Dr Claudine Crépin-Gilbert

Prof. Laïfa Boufendi

Prof. Khaled Hassouni

Prof Gilles Peslherbe

Dr Thierry Belmonte

Philippe Baranek

Isabelle Berbezier

Maurizio De Crescenzi

Luca Persichetti

Elena Ivanova

Qadir Timerghazin

Pere Roca

Erik Cabarrocas

Fatima Johnson

Gaël Bouanis

Enric Garcia Zucchi

Tatiana Caurel

Martin Novikova

Jean-Luc Foldyna

Jean Maurice

Denis Rehbinder

Eric Tondelier

Frédéric Paillassa

Jérôme Liege

Cyril Charlic

Jean- Eric Jadaud

Stanislas Bouree

Rasha Deby

Salomé Khoury

Christoph Forel

Jejune Lechner

Park

Laurence Gérot

Gabriela Medina

Keywords:

may come

New exotic nanostructured materials : Theoretical predictions and experimental verifications

Résumé

Cette thèse est consacrée à l'étude approfondie de formes exotiques de matériaux nano-structurés qui pourraient conduire à une avancée significative dans les nano-composants. Deux thèmes distincts ont été ainsi abordés. Le premier concerne les nano-clusters aromatiques de silicium synthétisés par plasma (SiNCs), tandis que le second est dédié aux structures silicium et germanium bi-dimensionnelles. Grâce à des simulations de type dynamique moléculaire et des calculs ab initio,a i n s iq u es u rd e sr e c h e r c h e se x p é r i m e n t a l e s ,n o u sn o u sp r o p o s o n sd ' e x p l o r e rl e sp r opriétés intrigantes, mais à fort potentiel, de ces matériaux exotiques. Dans la première partie, nous commençons par des études théoriques et montrons qu'il est possible d'obtenir un comportement aromatique pour des SiNCs hydrogénés ayant une taille de ⇠1nm.

Nous démontrons que les plasmas silane/hydrogène à basse température, proches de la formation de particules de poussière, présentent l'environnement idéal pour exploiter la tendance naturelle du silicium à la sur-coordination et donc pour la synthèse de structures à liaisons déficitaires en électrons. Ces nano-clusters qui se forment spontanément par auto-assemblage dans le plasma, ne possèdent pas de structure tétraédrique, sont plus stables que tous les autres SiNCs connus de cette taille et ont de fortes propriétés aromatiques dues à leur forte délocalisation électronique.

Nous montrons également que les SiNCs non tétraédriques présentent des modes de liaison à caractère métallique qui ressemblent fortement à celui d'un gaz d'électrons homogène dans des nano-clusters de métaux. Les SiNCs tétraédriques standards de cette taille ne peuvent absorber que dans l'ultraviolet, alors que nos calculs ont montré que des SiNCs purs, mais sur-coordonnés, absorbent dans la région spectrale ultraviolette, mais aussi dans le visible et l'infrarouge. Nous présentons ensuite une première preuve expérimentale pour nos prédictions théoriques. Nous avons mesuré in situ, dans un réacteur plasma l'absorption de la lumière visible des SiNCs. De plus, nos mesures réalisées en présence d'un champ électrique ont prouvées clairement que les SiNCs aromatiques possèdent un moment dipolaire permanent, que nous avons estimé entre 2,0 et 2,5 Debye, en excellent accord avec les calculs ab initio.E n fi n ,n o si m a g e sd em i c r o s c o p i eé l e ctronique à transmission des SiNCs, déposés dans des conditions de plasma optimisées, ont révélé la présence d'une autre forme exotique de silicium à structure hexagonale. Une telle structure se forme habituellement à des pressions extrêmement élevées appliquées sur des structures cubiques (diamant) de silicium. Nous souhaitons affirmer que c'est grâce à la "chimie au marteau" que ces conditions ont été atteintes.

