet with generic formula MX 2 , where M is a transition metal and X a chalcogen, tied by covalent bondings. The three-dimensional solid is obtained by stacking those MX 2 sheets, weakly bonded by van der Waals interactions. The family of group-VI TMD (i.e. TMD whose metallic element belongs to the 6th column of the periodic table like Cr,Mo or W) happens to have a moderate gap of the order of 1 eV suitable for optoelectronics [1]. In this thesis, we will restrict the study to this subclass of formula MX 2 where M is a transition metal (either molydbenum (Mo), or tungsten (W)) and X is a chalcogen (S, Se or Te). We will concentrate on WSe 2 , a compound slightly less studied than its counterpart MoS 2 . This restriction will not limit the discussion, as the physics of group-VI TMD family remains unchanged from MoS 2 to WTe 2 . In practice, this signifies that knowledge on one TMD is easily transferable to another.

In this chapter, we will focus on the crystallography and on the electronic structure of TMD (other properties of interest for optoelectronics will be briefly described in chapter 5). In the last part of this chapter, we will see how ARPES describes the evolution of the band structure of WSe 2 with an increasing number of layers.

. Crystallographic structure

Group-VI MX 2 compounds (referred to as TMD from now on) crystallize as layered systems. A unit sheet is organized as X-M-X, where chalcogenide X atomic sheets are disposed in mirror symmetry around the transition metal layer. The transition metal is covalently bonded with the six chalcogens in a trigonal prismatic coordination (see figure 4.1). When looked from above ([001] direction), the layer forms a honeycomb lattice and is best described with the hexagonal axis system. The corners of the hexagons are not equivalent so that there is only a threefold axis of rotation. Importantly, there is no inversion symmetry in monolayers, implying the space group P 6m2 (D 1 3h ). Multilayer TMD organize as weakly bonded van der Waals stacks of monolayers. In the 2H polytype, two successive layers are facing opposite directions and shifted in such manner that any transition metal of the top layer is aligned with two chalcogens of the bottom layer (see figure 4.1). Interestingly, odd and even numbered multilayers have different space groups [2]. Odd numbered structures do not exhibit inversion symmetry and therefore have P 6m2 (D 1 3h ) like in the monolayer case. Even numbered structures as well as bulk systems instead have P63/mmc (D 4 6h ) space group thanks to this additional symmetry [3]. Some other phases with octahedral coordination (see figure 4.1) exist giving the so called 1T and 1T' (distorted 1T) structures. In these phases, the crystal has widely different physical properties [4; 5; 6; 7]. hexagon that is rotated by 30 °with respect to the hexagons in real space. In the BZ there are several points of importance called high symmetry points. The Γ point is at the center of the BZ and the wave vector k = (0, 0) bears the full space-group (D 3h ). There are six K points at the corner of the hexagon, their distance to the center is d Γ-K is 4π 3a . Notably, the wave-vector group at ±K is reduced with only a threefold rotation axis, and a mirror symmetry (C 3h ). The K points are therefore not equivalent and are alternatively labelled K and -K. into ARPES data because one sees directly the band structure and the eventual deviations from the expected symmetry group. In the case of Raman or photoluminescence spectroscopy experiments, the group considerations become very important to label properly the observed peaks. Up to now, we have only discussed the in-plane structure. There is however the k z direction, which is of importance in bulk WSe 2 with additional high symmetry points. The full BZ for bulk WSe 2 is given in figure 4.3.

Distance Formula Value (Å -1 )

d Γ-A π/c 0.2424(2) d Γ-K 4π/3a 1.2762(4) d Γ-M 2π/a √ 3 1.1053(4)

. DFT calculations

Transition metal dichalcogenides band structure is very dependent on the number of layers. Figure 4.4c shows the density functional theory computed band structure for WSe 2 from monolayer to quadrilayer. The general features are a direct band-gap in the monolayer case with valence band maximum (VBM) at K. Each additional layer will add a new band at Γ. For the bilayer (2ML) case, the electronic gap already turns indirect and its value quickly converges to the value of bulk WSe 2 . The evolution of the gap was first studied with photoluminescence spectroscopy which confirmed the theoretical predictions: structures composed of 2ML or more see the intensity of the main photoluminescence peak largely quenched compared to the 1ML case. The value of the electronic gap also diminishes with increasing thickness as was shown with scanning tunneling spectroscopy [9] (STS) or later by ARPES [10]. One of the most important ingredients of the band structure of TMD is the spin-orbit coupling (SOC). In figure 4.4c, we see that the band at K is strongly affected by the inclusion of SOC. Without SOC, the number of bands at K increases with the number of layers. With SOC however, the structure at K remains essentially the same whatever the thickness of the sample with two spin-splitted bands. This effect is present for all TMD of the family and the splitting for different MX 2 compounds are reported in the table 3.1. This spin-splitting of the bands at K is particularly strong in WSe 2 because of the presence of high Z elements like W and Se with a difference as high as 480 meV. We noted that the splitting at K remains unaffected by the accumulation of layers. From a model perspective, the spin-orbit interaction seems to prevents the interlayer coupling. The first consequence is a strong spin polarization of the bands at K and -K as visible in figure 4.5. The second consequence is the so-called spin-layer locking where two successive layers have opposite spin-polarization at K.

Г K M Г Г K M Г Г K M Г Г K M Г

Without spin-orbit coupling

With spin-orbit coupling

Δs-o ~ 0.47 eV 0 -1 1 2 0 -1 1 2 Г -K -K K -K K K M Г K M Г Г K M Г Г K M Г Г K M Г

. Orbital content of the band structure

The features that were described in the previous section can be elegantly explained using a few orbitals and a tight-binding (TB) approach. The near Fermi-level band structure of WSe 2 is mostly the result of hybridization of the Se-4p orbitals (valence structure 3d 10 4s 2 4p 4 ) with W-5d orbitals (valence structure 4f 14 5d 4 6s 2 ). A crystal field approach with symmetry analysis of the W-5d orbitals shows that in the trigonal prismatic configuration, one can expect three groups of energy-degenerated orbitals: (d z 2 ), (d xz , d yz ) and (d x 2 -y 2 , d xy ) [12]. The state with lowest energy is |d z 2 ⟩ and the states with higher energy are |d x 2 -y 2 ⟩, |d xy ⟩ as represented in figure 4.6a). Orbital projected band structure calculation for WS 2 in figure 4.6c and spectral weights of the orbitals in the bands in figure 4.6d show that the bands at Γ should be mainly composed of d z 2 orbitals and bands at ±K should be mostly due to d x 2 -y 2 and d xy orbitals. For that matter, the visualization of the electronic distribution with maximally localized Wannier functions in figure 4.6b is particularly talkative. The d z 2 character of the Γ band appears very clearly. On the other hand, the mixing of multiple orbitals in the making of the bands at K makes the comparison with the original orbitals more difficult. All in all, inclusion of spin-orbit coupling is not fundamental to understand the valence band composition at K. The composition is principally the result of the constraints for the high-symmetry points. As an indication, this orbital composition for those points are summarized in table 4.3. where the bands at the high symmetry points are:

State

k = k = ±K Symmetry subgroup D 3h C 3h Majority orbitals M-d z 2 M-d x 2 -y 2 ,
|⟩ = |d z 2 ⟩ ⊗ (|↑⟩ + |↓⟩) (4.3) |±K⟩ = |d x 2 -y 2 ⟩ ± i |d xy ⟩ ⊗ (|↑⟩ + |↓⟩) (4.4)
The Hamiltonian of the system is then written with only the on-site contribution of the spinorbit coupling for the W-atom:

H SOC = λL • S (4.5) = λ 2 L z 0 0 -L z (4.6)
where:

L z = λ 2   0 0 0 0 0 2i 0 -2i 0   (4.7) H =    H 0 (k) + λ 2 L z 0 0 H 0 (k) - λ 2 L z    (4.8)
where H 0 (k) is the tight-binding Hamiltonian for three orbitals with the nearest/third-nearest neighbour hopping included. The splitting value λ is not predicted by the model and has to be retrieved from DFT calulations or experimental results. The model is also unable to recover the spin-splitting in the conduction band. Those finer effects need the inclusion of more orbitals (namely the neglected Se-4p and other W-5d orbitals) or even the full set of valence orbitals but at the cost of the readability of the model. Such treatments usually give results very similar to the experiment, as seen in figure 4.7 for bulk WSe 2 .

. Photoemission on WSe 2

. State of the art on HR-ARPES

In this section, we summarize the ARPES experiments performed on tungsten diselenide. The first ARPES measurements on WSe 2 were done in the late 1990s with three publications [18; 19; 20]. These publications contain, even to this day, valuable insights on WSe 2 and unique results. For instance, a nearly unique ARIPES (angular resolved inverse photoemission spectroscopy) determination of the conduction band can be found in the work of Finteis et al. [19]. This admirable work, using cutting edge developments of the era was however not pursued for a long twenty years gap. It is only when the field of 2D materials, and especially interest in MoS 2 , boomed that new characterisations were published. In comparison to older works, these ARPES studies benefited from the improvements of the light source (new generation synchrotron radiation sources) and the instrumentation (better electron analyzers with 2D detectors). A notable contribution is a complete characterization of bulk WSe 2 by Riley et al. in which not only ARPES but spin resolved measurements are reported [21]. Published in 2014, it is seemingly the first paper on the subject (excluding others TMD of the discussion) since the 1990s seminal work. The main results of these works is summarized in the table 4.4, where most important physical parameters of the valence band of WSe 2 from monolayer to bulk are reported. An illustration of those parameters is visible in figure 4.8. The table also contains important information such as: the substrate, the growth method, the type of transfer if the sample is measured on a different substrate than the one it was grown upon. The principal growth methods are:

• Chemical vapour transfer (CVT): mainly used to grow bulk crystals • Chemical vapour deposition (CVD): grows small scale flakes (nm -→ µm), usually with random orientation with respect to the substrate

• Molecular beam epitaxy (MBE): grows large scale thin-films (µm -→ cm), usually with good anisotropy (i.e. single orientation of the film)

• Exfoliation: not a growth method per se but which usually produces very clean few-layer films of intermediate scale (µm)

The transfer methods (PMMA based, dry and wet transfers) will not be described in this chapter. The interested reader will find a detailed review on this matter in reference [22].

There are only a few things to know at this point. First, those three denominations encompass several different transfer methods. Second, all of them involve ex-situ steps using chemicals that might induce deterioration of the films. The consequences can be an undesirable bonding between the WSe 2 film and the chemicals of the environment, changing locally its properties, or from the presence of defects and impurities which can cause additional damage to the film. The main conclusions that can be extracted from the table 4.4 are the following:

• ARPES measurements confirm the main properties expected from the DFT calculations, namely: the change of VBM from Γ to K when the films are more than one layer thick, more and more bands at Γ when the thickness increases, and the strong splitting ∆ s-o at K (see figure 4.9)

• There is no such thing as a free-standing 2D material. The binding energy of the topmost Γ band E Γ shows significant energy shifts when different substrates are used.

For instance, the charge transfers occurring with WSe 2 using monolayer graphene (MLG) or bilayer graphene (BLG) as buffer layers are very different. The substrate can also induce strain effects which are known to modify the band structure and should therefore not be overlooked [23; 24; 25] • The growth technique matters. Exfoliated samples tend to have a larger ∆ s-o splitting than samples obtained from MBE or CVD of same thickness Finally, one should always be cautious with the electronic band gap values given in the literature.

It can be extracted using several methods: local charge injection with scanning tunneling microscope tip [26], direct gate bias [10], surface doping [26; 11]... In these works, it is shown that bandgap renormalization (reduction) occurs when the doping increases.

Г K E Г ∆ s-o E G E K 1 E K 2 Valence bands Conduction band
-1 0 1 -4 -2 0 k || ( Å-1 ) E -E F (eV) -1 0 1 k || ( Å-1 ) -1 0 1 k || ( Å-1 )

Substrate

Growth method Transfer method E

Γ (eV) E K 1 (eV) ∆ s-o (eV) E Γ -E K 1 (eV) E G (eV

. Photon energy dependence of the spectral function 4.5.1 . Motivation of the study

TMD are well known 2D systems and, up to now, we have presented the main characteristics of their band structure. One of the defining elements was the number of bands at Γ which was proved to be a reliable way to determine the thickness of a sample as can be seen in figure 4.9. However as reported in our work on WSe 2 (presented in the chapter 5) [33] it seems, the number of bands at Γ do not match the known thickness from AFM characterization. Our MBE grown layer of mixed 1ML and 2ML WSe 2 was presenting the photoemission spectrum of a monolayer sample with only one band at Γ. However, the sample had significant 2ML coverage and one would expect that, with the average contribution of 1ML and 2ML, the topmost Γ band would be visible. It was only by cranking up the contrast unreasonably that the second band could appear. Changing the photon energy from 50 eV to 56 eV also helped to have the band visible (see figure 5.13), indicating that cross-section effects play an important role. The system studied in this paper being too complex (WSe 2 transfered on BiFeO 3 ), we resorted to study this effect in more details with a simpler heterostructure: WSe 2 on a graphene substrate. At this point of the analysis, the important variables seem to be the substrate, the photon-energy hν and the thickness of the layer.

-1 0 1 -4 -3 -2 -1 0 k || ( Å-1 ) E -E F (eV) -1 0 1 k || ( Å-1 ) -1 0 1 k || ( Å-1 ) -1 0 1 k || ( Å-1 )

. Methods

Fabrication of the samples Centimeter scale (1 cm x 1 cm) 2ML (two samples "A" and "B") and N-layer (N-ML, with N≈ 6, 7) WSe 2 thin films were grown on graphene on Silicon carbide (Gr-SiC) substrates by our collaborator team, SPINTEC (M. Jamet and C. Vergnaud) in Grenoble. To complete the experiment, a (5 mm x 5 mm) sample of 3ML WSe 2 was grown on Mica and then wettransferred onto a Gr-SiC substrate. The SiC substrates were slightly doped.

Sample preparation for photoemission experiments

The samples were mounted on tantalum plates and held with Mo clips (see figure 4.11). Before transferring to the measurement chamber, they were annealed at 300 °C until the pressure stabilised and reached down P ≈ 10 -9 mbar (about three hours). The annealing aimed at eliminating most of the contamination adsorbed on the surface.

ARPES measurements

Photon energy dependent measurements were performed at the CASSIOPEE beamline using the HR-ARPES endstation equipped with a Scienta R4000 electron analyzer. All measurements were done at room temperature (c.a. 300K). The samples were first aligned with the Γ -K direction along the slit of the analyzer. The dispersion (k ∥ , E) can then be measured in one shot. We measured it for photon energies between 20 and 90eV with a 1eV step. The absolute values of the photon energies are not perfectly known. To calculate as precisely as possible the relative binding energies from one measurement to the other, the Fermi-level energy was measured every 5eV (20,25...90eV) using the Mo clips holding and connecting the samples to the ground. The other Fermi levels E F (hν) were linearly interpolated from the data set to find the binding energy for all spectra. To allow more quantitative analysis, the intensities of the spectra were normalised to the secondary electron background intensity above the Fermi level (excited by higher harmonics of the undulator), correcting both the detector background and the differences in flux between two photon energies at the beamline.

E k (E F ) -hν (eV) 2ML -A 2ML -B 1 2ML -B 2 3ML N -ML Figure 4
.12: Fermi level kinetic energy E k (E F ) relatively to the photon energy (monochromator reading). In an ideal experimental setup, the curve should be constant. The 2ML-B sample was measured at two different locations.

Spectral function calculations

Theoretical determinations of the spectral function for 1ML, 2ML, 3ML and N-ML WSe 2 were made to compare with the results of the experiments. The calculations were done by our collaborator Ján Minár (University of West Bohemia, Czech Republic) with the SPR-KKR code. This code is a one-step model implementation with fully relativistic approach using multiplescattering determination of the spectral function as detailed in section 1.8. The calculation was done according to the experimental setup at CASSIOPEE (see figure 4.13) so that the matrix elements are the same.

. Results

Preliminary ARPES characterisation

We first checked the samples to assess their quality and their band structure. Because ARPES is a reciprocal space resolved technique, we can extract qualitative information about the crystallography of the sample like symmetries or surface reconstructions. These will manifest in the band structure symmetry and band duplication. Chemical homogeneity can be checked as well by looking at binding energy shifts and sharpness of the bands. We review here the evidences collected by the mean of angle resolved photoemission on all the samples. The method we employ is the same described in section 2.4.2. We measured a data block by varying the rotation angle θ to obtain the perpenticular k y direction. In such a block of data, one can see arbitrary dispersions or isoenergy planes which show the symmetries of the sample (see figure 4.14).

2ML-A

The first sample is the 2ML-A (2ML WSe 2 grown on graphene). The constant energy cuts shown in figure 4.15a show a relatively well defined band structure with K directions clearly visible for both graphene and WSe 2 . The alignment K W Se 2 and K Gr also tells us that the epitaxy was successful. The image does not have excessive azimutal smearing meaning that the probed WSe 2 is essentially single domain. The second derivative highlights the two nested bands at K W Se 2 without other contributions. We however see that the intensity of those bands is strongly reduced near the K Gr points: instead of seeing two complete nested triangular shapes, their intensity fades off, and we only see two open triangles. The two pictures in figure 4.15b are a zoomed view of the K Gr point at two binding energies. In the top picture of figure 4.15c at energy E B = -1.6 eV, we see that the graphene band is actually doubled. This means that the substrate ends with two graphene layers (=bilayer graphene, BLG). This is confirmed by the top ARPES cuts in figure 4.15c. In the bottom picture at energy E B = -0.3 eV, near the Dirac point E D = -0.32 eV, we see six other points at a distance of 0.4 Å -1 from the center K Gr . This shows that there is a surface reconstruction of the graphene. The bottom ARPES cuts in figure 4.15c shows that a standard Dirac cone seems to be present in between the doubled cones. It is highlighted with a red dashed line. The underlying graphene is a mix of single layer graphene (SLG) and BLG. The magnitude of the charge transfer being dependent on the number of graphene layers [28], the photoemission spectrum of the WSe 2 is duplicated.