Dans la seconde partie, nous avons entrepris des études théoriques et expérimentales approfondies sur la croissance d'une nouvelle forme allotropique de silicium et de germanium: le silicène et le germanène, à savoir, une mono-couche d'atomes intégrée dans un réseau hexagonal qui ressemble fortement au graphène. Afin d'exclure tout mélange entre les atomes de silicium ou de germanium avec le substrat et de conserver leurs caractéristiques prometteuses comme de nouveaux matériaux de Dirac, nous avons effectué nos dépôts, sur un substrat de graphite chimiquement inerte. Une de nos découvertes cruciales est que les mono-couches de silicène ou de germanène interagissent avec le substrat de graphite uniquement via des forces de van der Waals. Cette interaction est suffisamment forte pour stabiliser les mono-couches, déposées même au-dessus de la température ambiante, mais suffisamment faible pour empêcher toute hybridation ou alliage entre le silicium ou le germanium et les atomes de carbone du substrat. Par conséquent, les propriétés électroniques exceptionnelles du silicène et du germanène, tels que les cônes de Dirac et les électrons sans masse, sont préservées même après leur dépôt sur les surfaces de graphite.

List of Figures

Introduction

Silicon, an element that is everywhere Silicon, an abundant and nontoxic element, has become a technologically paramount material in the last century. Silicon lies at the heart of our modern technology as the material for integrated circuits and solar cells. In short, silicon is to technology as carbon is to biology. Thanks to the favorable properties of its amorphous and diamond-cubic lattice structures; such as stability at high temperature, ability to be both p and n doped, and ability to grow an oxide surface layer, that silicon has gained such a technological supremacy. Fortunately, silicon possesses a rich and complex free energy landscape, with many local energy minima, allowing for the formation of a variety of unusual new structures. The so far discovered exotic structures of silicon have shown to exhibit desirable properties so that one or more of those structures could become the basis for the next revolution in nanotechnological devices.

Exotic forms of silicon

Silicon with a variety of three-dimensional crystal structures

Several exotic allotropes of Si were formed under the application of extremely high pressure using different methods. [1] At high pressure, Si takes on numerous dense structures with increasing coordination and tends to favor a metallic behaviour. Under the compression of the standard 4-fold coordinated diamond-cubic Si (referred to as Si-I) to ⇠12 GPa, it undergoes metallization into Si-II (body centred tetragonal) and becomes 6-fold coordinated. [2]A b o v e⇠13 Gpa, Si-II distorts into a body-centred orthorhombic structure. [3] Yet, with the application of more pressure, this polymorph transforms into an 8-fold coordinated simple hexagonal structure (Si-V) at ⇠16 GPa. [4] Another orthorhombic structure (Si-VI) is found at ⇠38 GPa, with a mixture of 10and 11-fold coordination. [4,[START_REF] Hanfland | Crystal structure of the high-pressure phase silicon VI[END_REF] Compression to ⇠42 GPa yields the 12-fold coordinated hexagonal close-packed structure (Si-VII). [4]F i n a l l y ,t h ef a c e -c e n t r e dc u b i cs t r u c t u r e( S i -X )f o r m sa t⇠79

Silicon with multiple bonds

Ad i fferent, yet exotic, form of silicon was realized by West et al. in the early 1980s when they succeeded in stablizing a compound containing a Si-Si double bond,t e t r a m e s i t y l disilene. [START_REF] West | Tetramesityldisilene, a stable compound containing a silicon-silicon double bond[END_REF] There work was, by itself, a major breakthrough and had proved controversial to the "double bond-rule". According to this rule, multiple bonds become less stable for second-and third-row elements; it was justified to be the result of the large radii of these atoms, leading to poor overlap between adjacent p orbitals. The secret for their success was the kinetic stabilization by protecting the molecule's reactive sites with bulky substituents that form a protective shell around the highly reactive Si-Si double bond. Since then, about 100 disilenes [START_REF] Kira | Progress in the chemistry of stable disilenes[END_REF]a sw e l la s compounds with a Si-Si triple bond have been isolated. [START_REF] Sekiguchi | A stable compound containing a silicon-silicon triple bond[END_REF]