2ML-B

The second sample is the 2ML-B (2ML of WSe 2 on graphene). The remarks done for the sample 2ML-A hold for this one although the azimutal dispersion is a bit stronger as shown in the constant energy cuts in figure 4.17 (K points are a little bit less pronounced). The WSe 2 is epitaxial and the graphene has the same reconstruction as before. The ARPES cut of the Dirac cone is doubled as before, hinting at BLG. Figure 4.18a is a photoemission spatial map of the sample (0.1 mm step, intensity of the top of the valence band). It shows that the sample is rather homogeneous in the sub-millimeter scale (much bigger than beam caracteristic length ≈ 50 µm). At the mm scale however, the sample is inhomogeneous. There are two types of areas, showing a slight shift of the VB from one to another. Their respective photoemission spectra are shown in figure 4.18b. In the first one (2ML-B 1 ), the top of the Γ band is near E B ≈ -1 eV while in the second one (2ML-B 2 ) the top of the Γ band energy is at E B ≈ -0.8 eV. The two ARPES spectra also evidence a slight difference in the cross-section at hν in the two cases: the top band is brighter in 2ML-B 2 than in 2ML-B 1 . A more detailed analysis of the band structure shows that in the 2ML-B 1 region, they are splitted in a similar fashion to 2ML-A. In figure 4.19, the lowest Γ valence band is splitted into two Γ 1 and Γ ′ 1 bands with shift 0.12 eV. Using, a trained eye and second derivative analysis, there is another splitted band Γ ′′ 1 . The top Γ band should also present similar or same splittings but they remain hard to resolve with the present data. In the case of 2ML-B 2 , there is no trace of splitting in the second derivative, suggestive of a more uniform substrate.

3ML

The third sample is a 3ML WSe 2 grown on mica then transferred on a graphene/SiC substrate. The constant energy cuts in figure 4.20a show a well defined band structure with K points with little azimutal smearing. There are no any visible ring pattern as there was in the previous samples but the secondary electron background is higher. In spite of that, the second derivative highlights nested bands at K W Se 2 appearing very clearly. Contrarily to the two previous cases, the K W Se 2 and K Gr points are not aligned but separated by an angle of 13 • . This is because the sample was grown on a mica substrate first before performing a transfer onto the graphene substrate. The two structures have therefore no reason to be aligned one with respect to the other. The details on the bottom of figure 4.20b show that the graphene has the same surface reconstruction as before, although this time there is only one cone. This is visible both in the zoomed constant energy cut (E B = -1.6 eV) and the ARPES cut in figure 4.20c implying that the substrate is SLG. E -E F (eV)

N-ML

The last sample is the N-layer WSe 2 (N-ML). The constant energy cut in figure 4.22 show only the band structure of WSe 2 film. In this sample, the graphene is not visible anymore because of the high number of WSe 2 layers (thick sample). The definition of the ARPES image suggests that the sample is of very high quality with very low azimutal dispersion. Looking closely, it is possible to see ring patterns that hints at some disorder. The overall sharpness of the bands, however, is a strong indicator of the quality of the sample (see figure 4.23). 120

Г K WSe 2 k x ( Å-1 ) k x ( Å-1 ) k y ( Å-1 )
-1 0 1 -4 -2 0 k || ( Å-1 ) E -E F (eV)

Comparison of the samples

Now that the sample are characterized, let's try a comparison of their band structure. We saw that 2ML-A and 2ML-B 1 had more or less the same properties with the striking duplicated bands feature. This indicated that the substrates were locally non uniform with a mixture of graphene thicknesses (BLG and SLG mainly). On the other hand 2ML-B 2 sample did not show any of those. For this reason we will proceed to the discussion by first comparing the photon energy dependence of the spectral function for 2ML-B 2 (from now on referred to as 2ML), 3ML and N-ML samples to bring to light a general trend.

Photon energy dependent characterisation

Let us remind that we started this study to try to explain the photon energy dependence of the band structure presented in figure 4.10. With this in mind, we measured the dispersion along Γ-K for all the samples and for photon energies between 20 and 90 eV. As explained in chapter 2, this corresponds to changing k ⊥ (see formula (1.99)) and to travel along the Γ -A direction of the reciprocal space. The top part of figure 4.26c shows the results of photon energy scans for 2, 3 and N-ML samples. Following the brightest bands from low to high photon energy, we observe that the two topmost levels of the 2ML sample appear and disappear when varying the photon energy. Near photon energies 20 eV and 43 eV applying the rule 1ML = one band at Γ, the sample is indistinguishable from a monolayer system with only one visible band at Γ. The same phenomena is measurable on the 3ML sample with instead three bands appearing and disappearing. The binding energies are reported in table 4.5. Finally, we see clear band oscillations in the N-ML case. We see that the oscillation pattern is actually twofold. This repetition is especially visible between 60 and 90 eV. 2ML 3ML

Γ 3 - -1.07 Γ 2 -0.83 -1.34 Γ 1 -1.39 -1.80

Modeling the effect

The strong band dispersion in the k z direction is expected for bulk samples. As stated before in the section 4.3.2, the dispersion at Γ emerges from the strong interlayer overlap of the orbitals W-d z 2 and Se-p z . The phenomenon is well characterized (see references [11; 18; 21; 34]) in WSe (right). The conversion hν -→ k z is done using k z = 2m h2 (E k cos 2 θ + V 0 ) with θ = 0 and V 0 = 13 eV from reference [21].

A minimal style model could be a tight-binding approach, since the effect has an hybrid character (2D system going 3D) with the interlayer interactions. Following the argument of [11], we observe in figure 4.6 that the W-5d z 2 and Se-4p z orbitals in the Γ state are not strongly hybridized keeping their shape. The p z orbitals on Se atoms are expected to contribute the most to interlayer hopping.

We will follow a general procedure described in reference [37]. The tight-binding hamiltonian is first defined for a ML, using 11 basis states (five W-5d, three Se-4p top and three Se-4p bot orbitals) namely:

d x 2 -y 2 , d xy , d xz , d yz , d z 2 , p top(bot) x , p top(bot) y , p top(bot) z

(where p top(bot) i

indicates the orbitals from the Se atom above (below) the W atom) and allowing for nearest neighbour hoppings. They then define this hamiltonian to be:

h M L (R) = 6 n=0 c † R+an h n c R (4.9) with c † R = c † R,d x 2 -y 2 , c † R,dxy , c † R,dxz , c † R,dyz , c † R,d z 2 , c † R,p top(bot) x , c † R,p top(bot) y , c †

R,p top(bot) z

the vector containing creation operators. The creation operators add an electron in the α orbital at

M L l=2n = h M L E , h M L l=2n+1 = h M L O . The h M L E(O)
can be calculated by rotating the hamiltonian h M L (position vectors + wavefunctions) as defined in [37]. The interlayer hopping matrix h int is defined using the two center Slater-Koster integrals [38]:

h int p bot i ,p top j = V ppσ (r) r i r j r 2 + V ppπ (r) δ ij - r i r j r 2 (4.10)
In which the vector r being the relative position of the two orbitals. Interestingly, Fang et al. found an universal behaviour of the functions V ppb (b = σ, π) as they depend only on the two-site distance [39]. They fitted with the model:

V ppb (r) = ν b e - ( r R b ) η b (4.11)
With these ingredients, one can write the N-layer tight-binding matrix as follows:

     h M L 1 h int h int h M L 2 h int h int . . . h int h int h M L N      (4.
12)

The discrete nature of the k z dispersion should then be computable in the spirit of the work of Moser (see figure 4.29) [40]. This TB-calculation performed on a monoatomic chain for an increasing length shows well the same kind of discrete to dispersive transition of the electronic structure.

125 Extracted from [40].

Evolution of the cross-section with photon energy: detailed comparison with onestep model

The one-step model used in this work was powerful enough to predict the band dispersion for an increasing number of layers as well as their intensities. In an opposite spirit to the minimal tight-binding model presented earlier, we seek to characterize how close the calculation came to the experimental results. To do so, we compare the intensities of a given energy level along the Γ -A line. We establish the methodology with figure 4.30. On the experimental data, we integrated the signal over a range ±25 meV around the discrete energy levels to account for their smearing. This gives an estimate of the overall cross-section profiles σ E (hν), E being the binding energy in question. By doing the same thing on the theoretical results of the one-step model (shown in figure 4.26) we can compare the theoretical and experimental cross-sections . from one-step model calculations and experiments. Overall, the computed cross-sections have good qualitative agreement with experimental data: in the case of the 2ML the contrast inversions (one band becomes brighter than the other) and high intensity peaks roughly match. This holds true in the 3ML version even though the agreement is slightly worsened. In general, the evolution of the cross-sections in the experimental data is much smoother compared to the theory.

Effect of the substrate on the cross-section on bilayer samples

Finally, we study the effect of the substrate on these cross-sections. As explained earlier, we performed several times the photon-energy dependent mapping on bilayer samples: 2ML-A, 2ML-B 1 , 2ML-B 2 and 2ML-BFO. Each of them has a different substrate. Figure 4.32 shows the hν maps for each of these systems, their second derivatives as well as the band-resolved cross-section profile. The photon energy dependences look more or less the same. The splitted levels due to the mixed-thickness graphene substrates (2ML-A, 2ML-B 1 ) are still visible but do not change much the general shape (second derivative). A quick look at the cross-sections tells the same story. The charge transfers from the graphene-layers do not impact the profiles. On the BFO, the maxima of the cross-section are seen at lower photon-energies compared to samples on graphene substrate. This could indicate a slight strain on the z axis. The overall behaviour is however the same.

. Conclusion

In this chapter, we reviewed the most important properties of TMD (in particular WSe 2 ) from the point of view of photoemission spectroscopy: crystallographic considerations and its band structure as well as several models of increasing complexity. We reviewed the state of the art of ARPES on WSe 2 and then studied in detail the photoemission spectra by varying the photon-energy (k z mapping), to explore the 3D character of these materials. It showed that even with extremely thin samples, the hybridization between layer is already measurable. In some sense, few-layer TMD lie at the frontier of 2D and 3D materials. The valence bands display both 2D (K point dispersion and spin-polarization are independent on k z ) and 3D (Γ point dispersion evolves dramatically with k z ) characters: the spin-orbit coupling is robust. Those properties are at the origin of the many interesting optical/transport properties of TMD. However, they are not all static, and TMD have proven to be very responsive to stimulation (strain, electric/magnetic fields, temperature...) expanding further the realm of possible applications. In the chapter 5 we will explore how to tune their band structure using proximity effects with ferroelectric materials.

-2D Materials and Ferroelectrics Heterostructures

. Introduction

TMD hold many promises regarding the modulation of their properties. Their ultrathin character allows their band-structure to be easily modified by external stimuli and proximity effects. From this, stems the idea to incorporate them with ferroelectrics. Taking advantage of ferroelectrics' built-in polarization, one could induce substantial changes their physical properties. In this chapter, after a review on these topics, we will present results on large areas thin WSe 2 layers on BiFeO 3 and the variation of their electronic structure when changing the polarization of the ferroelectric. Last, we will study micrometric ML flakes of WSe 2 deposited on domains with different polarization of patterned LiNbO 3 .

. State of the art of TMD-Ferroelectrics integration

As presented in the chapter concerned about ferroelectric materials, perovskite oxides remain the most relevant ferroelectrics to be integrated with TMD. For this reason, most attempts to study these heterostructures have used BaTiO 3 (BTO), Pb(Zr, Ti)O 3 (PZT), LiNbO 3 (LNO) and BiFeO 3 (BFO). A significant portion of the work on TMD-ferroelectric also involves the polymer P(VDF-TrFE). All things considered, the ferroelectric layer is used to dope the TMD. To this day, two effects have been seriously studied: the effect of ferroelectric domain orientation on the photoluminescence (PL) spectra (photonics oriented application), the effect of the induced doping on transport properties and even the fabrication of some devices (microelectronics oriented applications).

. Photoluminescence of TMD-Ferroelectrics heterostructures

Photoluminescence in TMD is shaped by the radiative recombination of excitons, that are bound states of hole and electron. These excitons are the main reason for the differences between optical band-gap measurement and electronic band-gap. As a general rule, in TMD, the PL intensity decreases with the number of layers [1]. ML samples have relatively higher intensity because of the direct band-gap at K. For this reason, we will mostly talk about ML photoluminescence. The main exciton line (X 0 ) for WSe 2 on SiO 2 is around 1.75 eV [2; 3] but it is sensitive to the substrate. Courtade et al. reported a value close to 1.72 eV when WSe 2 is encapsulated in hBN [4]. On that matter, ferroelectric substrates have two main effects on the PL spectrum: different domain orientations tend to shift the main emission line by tens of meV and the PL intensity is strongly connected to the underlying domain polarization (see figure 5.3) with relative intensities changing from twice-fold to hundred-fold. Those effects were observed in MoS 2 /LNO [5], MoS 2 /BFO [6], MoS 2 /PZT [6], WS 2 /PZT [7], WSe 2 /BFO [8], MoSe 2 /LNO [9],and WSe 2 /LNO [9]. Similar results were obtained with WSe 2 , MoTe 2 , WS2 on P(VdF-TrFE) [10]. At the exception of reference [5] that attributes the change of energy of the PL emission line to the strain exerted on the flake, the other references show that the induced doping is the source of this modification. Notably, Ko et al. (reference [6]) were able to switch the PL characteristics by switching the polarization. Those results are consistent with experiments where ML WSe 2 is subjected to gate voltage [3; 4; 11]. Figure 5.2d and e. For strong n-doping, we see that the X 0 line disappears (Figure 5.2a and b) and is replaced by another line X -. On the other hand, for strong p-doping the line X + appears instead. X -and X + are respectively the signatures of electron bounded trions and hole bounded trions. In ML TMD, the excitons and trions live on the K and -K points of the band structure. They are illustrated in figure 5.1. Because of the quantum numbers involved (valley index (i.e. sitting on K or -K point), spin-polarization) some recombinations are not allowed (spin-forbbiden/momentum forbidden), it is possible to categorize different kinds of trions as discussed in great detail in [4]. Interestingly, symmetric negative trions have far more optically active configurations (6) than symmetric positive trions (2). This could be the origin of the intensity imbalance between X + and X -visible in figure 5.2c. In reference [9], Wen et al. showed that the ferroelectric polarization had an opposite effect on the emitted intensity, whether n-doped MoSe 2 or p-doped WSe 2 was used. In n-doped MoSe 2 , the part of the TMD flake on P up (more n-doped) had reduced intensity comparatively to part flake on P down (more neutral). In p-doped WSe 2 however, the part of the flake on P up (more neutral) had higher intensity comparatively to part flake on P down (more p-doped).

Peak intensity (cts s

. Transport properties

The transport properties of TMD on ferroelectrics have been studied in the two geometries described in figure 5.4. The Ferroelectric Tunnel-Junction (FTJ) and the Ferroelectric Field-Effect-Transistor (FeFET). The FTJ is most of the time a stack of the 2D material on the ferroelectric with a top electrode being graphene layer or the tip of an AFF/PFM/KPM... The FeFET is a variation on the standard FET geometry, that allows a state to be written into the ferroelectric polarization. This version is adapted for 2D semiconductors, with the ferroelectric under the 2D channel. Transport experiments have demonstrated resistive switching associated with the polarization in both geometries (measuring I DS in the FeFET geometry, and the tunnel current in FTJ). Such behaviour was reported on MoS 2 /PZT [13; 14], MoS 2 /LNO [15], MoS 2 /BTO [16; 17], MoS 2 /BFO [18],MoSe 2 /BTO [19], MoTe 2 /P(VdF-TrFE) [20; 10]. The two polarizations define a high and low resistivity state. The ratio between the high resistivity and low resistivity states (also known as ON/OFF ratio) ranges from one order of magnitude to five. Li et al. showed that the thickness of the TMD layer affected the performance (ON/OFF ratio) of the device. Overall, the ratio globally decreases with increasing thickness. However, the behaviour was not monotonous and ON/OFF ratio could be maximised with a near 5ML-thick sample [17]. These heterostructures show great potential for nanoengineering: Lipatov et al. were able to combine both FTJ and FeFET geometries to build a programmable transistor at the nanoscale, drawing the circuit on the ferroelectric domain orientation [14]. Other configurations involved PN junctions [8; 9; 10] to test the rectifying behavior of a ferroelectric driven junction.

The works we have presented demonstrate the extent of the integration of TMD with ferroelectrics. The effects on resistivity and PL have good reproducibility. Furthermore, they are robust upon change of the TMD and/or the ferroelectric. Despite these encouraging results, there remains the issue of the scale: to this day the typical TMD-Ferroelectric device is no bigger than a few 10 µm. In addition, the measurement of the effects of the ferroelectric on the band structure to this day are all indirect. In the next section, we will present our efforts towards bigger devices and hopefully a better understanding of the heterostructure using photoemission spectroscopy.

. WSe 2 -BiFeO 3 heterostructure

. Introduction

In this section, we present the results of our collaborative work on WSe 2 on BiFeO 3 heterostructures published in reference [21]. This work consists in a direct measurement of the modulation of the electronic band structure of a TMD by ferroelectric polarization using angle-resolved photoelectron emission spectroscopy (ARPES). We worked with high-quality WSe 2 grown by molecular beam epitaxy (MBE) and transferred using chemical methods onto BFO films grown by pulsed laser deposition (PLD) with different as-grown polarization states. First, we demonstrate that the single crystalline character of WSe 2 layers are preserved after transfer on BFO. Then, we show that 1 to 3 monolayers of WSe 2 exhibit a giant rigid band shift (ca. 750 meV for a trilayer) when deposited on upward vs downward polarized BFO. This unprecedented value offers new opportunities to manipulate the electronic properties of TMDs by proximity effects.

. Sample Fabrication

For the experiment, two sets of samples were prepared. The first batch combines WSe 2 grown up to three layers thick transfered onto two BFO substrates (UP and DOWN), these samples are labeled '3L' samples. The second batch combines WSe 2 grown up to two layers thick transfered onto two BFO substrates (UP and DOWN), these samples are labeled '2L' samples. In addition a reference sample of 3L on Graphene/SiC was also produced to compare WSe 2 in a known configuration against WSe 2 on BFO. The fabrication process is described hereafter. Growth of BFO thin-films by PLD. The growth and characterization of the BFO was performed by our collaborators (Sara Varotto, Stephane Fusil, Vincent Garcia and Manuel Bibes) of the Oxitronics team at Unité Mixte CNRS-Thalès. Epitaxial thin films of BFO (001) were prepared by PLD on (110) o -oriented DyScO 3 (DSO) (the "o" subindex indicates orthorhombic notation) covered with bottom electrodes of SrRuO 3 (SRO) or La 0.7 Sr 0.3 MnO 3 (LSMO). Following Yu et al., [22] the choice of the bottom electrode sets the direction of the out-of-plane component of the ferroelectric polarization of the BFO film (see section 3.4.3). The SRO(LSMO) electrode was epitaxially grown to 5 nm thickness on the DSO substrate by PLD under a pressure of 0.2 mbar (0.4 mbar), a laser frequency of 2 Hz with the substrate maintained at 650°C. The BFO was epitaxially grown (see X-ray diffraction in figures 5.5c and 5.5f) to 40 nm thickness on the electrode by PLD under a pressure of 0.4 mbar, a laser frequency of 5 Hz with the substrate now maintained at 700°C. The samples were then slowly cooled to room temperature (-10°C/min). Figure 5.5a,b and d,e presents the AFM of the BFO surface and PFM phase images of the out-of-plane piezoresponse signal. In the virgin state, the contrast was homogeneous for both samples and could be switched reversibly by applying dc voltages higher than the coercive voltages (cf. concentric square patterns). Before switching, the piezoresponse occurred in phase with the excitation for the LSMO sample and with a phase shift of 180°for the SRO samples, indicating that the out-of-plane component of the polarization was pointing up and down, respectively. DOWN(UP) BFO samples were therefore grown on DSO(110) substrates on top of a SRO(LSMO) electrode. Growth of WSe 2 films by MBE. The growth, structural characterization and transfer of the WSe 2 thin films was realised by our collaborators of SPINTEC in CEA Grenoble (Céline Vergnaud, Matthieu Jamet, Alain Marty, Frédéric Bonell). Because of the temperature incompatibility for the direct growth of good quality WSe 2 films on BFO, we first grow WSe 2 films on mica substrates [23; 24]. The 15×15 mm 2 mica substrates from Ted Pella Inc. were first exfoliated mechanically using scotch tape to obtain a clean surface right before the introduction into the ultrahigh vacuum (UHV) growth chamber where the base pressure was kept in the low 10 -10 mbar. The substrates were then outgassed in the MBE chamber at 700 • C for 5 minutes to remove all the contaminants. The growth temperature as given by a thermocouple in contact with the sample holder was 920 • C [24]. WSe 2 layers were grown by co-evaporating W from an e-gun evaporator at a rate of 0.15 Å/min and Se from an effusion cell. The Se partial pressure measured at the sample position is fixed at 10 -6 mbar. In-situ reflection high energy electron diffraction (RHEED) is used to monitor the WSe 2 crystal structure during the growth. The RHEED results indicate the epitaxial growth on the single-crystalline mica substrate (figures 5.6a and 5.6b). For the trilayer sample (3L), the atomic force microscopy (AFM) image of figure 5.6d sample reveals a continuous coverage with roughly 24% of the surface corresponding to 3 ML (trilayer), 57% to 2 ML (bilayer) and 17% to 1 ML (monolayer). The area fraction of uncovered mica substrate (0 ML) and 4 ML is negligible, accounting for about 2%. The line profile extracted on figure 5.6e shows that the layers are 0.7 nm thick as expected for bulk WSe 2 . For the bilayer sample (2L), AFM images give a composition of 74% 1 ML and 26% 2 ML (figure 5.7a-d). Wet-transferring of the WSe 2 films. The WSe 2 films were then wet-transferred on the BFO samples [23]. To transfer the WSe 2 layers from mica to BFO, we used a wet transfer method starting from the spin coating of a varnish on the sample. After evaporation of the solvent, the varnish film becomes solid and ensures the mechanical stability of the layers during the transfer process. The sample is then dipped into deionized water where the water penetrates at the interface between WSe 2 and mica. It progressively lifts off the varnish/WSe 2 stack which floats at the water surface. The floating stack is then gently fished using the target BFO substrate. The transferred layers are finally baked out on a hot plate at 80 • C for few minutes.

In the case of the 3L sample, the whole surface was covered (cf. figure 5.8d). X-ray diffraction (figure 5.8(b,c) before and after the transfer shows that the WSe 2 keeps its crystalline properties, i.e. the substrate and the transfer procedure do not induce additional strain or defects in the layer. The in-plane lattice parameter for WSe 2 on BFO is a = 0.3284 nm, compared to a = 0.3272 nm on mica. A value of a=0.3282 nm for bulk WSe 2 is reported in reference [25]. This implies a 0.06% strain for WSe 2 on BFO comparatively to a 0.3% strain on mica, meaning that the WSe 2 layer is nearly relaxed on BFO. Azimuthal X-ray diffraction demonstrates that the WSe 2 was grown with little angular dispersion and was transferred in exact conformity as only a little fraction of misoriented grains indicated by black arrows are detected before and after the operation as shown in figure 5.8(b,c) (note that the diffraction intensity is in log-scale). PFM also confirmed that the BFO keeps the same polarization orientation after being covered by WSe 2 (see figure 5.8(d-g)).

. Methods

X-ray diffraction. X-ray diffraction analysis was done with 5-axis Smartlab Rigaku diffractometer. The source was a Copper rotating anode (K α = 1.54 Å operated at 45 kV and 200 mA). It is followed by a parabolic mirror and a parallel in-plane collimator resolving up to 0.5°. They constitute the primary optics, a second parallel collimator is used on the other side. All measurements were performed using a K β filter. Piezoresponse force microscopy. The experiments were conducted with an atomic force microscope (Nanoscope V multimode, Bruker) and two external lockin detectors (SR830, Stanford Research) for the simultaneous acquisition of in-plane and out-of-plane responses. An external ac source (DS360, Stanford Research) was used to excite the bottom electrode at a frequency of 35 kHz while the conducting Pt-coated tip was grounded. Surface preparation for ARPES. The samples were mounted on Tantalum Omicron plates and electrically connected from the top with Molybdenum plates, i.e. grounded to the manipulator. The samples stayed a few hours in UHV in order to outgass as much surface contaminants as possible. We did not perform any kind of annealing prior to photoemission measurements for the WSe 2 on BFO samples to avoid damaging the BFO and its ferroelectric domain structure. Angle-resolved photoelectron emission spectroscopy. Unless specified, we used horizontally polarized photons with energy hν = 50 eV at the Cassiopee beamline. The electron detector was a Scienta R4000 hemispherical Analyzer used at pass energy 20 eV, with an angular resolution of 0.1°and instrumental energy resolution of 18 meV. The analyzer slits were vertical and curved. The Fermi energy E F was measured using a gold sample in equilibrium with the manipulator. The manipulator was at room temperature to prevent charging effects due to the semiconducting properties of both WSe 2 and BFO. We checked that the charging effect due to the beam flux was negligible. Figure 5.9a shows the photoemission of the valence band of WSe 2 on BFO from 15K to room temperature. In figure 5.9b we track the binding energy of the VBM (also the highest intensity band at hν = 30 eV) of WSe 2 on BFO from 15K to room temperature. We see that the binding energy converges at room temperature, indicating that the charges are properly evacuated. The measurements of the binding-energy being the same at 30 and 50 eV, supports this statement: the photon yield of the beamline at 30 eV is nearly twice the photon yield at 50 eV (see figure 5.9c). In addition, the cross-section of the top Γ band is much higher at 30 eV than 50 eV (see figure 5.9(d,e) or W-5d core-levels predicted cross-section in figure 1.5). This shows that the verification was done in the worst possible configuration (maximize charging).

In the 3L sample, ARPES measurements correspond to the combination of the band structures of 3ML, 2ML and 1ML with weights in table 5.1. Similarly, in the 2L sample, ARPES measurements correspond to the combination of the band structures of 2ML and 1ML. The 3L and 2L labels were attributed on the basis of the number of visible bands at Γ, three for the 3L sample and two for the 2L sample. The fine discussion about the relevance of such criteria was discussed in previous chapter since the number of bands varies with photon energy.

144

. Results

The first two stacks studied by ARPES are schematically shown in figure 5.10a and consist in 3L WSe 2 transferred on DOWN and UP BFO respectively. X-ray photoemission spectroscopy (XPS) of We 4f and Se 3d core levels are displayed in figure 5.10b. For both, we observe a single pair of peaks: 4f 7/2 and 4f 5/2 for W and 3d 5/2 and 3d 3/2 for Se with no evidence of secondary phase like WO x or SeO x . Moreover the spin-orbit splittings (2.15 eV for W and 0.82 eV for Se) are in good agreement with tabulated value [26]. The only difference between DOWN and UP BFO polarities is a rigid energy shift of core level peaks by -0.6 eV which is characteristic of the electrostatic doping effect. In figure 5.10c, the constant energy contours 500 meV below the valence band maximum (VBM) show the two opposite valleys K and -K as well as the Γ zone center at the bottom right of the image. WSe 2 clearly exhibits a single crystal orientation. At this energy, the lowest Γ band already deviates from the radial symmetry to conform to an hexagonal one. The two nested bands at K and -K show the characteristic trigonal warping due the crystal symmetry. We conclude that the 3L WSe 2 transfer on BFO fully preserves the electronic structure and single crystalline character of WSe 2 grown by MBE. Figure 5.11 displays ARPES measurements of the 3L films transferred on BFO and on a reference graphene substrate. figure 5.11a and 5.11c present the Γ -K slices of 3L WSe 2 on UP and DOWN BFO showing the features of the band structure from the top of the valence band down to 4 eV below the Fermi energy. Considering AFM images, black the observed band structure corresponds to the superposition of those of 3 ML, 2 ML and 1 ML of WSe 2 . As a comparison, we provide in figure 5.11e the same ARPES data for 3L WSe 2 transferred on graphene/SiC. In the valence band we see the two spin-splitted bands at K joining and separating into a threefold band at Γ. One Γ branch culminates to define the VBM. The general shape of these Γ -K slices corresponds to the band structure predicted and experimentally measured for WSe 2 [11]. After Laplacian normalization, the bands are ). The splitting at K is consistent with what can be found in the literature [11; 27; 28; 29] and equal to ∆ S-O =460 meV for WSe 2 on DOWN BFO, 475 meV on UP BFO and 480 meV on graphene. Comparing the band shifts for the three samples 3L WSe 2 on DOWN BFO, UP BFO and Gr/SiC we find that the VBM is located -0.215 eV, -0.985 eV and -1.222 eV below the Fermi energy, respectively. The overall band structure of 3L WSe 2 is the same on BFO and graphene (the additional band appearing in the bottom left of figure 5.11f corresponds to the graphene Dirac cone) and the principal effect of ferroelectric polarization is a rigid shift of the band structure, expected from electron accumulation (UP) or depletion (DOWN) from the BFO. We measure an average rigid shift between the two samples of ∼ 750 meV. This is the major result of this study since this shift value is significantly higher than the 450 meV step measured by Kelvin force microscopy in reference [8], which attests of the better coupling between the ferroelectric and the TMD in our samples. We note that it is also close to the value predicted from density functional theory for MoS 2 /BiFeO 3 [30]. Figure 5.12 shows the ARPES measurements of 2L WSe 2 on DOWN and UP BFO in a Γ-K slice. As expected from AFM images, the data share similarities with the band structure of a standard 1L sample [11] in particular near Γ. A comparison between the data for the DOWN and UP samples again reveals -

146 -K -Q Γ -3 -2 -1 0 E -EF (eV) -K -Q Γ Γ -Q -K -1.34 -0.67 0 -3 -2 -1 0 k || ( Å-1 ) E -EF (eV) -1.34 -0.67 0 k || ( Å-1 ) k || ( Å -1 )
-640 meV -K -Q Γ Q K -3 -2 -1 0 E -EF (eV) -K -Q Γ Q K -1.
Q Q -3 -2 -1 0 k|| (Å -1 ) E -EF (eV) Γ Q Γ -Q k|| (Å -1 ) Q Γ -Q k|| (Å -1 ) Q Γ -Q k|| (Å -

. Discussion Reproducibility of the shift

In the previous part we showed that the BFO polarization could shift the band structure of the WSe 2 . We actually performed the experiment several times with the same results. Here we show additional ARPES results on another WSe 2 /BFO heterostructure. The fabrication methods are essentially the same as for the 2L samples with an even lower 2ML coverage deported on UP and DOWN BFO substrates. We will label these as 2L'. Figure 5.14 shows the ARPES measurements of this 2L' samples. The measurements are centered on Γ. The bands are less visible than the 2L samples but one still recognizes the band structure of WSe 2 . As previously, we observe the shift of the band structure: UP polarization induces n-doping while DOWN polarization induces p-doping. From those elements, we can say confidently that the polarization dependent band shift is definitely reproducible (this will be again shown for WSe 2 on LiNbO 3 in section 5.4). This said, the results are not exactly the same for each experiments. To discuss the reproducibility in more details, we reported in table 5.2 the energy of the bottom-most Γ band, E Γ . This allows to compare WSe 2 samples of different thickness since this band is always present, provided the right photon energy is used. For the DOWN samples, the mean position of E Γ is -1.03 eV (standard deviation 150 meV). For the UP samples, the mean position of E Γ is -1.66 eV (standard deviation 100 meV). The mean shift is of 630 meV (standard deviation 150 meV). There is definitely a trend. The variations between different samples are not enormous but still significant. At this point, they remain difficult to explain. Here are some of the most probable explanations :

-3 -2 -1 0 E -E f (eV) -0.5 0 0.5 -4 -3 -2 -1 0 k || ( Å-1 ) E -E f (eV) -0.5 0 0.5 k || ( Å -
• The interface between the BFO and the WSe 2 plays a great role and the impact of the transfer process on it is not yet understood.

• The quality of the WSe 2 film determines the quality of the charge transfer with the BFO. Films with poorer crystalline quality such as 2L' samples could limit the amount of transferred charges because of defects.

• The thickness of the film (3L > 2L > 2L' in the range studied here) impacts the ability of the film to screen charges from the BFO (low thickness limits the charge transfer). Here E Γ is the energy of the bottom-most band at Γ since it is the only one common to all thicknesses.

Sample

Effect of the annealing on the heterostructure

Low temperature (250-300 °C, 1h) UHV annealing is a standard procedure in surface science. It usually helps cleaning the sample by desorbing pollution from the surface. After such treatment, the secondary electron background is reduced since less pollution is there to scatter electrons away from their trajectory. We did such a treatment to the 3L samples and the expected effects were observed (see figure 5.15). Comparing the images 5.15a and b or 5.15c and d, we see that the bands are definitely sharper. The annealing had however a surprising effect: we observed that the band structure of the annealed DOWN sample was 710 meV away from its pre-annealing position. For all that can be seen by ARPES, the spectral response of the sample had become identical to the UP sample. Interestingly, the band structure of the WSe 2 on the UP sample did not move a lot (down by 120 meV from its original position).

After analysing ARPES, we switched to XPS to see if the effect was measurable beyond the valence band. Indeed, the whole core-level spectra had shifted in a similar fashion. Figure 5.16 shows the survey of the main lines for the two 3L samples before and after annealing. Even though the scale is large as compared to the shift, we see that the straight black lines centered at Se3d 5/2 and W4f 7/2 for the DOWN survey are off for all the other spectra. We will not discuss the effect of the annealing on the photoemission lines associated to WSe 2 because their shape were unaffected by the operation. On the Bi4f core-levels however, we could observe some changes. Figure 5.17 shows the XPS spectra of Se3p and Bi4f before and after annealing for UP and DOWN samples. The most prominent peak is the Se3p doublet (∆ s-o = 5.70 eV, green dashed line). Two other contributions coming from Bi4f can be seen. From an angle dependent analysis done on another set of samples (Bi5d orbitals however, not shown) we attribute a first doublet to the Bi-atoms located in the bulk (∆ s-o = 5.30 eV, blue dashed line) and a second to the surface (magenta dashed line). In this figure, it is easier to see the shift of the core levels after annealing. The second striking fact is the intensity ratio of surface and bulk Bi4f peaks, in BFO DOWN it is much higher than in BFO UP. This stays true even after annealing, even though the surface contribution becomes much weaker for both polarization. We quantify this using the peak areas from our fits. The values are reported in the table 5.4. Additionally, we reported in table 5.3 the positions of the surface and bulk Bi4f 7/2 peaks. ∆ B-S quantifies the difference between the binding energy of the bulk and surface components of the Bi4f 7/2 peak. We remark that this difference is higher in UP sample than DOWN sample. If we assume that the bulk component does not move in energy, that means the environment of surface Bi-atoms is different for the UP and DOWN samples. After annealing, this stays true but ∆ B-S is reduced of 0.39 meV for both samples.

The effect of the annealing on the Bi4f core levels is very similar for both polarization. The surface contribution diminishes and the bulk-surface separation is reduced of the same amount. The position of the surface peak takes in both cases a nearly identical value (see table 5.3). From the point of view of ARPES and XPS for the most part, it is almost as if the polarization of the DOWN sample had switched to UP. This is not a one time issue since annealing on samples 2L' had the exact same effect. Even more surprisingly, this did not have any impact on the PFM measurements done after the beamtime (see figure 5.8) that confirmed the expected polarization. At this point we cannot explain this effect but it proves that the physics of the WSe 2 -BFO heterostructure is rich.

Detailed interaction mechanism between WSe 2 and BFO

This whole discussion of the previous section opens the question of the detailed interaction of the WSe 2 with BFO. Before annealing, the dipole picture can easily explain the shifts of the band structure. The annealing revealed that the interaction is more complex than it seemed and that knowing the surface state of the BFO is critical. This is what suggests the work of Spaldin et al. (reference [31]) as well as Jin et al. (reference [32]). According to Jin et According to Dai et al. the surface state of the BFO is a major actor of the interaction with a TMD. In their work, they study the interaction of MoS 2 on BFO(111) with first principles calculations. They found that the relative Pauling's electronegativities of the Bi,O and S elements (Se in our case) was a good predictor of the direction of the charge transfers [30] and explained the shifts of the band structure of MoS 2 . While the applicability of their results is limited in our case, since BFO(111) UP and DOWN terminations are neutral (-Fe-O 2 -Bi or BiO 2 ), they still argue in favour of a better knowledge of the BFO's surface state.

Conclusion

In this section we discussed in detail the effect of the BFO polarization state on the band structure of WSe 2 for different thicknesses. The fabrication method proved itself ARPEScompatible which was not guarantee given the surface sensitivity of the technique, the number and the nature of the ex-situ steps for the sample preparation. Using ARPES, we were able to quantify sizable shifts of the band structure and were able reproduce the effect several times. However, the annealing of the samples proved that the interface chemistry of the WSe 2 and the BFO was an important factor to be taken in account for future investigations. Our results on WSe 2 /BFO heterostructures show noticeable effect of the ferroelectric polarization on the electronic structure of the TMD. These results were obtained on 2 and 3 layers TMD samples. On the course of these experiments, we could not reach the monolayer limit. Would these effects be more pronounced on a monolayer ? Does the relative thickness inhomogeneity of our samples hinder the interaction in some way ? To adress these questions, we devised to measure ML flakes of WSe 2 transferred onto LiNbO 3 substrates. At the price of the scale, the use of flakes guarantees the homogeneity of the thickness. Secondly, while we used separated BFO samples to study the effect of the polarization, with patterned LNO we get to have both polarizations on a single sample. In addition, with the used LNO cut, the polarization is purely out-of-plane. This would hopefully maximize the effect of the LNO on the WSe 2 . In this section, we will first present the preparation of these heterostructures. We will then explain how we took advantage of the NanoESCA endstation (Elettra Sincrotrone, Trieste, Italy) to perform a combined microscopic and spectroscopic characterization of our samples.