Aromaticity

The concept of aromaticity has fascinated chemists since the 1860s when the architect of chemistry,A u g u s tK e k u l é,fi r s ti n t r o d u c e dt h et e r ma r o m a t i c i t yi nh i sf a m o u sp a p e r s , [START_REF] Kekulé | Sur la constitution des substances aromatique[END_REF][START_REF] Kekulé | Note sur quelques produits de substitution de la benzine[END_REF][START_REF] Kekulé | Untersuchungen uber aromatische verbindungen[END_REF]a n d explained the particular stability and low reactivity of a class of molecules related to benzene and its derivatives. In 1931, Hückel formulated his famous 4n+2 rule. [START_REF] Hückel | Quantum-theoretical contributions to the benzene problem. I. the electron configuration of benzene and related compounds[END_REF][START_REF] Hückel | Quantum theoretical contributions to the problem of aromatic and non-saturated compounds[END_REF] It states that if an unsaturated monocyclic hydrocarbon molecule has the right number of ⇡-electrons obeying this rule, then the molecule is aromatic and has high stability and low reactivity. Aromaticity was primarily limited to the branch of organic chemistry. However, with the high-level theoretical treatments and advances in experimental techniques to create novel molecules and clusters, the concepts of aromaticity have found wider applicability far beyond benzene. [START_REF] Iwamoto | Persilastaffanes: Design, synthesis, structure, and conjugation between silicon cages[END_REF][START_REF] Boldyrev | Beyond organic chemistry: Aromaticity in atomic clusters[END_REF]

Silicon goes aromatic

As a consequence of the ⇡ bonding capability of Si and the crucial role of benzene in the field of aromatic chemistry, aromaticity in compounds involving Si shifted into focus. Considerable research interests have been directed toward the synthesis and characterization of aromatic Si compounds. Such unconventional structures have proved to be stable. However, their stability relies on an extremely complicated atomic structure. [START_REF] Abersfelder | A tricyclic aromatic isomer of hexasilabenzene[END_REF]At r i c y c l i ca r o m a t i ci s o m e ro fh e x a s ilabenzene, was recently reported by Scheschkewitz et al. The structure they had obtained had six mobile electrons cyclically delocalized over four cenrtal Si atoms (see Figure 1.3). Theoretical analyses have revealed that the mobile electrons were of ⇡-, σ-andnon-bondingtypeelectrons,in contrast to the renowned six ⇡-electrons of Hückel aromatic systems. The attachment of highly bulky molecules (see the "Tip"in Figure 1.3)i si n e v i t a b l et og i v et h es t r u c t u r et h ek i n e t i ca n d thermodynamic stability. Such a discovery, by its very nature, brings with it the amelioration to the field of aromaticity. However, the cumbersome structure of those synthetic aromatic Si compounds prevents their use as building blocks for the construction of bigger structures, as it is the case in organic chemistry. Most of the results of this thesis are presented by means of papers that have been published in peer-reviewed journals (a list is provided at the end of this thesis), while the remaining results are under preparation for publication. We hope that we have succeeded in presenting our work in a comprehensive and attractive way and that you will enjoy reading this thesis.

Si Si Si

At this point, we would like to take the opportunity and clarify that the research work was carried out collaboratively as follows:

• All experiments in Part I of the thesis were performed by us in LPICM. The cavity enhanced absorption measurements were the result of a successful collaboration with experts in this field of research, Andy Ruth and Bryan Keary from University College Cork (UCC), Cork, Ireland. We combined our plasma reactor with their cavity enhanced absorption set-up and performed the absorption measurements together.

• Nudged elastic band calculations in Part I of the thesis were performed by L. Ivanova and Q. Timerghazin in Marquette University, Milwaukee, USA.

• All other theoretical calculations and simulations have been accomplished by us at LPICM. Chapter 2