. LiNbO 3 surface preparation Initial microscopic characterization

For this experiment we used industrial z-cut ([0001] h direction) single crystals periodically poled (see section 3.4.2) lithium niobate from HC photonics. The crystals were of stoechiometric composition with 1.2 mol% MgO doping so that they had already some conductivity. The physical dimensions were 10x5x0.5 mm 3 . The period of the poling was 60 µm alternating up and down domains with no in-plane component (because of the direction of the cut, the polarization is perpendicular to the surface of the sample) as illustrated in figure 5.20b. The rest of figure 5.20 shows the initial state of the LNO substrate characterized by different microscopic techniques. Figure 5.20a shows the surface state of the LNO as seen with an optical microscope, the period is indeed 60 µm but one kind of domain is systematically larger than the other. The PFM inset confirms that the wide domains are always P down and the narrow domains P up . The optical microscopy also evidences that the border between domains are not clean. AFM characterization confirms the poor state of the domain borders: first of all, the two domains have a height difference of about 0.4 µm. Secondly, a 4 µm wide uneven area always lies in-between the two domains. The substrate has a pronounced topography, probably originating from the poling process. When looking at a the topography of a single domain as in figure 5.20d, we can see that the surface is extremely flat with a RMS rugosity of 320 pm. We checked the poling on the same area with the PFM as shown in figure 5.20f, domain with outward (inward) polarization have the lowest(highest) height. We did not try to change the polarization with the tip of the PFM because of the high coercive field of LNO (see the discussion in chapter 3).

Vacuum annealing of the surface and XPS characterization

Because lithium niobate is strongly insulating, it cannot be measured directly by photoemission. We resorted to the creation of vacancies using the method in [33; 34]. For this step we used the MesoXScope station at CEA Saclay with the help of our collaborators (Nick Barrett, Grégoire Magagnin, Christophe Lubin). We first checked the sample with XPS (see figure 5.22a, top-most yellow spectrum) and then performed a soft vacuum annealing of 2h at 300°C, till the pressure reached 10 -9 mbar . We then measured a second overview spectrum of the cleaned sample (see figure 5.22a, middle orange spectrum). We then slowly increased the temperature from 300°C to 600°C keeping the pressure under 10 -8 mbar . The last annealing step was to keep the sample above 600°C for 2h. Because of the long total annealing time, the pressure was not higher than 10 -9 mbar. We then measured the XPS spectrum of the fully annealed sample (see figure 5.22a, bottom red spectrum). After this sequence, it was possible to get a photoemission signal on the PEEM available at the MesoXscope endstation.

The final spectrum after annealing is shown in detail in figure 5.22b-d. Those spectra are similar to the measurements of Steiner and Höchst [35]. We see that the main effect of the annealing is to shift the peaks towards the right (see the grey arrows in figure 5.22a). Table 5.5 shows the binding energies of the Nb3p 3/2 and Nbdp 5/2 . The difference of binding energy before and after the 600 °C annealing is close to 10 eV. This shows that the substrate was charging under the beam. In addition to that, satellite like peaks appear in the spectrum near Nb3p and Nb3d doublets, showing that the annealing has a significant effect on the surface. We will not comment further on this because we do not understand what happens in detail.

The main features of the annealed LNO spectrum are summarized in table 5.6. Remarkably, all the peak positions are 2 eV closer to the Fermi-level than in the results of Steiner and Höchst [35]. Using the theoretical Scofield sensitivity factors [36] (cross-section), we calculated twice (after soft annealing and after hard annealing) the ratio η = [Li]/ [Nb]. In both cases it was equal to 0.97.

One of the biggest effects of the annealing is the color change of the sample. Initially, the LNO sample is completely transparent but the more it is (vacuum)-annealed, the more it takes a dark blue color (see figure 5.21). This is in line with the observations done by Courths et al. [37]. With the recipe for the substrate preparation, we reproduced the preparation on another LNO substrate at CASSIOPEE beamline that was used for the rest of the experiment. The substrate took a little longer to anneal (3h above 600 °C in UHV) but ended up with the same blue tint.

After annealing Before annealing In this section, we present the method that was used to prepare and transfer the WSe 2 monolayers onto the LNO substrates. All this part was done in the clean-room lab of the Centre de Nanosciences et de Nanotechnologies (Palaiseau) (C2N), and the different steps were largely defined by Julien Chaste and Cléophanie Brochard (C2N). Because of the sensitive nature of ferroelectric substrates, we had to adapt the method. The whole procedure is summarized in the figure 5.25. The reader should refer to this figure for the labels of the steps that will be described below. The transfer method is a deterministic dry transfer technique that allows a limited amount of WSe 2 flakes to be transported from one substrate to another. In this method, the flakes are picked-up with a polydimethylsiloxane bubble (PDMS). While it is labelled as a dry transfer method, it still uses a lot of UHV-unfriendly chemicals.

Growth of the WSe 2

The WSe 2 flakes were grown by Fabrice Oehler (C2N) by CVD on a Si/SiO 2 substrate of 258 nm thick (step I.1). The flakes were typically 10 to 20 µm large. Among them, we selected monolayer flakes to perform our experiment. The flakes were chosen on the basis of their size (≈ 20 µm wide) and Raman spectrum confirming their thickness (1ML). To prepare the separation from the substrate, the flakes were coated with PMMA using spin-coating and then left inside liquid nitrogen for 15 minutes (step I.2). In this step, the flakes are strained because of the thermal contraction of the PMMA, weakening their bonding to the substrate. The sample was then cleaned bathing into acetone first for 30 min and then isopropanol for 2 min (step I.4).

Preparation of the transfer bubble

A bubble of PDMS was prepared with a dome shape in the middle, allowing for precision pick-up (figure 5.23a). The bubble was put inside a O 2 plasma for 30s (pressure 0.8 mbar of O 2 in the chamber). This step facilitates the spin-coating of PPC (spin coating parameters: 2000 rpm for 30s), since the PPC wets much better the PDMS surface after such treatment.

Transfer of the WSe 2 on the substrate

The LNO substrate was cleaned in an ethanol ultrasound bath for 10 min prior to the transfer. The transfer is done using the PPC coated PDMS bubble, taking the flake from the Si/SiO 2 substrate to the LNO. The transfer setup is described in figure 5.24. The layout allows for deterministic transfer, meaning that the desired flakes are taken and transferred with the exact location and orientation to the new susbtrate. The technique is limited to a few flakes per transfer bubble, that can be used only once. In our case, it was necessary to have monolayer flakes transferred on different domains to observe the effect of the polarization. A drop of deionized water was put on the sample were the flakes were grown (step II.2). The PDMS+PPC complex was then brought in contact to the surface. To pick-up the WSe 2 , the temperature was raised to 35 °C for a minute and decreased back to 25 °C (step II.3). Once this was done, the flakes were held onto the PDMS bubble (step II.4). The bubble was then aligned with the LNO and brought to the surface of the new substrate. The temperature then was slowly risen to 100 °C expanding the contact area of the bubble with the substrate (step II.5). After this, it was possible to separate the PDMS from the surface leaving the WSe 2 there (step II.6). In figure 5.26, we see that after the transfer, the surface of the new substrate has been spoiled by the PDMS+PPC bubble. It left a visible dark spot on the sample, and at this stage it is impossible to tell if the transfer was successful. The sample was cooled down to 25 °C and the next step was to clean the surface. This was done by vacuum annealing: the sample was heated to 300 °C at a pressure of 10 -6 mbar for a few hours (step II.7). After this we checked the surface with as optical microscope and found the flakes transferred onto the domains. Figure 5.26 shows that the flakes were transported without deformation from the Si/SiO 2 to the LNO. To get a primer idea of the surface state of the substrate, we made work function maps calculated from energy-filtered PEEM measurements on different locations (several 100 µm away from each other) on the substrate (see method section 5.4.4). Figure 5.27a and b show these maps with a common color scale. Brighter colors indicate higher work function. We see that the work function on the two locations is very different. It is correlated with the topography of the sample and most defects have higher work functions. From [39; 38; 33], we expect that differently polarized domains have different work functions. While this difference is visible in 5.27a, in 5.27b the successive domains have nearly identical work functions. The average work function along the red and blue lines is shown in figures 5.27c and d. There are variations even along the x direction inside a domain. We study the domain average work function for each domain in figure 5.27e. The absolute value seem to drift along the x direction. We calculate the average (along x) absolute variation of the work function δ

W F = 1 N N i=1 δ i,i+1 = 1 N N i=1 |Φ i -Φ i+1 |.
Overall, this difference between domains amounts to 13 meV for the first picture. In the picture 5.27b, it is reduced to 3 meV, almost insignificant since there is more intra-domain variation than between two successive domains.

. Spectroscopic characterization of the heterostructure: k-PEEM

The second part of this work was the characterization with k-PEEM. Figure 5.32 shows the general results with constant energy cuts in the band structure of ML WSe 2 . Before doing the measurements, we took care of aligning the high symmetry direction with respect to the analyzer by rotating the sample. The energy cuts presented in 5.32 were chosen to maximise visibility, one at the top of the valence band and one at the bottom as indicated in figure 5.32a. The quality of the spectra was exceptionally good for flake 3 with very little secondary electron background but we were able to resolve properly the band structure nicely for all the flakes. Looking at the middle column of the figure 5.32 (c-f-i) it seems that, for these cuts at the top of the VB the usual hexagonal symmetry of the first BZ was reduced. We see a triangular pattern (see visual help in figure 5.32a) where the K/-K points seem to form groups of two. The effect is independent on the ferroelectric polarization since it is visible of each flake. We compared with measurements using the same photon energy hν = 60 eV but with linear horizontal (LH) polarization (see figure 5.33). The pattern seemed different, with the triangular shape reversed. Super-imposing LV and LH measurements, we see that the full hexagonal symmetry is restored with evenly placed K points (not shown). This shows that this is an effect of selection rules: only half of the bands at K have the proper symmetry in the configuration we adopted. The central Γ point is also largely extinct in the middle of the image, more generally we see that around k x = 0 almost no electron are detected indicating an additional forbidden transition with this detection geometry. In contrast, for the lower valence band part, the hexagonal symmetry is manifest. Using this, we look at the dispersion along Γ -K (black cuts in figures 5.32 (b-e-f). Figure 5.34 shows the result of these cuts its top part. The dispersion in this high symmetry direction shows clearly the unique Γ band. While the band is undoubtedly present, we do not resolve the splitting at K, even with the second derivative. When comparing the three flakes, the photoemission spectrum of flake 3 is clearly lower. To quantify the shift, we integrated the whole photoemission signal obtained (over k x and k y ) to have the density of states. The result is shown in figure 5.34g. The density of states is clearly dependent on the polarization. The position of first peak (the valence band) is shifted by 150 meV towards higher binding energies when the substrate is P up relatively to when the substrate is P down .

. Discussion

Overall, we fabricated a truly 2D-3D heterostructure composed of ferroelectric material (LNO) and monolayer WSe 2 using a deterministic dry transfer method. Despite abrupt domain walls, the LNO surface was very flat (300 pm RMS roughness) which ensured a very good contact with the WSe 2 . On a single sample, we were able to measure successfully the band structure of WSe 2 as a function of the LNO polarization despite the insulating nature of LNO and the ex-situ fabrication. The LNO substrate was made slightly conductive with high temperature UHV annealing which made the study possible altogether. This resulted in a measurable shift of the band structure of WSe 2 of 150 meV between oppositely polarized domains. First of all, the shift goes in the expected direction: P down polarization induces p-doping and P down polarization n-doping. Despite this success, the relative shift of the band structure was lower than in the WSe 2 /BFO heterostructure. This is surprising considering their similar crystallography and P s value (see chapter 3). One explanation could stem from the initial MgO doping and the surface preparation of the LNO substrate. Having a more conductive material is detrimental to ferroelectricity because of the induced screening by the free charges. The control of the effect of the annealing could be improved seeing the local differences between adjacent domains in the work function from one zone to another. Some regions retained a significant relative work function between domains while others were indistinguishable on that matter. This calls for a better understanding/mastery of the high temperature annealing step. This said, even though the value of the work function varied on the surface, there was a significant elevation of the work function relatively to the substrate for all monolayers of the study. The values were in par with the 50 mV surface potential drop in KPFM measurements on MoS 2 /LNO of the reference [15], but we could not correlate them to the position of the valence band. Positively, this means that the surface state of the LNO has less impact than the bulk effect of the polarization. This promises a robustness of the shift that was partly tested with an annealing step prior to the PEEM measurement. The robustness was also tried in terms of reproducibility: at least for P down domains we proved that even though the size of the flakes, their location (hundred µm away) and the local work function of LNO were different, the effect of the polarization on the band structure was the same.

-General conclusion

General outlook on the thesis

This thesis was an opportunity to play with TMD in a wide number of settings. We performed a fundamental study of the band structure of TMD as a function of their thickness. This study revealed the profound 2D-3D hybrid character of these materials beyond the monolayer limit. Indeed, using angle resolved photoemission spectroscopy and state of the art MBE grown high quality thin-films, we could see how increasing the thickness of WSe 2 affected its band structure. We could observe how the 3D character of the material is building up, offering a textbook like example of starting from discrete energy levels to a continuum of state as studied in all courses of solid-state physics. This study alone confirms, if it was needed at all, the interest of synchrotron radiation for condensed matter systems. It is the only way providing, to date, a continuously tunable photon source in the UV/soft X-ray range with sufficient flux to perform k z resolved ARPES.

As we were able to show in this thesis, the integration of TMD with ferroelectrics is only at its beginnings. Despite the interesting proof of concept for applications, no one have studied in detail the inner workings of such heterostructures. This work constitutes a first step towards this goal. We were able to produce hybrid TMD-ferroelectric and analyze them by the means of photoemission despite unfavourable conditions. Indeed, ferroelectrics materials are for the most part insulators, and therefore unsuitable to study by photoemission.

We used two approaches with "opposite" methodologies: -The first used two to three layer thick MBE grown large scale WSe 2 . These films were wet-transferred on separate samples of BiFeO 3 thin-films with different electrical polarizations. The whole heterostructure was studied in a purely spectroscopic fashion by ARPES. We could see a relative rigid mean shift as high as 750 meV between the band structure of WSe 2 on outward and inward polarized BFO. The experiment was performed several times with different film thickness and the results were consistent each time, proving a correct reproducibility of our process. Surprisingly for such a standard procedure, these sample were not vacuum-annealing resilient. We found that a simple 250°C annealing affected DOWN BFO in such a way that the ARPES spectrum became indistinguishable from UP BFO. Despite our efforts studying the core-level spectrum before and after annealing, we could not find the exact cause of the phenomenon. This highlighted that, in the case of WSe 2 -BiFeO 3 heterostructures, a precise knowledge of the surface state of the BFO is crucial.

-The second one used micrometer sized WSe 2 flakes dry-transferred on a single patterned crystal of bulk LiNbO 3 . With the methods developed by Rault et al. on BaTiO 3 , we could prepare the LiNbo 3 substrate to be compatible with photoemission study. We used a combined spectroscopic and microscopic approach to study the system. The effect on the band structure was less impressive than on BFO, with a mere 150 meV of difference between oppositely polarized domains. The sample, however, proved was more robust upon, e.g., a mild annealing. Additionally, from work-function study, it seems that the surface state of LNO is less critical than for BFO. This would mean that the shift is mostly bulk controlled, ensuring stability of the fabrication.

Scientific perspectives

The initial goal of this thesis was to observe an anisotropic Rashba spin-splitting at Γ. Some theoreticians calculated that, monolayer TMD subjected to a sufficient electric field could see their band at Γ splitted by virtue of broken inversion symmetry (see C. Cheng et al., Nanoscale 8, 17854-17860 (2016)), an effect similar to the Rashba effect as sketched in figure 6.1. In their model, the spin-splitting was anisotropic meaning that the majority spins were different in the Γ -K and Γ -M directions. Additionally, the effect of the electric field was to raise the bands centered at the Γ point above the zero field VBM (originally located at the K points). Taking advantage of the polarization of a ferroelectric substrate, it would be possible to get this effect passively. This would constitute a definite step towards extremely modular spintronic devices since such anisotropy could be used in spin-filters or spin-injection devices. Despite our efforts, we never saw these effects. On monolayer samples, the resolution was always too poor to see any measurable modification at Γ. Either much higher field values are needed (improvement of the samples), or the effects were overestimated by the model. Nevertheless, we were able to observe large binding energy shifts of the band structure of WSe 2 millimetric films or micrometric flakes under the influence of the ferroelectric polarisation (BFO or LNO). From this point on, there are some obvious scientific perspectives. The first one is towards a better understanding of the interaction of WSe 2 and BFO. On micrometric-sized flakes of WSe 2 transferred on BFO, we undertook spatially resolved photoemission spectroscopy at ANTARES. We hope to elucidate the interaction between WSe 2 and BFO. The second one would be to produce a theory of screening of ferroelectrics using semiconductor electrodes. Screening by metallic materials is relatively well understood: the progresses in the fabrication of ferroelectric oxides with pulsed laser deposition now allow for incredibly clean interfaces. With 2D semiconductors the pictures is unclear. If there is charge transfer at the interface, does the thickness of WSe 2 limit or enhance in some way the ability of the semiconductor to screen? Answering these questions might help us to improve of our methods and eventually extend them to other materials.

Non-scientific perspectives

Outside of science the perspective of this work are sadly, less appealing. While devices integrating TMD show a lot of potential applications, I see two shortcomings:

-From a geopolitical point of view, most of the raw materials we used are listed as "critical materials" according to the 2020 Study on the EU's list of critical raw materials (2020) published by the European Commission (Directorate-General for Internal Market, Industry, Entrepreneurship and SMEs). While the definition of "critical" is always context dependent, the perspective of the report is to balance the supply risk of certain raw materials against their potential utilization in current or developing technologies and industries in a near future. Among the materials used we have :

• Tungsten (main global supplier China, share 69%)

• Bismuth (China, 86%)

• Dysprosium (China, 86%)

• Scandium (China, 66%)

• Strontium (Spain, 31%)

• Titanium (China, 45%)

• Lanthanum (China, 86%)

• Ruthenium (South Africa, 93%)

• Niobium (Brazil, 89%)

• Lithium (Chile, 44%) Additional details on the market share and estimated resources per material and per country can be found in the Mineral Commodity Summaries 2022 from the U.S Geological Survey. As can be seen easily for our systems only, the main supplier of critical materials is China. Seeing how the geopolitical making is nowadays subject to instability, the rise of autocratic governments worldwide... what would be the strategical decision ? Is it to develop technologies that in the end, will put the European Union in a dependence relationship with a country with notably divergent interests ? -From an ecological point of view, the creation of new energy sources has not resulted, to this day, to a reduction of older ones. For instance, material flow analysis shows that coal consumption did not decrease with the democratization of petrol, nor did nuclear energy. The relative use decreased, but the absolute consumption increased. Instead, all of them were added on top of each other, and other usages were found for older sources of energy (see, L'anthropocène est un « accumulocène » Jean-Baptiste Fressoz, CNRS, EHESS). Similarly, thanks to rebound effect, the improvement of motor technology did not result in a global reduction of petrol consumption but in a democratization of the use of the car. In what are the technologies that we develop different ? Will hybrid TMD-ferroelectric low energy consumption memories replace older hard drives or SSD or will the two industries cohabit with each other for an undetermined period of time ? Will TMD transistors replace standard silicium based technology ? Or will the impact of the research just be an incentive to promote even more the mining industry for rare earths ?

A -Methods for data Analysis

A.1 . Introduction

In this part we present some of the methods that were implemented in a transverse program to analyze photoemission data. ARPES and PEEM data are visual in nature. For this reason analysing them involves a lot of image treatment. Even if it is possible to use curve fitting procedures for energy distribution curves or momentum distribution curves in a fashion similar to standard core-level photoemission; most of the analysis will not go to these length. Indeed, it can be a cumbersome work to extract physical quantities other than binding energy at discrete points of the spectrum. On the other side, many of the igor procedures available and used routinely do not allow for more than visualization of the simplest data sets. However, in a beamtime, it is sometimes required to analyze large data sets on the fly just to get an idea of the direction to take.

The development of CASSIOpy was an effort to get a centralized software where I could analyze the data with standard image processing tools. The idea was to be able to take data from different endstations (CASSIOPEE, ANTARES, BADELPH, NANOESCA, MESOXSCOPE). Due to the location of the thesis, the software is optimized to the workflow on CASSIOPEE and extend the capabilities of the beamline to the maximum. More details can be found on the page: https://gitlab.com/SLZ_Raph/cassiopy.

A.2 . General tools for ARPES data analysis

A.2.1 . Filters Second derivative

The second derivative here is defined as a weighed Laplacian. It is a requirement since ARPES data is collected on an inhomogeneous grid (angular x energy) both in units and scale.

∇ 2 ≡ C x 1 ∂ 2 ∂x 2 1 + C x 2 ∂ 2 ∂x 2 2 (A.1)
With C -1 x i = (dx i ) 2 the step for axis i. The filter in actual use is defined as follows:

∇2 ≡ max{-∇ 2 , 0} (A.2)
The restriction to positive values avoids the apparition of double peaks in the data.

2D mean curvature

H ≡ 1 + C x 1 ( ∂ ∂x 1 ) 2 C x 2 ∂ 2 ∂x 2 2 -2C x 1 C x 2 ∂ ∂x 1 ∂ ∂x 2 ∂ 2 ∂x 1 ∂x 2 + 1 + C x 2 ( ∂ ∂x 2 ) 2 C x 1 ∂ 2 ∂x 1 2 1 + C x 1 ∂ 2 ∂x 2 1 + C x 2 ∂ 2 ∂x 2 2 3 2 (A.3) With C -1 x i = (dx i ) 2
the step for axis i.

A.2.2 . Data normalisation

One common way to improve significantly the quality of the data is to remove the detector background. At CASSIOPEE, the detector leaves a trace on the data that can make the reading of the data difficult when the count rate is low. The background is shown in figure A.1. In swept-mode the background only depends on θ since a single energy line is measured. One common way to normalise the data is to integrate out the secondary electron background from the photoemission signal I(θ, E) from the energy range above the Fermi-level (or within the gap for semi-conductors) (see equation (A.4)). This background is then smoothed using a gaussian function g σ so that no noise is added to the spectrum.

I BKG (θ) = E 2 E 1 I(θ, E)dE (A.4)
We then obtain the normalised spectrum using:

I norm (θ, E) = I(θ, E) (g σ * I BKG )(θ) (A.5)
An example is given in figure A.2c. The corrected spectrum is clearly uniform as expected. The method is very efficient and can be readily extended to multi-dimensional measurements like 3D-band structures (coordinates E,k x ,k y ). -1 0 1

-1 0 1 k x ( Å-1 ) k y ( Å-1 ) -1 0 1 -1 0 1 k x ( Å-1 ) k y ( Å-1 )

A.3 . Computing work function maps from PEEM data

The aim of a PEEM measurement is usually to compute work functions. To do so, the first step is to acquire a stack of images at different kinetic energies (see figure A.4a). This constitutes a data block with coordinates (x, y, E k ). Usually, the resolution of the images (in pixel) is 600x600. At this resoltution already, there are too many points to be analyzed manually. For this reason, J. Rault introduced a method to calculate automatically the work function on each pixel of the image. The idea is to fit the secondary electron edge with a complementary error function (see equation (A.6)). In figure A.4b we show different points on the surface and in figure A.4c the corresponding kinetic energy spectra. The raising part of the peak can be accurately described by an error function.

I(E) = I 0 1 -erf E -Φ S √ 2σ + I B (A.6)
where I 0 is the intensity, Φ S the work function, σ is related to the steepness of the error function and I B a possible constant background.

A.3.1 . Energy dispersion correction

As explained in chapter 2, for geometrical reasons, the analyzer is not able to sort all electrons perfectly by their kinetic energy and position of origin. In photoemission microscopy in particular, this induces a dispersion along the y direction of the analyzer. The electrons at the center y 0 of the image will not have the same kinetic energy as the one at the top or bottom edges of the image. The equation of the dispersion giving the energy correction ∆E(y) for an electron detected at position y is given by:

∆E(y) = E p M 1 (y -y 0 ) f (A.7)
where E p is the pass energy of the analyzers, M 1 the lateral magnification and f the focal length of the extractor. In principle the correction could be calculated from the experimental settings of the analyzer. However, it is also possible and much simpler to have the nonisochromaticity of the image extracted directly from the measurements in a self-consistent fashion. The curve is obtained by averaging every spectrum extracted from the same y value (effectively integrating out x). This gives a spectrum that is itself fitted with equation A.

A.3.2 . Estimation of the parameters

Because the correction and the fit are lengthy processes (easily 30 min per map), we created an estimator of the work function maps that allows to see the results in a matter of minutes (2 min) and shorten the fitting time. We describe the procedure thereafter. The first step is to find the maximum of the curve. The energy of the maximum gives an approximate limit of the region that can be fitted. The error function can only represent accurately the edge and the quality of the fit will break down if we try fitting way past after the peak. To ensure to get the plateau, we take a small fixed number of points N (fine tuned to 5 in our algorithm) after the maximum. We therefore have knowledge of (E max , I max ) and deduce E lim = E max + N δE with δE the energy step of the experiment (10 -20 meV typically). From the value of I max , we estimate I 0 = I max /2.4. This last factor was obtained after fine tuning the algorithm. We then find the value I min in the range [0,E lim ] which gives us the estimator for the offset I B .

The data are then smoothed and we calculate the derivative. In principle, in the selected range [0,E lim ] the function is monotonous with the maximum of the derivative located at the steepest point (i.e. middle of the error function). This allows to find the estimation for the work function E W F .

B -Characterization of the HU256 Undulator at CAS-SIOPEE

One recurrent issue when performing photon energy dependent measurement is the relative "unreliability" of the monochromator (see figure B.1). In addition to that, the number of photons produced by the undulator for two different settings is not the same. With HU256 undulator, the "flux" of photons (number of photons produced at a given energy hν by unit of time) is known to decrease with increasing hν. We decided to characterize this aspect of the beamline. The results were used to have an accurate determination of the (relative) binding energy for hν-dependent measurements.

E k (E F ) -hν (eV) 2ML -A 2ML -B 1 2ML -B 2 3ML N -ML

B.1 . Methods

Measurement parameters

The beamline being well conceived, there is a retractable 10x10 mm 2 photodiode (Opto Diode -AXUV100) [1] after the monochromator (see figure B.2). We varied the photon energy changing the monochromator simultaneously with undulator from 20 to 150 eV with a 10 meV step and measured the current inside the photodiode with a Kethley 617 ammeter. The beamline resolution (monochromator slits) was kept constant at 9 meV. The current in the storage ring was 500 mA. Bibliography [1] Opto Diode Corp., Sensor Photodiode AXUV100.

B.2 . Results

D -Résumé de la thèse en Français

D.1 . Introduction

Cette thèse a pour objet l'étude des hétérostructures composées d'une part de dichalcogénures de métaux de transition, et d'autre part de matériaux ferroélectriques. Les premiers sont une classe de matériaux qui ont attiré l'attention de nombreux physiciens spécialistes de la matière condensée. De formule MX 2 , ils sont composés d'un métal de transition (M = Mo, W) relié avec deux chalcogènes (X = S, Se, Te). Longtemps à l'arrière garde de l'intérêt des physiciens, ces matériaux ont émergé comme une sérieuse alternative semi-conductrice au graphène au début des années 2010. En effet, en plus de partager avec celui-ci une structure hexagonale, un crystal de MX 2 consiste aussi en un empilement de couches X-M-X légèrement reliées entre elles par des interactions de van der Waals. Forts des ces similarités, les TMDC se prêtent volontier à toutes formes d'applications. Leur bande interdite de l'ordre de l'électron-volt en fait des matériaux de choix pour des applications photovoltaïques. De plus, ils transitent d'un gap direct à un gap indirect en passant d'une épaisseur d'une monocouche à plusieurs. La présence d'éléments à numéro atomique élevé implique un fort couplage spin-orbite qui se retrouve dans la structure de bande du matériau et impacte directement ses propriétés électroniques et optiques. Il serait même possible de se servir de ces matériaux pour la spintronique. Ces exemples suffisent à démontrer la modularité de ces matériaux dont la physique est riche malgré une structure trompeusement simple. En particulier, leur nature bidimensionnelle implique une sensibilité exacerbée aux effets de proximité et autres stimuli extérieurs. Les matériaux ferroélectriques, deuxième composant de nos hétérostructures, ont prouvé leur intérêt bien plus tôt. Découverts dans les années 1920, ils possédent d'étonnantes propriétés. Un matériau ferroélectrique est intrinsèquement doté d'une polarisation électrique qui peut être modifiée de manière pérenne avec l'application d'un champ électrique. Dans le cas le plus simple, le cristal possède deux états stables de polarisation opposée. Ces dernières années ont vu la maîtrise croissante de la fabrication de perovskites ferroélectriques. Il est désormais possible de contrôler leurs propriétés à l'échelle atomique. Emerge alors naturellement l'idée de combiner ces deux classes de matériaux pour produire des dispositifs. L'une des voie d'application étant la fabrication d'hétérostructures hybrides TMD-Ferroélectriques pour produire des FeFET (transistors à effet de champ basés sur des ferroélectriques) extrêmement modulables et passifs. Dans cette thèse, nous tentons de contrôler la structure de bande des TMDC par l'intermédiaire de ferroélectriques. L'étude est réalisée principalement en utilisant des techniques basées sur la photoémission. Grâce à elles, nous avons accès à l'état chimique des matériaux (spectroscopie de photoémission par rayons-X), la structure de bande (spectroscopie de photoémission résolue en angle) et l'état de la surface (microscopie de photoémission d'électrons). Les mesures ont été réalisées en synchrotron, pour la plupart sur la ligne de lumière CASSIOPEE du Synchrotron SOLEIL. D'autres lignes de lumière ont été utilisées notamment NanoESCA (Elettra Sincrotrone, Italie), ANTARES (Synchrotron SOLEIL, France).

D.2 . Photoémission et matière condensée

La photoémission, technique reine dans cette thèse, est une méthode de mesure basée sur l'effet photoélectrique. En illuminant un solide avec une source de photons, les électrons se retrouvent excités par la lumière. Pour certains, l'apport énergétique est tel qu'ils s'échappent du matériau. Ces "photoélectrons" peuvent être mesurés à l'aide d'un analyseur hémisphérique qui permet de déterminer leur énergie cinétique, leur impulsion (et même parfois leur spin). Selon la situation, il est très difficile ou très aisé d'interpréter un spectre de photoémission. Ce chapitre introduit les outils mathématiques nécessaires pour comprendre la photoémission en détail.

Dans un premier temps, sont exposés les rappels de matière condensée avec le hamiltonien général pour un système à N-électrons. Nous reformulons dans le langage des opérateurs de création et d'annihilation cet hamiltonien. Par la suite, nous introduisons les notions de fonction de Green / propagateurs dans le cas d'un hamiltonien simple (une particule). Nous obtenons les formulations de la fonction spectrale du système. D'une manière similaire nous généralisons ces concepts pour un hamiltonien à N-électrons. Ces outils interviennent par la suite pour présenter différents modèles du phénomène. Les modèles sont présentés par ordre de complexité croissante. Tout d'abord le modèle à trois une étapes qui découpe la photoémission de la manière suivante : photoexcitation, transport, émission. Cette formulation permet de mettre en évidence les différentes contributions au photocourant : éléments de symétrie, section-efficace et géométrie du système qui pondèrent la fonction spectrale intrinsèque au système. Avec des hypothèses supplémentaires sur l'état final des photoélectrons nous arrivons aux formules classiques de photoémission qui relient l'angle d'émission θ et l'énergie cinétique E k des électrons à leur vecteur d'onde k ∥ = 2me h2 E k sin θ. Enfin nous décrivons le modèle à une étape, qui traite de manière consistante le système. Cette approche utilise les fonctions de Green du système introduites au début du chapitre pour calculer le courant de photoémission. Le code SPR-KKR pour le modèle à une étape est brièvement évoqué dans la mesure où il sert à l'étude des TMDC plus tard dans le manuscript.

D.3 . Photoémission expérimentale

Ce chapitre décrit les aspects expérimentaux de la photoémission en synchrotron. Sont décris en détail la source de lumière (rayonnement synchrotron), les appareils de détection (analyseur hémisphérique) avant d'aborder les différentes manières d'utiliser la photoémission. Le synchrotron est un accélérateur d'électrons optimisé pour la production de rayonnement synchrotron. Ce rayonnement est produit lorsque des électrons sont accélérés à des vitesses relativistes et possède plusieurs propriétés intéressantes : il est intense et focalisé, il couvre une large portion du spectre électromagnétique (des infrarouges aux rayons gamma), il est polarisé et pulsé (temps caractéristique de l'ordre de la picoseconde) en raison du regroupement des électrons en paquets dans l'anneau. Cette dernière propriété peut être utilisée pour réaliser des expériences résolues en temps. L'anneau du synchrotron est composé de plusieurs sections : les sections droites et les aimants de courbure ou du rayonnement est émis. Typiquement, on retrouve d'autres types éléments d'insertion pour produire du rayonnement synchrotron. Les premiers sont des onduleurs et les seconds sont des wigglers. Les deux sont composés d'une suite d'aimants qui font onduler la trajectoire des électrons. A chaque ondulation, du rayonnement est produit. Bien qu'ayant des structures similaires, la différence entre un wiggler et un onduleur tient à leur caractéristiques géométriques. Les caractéristiques du rayonnement émis sont dépendentes des détails de la configuration (période des aimants, longueur de l'onduleur).

La lumière produite est utilisée pour extraire les électrons de la matière, typiquement des énergies de 20 à 1500 eV sont utilisées sur CASSIOPEE. Ces électrons sont mesurés à l'aide d'un analyseur hémisphérique. Cet appareil utilise deux hémisphères concentriques réglés entre lequels une différence de potentiel est imposée. Les électrons évoluant dans le potentiel de l'instrument sont triés en fonction de leur énergie cinétique initiale, ce qui permet d'obtenir la distribution énergétique des photoélectrons. La géométrie du détecteur conservant l'angle d'émission donne l'information nécessaire pour calculer le vecteur d'onde et l'énergie cinétique. Des considérations sur les fentes de sorties et l'impact sur la résolution sont étudiées.

Une fois l'analyseur hémisphérique décortiqué, nous expliquons dans les grandes lignes les principes qui permettent d'interpréter différents types d'expériences à commencer par la spectroscopie de photoémission par rayons-X. La technique peut être utile pour: identifier des espèces chimiques, étudier les différences d'environnement chimique ou les effets électrostatiques sur les niveaux de coeur (charge). Nous abordons ensuite l'ARPES, sous l'angle pratique, à savoir, comment utiliser les angles et l'énergie des photons incidents pour explorer l'espace réciproque. Nous décrivons les limites expérimentales: taille mesurable d'espace réciproque contre résolution en vecteur d'onde. Enfin, nous expliquons une technique complémentaire de la spectroscopie de photoémission : la microscopie d'émission de photoélectrons. Cette technique permet l'imagerie directe ou l'imagerie en espace réciproque, au prix d'une résolution spectrale au rabais. Nous prenons l'exemple du double analyseur hémisphérique utilisé à NanoESCA en détaillant notamment l'optique de mesure.

D.4 . Ferroélectricité

Ce chapitre résume les informations importantes sur la ferroélectricité. Sont résumés en premier lieu les aspects élémentaires de la ferroélectricité en rapport avec le type de transition de phase. Dans un premier cas, la ferroélectricité émerge d'une transition displacive dans laquelle un dipôle émerge de la cellule unitaire qui n'est absolument pas polarisée dans la phase paraélectrique. Dans le deuxième cas, la cellule unitaire est polarisée dans une direction aléatoire mais le cristal en moyenne n'a pas de polarisation nette. Ces deux mécanismes constituent des extrêmes et la plupart des matériaux ont des comportements intermédiaires. Nous ajoutons des considérations sur les phénomènes hystérétiques observables dans les ferroélectriques, dans leur fonction diélectrique en l'occurence, pour relier les propriétés optiques de ces matériaux à leur structure crystallographique et modes de vibration. Nous présentons en second lieu divers modèles de la ferroélectricité, en commençant par des approches phénoménologiques type Landau. Nous exposons ensuite le modèle en vigueur de la ferroélectricité qui offre la possibilité d'une approche microscopique au phénomène. Cette approche permet de comprendre les limites de la vision purement displacive dans laquelle les électrons sont parfaitement localisés sur certains sites atomiques. Nous discutons ensuite d'aspects plus pratiques comme les conditions aux limites d'un ferroélectrique (écrantage des charges liées) qui contribuent à leur stabilité. En effet, ces charges induisent un champ électrique opposé à la polarisation, appelé champ dépolarisant. Dépendant de la géométrie (plus fort dans les films fins) et des conditions aux limites, il peut nuire à la stabilité de la ferroélectricité. Nous prenons l'exemple du modèle de Thomas-Fermi qui donne une première idée des effets d'écrantages sur le champ dépolarisant, et illustre l'importance d'un bon écrantage. Sont ensuites présentées les perovskites ferroélectriques d'abord de manière générale (structure cristallographique, facteur de Goldschmidt). Nous développons plus sur les exemples du niobate de lithium (LiNbO 3 , LNO) et du ferrite de bismuth (BiFeO 3 , BFO). Dans le cas du BFO nous expliquons en détail les techniques qui permettent l'ingéniérie de structure, qui permettent de contrôler avec le type de substrat à la fois : la structure crystallographique du BFO et sa croissance, la valeur absolue de sa polarisation électrique, l'orientation de celle-ci et le type de variants (configurations cristallographiques équivalentes dans le même cristal). Nous expliquons enfin les méthodes de mesure principales pour caractériser les ferroélectriques, la microscopie à force atomique d'abord et la microscopie de photoémission d'électrons de l'autre.