Introduction

Electron-deficiency aromaticity in silicon nanoclusters

In 2012, Vach introduced a new concept of aromaticity, namely "electron-deficiency aromaticity". [START_REF] Vach | Electron-deficiency aromaticity in silicon nanoclusters[END_REF] In his work, Vach had shown the possibility of obtaining aromatic-like behavior with very simple and extremely stable systems. Unlike other strategies that rely on the formation of multiple bonds in order to achieve aromaticity in Si compounds, and that become impossible without binding the Si atoms to bulky stabilizing molecules (see Figure 1.3), the systems that Vach found are comp osed of only Si and H atoms and have a size of ⇠1nm. In his approach, Vach had exploited the natural tendency of Si to overcoordination for the formation of electrondeficient systems in a silane-based plasma. Overcoordination of Si does not only appear in systems subjected to extremely high pressure, yet it has also shown to occur in liquid Si where the first nearest-neighbor atomic coordination number is found to be 5.5-6. [START_REF] Okada | Persistence of covalent bonding in liquid silicon probed by inelastic x-ray scattering[END_REF] The deficiency of electrons in the bonds between Si atoms will induce extensive electron delocalization and will, in turn, provide enhanced stability to the system, as well as other aromatic properties. This novel phenomenon was the outcome of extensive studies on the nucleation and growth of hydrogenated silicon (Si m H n )n a n o c l u s t e r si nap l a s m ab a s e do nav a r i e t yo ft h e o r e t i c a lm e t h o d s . [START_REF] Vach | Controlled growth of silicon nanocrystals in a plasma reactor[END_REF][START_REF] Vach | Ultrasrable silicon nanocrystals due to electron delocalization[END_REF][START_REF] Vach | Growth dynamics of hydrogenated silicon nanoparticles under realistic conditions of a plasma reactor[END_REF] The growth of Si m H n nanoclusters in a silane plasma was investigated using first-principles molecular dynamics simulations. The properties of those self-assembled crystalline (symmetric) and amorphous (asymmetric) nontetrahedral nanoclusters were studied at a high-level of theory: using ab initio methods. Among the reported nanoclusters, we will focus here on one precise example that combines overcoordination with highly symmetrical ring structures, Si 19 H 12 (isomerA) shown in Figure nent dipole moment. Furthermore, calculations on the magnetic shielding and the magnetically induced ring currents have shown that this nontetrahedral silicon nanocluster acquires some aromatic properties that are 3 times stronger than that of benzene. [START_REF] Vach | Ultrasrable silicon nanocrystals due to electron delocalization[END_REF] Indeed, Si manifests remarkable changes in its optical and electronic properties when its size approximates the exciton Bohr radius. Those particular properties derive from the quantum confinement effect. Under quantum confinement conditions, the band gap becomes size-dependent, increasing as the size decreases. Therefore, tuning the absorption and emission spectra, by our experimental part in Chapter [START_REF] Hanfland | Crystal structure of the high-pressure phase silicon VI[END_REF] with an overview of capacitively coupled plasma (CCP)

processes and the principles of cavity-enhanced absorption spectroscopy. In Chapter [START_REF] Duclos | E x p e r imental study of the crystal stability and equation of state of Si to 248 GPa[END_REF], we describe our experiments and diagnostics. In Chapter [START_REF] Wippermann | Novel silicon phases and nanostructures for solar energy conversion[END_REF], we present the experimental results along with discussions.

Chapter 3

Model and Methods

This chapter describes the computational methods used throughout our study on aromatic silicon nanoclusters, focusing on one isomer with a Si 19 H 12 stoichiometry and with an asymmetric configuration. We will refer to it as isomer D in the text. A theoretical background covering molecular dynamics simulations, density functional theory (DFT) and time-dependent DFT (TD-DFT) is presented in Appendices A, B,a n dC,r e s p e c t i v e l y .

Molecular dynamics simulations

Ab initio molecular dynamics (AIMD) simulations were performed with the Vienna Ab initio Simulation Package (VASP-5.3.5) [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using aplane-w a v ebasisset[END_REF]u s i n gap r oj e c t o r -a u g m e n t e dw a v e( P A W )p s e u d o po t e n t i a l [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] in conjunction with the Perdew-Burke-Ernzerhof (PBE) [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] exchange-correlation functional. The planewave basis set was restricted to a cut-off energy of 500 eV and the time-step was 0.1 femtoseconds. The convergence criterion for the electronic iterations was chosen to be 10 -5 eV.

Geometrical and electronic structure optimizations

The geometrical and electronic structure optimizations of the hydrogenated silicon nanocluster under study1 were systematically performed with DFT using the hybrid, non-local exchange and correlation functional (B3LYP) with the Gaussian 09 software. [32] The 6-311++G (d,p) basis set with diffuse and polarisation functions was used for both Si and H atoms. We have used an open shell system to optimize the structures for different multiplicities and found that the lowest energy structures are obtained for the singlet state that can well be described as a closed shell system. In all cases, a frequency analysis was performed to verify vacuum stability.