D.5 . Structure électronique des dichalcogénures de métaux de transition

Ce chapitre est composé de deux parties. La première est dédiée à poser les bases sur les dichalcogénures de métaux de transition à savoir leur structure cristallographique et leur structure de bande. Nous présentons les résultats théoriques de calculs de structure de bande réalisés avec la théorie de la fonctionnelle de la densité (DFT) qui montrent l'importance du couplage spin-orbite sur la forme et la polarisation (en spin) des bandes. Nous détaillons ensuite la composition orbitale des bandes. Dans un modèle simplifié, les bandes à Γ sont principalement décrites par les orbitales d z 2 et p z alors que les bandes à ±K sont décrites par une combinaison des orbitales d xy , d x 2 -y 2 et p x , p y . L'état de l'art de la photoémission sur du WSe 2 est présenté, mettant en évidence l'importance du substrat, du nombre de couches de WSe 2 et des méthodes de fabrication sur la structure de bande. La deuxième partie montre les premiers résultats ARPES sur une cohorte de WSe 2 sur des substrats de Gr/SiC. Nous mesurons la structure de bande de WSe 2 bi-, tri-, et multicouche en fonction de l'énergie de photon. L'étendue mesurée va de 20 à 90 eV avec un pas de 1 eV. Avec à l'appui des simulations type modèle à une étape ainsique ces mesures nous montrons que le WSe 2 a une structure 3D qui émerge rapidement avec le nombre de couches. Nous proposons ensuite un modèle de liaisons fortes, avec transitions entre couches dominées par les sauts entre orbitales p z qui explique la forte hybridation des orbitales dans cette direction. Nous étudions ensuite en détail les fonctions spectrales associées à différentes bandes en comparaison avec les calculs du modèle à une étape.

D.6 . Hétérostructures de Matériaux 2D et de ferroélectriques

Ce dernier chapitre compile les résultats obtenus sur la thématique des hétérostructures TMD-Ferroélectrique. Nous présentons tout d'abord l'état de l'art sur la question. En l'occurence, la plupart des systèmes existants dans la littérature sont orientés dispositifs et les principales mesures relèvent de la spectroscopie de photoluminescence ou des mesures de transport. La plupart des ferroélectriques représentés sont des perovskites (BaTiO 3 , LiNbO 3 , BiFeO 3 , PbZr x Ti 1-x O 3 ...) avec une utilisation notoire de P(VdF-TrFE). La polarisation ferroélectrique agit comme une source de dopant pour la couche de TMDC qui la recouvre. En photoluminescence, cela décale la position du pic de ainsi que son intensité. En transport, cela donne lieu à des ratio de résistance entre les deux états de conduction très élevés. Nous étudions ensuite la première hétérostructure : WSe 2 sur BiFeO 3 . Les couches de WSe 2 ont été préparées par épitaxie de jet moléculaire d'une part, et caractérisées par la diffraction aux rayons X, AFM et diffraction rasante d'électrons de haute énergie (RHEED). De l'autre côté, des couches minces de BiFeO 3 ont été fabriquées par ablation laser pulsée sur des substrats de DyScO 3 eux même recouverts d'une couche intercalaire de SrRuO 3 pour la polarisation down, et La 1-x Sr x MnO 3 . Après transferts humides du substrat d'origine vers les BFO, nous mesurons la structure de bande de tricouche et de bicouches de WSe 2 en ARPES. Nous mesurons un décalage relatif des bandes de l'ordre de 750 meV dans le cas tricouche et de 640 meV dans le cas bicouche. Nous abordons ensuite les limites de ces mesures, en effet, un recuit sous ultra-vide léger (250 °C, 1h) suffit à rendre les hétérostructures sur BFO down équivalentes aux hétérostructures sur BFO up du point de vue de la spectroscopie. Cela emmène à la conclusion que l'interface du BFO et du WSe 2 est sensible et mérite d'être étudiée en détail. Puisque la terminaison du BFO semble importante, nous exposons les arguments de Spaldin et al. sur les possibles configurations de surface du BFO. Nous étudions enfin le second système, des hétérostructures de WSe 2 sur LiNbO 3 . Dans ce cas les flocons de WSe 2 sont préparées par dépôt chimique en phase vapeur (CVD), ce qui permet un contrôle précis de leur épaisseur. Le substrat de LNO provient d'un fournisseur industriel et possède des domaines périodiques d'une largeur de 30 µm. Les flocons de WSe 2 sont transférés à sec sur les substrats avant d'être recuits sous vide pour nettoyer le système. Nous mesurons les hétérostructures avec différentes techniques de microscopie (AFM, PFM, microscopie optique, photoluminescence et PEEM). Des mesures de microscopie de photoémission nous tirons les cartes de travail de sortie qui informent de l'état de surface et la structure de bande de trois flocons chacun sur des domaines différents. Les mesures de spectroscopie indiquent que la structure de bande du WSe 2 a un décalage relatif dû à la polarisation sous-jacente de 150 meV. Au vu des inhomogénéités de surface sur le LNO, il semble que l'état de surface n'a pas d'incidence directe sur le décalage mesuré près du niveau de Fermi.

List of Figures

1.1 direction. (c) Effect of the crystal field on the binding energy of d orbitals for a perfect octahedron, lifting the degeneracy into two levels. The high energy level e g contains orbitals pointing to the corners of the octahedron (oxygen), the low energy levels t 2g contains orbitals pointing away. Adapted from [38]. .

Figure 1 . 1 :

 11 Figure 1.1: (a) k-dependent dispersion for the non interacting system. (b) k-dependent spectral function of the many body system. Adaptated from[8]

Figure 1 . 3 :Figure 1 . 4 :

 1314 Figure 1.3: (a) Real space picture of photoionization (b) Energy picture (c) Photoemission spectrum adapted from ref[22]

Figure 1 . 5 :

 15 Figure 1.5: (a) Interaction geometry of a d x 2 -y 2 orbital with light.The interaction plane contains the wave vector of the electromagnetic plane wave and the impulsion of the outgoing electron, adaptated from[8]. (b) Total photoionization cross section of a W -atom with partial photoionization cross section of W 5d valence state and W 4f core level. Data taken from[28].

Figure 1 . 6 :

 16 Figure 1.6: (a) Different scattering processes after excitation, i -primary electron, ii -secondary electron scattered through the material then emitted, iii -reabsorbed electron. (b) Universal Inelastic Mean Free path for electrons with data points from reference [30], the fitted curve in plain color has the equation λ IM F P = 1430E -2 + 0.54 √ E [31]. (c) Emission geometry and contribution to the photoemitted signal (in %) of the sample as a function of the depth z.

Figure 1 . 7 :

 17 Figure 1.7: (a) Potential seen by an electron in a solid, (b) Model of the potential step, (c) Refraction of the electrons at the surface of the sample.

Figure 1 . 8 :

 18 Figure 1.8: (a) Geometry of an ARPES experiment, (b) three-steps model for a valence electron: optical excitation to a higher energy band, matching with free electron model, and the resulting measured spectrum.

Figure 1 . 9 :

 19 Figure 1.9: One step-model photoemission: the electron is excited from a valence state inside the crystal into a high energy state outside.

Figure 1 . 10 :

 110 Figure 1.10: Building up scattering.

Figure 2 . 1 :

 21 Figure 2.1: Left: Schematic view of a synchrotron radiation facility, with its linear accelerator, its booster and the storage ring. As we will see later, the produced light is used by beamlines dedicated to various photon-based techniques (X-ray diffraction, X-ray absorption, photoemission, imaging…). The electrons which lost energy by radiating are re-accelerated at each turn by an electric field created by a radio-frequency cavity synchronized with the electron bunches. Right: View of the inside of the storage ring at Synchrotron SOLEIL.

Figure 2 . 2 :

 22 Figure 2.2: First observation of synchrotron radiation at the General Electric laboratory in 1947. Extracted from [1].

Figure 2 . 3 :

 23 Figure 2.3: (a) Emission from a bending magnet and (b) from an insertion device.

Figure 2 . 4 :

 24 Figure 2.4: The different light sources on a synchrotron radiation facility: (a) a bending magnet gives a continous emission spectrum up to a critical photon energy . (b) In a wiggler, electrons oscillates with large deviation angles : the emission spectrum is similar to a bending magnet, although more intense and possibly shifted in energy. (c) In an undulator, electrons oscillate close to the insertion device axis. The interferences between the different emission points gives a emission spectrum made of a main line (first harmonics) and its harmonics. Remade from [2].

Figure 2 . 5 :

 25 Figure 2.5: The CASSIOPEE HU256 undulator. The left picture was taken where it was built, in Novossibirsk (Russia). The right picture shows the HU256 undulator installed in the storage ring.

Figure 2 . 6 :

 26 Figure 2.6: Principle of an APPLE 2 undulator.

Figure 2 . 7 :

 27 Figure 2.7: The CASSIOPEE APPLE 2-type HU60 undulator.

Figure 2 . 8 :

 28 Figure 2.8: 3D view of the CASSIOPEE beamline. Red and blue areas after the deviation mirror (number 4) highlight the paths toward High Resolution (HR) and spin-resolved ARPES stations respectively.

Figure 2 . 11 :Figure 2 .

 2112 Figure 2.11: Diagram illustrating the slit geometry relative to the sample and the effect of the shape of the slit on momentum resolution.

Figure 2 . 15 :

 215 Figure 2.15: Illustration of core and valence levels in a solid. Adapted from Kai Siegbahn's 1981 Nobel lecture.

Figure 2 .

 2 Figure 2.16: (a) XPS of C-1s and O-1s lines for different chemical species. The main line moves as a function of the kind of bonding and the chemical environment (b) Calculated chemical shifts for the C-1s line in different chemical compounds with the model (2.12).Figures extracted from[11]

  .18).

Figure 2 . 17 :

 217 Figure2.17: Geometry of the a 3-rotation manipulator with angles φ, θ and τ and their k-space effect. The bottom diagrams show the behavior of the reciprocal space upon rotations with respect to the fixed axis of the detector.

Figure 2 . 18 :Figure 2 . 19 :Figure 2 . 20 :

 218219220 Figure 2.18: Effect of the photon energy on the accessible size of the reciprocal space. The data shown is the band structure of N-layer (≈ 6-8) WSe 2 at two photon energies.

Figure 2 .Figure 2 . 22 :

 2222 Figure 2.21: (a) Diagram of PEEM with double hemispherical analyzer, (b) Detail of the immersion lens.

  k y ) with k x and k y the momentum. An example can be seen in the figure 2.23.b. In x-PEEM mode, the measurements shape the data as I = f (E k , x, y). The figure 2.23.a shows an example of an area measured at different kinetic energies.

Figure 2 .

 2 Figure 2.23: (a) Example of energy filtered real space microscopy, (b) Example of energy filtered momentum microscopy, (c) Free electron dispersion obtained in k-PEEM from low kinetic energy electrons.

Figure 2 . 24 :Figure 2 . 25 :

 224225 Figure 2.24: Detail of figure 2.23. The contours of the multilayers are highlighted with red dashed lines.

Figure 3 . 1 :

 31 Figure 3.1: Top: Order-disorder type transition of strained SrTiO 3 adaptated from [1]. Bottom: Displacive type transition between cubic paraelectric BaTiO 3 and two (out of four) of its tetragonal counterparts for up and down polarization.

Figure 3 . 2 :

 32 Figure 3.2: Dielectric function of BTO according to ref[3]. The function is discontinuous near the transitions at critical temperatures T c = 120,0,-90 °C.

Figure 3 . 3 :

 33 Figure 3.3: Typical P-E (polarization/electric field) loop for ferroelectric materials.

Figure 3 . 4 :

 34 Figure 3.4: (a) Change of the shape of the Landau free-energy density around the critical temperature T c . (b) Polarization-Temperature phase diagram predicted from the the model. (c) Explanation of the P-E loop hysteresis with the biased Landau model.

Figure 3 . 5 :

 35 Figure 3.5: Dependence of the calculated polarization on the location of the unit cell. Extracted from [10].

Figure 3 . 6 :

 36 Figure 3.6: Comparison of Bloch waves states for a band n at different wavevectors k with the Wannier functions of the same band n localised on the centers of the lattice. Adapted from [11].

Figure 3 . 7 :

 37 Figure 3.7: Polarization induced surface charges

Figure 3 . 8 :

 38 Figure 3.8: Different screening mechanisms in ferroelectrics: (a) Reduction of the depolarization by formation of ferroelectric domains stabilizing each others. (b) Intrinsic screening by defects. (c) Extrinsic screening by adsorbates. Adapted from [22].

Figure 3 . 9 :

 39 Figure 3.9: (a) Ferroelectric material in between two electrodes fixing the potential to the ground, (b) Charge distribution induced in the electrodes over a screening length λ ef f due to imperfect screening, (c) Induced voltage drop across the sample.

Figure 3 .

 3 Figure 3.10: (a) Cubic perovskite lattice. (b) Stacking of AO-BO 2 planes in the <001> direction. (c) Effect of the crystal field on the binding energy of d orbitals for a perfect octahedron, lifting the degeneracy into two levels. The high energy level e g contains orbitals pointing to the corners of the octahedron (oxygen), the low energy levels t 2g contains orbitals pointing away. Adapted from [38].

Figure 3 . 11 :

 311 Figure 3.11: Tolerance factor contour plot calculated for several perovskites. Ionic radii from [42].

  rh ) direction, LNO looks like a stack of distorted oxygen octahedra, embedding in a repeating sequence Li, Nb ions and vacant sites (see figure 3.12b). In the paraelectric phase, Li and Nb atoms are separated by c hex 4 with Nb atoms in octahedral sites and Li atoms inside oxygen triangles (figure 3.12a). Ferroelectricity in LNO emerges from the relative displacement of the Li-and Nb-cations along the [0001] hex direction [44]. The Li-cations step out of the triangles and the Nb cations are not perfectly centered in the octahedra leading to a net polarization.

Figure 3 .

 3 Figure3.12: (a) Structure of paraelectric LNO in the rhombohedral system and view from the [0001] direction adapted from[24]. (b) Ferroelectric LNO along the direction [0001] adapted from[START_REF] Sanna | LiNbO 3 surfaces from a microscopic perspective[END_REF]. Nb filled octahedron are highlighted in red.

31 )( 3 . 32 )Figure 3 .Figure 3 .

 3133233 Figure 3.13: (a) Phase diagram of LNO adapted from [49]. (b) Li vacancy compensation adapted from [45].

Figure 3 . 15 :

 315 Figure 3.15: Poling procedure of LNO as defined by Myers et al. (a) Insulator on conductor scheme, (b) Conductor on insulator scheme. Remade from [52].

Figure 3 .

 3 Figure 3.16: pseudo-cubic representation of BFO, rhombohedral phase with antiferroelastic tilt of the octahedra and view from [111] pc direction. Adapted from reference [58].

Figure 3 . 18 :

 318 Figure 3.18: Polarization of BFO in the pseudo-cubic and monoclinic M A and M B lattices in relation with the reference rhombohedral system. Adaptated from reference [60].

Figure 3 . 19 :

 319 Figure 3.19: Comparison between lattice parameters of common substrates and materials grown as thin films. Taken from reference[START_REF] Schlom | Strain tuning of ferroelectric thin films[END_REF].

Figure 3 . 20 :

 320 Figure 3.20: Evolution of the Curie and Néel temperatures with different substrates. STO and DSO are highlighted. Adapted from [65].

cFigure 3 .

 3 Figure 3.21: (a) Equivalent directions for the polarization. (b) Resulting variants taken from [66]. (c) Reduction of the number of possible variants depending on the miscut and the substrate orientation, taken from [30].

7 (

 7 La,SrO)+0.7

Figure 3 . 22 :

 322 Figure 3.22: Epitaxial matching of LSMO and BFO cells. Adapted from [67].

Figure 3 . 23 :

 323 Figure 3.23: Polar discontinuity model with punctual charges Q, electric field E, and potential V of MnO 2 terminated LSMO and SrO terminated SRO leading to UP and DOWN polarization states that were used in this thesis work.

Figure 3 .

 3 Figure 3.24: (a) Diagram of a standard AFM setup. (b) Effect of the movement of the cantilever on the detector.

UFigure 3 . 25 :

 325 Figure 3.25: Lennard-Jones potential

igure 3 . 26 :

 326 Response of oppositely polarized ferroelectrics domains to the alternative voltage V AC cos(ωt).

Figure 3 .Figure 3 .

 33 Figure 3.27: Dead-layer slowing down photoemitted electron. Reproduced from [22].

2 Figure 4 . 1 :√ 3 .

 2413 Figure 4.1: Crystallographic structure of 2H-MX 2 compounds.

  Finally the three points M i (i ∈ {1, 2, 3}) are in the middle of each edge of the BZ. Their distance to the center d Γ-M is 2π a √ 3 and their symmetry group further reduced to only a 180°rotation and a mirror symmetry (C 2v ) [8]. A diagram summarizing this discussion is visible in figure 4.2. Strictly speaking, these considerations are only valid for a monolayer sample since multilayer have different point groups. It is of less importance when navigating

3 Figure 4 . 2 :Figure 4 . 3 :

 34243 Figure 4.2: (Left) MX 2 plane with lattice vectors and reciprocal vectors. (Right) Corresponding Brillouin zone.

MX 2 2 2 WS 2 2 ∆

 22222 MoS MoS WSe

Figure 4 . 4 :Figure 4 . 5 :

 4445 Figure 4.4: DFT calculated WSe 2 band structure for 1ML, 2ML, 3ML and 4ML WSe 2 . Calculations on top (bottom) neglect (include) the SOC. Adapted from [2].

Figure 4 . 6 :

 46 Figure 4.6: (a) Crystal field splitting of d-orbitals in trigonal prismatic configuration. (b) Calculated charge density for Γ and ±K states in WS 2 taken from [13]. (c) Monolayer WS 2 band structure with SOC included and orbital projection, adapted from [14]. Remarkably, the relative position between the bands maxima at K and Γ is in agreement with the crystal field arguments. (c) Band compositions (in %) of the valence and conduction bands of monolayer WSe 2 with SOC taken into account. Adapted from [15].

Figure 4 . 7 :

 47 Figure 4.7: Spectral weights of orbitals in the valence band of bulk WSe 2 with SOC included. Adapted from [17].

Figure 4 . 8 :

 48 Figure 4.8: Schematic band diagram of WSe 2 and its important parameters (we note E Γ the topmost visible band at Γ on multilayer samples.)

Figure 4 . 9 :

 49 Figure 4.9: ARPES along Γ -K direction of 1ML, 2ML and 3ML WSe 2 , measured with photons of energy hν = 27 eV. Remade with accessible data from [10].

Figure 4 . 10 :

 410 Figure 4.10: ARPES of 2ML WSe 2 at several photon energies. Depending on the photon energy, one or two bands are visible at Γ.

Figure 4 . 11 :

 411 Figure 4.11: Mounted sample.

Figure 4 . 13 :

 413 Figure 4.13: Top and side view of the experimental geometry. ε is the polarisation vector of the beam (LH) oriented at 45°from the sample normal facing the analyzer. The slit of the analyzer is vertical (along z).

Г

  Isoenergy cut (k x ,k y ) K Dispersion cut (E,k)

Figure 4 .

 4 Figure 4.14: Example 3D data block (E,k x ,k y ) of the band structure of the N-ML (multilayer) sample.

Figure 4 .

 4 16 shows how the bands at Γ are duplicated due to the mixture of different thicknesses of graphene. The shifts between the splitted lines, namelyδΓ 1 = |Γ 1 -Γ ′ 1 | and δΓ 2 = |Γ 2 -Γ ′ 2 | (see definitions on figure 4.16), were equal with value 0.12 eV.

1 )Figure 4 . 15 :

 1415 Figure 4.15: Sample 2ML-A: (a) Constant energy cut of the band structure (left) and second derivative (right). The thick black line on top indicates the scale 1 Å -1 . (b) Zoom on the area surrounded by the red frame in (a) of the constant energy cut near the graphene cone at high symmetry point K Gr at two energies. The thick white line at the bottom indicates the scale 0.2 Å -1 . (c) Γ -→ K Gr -→ -K Gr ARPES cuts with logarithmic color scale. The top cut (red arrows) is made along the thick red line/dash line highlighted in b. The bottom cut (cyan arrows) is made along the cyan line/dash line highlighted in b. All measurements are done with hν = 90 eV, LH polarization.

Figure 4 . 16 :

 416 Figure 4.16: Sample 2ML-A: Γ -K W Se 2 ARPES cut and second derivative, hν = 33 eV, LH polarization. The photon energy was selected to maximize the visibility of the bands, evidencing the splittings.

Figure 4 . 17 :Figure 4 . 18 :

 417418 Figure 4.17: Sample 2ML-B 1 : (a) Constant energy cut of the band structure and second derivative. The thick black line on top indicates the scale 1 Å -1 . (b) Zoom on the area surrounded by the red frame in (a) the constant energy cut near the graphene cone at high symmetry point K Gr at two energies. The thick white line at the bottom indicates the scale 0.2 Å -1 . (c) Γ -→ K Gr -→ -K Gr ARPES cut with logarithmic color scale. The cut is made along the thick red line/dash line highlighted in b. All measurements are done with hν = 90 eV, LH polarization.

Figure 4 . 19 :

 419 Figure 4.19: Sample 2ML-B: Γ -K W Se 2 ARPES cuts and second derivatives on locations representative of the two B 1 and B 2 zones, hν = 33 eV, LH polarization.

Figure 4 . 20 :

 420 Figure 4.20: Sample 3ML: (a) Constant energy cut of the band structure and second derivative. The thick black line on top indicates the scale 1 Å -1 . (b) Detail of the constant energy cut near the graphene cone at high symmetry point K Gr at two energies. The thick white line at the bottom indicates the scale 0.2 Å -1 . (c) Γ -→ K Gr -→ -K Gr ARPES cut with logarithmic color scale. The cut is made along the thick red line/dash line highlighted in b. All measurements are done with hν = 90 eV, LH polarization.

Figure 4 .

 4 Figure 4.21: Γ -K W Se 2 Band dispersion of the 3ML sample. hν = 60 eV, polarization LH.

Figure 4 . 22 :

 422 Figure 4.22: Sample N-ML: (a) Constant energy cut of the band structure and second derivative. The thick black line on top indicates the scale 1 Å -1 .

Figure 4 .

 4 Figure 4.23: Γ -K W Se 2 Band dispersion of the N-ML sample. hν = 60 eV, polarization LH.

Figure 4 . 24 :

 424 Figure 4.24: Top row: Comparison between the ARPES spectra near Γ for WSe 2 samples of all the thicknesses. Bottom row: Second derivatives. Measured with photon energy hν = 60 eV, polarization LH.

Figure 4 .Figure 4 . 25 :

 4425 Figure 4.25: Data block for hν dependent photoemission on N-ML sample.

Figure 4 .

 4 Figure 4.26: (a) 3D BZ of WSe 2 . (b) One-step calculation of the bands intensity variations at Γ for 1ML along the Γ -A direction (c) Experimental bands intensity variation at Γ as a function of the photon energy, second-derivative and one-step calculation results for 2, 3 and N-ML of WSe 2 .

  2 and our measurements on the N-ML sample agree well with the literature. The dispersion compares well with the results of Riley et al. as shown in figure 4.27. The dispersion in the few-layer systems, however, is less expected because of their supposed 2D nature. Most DFT calculations assume the 2D character of TMD and obtain results on a mesh with only one k z point. What we see in 2ML and 3ML samples is the emerging k z dispersion: despite the interlayer hybridization, the states probed by photon energy dependence remain strongly localized, giving rise to the staggered pattern. The same kind of pattern was seen by Miwa et al. on 1ML and 2ML MoS 2 [35] and Ohta et al. on increasingly thick graphene samples (1ML to 4ML) [36].

Figure 4 . 27 :

 427 Figure 4.27: Band intensity along the Γ -A direction for the N-ML sample with hν from 20 to 90 eV (left) compared to a measurement on a bulk sample with hν from 20 to 130 eV [21]

Figure 4 .

 4 Figure 4.28: (a) Geometry of in-plane hopping. (b) Out-of-plane hopping through Se-p z states.

Figure 4 . 29 :

 429 Figure 4.29: Calulated photoemission spectra along a monoatomic linear chain where the band becomes more and more continuous with the length of the chain (the number of atoms in the chain is given at the bottom of each panel). Extracted from [40].

Figure 4 .Figure 4 .

 44 Figure 4.30: (top) 2ML and 3ML hν dependent dispersions at k || = 0. (bottom) corresponding intensities after integration around the energy level in the area of same color above.

Figure 4 . 31 :

 431 Figure 4.31: Comparison between one-step model and experimental cross sections.

Figure 4 .

 4 Figure 4.32: (a) Energy dependent photoemission cut at Γ for different 2ML samples and the second derivative. The white dashed zones define the borders of the two areas (top and bottom) where the signal is integrated. The result of the integration is shown in (b) for the top and bottom bands for each sample.

Figure 5 .Figure 5 . 1 :

 551 Photoluminescence in TMD is shaped by the radiative recombination of excitons, that are bound states of hole and electron. These excitons are the main reason for the differences between optical band-gap measurement and electronic band-gap. As a general rule, in TMD, the PL intensity decreases with the number of layers[1]. ML samples have relatively higher intensity because of the direct band-gap at K. For this reason, we will mostly talk about ML photoluminescence. The main exciton line (X 0 ) for WSe 2 on SiO 2 is around 1.75 eV [2; 3] but it is sensitive to the substrate. Courtade et al. reported a value close to 1.72 eV when WSe 2 is encapsulated in hBN[4]. On that matter, ferroelectric substrates have two main effects on the PL spectrum: different domain orientations tend to shift the main emission line by tens of meV and the PL intensity is strongly connected to the underlying domain polarization (see figure5.3) with relative intensities changing from twice-fold to hundred-fold. Those effects were observed in MoS 2 /LNO[5], MoS 2 /BFO[6], MoS 2 /PZT[6], WS 2 /PZT[7], WSe 2 /BFO[8], MoSe 2 /LNO[9],and WSe 2 /LNO[9]. Similar results were obtained with WSe 2 , MoTe 2 , WS2 on P(VdF-TrFE)[10]. At the exception of reference[5] that attributes the change of energy of the PL emission line to the strain exerted on the flake, the other references show that the induced doping is the source of this modification. Notably,Ko et al. (reference [6]) were able to switch the PL characteristics by switching the polarization. Those results are consistent with experiments where ML WSe 2 is subjected to gate voltage[3; 4; 11]. Figure5.2 shows the effect of electrostatic gating on the PL spectrum, the geometry of the gating is indicated in

Figure 5 Figure 5 . 3 :Figure 5 . 4 :

 55354 Photon energy (eV)

Figure 5 . 5 :

 55 Figure 5.5: Characterization of the BFO surfaces used for 3L samples. (a,d) AFM of the BFO samples (b,e) PFM phase of DOWN BFO grown on SRO (blue square) and UP BFO grown on LSMO (red square). White arrows represent the UP out-of-plane electric polarization, black arrows represent the DOWN out-of-plane electric polarization. (c, f) Out-of-plane θ -2θ scan of XRD intensity in log-scale of epitaxially grown BFO on DSO with SRO(LSMO) electrode -BFO DOWN(UP) -before WSe 2 transfer.

Figure 5 . 6 :Figure 5 . 7 :

 5657 Figure 5.6: (a-b) RHEED patterns of 3L WSe 2 along the directions [210] (azimuth φ = 30 • ) and [100] (azimuth φ = 0 • ). (c) Crystal structure of a single WSe 2 layer. the vector a is the [100] direction, the vector b is the [010] direction. (d) AFM image of grown 3L WSe 2 on mica. (e) Line profile of the A-B segment in d, the height difference between two layers is 0.7 nm. (f) Bar chart of the surface coverage (%) of WSe 2 with different thicknesses, according to the AFM data in d. Precise data in table 5.1.

Figure 5 . 8 :

 58 Figure 5.8: (a) Diagram of the transfer of MBE grown WSe 2 onto BFO (reproduced from [23]) (b) Azimuthal XRD intensity in log-scale before and after transfer on opposite polarized BFO substrates. The black arrows indicate the intensity from disoriented WSe 2 regions, invisible in linear scale. (c) In plane radial θ-2θ scan of XRD intensity in log-scale before and after transfer on opposite polarized BFO samples. (d) AFM image BFO UP after transfer of WSe 2 . (e) Top: Out-of-plane PFM phase signal of BFO UP after transfer, Bottom: in-of-plane PFM phase signal of BFO UP after transfer, showing stripes, 2 variants. (f) AFM image BFO UP after transfer of WSe 2 . (g) Out-of-plane PFM phase signal of BFO DOWN after transfer, Bottom: in-of-plane PFM phase signal of BFO DOWN after transfer, showing stripes, 4 variants.

Figure 5 . 9 :Figure 5 .

 595 Photon energy hν (eV)Beamline flux photons•s -1 )

Figure 5 .

 5 Figure 5.11: (a) ARPES raw data and (b) second derivative of Γ -K slice of the valence band of 3L WSe 2 on DOWN BFO. (c) ARPES raw data and (d) second derivative of Γ -K slice of the valence band of 3L WSe 2 on UP BFO. (e) ARPES raw data and (f) second derivative of Γ -K slice of the valence band of 3L WSe 2 transferred on graphene/SiC.

Figure 5 .

 5 Figure 5.12: (a) ARPES raw data and (b) second derivative of Γ -K slice of the valence band of 2L WSe 2 on DOWN BFO. (c) ARPES raw data and (d) second derivative of Γ -K slice of the valence band of 2L WSe 2 on UP BFO. The ARPES was measured with hν = 50 eV and linear horizontal polarization.

Figure 5 .

 5 Figure 5.13: (a) ARPES raw data of Γ -K slice of the valence band of 2L WSe2 on UP BFO at photon energy hν = 50eV. (b) Second derivative of Γ -K slice of the valence band of 2L WSe2 on UP BFO at photon energy hν = 50eV, in log-colors. (c) ARPES raw data of Γ -K slice of the valence band of 2L WSe2 on UP BFO at photon energy hν = 56eV. (d) Second derivative of Γ -K slice of the valence band of 2L WSe2 on UP BFO at photon energy hν = 56eV.

Figure 5 .

 5 Figure 5.14: (a) ARPES measurement (hν = 50 eV, LH polarization) and (b) second derivative for the 2L' UP sample. (c) ARPES measurement (hν = 50 eV, LH polarization) and (d) second derivative for the 2L' DOWN sample.

Figure 5 . 15 :

 515 Figure 5.15: ARPES measurements of 3L DOWN sample before (a) and after (b) UHV annealing. ARPES measurements of 3L UP sample before (c) and after (d) UHV annealing. hν = 50 eV, polarization LH.

Figure 5 . 16 :Figure 5 . 17 :

 516517 Figure 5.16: XPS survey of 3L samples (UP and DOWN) before and after annealing. The main core levels are indicated for the UP sample. The photon energy was hν = 500 eV and the polarization LH.

bFigure 5 .

 5 Figure 5.18: (a) Diagram of the crystallographic structure of BFO. (b) Possible configurations of the surface. Taken from [32].

Figure 5 .

 5 Figure 5.20: (a) Optical microscope photography of the surface of the LNO substrate as received. Inset of PFM phase associated with the photography. The red frame highlights domain seen in the next images. (b) Diagram of the specifications of the LNO substrate. (c) Topographic characterization of a domain wall by AFM. (d) Zoom-in topography on one of the domain. (e) Height profile across a domain (A-B) segment on the image (c). (f) Top: Phase out-of-plane PFM signal of the LNO substrate across the area seen in (c). Bottom: Amplitude PFM signal of the same area.

Figure 5 . 49 Table 5 . 5 :Figure 5 .

 549555 Figure 5.21: LNO substrate before and after annealing.

Figure 5 .

 5 Figure 5.23: (a) PDMS bubble on a glass slide. (b) O 2 plasma treatment of the PDMS. (c) Preparation of the spin-coating, applying PPC on the bubble.

Figure 5 . 24 :

 524 Figure 5.24: Left: apparatus used for the transfer from Si/SiO 2 to LNO. The setup is composed of a micrometric displacement plates. The global stage (in blue) supports two smaller stages for the deterministic transfer. The "bubble stage" (in red) has movements in the three directions plus an in plane rotation. The "substrate stage" (in yellow) has the same controls. The whole setup is connected to an optical microscope aligned above the substrate stage. Right: zoom-in on the substrate stage before picking-up WSe 2 flakes. The temperature control helps for the pick-up and transfer operations.

Figure 5 . 26 :Figure 5 .

 5265 Figure 5.26: Optical microscopy of the WSe 2 flakes in the transfer process. The monolayer flakes are framed in black.

5. 4 . 6 .

 46 Microscopic characterization of the heterostructureThis section presents the microscopic characterization performed on the monolayer flakes. As a reminder, figure5.28 shows a HR photography of the area of LNO supporting the flakes. The labels (flakes 1,2 and 3) that we will use from now on to refer to the different flakes in the picture. PFM characterization (figures 5.29, 5.30 and 5.31 (c-d)) show the PFM phase on the different locations. The domain structure is extremely clear and shows that flake 1 and 2 were transferred on inwardly polarized domains (P down ) while flake 3 was deposited on outwardly polarized domain (P up ). Additionally, these images indicate that the heterostructure is robust against the transfer process in a first step and moderate UHV annealing (see methods). AFM characterization (figures 5.29 and 5.30 (f)) confirmed that the flakes are monolayer, as hinted by their low reflectivity in optical microscopy (figures 5.29, 5.30 and 5.31 (e)). The height profiles (figures 5.29 and 5.30 (g)) indicate 0.7nm thick layers as expected from the previous work with WSe 2 /BFO. The main result of this part are the work function maps calculated from energy-filtered PEEM images. The work function maps (figures 5.29, 5.30 and 5.31 (a)) show that the work function of electrons in WSe 2 is higher relative to the domain it lies on. The line profiles in figures 5.29, 5.30 and 5.31 (b)show that the order of magnitude is always tens of meV with 24 meV of difference in flake 1, 60 meV for flake 2 and 41 meV for flake 3. We will comment this in the discussion later.

Figure 5 . 28 :Flake 1 Figure 5 .Flake 2 Figure 5 .

 5281525 Figure 5.28: HR photography (enhanced contrast) of the area where the flakes were transferred.

Flake 3 (Figure 5 .

 35 Figure 5.31: (a) Work function map around flake 3. (b) Line profile of the work function between A-B points in (a). (c) PFM phase of the same area. (d) PFM amplitude of the area. (e) Optical microscopy (enhanced contrast) of the flake.

Figure 5 .

 5 Figure 5.32: (a) Diagram of the energy cuts with reading help (right) for the position of the high symmetry points. The image is centered on the Γ point. The points on the dotted inner circle are the K points. (b-c, e-f, h-i) Constant energy cuts for the three flakes, left column: bottom of the VB (black lines along Γ -K), middle column: top of the VB as indicated in (a). All measured with LV-polarized photons of energy hν = 60 eV, LV. (d-g-j) PEEM images of the areas studied in spectroscopy with iris location indicated in yellow.

Figure 6 . 1 :

 61 Figure 6.1: (a) Schematisation of the Rashba effect at Γ in a TMD layer deposited on a ferroelectric substrate as predicted by Cheng et al. with (b) zoom-in on the spin-splitted bands at Γ.

Figure A. 1 :

 1 Figure A.1: Fixed mode background of the detector in an energy region without signal (above Fermi-level).

  Figure A.3 shows a constant energy cut before and after individual normalization of each spectra (one ARPES spectrum for every k y value). The four graphene pockets near the corners are more visible after normalization and the gradient in the secondary background has completely disappeared from the image.

Figure A. 2 :

 2 Figure A.2: (a) Angle dependent background (blue = smoothed, red = raw) calculated from the area with black stripes in the Fermi level measured at hν = 22.5 eV (b). (c) Normalised spectrum.

Figure A. 3 :

 3 Figure A.3: Left: Raw constant energy cut in the band structure of 3ML WSe 2 /Gr measured at hν = 90 eV, LH polarization. Right: Same normalised constant energy cut.

Figure A. 4 :

 4 Figure A.4: (a) Stack of images obtained at different kinetic energies for energy filtered PEEM. (b) PEEM image with colored crosses.(c) The photoemission spectra near the secondary electron edge on the locations highlighted in (b). (d) Work function map without correction of the energy scale. (e) Same map after correction. Both images share the energy scale.

Finally, σ is approximated as σ = 1 2

 2 |E max -E W F | with the factor 1 2 compensating for the √ 2 factor in the definition of the error function. The different parameters that we used are illustrated in figure A.5. The figure also shows the results of the actual work function map computation against the estimator. We see that the estimator is able to provide an accurate picture of the work function. For instance, we can see the defects and local variations on the LNO surface and recognize that different thickness of WSe 2 have different work functions. The estimator systematically estimates the work function of WSe 2 lower than the actual fit.

Figure A. 5 :

 5 Figure A.5: Top: Important parameters extracted from the photoemission spectra to estimate the parameters of the fit. The outset shows a diagram of the derivative of the edge. Bottom: Work function map calculated and estimated work function map. Both maps share the color scale on the right.

Figure B. 1 :

 1 Figure B.1: Fermi level kinetic energy E k (E F ) relatively to the photon energy (monochromator reading) measured on a Mo clip in equilibrium with the manipulator and connected to the ground. In an ideal experimental setup, the curve should be constant. Additionnaly it varies with time. 2ML-A and B were measured with few days of difference and stay very close. The measurements were months appart from 3ML and N-ML.

Figure B. 3 :

 3 Figure B.3: Top: Efficiency of the photodiode. Middle: Current measured on the photodiode at beamline resolution 9 meV. Bottom: Deduced photon flux.

  (a) k-dependent dispersion for the non interacting system. (b) k-dependent spectral function of the many body system. Adaptated from[8] . . . . . . . . 1.2 (Left) Hertz experiment of spark generation. Many windows were tested: quartz, glass, metal sheets etc... as described in[12; 13] -(Right) Apparatus for Lenard photoelectric effect experiment reproduced from[15] . . . . . . 1.3 (a) Real space picture of photoionization (b) Energy picture (c) Photoemission spectrum adapted from ref [22] . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The three-step model, figure extracted from [23] . . . . . . . . . . . . . . . . . 1.5 (a) Interaction geometry of a d x 2 -y 2 orbital with light. The interaction plane contains the wave vector of the electromagnetic plane wave and the impulsion of the outgoing electron, adaptated from [8]. (b) Total photoionization cross section of a W -atom with partial photoionization cross section of W 5d valence state and W 4f core level. Data taken from [28]. . . . . . . . . . . . . . . . . . 1.6 (a) Different scattering processes after excitation, i -primary electron, iisecondary electron scattered through the material then emitted, iii -reabsorbed electron. (b) Universal Inelastic Mean Free path for electrons with data points from reference [30], the fitted curve in plain color has the equation λ IM F P = 1430E -2 + 0.54 √ E [31]. (c) Emission geometry and contribution to the photoemitted signal (in %) of the sample as a function of the depth z. . 1.7 (a) Potential seen by an electron in a solid, (b) Model of the potential step, (c) Refraction of the electrons at the surface of the sample. . . . . . . . . . . .1.8 (a) Geometry of an ARPES experiment, (b) three-steps model for a valence electron: optical excitation to a higher energy band, matching with free electron model, and the resulting measured spectrum. . . . . . . . . . . . . . . . . . . 1.9 One step-model photoemission: the electron is excited from a valence state inside the crystal into a high energy state outside. . . . . . . . . . . . . . . . . 1.10 Building up scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Left: Schematic view of a synchrotron radiation facility, with its linear accelerator, its booster and the storage ring. As we will see later, the produced light is used by beamlines dedicated to various photon-based techniques (X-ray diffraction, X-ray absorption, photoemission, imaging…). The electrons which lost energy by radiating are re-accelerated at each turn by an electric field created by a radio-frequency cavity synchronized with the electron bunches. Right: View of the inside of the storage ring at Synchrotron SOLEIL. . . . . . . . . . 2.2 First observation of synchrotron radiation at the General Electric laboratory in 1947. Extracted from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 (a) Emission from a bending magnet and (b) from an insertion device. . . . . . 2.4 The different light sources on a synchrotron radiation facility: (a) a bending magnet gives a continous emission spectrum up to a critical photon energy . (b) In a wiggler, electrons oscillates with large deviation angles : the emission spectrum is similar to a bending magnet, although more intense and possibly shifted in energy. (c) In an undulator, electrons oscillate close to the insertion device axis. The interferences between the different emission points gives a emission spectrum made of a main line (first harmonics) and its harmonics. Remade from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The CASSIOPEE HU256 undulator. The left picture was taken where it was built, in Novossibirsk (Russia). The right picture shows the HU256 undulator installed in the storage ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Principle of an APPLE 2 undulator. . . . . . . . . . . . . . . . . . . . . . . . 2.7 The CASSIOPEE APPLE 2-type HU60 undulator. . . . . . . . . . . . . . . . . 2.8 3D view of the CASSIOPEE beamline. Red and blue areas after the deviation mirror (number 4) highlight the paths toward High Resolution (HR) and spinresolved ARPES stations respectively. . . . . . . . . . . . . . . . . . . . . . . 2.9 Diagram of a HDA. R 0 is the radius where the entry slit is located, r 0 the radius of the incoming electron. R π is the radius where the exit slit is located, r π the radius of the outgoing electron. . . . . . . . . . . . . . . . . . . . . . 2.10 Sensitivity of a HDA of mean radius R 0 = 20 cm from equation (2.8). Linear approximation in dashes. The measured kinetic energy range for a 2D detector is usually E k ∈ [0.95E p , 1.05E p ] where the relation between r π and E k is close to linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Diagram illustrating the slit geometry relative to the sample and the effect of the shape of the slit on momentum resolution. . . . . . . . . . . . . . . . . . . 2.12 (a) Geometry of the HDA with radial slit and corresponding dispersion obtained on the detector side for different kinetic energies (b) mapping obtained for straight slit (c) for radial slits (d) for curved slits. E k ∈ [0.95E p , 1.05E p ] and angular opening ±15 • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Electrical connection between the sample and the analyzer. . . . . . . . . . . . 2.14 Typical photoemission spectrum when the analyzer is calibrated. The electrons with lowest kinetic energy are secondary electrons that have lost almost all their energy in collisions. They are useful to measure the work function. The electrons with the largest kinetic energies are valence band electrons. Adapted from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Illustration of core and valence levels in a solid. Adapted from Kai Siegbahn's 1981 Nobel lecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 (a) XPS of C-1s and O-1s lines for different chemical species. The main line moves as a function of the kind of bonding and the chemical environment (b) Calculated chemical shifts for the C-1s line in different chemical compounds with the model (2.12). Figures extracted from [11] . . . . . . . . . . . . . . . 2.17Geometry of the a 3-rotation manipulator with angles φ, θ and τ and their k-space effect. The bottom diagrams show the behavior of the reciprocal space upon rotations with respect to the fixed axis of the detector. . . . . . . . . . . 2.18 Effect of the photon energy on the accessible size of the reciprocal space. The data shown is the band structure of N-layer (≈ 6-8) WSe 2 at two photon energies. 2.19 Transformation from (E k , β) to (E k , k) coordinates using the relation 2.18. The spectrum was measured with photon energy hν = 27 eV. . . . . . . . . . 2.20 Example of a 3D block of data from ARPES with coordinates (k x ,k y ,E). The data shown is the band structure of N-layer (≈ 6-8) WSe 2 measured with photons of energy hν = 90 eV to get most of the Brillouin zone on the detector. 2.21 (a) Diagram of PEEM with double hemispherical analyzer, (b) Detail of the immersion lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22 Detail of the PEEM column with electron trajectories for real space (top) and reciprocal space (bottom) imaging. Remade from [17]. . . . . . . . . . . . . . 2.23 (a) Example of energy filtered real space microscopy, (b) Example of energy filtered momentum microscopy, (c) Free electron dispersion obtained in k-PEEM from low kinetic energy electrons. . . . . . . . . . . . . . . . . . . . . . . . . . 2.24 Detail of figure 2.23. The contours of the multilayers are highlighted with red dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.25 Examples of contrast due to topography and work function. . . . . . . . . . . 3.1 Top: Order-disorder type transition of strained SrTiO 3 adaptated from [1]. Bottom: Displacive type transition between cubic paraelectric BaTiO 3 and two (out of four) of its tetragonal counterparts for up and down polarization. . . . 3.2 Dielectric function of BTO according to ref[3]. The function is discontinuous near the transitions at critical temperatures T c = 120,0,-90 • C. . . . . . . . . . 3.3 Typical P-E (polarization/electric field) loop for ferroelectric materials. . . . . . 3.4 (a) Change of the shape of the Landau free-energy density around the critical temperature T c . (b) Polarization-Temperature phase diagram predicted from the the model. (c) Explanation of the P-E loop hysteresis with the biased Landau model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Dependence of the calculated polarization on the location of the unit cell. Extracted from [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Comparison of Bloch waves states for a band n at different wavevectors k with the Wannier functions of the same band n localised on the centers of the lattice. Adapted from [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Polarization induced surface charges . . . . . . . . . . . . . . . . . . . . . . . 3.8 Different screening mechanisms in ferroelectrics: (a) Reduction of the depolarization by formation of ferroelectric domains stabilizing each others. (b) Intrinsic screening by defects. (c) Extrinsic screening by adsorbates. Adapted from [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 (a) Ferroelectric material in between two electrodes fixing the potential to the ground, (b) Charge distribution induced in the electrodes over a screening length λ ef f due to imperfect screening, (c) Induced voltage drop across the sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 (a) Cubic perovskite lattice. (b) Stacking of AO-BO 2 planes in the <001>

  The following years, the theory of

	Condenser
	plate	Window	Electrometer
		Emitting cathode	Magnetic Field	Electrometer
	Ruhmkorff
	coil	To pump
	Condenser
	plate
	Spark Generator	Detector ring	UV Source

Figure 1.2: (Left) Hertz experiment of spark generation. Many windows were tested: quartz, glass, metal sheets etc... as described in [12; 13] -(Right) Apparatus for Lenard photoelectric effect experiment reproduced from

[15]

table 1

 1

	hν (eV) λ (nm)
	20	62
	50	24.8
	120	10.4
	1487	0.84
	Table

.1). Since λ/a 0 ≪ 1, the vector potential is nearly constant over atomic lengths giving A = A 0 .

Table 2 .

 2

		1/2	4/2	6
	d	5/2, 3/2	6/4	10
	f	7/2, 5/2	8/6	14

1: Summary of the properties of total angular momenta for s,p,d and f shells in core level photoemission.

Table 3 .

 3

1: Overview of ferroelectric materials and merit figures.

Table 4 .

 4 1: Distance of the main symmetry points to Γ, using crystallographic data a = 3.282(1) Å and c = 12.96(1) Å from[3].

Table 4 .

 4

	Energy (eV)
	Monolayer (1ML)	Bilayer (2ML)	Trilayer (3ML)	Quadrilayer (4ML)
	Energy (eV)

2: Theoretical spin orbit couplings for MX 2 compounds (largely confirmed by experiments)

[11]

Table 4 .

 4 3: Table of orbital contributions at the high symmetry points for a monolayer MX 2 .Adapted from[16] The minimal model for the valence band structure of WSe 2 is defined by Liu et al.[16] by neglecting the contribution of Se-4p orbitals and only keeping the following set of W-5d orbitals: (d z 2 , d xy , d x 2 -y 2 ). The two up (↑) and down (↓) spin states are added when SOC is taken into account. This defines the following six-state basis:|d z 2 ,↑⟩ , |d xy , ↑⟩ , |d x 2 -y 2 , ↑⟩ (4.1) |d z 2 , ↓⟩ , |d xy , ↓⟩ , |d x 2 -y 2 , ↓⟩ (4.2)

	d xy

Table 4 .

 4 ) Reference 4: Summary of parameters of the band structure extracted from ARPES measurements performed on WSe

	2
	Monolayer WSe

2 . MLG: Monolayer Graphene, BLG: Bilayer Graphene, TLG: Trilayer Graphene...

Table 4 .

 4 5: Binding energies at Γ of 2ML and 3ML samples.

Table 5 .

 5 1: Table of data summarizing the surface state of the heterostructure for both 2L and 3L samples. The coverage percents were both calculated on a 25 µm 2 surface.

		Sample	2L	3L
	RMS Rugosity	UP	0.2	0.853
	BFO (nm)	DOWN	0.1	0.48
	Ferroelastic	UP	stripes, 2 variants stripes, 2 variants
	in-plane variants	DOWN stripes, 4 variants stripes, 4 variants
		0ML	43.307	0.138
	Coverage WSe 2 (%)	1ML 2ML 3ML	41.867 14.589 0.237	16.942 57.008 24.921
		4ML	0	0.991

Table 5 .

 5 2: Summary of the properties extracted from the band structure of the WSe 2 /BFO.

Table 5 .

 5 3: Fitted binding energies for the surface and bulk Bi4f core levels from the spectra in figure5.17.

	Sample	DOWN	DOWN (annealed)	UP	UP (annealed)
	Area ratio A B /A S	5.9	14.1 4.8	12.4

Table 5 .

 5 4: Area ratio A B /A S of the bulk and surface peaks from the spectra in figure5.17.

Table 5 .

 5 6: Table of the main extracted parameters from XPS, peak position, spin-orbit coupling ∆ s-o , peak area and used Scofield sensitivity factors.

		E B (eV) ∆ s-o (eV) Area	Scofield Sensitivity Factor (at hν = 1486.6 eV)
	Li1s	52.7	-	139.51	0.057
	Nb3s	468.6	-	3646.8
	Nb3p 3/2	363.5	15.4	24653
	Nb3d 5/2	205.4	2.69	19962
	Nb4s	58.7	-	1007	0.402
	Nb4p 3/2	33.2	1.89	4588.5

CVD growth of WSe 2 and dry transfer to LiNbO 3

  Step by step procedure for PDMS based dry transfer.

	Preparation of the WSe 2
	I.1		I.2	15 min	I.3		I.4	>30 min (acetone) 2 min (isopropanol)	I.5
				PMMA		PMMA		PMMA
		WSe2		WSe2		WSe2		WSe2	WSe2
		Si/SiO 2		Si/SiO 2		Si/SiO 2		Si/SiO 2	Si/SiO 2
			Liquid nitrogen (77K)		Acetone + Isopropanol clean
	Transfer to new substrate			35°C
	II.1	PDMS	II.2	PDMS	II.3	25°C	II.4	PDMS	II.5
		PPC		PPC		PDMS		PPC	PDMS
		WSe2		WSe2	Water	PPC WSe2	Water	WSe2	PPC WSe2
		Si/SiO 2		Si/SiO 2		Si/SiO 2		LiNbO 3	LiNbO 3
						Heating 25°C			Heating 100°C
	II.6	PPC PDMS		II.7		PPC	T ~ 300°C P ~ 10 -6 t ~ 6h UHV Annealing	II.8
		WSe2				WSe2			WSe2
		LiNbO 3				LiNbO 3			LiNbO 3
		Heating 100°C
	Figure 5.25: WSe 2 /Si/SiO 2 Before pick-up (step II.1)		LNO surface after WSe 2 transfer (step II.6)	LNO surface after WSe 2 transfer and cleaning (step II.8)
	60 µm				60 µm			60 µm

Remerciements appuyés à Manuel Bibes (unité mixte de Physique CNRS-Thalès), conjointement à Céline Vergnaud et Matthieu Jamet (laboratoire IRIG-SPINTEC au CEA Grenoble), pour tout ce temps passé à m'aider dans la préparation des expériences, dans la rédaction du papier et du soutien lors de ma troisième année... A l'unité mixte de Physique CNRS-Thalès, Sara Varotto (straordinaria), Luis Moreno (maestro de la MBE) et Srijani Mallik (উত্তম) la dream-team de l'ARPES sous la direction de Manuel Bibes. Merci pour ces moments en temps de faisceau. Au Centre de Nanosciences et Nanotechnologies : Thomas Maroutian de l'équipe Oxides, le MVP de la ferroélectricité et du PFM. De l'équipe matériaux 2D, Abdelkarim Ouerghi et Julien Chaste présents depuis le début et sources de conseils, ainsi que

This is a slight abuse of terminology as the object we will study does not coincide anymore with the mathematical Green function defined earlier

Some modern estimators like the model of Bartel et al. were recently discovered using machine learning methods[41]

a sizeable energy shift, here corresponding to 640 meV, slightly lower than but consistent with the value found for the 3L sample, confirming the influence of the BFO polarization direction on the electronic structure.

A.5 Top: Important parameters extracted from the photoemission spectra to estimate the parameters of the fit. The outset shows a diagram of the derivative of the edge. Bottom: Work function map calculated and estimated work function map. Both maps share the color scale on the right. . . . . . . . . . . . . . . .

Remerciements

. Methods X-ray photoemission spectroscopy

The XPS spectra for the preparation of the LNO were performed at on MesoXScope experimental station with the help of Grégroire Magagnin, Antony Boucly, Christophe Lubin and Nick Barrett (CEA, SPEC). We used a Al-K α hν = 1486.6 eV source oriented about 54°from the normal of the sample. The detection was normal to the sample surface. For the analysis of the peak areas, the single peaks were fitted with gaussian functions (Li1s, Nb4s, Nb3s) and the spin-orbit splitted doublets with Voigt doublets (Nb3p, Nb3d, Nb4p). The secondary electron background was a Shirley function. All the fits were done using the software KolXPD.

Photoemission microscopy

Photoemission microscopy was done at the NanoESCA beamline at Elettra with the help of Daniel Baranowski, Iulia Cojocariu and Vitaliy Feyer. The NanoESCA beamline is equipped with a Focus double-hemisphere IDEA analyzer described in chapter 2. The sample was mounted on iron omicron plates and put in the UHV chamber. It was then annealed above 200 °C for one hour until the pressure in the chamber reached below 10 -9 mbar. All the following measurements were done at room-temperature. x-PEEM measurements were done using a broad spectrum Hg-source (hν ≈ 4.9 eV) and Field of View (FoV) 67µm. The kinetic energy scale was corrected using the known work-function of the analyzer ϕ ana = 4.25 eV. Unless specified otherwise, k-PEEM measurements were done using LV polarized photons of energy hν = 60 eV. The size of the beam was 10 µm wide for the main spot. To limit the background from the substrate, the iris aperture was used and centered on the individual flakes.

Calculation of the work function map from x-PEEM data

The work function maps were calculated from energy filtered PEEM (x-PEEM) data using the procedure defined in [38; 39]. We briefly summarize the steps here. We collected energy filtered PEEM images on a range of 3 eV with a 10 meV step. This gave us a data block with coordinates (x,y,E k ). First the non-isochromaticity of the analyzer was corrected numerically on each image. We then fitted the secondary electron edge (see figure 2.14) on each point (x,y) of the map with a complementary error function. This gave a value of Φ W F for each coordinate. Those composed the new work functions images. The code used was adapted from the work of J. Rault (IgorPro version) and G. Magagnin (Python version) to be fully compatible with data from NanoESCA beamline.

Piezoelectric force microscopy

The PFM characterization on individual flakes was done with Thomas Maroutian (C2N) after the beamtime. The AC voltage was 2V and the lock-in frequency 360kHz.

Calculation of the number of photons

The detector has a power-efficiency S(hν) as indicated in the datasheet in A/W [1]. We measure the current I(hν) in A. Given a photon energy hν (in eV), the flux Φ(hν) in photons/s is given by:

List of Tables

List of Abbreviations