Electron localization function

The Electron localization function (ELF) has emerged as a powerful tool to understand in a qualitative way the behavior of electrons in a nuclei system. It is possible to explain a great variety of bonding situations ranging from the most standard covalent bond to the metallic bond. In 1990, Becke and Edgecombe [START_REF] Becke | A simple measure of electron localization in atomic and molecular systems[END_REF]i n t r o d u c e dt h eE L Fa sa" s i m p l em e a s u r eo fe l e c t r o n localization in atomic and molecular systems". They associated the localization of an electron with the probability density for finding a second like-spin electron near a reference point

D(r)= X i |rφ i (r)| 2 - 1 4 |r⇢(r)| 2 ⇢(r) (3.1)
For small probability densities, D(r)i sa l s os m a l la n dt h ee l e c t r o ni sm o r el o c a l i z e d . B e c k ea n d

Edgecombe proposed two additional scaling rules.

1) The homogeneous electron gas is used as a reference, the kinetic density of which is given by

D 0 (r)= 3 10 (3⇡ 2 ) 2/3 ⇢(r) 5/3 (3.2)
2) For numerical convenience the function is mapped to one; i.e. the ELF is defined between 0 and 1. Hence, they proposed the following ELF:

ELF = 1 (1 + χ 2 ) (3.3)
where

χ = D D 0 (3.4)
Becke and Edgecombe attributed the ratio χ to a dimensionless localization index, calibrated with respect to the uniform electron gas. In this way, the following inequality is obeyed

0  ELF  1 (3.5)
and a value of the ELF close to 1 corresponds to a region in space where there is a high probability of finding electron localization, whereas an ELF value close to one-half corresponds to a region of electron gas-like behavior. [START_REF] Savin | Electron localization in solid-state structures of the elements -the diamond structure[END_REF]M o r e o v e r ,l o c a lE L Fm a x i m aa r ee n t i t l e d" a t t r a c t o r s " .

In this sense, covalent bonds can normally be identified by ELF attractors in between the two corresponding atoms.

ELF Plots: The ELFs were plotted using the Multiwfn program [START_REF] Lu | A multifunctional wavefunction analyzer[END_REF]u s i n gac o l o r -fi l l e dm a p . This type of map uses different colors to present real space function value in different regions.

By default, if the function value exceeds lower (upper) limit of color scale, then the region will be filled by black (white).

optimizations were also used in NEB calculations. The chemical nature of all transition states was confirmed by intrinsic reaction coordinate (IRC) reaction path calculations. [START_REF] Hratchian | Using Hessian updating to increase the efficiency of a Hessian based Predictor-Corrector reaction path following method[END_REF][START_REF] Hratchian | Accurate reaction paths using hessian based predictor-corrector integrator[END_REF] 3.6 Calculation of the absorption spectra

The absorption spectra of nanoclusters is characterized by the vertical excitation energies, according to the Franck-Condon principle. The Franck-Condon principle states that, because nuclear masses are so much larger than the mass of an electron, an electronic transition occurs within a stationary nuclear framework. As a result, the nuclear locations remain unchanged during the actual transition, but then readjust once the electrons have adopted their final distribution. [START_REF] Atkins | r s i t yP r e s s[END_REF] The vertical excitation energies correspond to the energy difference between the excited state and the ground state potential energy curves at the minimum point of the ground state geometry.

The oscillator strength, f,expressestheprobabilit yofligh tabsorptioninelectronictransitions between the energy levels of an atom or molecule. From Fermi's golden rule, within the dipole approximation,t h eo s c i l l a t o rs t r e n g t ho fa ne l e c t r o n i ct r a n s i t i o ni s

f = 2 3 m e ~2 (E f -E i )|µ| 2 (3.6)
where m e is the mass of the electron, (E f -E i )i st h ee l e c t r o n i ce x c i t a t i o ne n e r g y ,a n dµ is the transition dipole moment. The total transition dipole moment of an electronic transition from an initial state to a final state, within the Franck-Condon and Born-Oppenheimer approximations, is:

µ =< Ψ e f |µ e |Ψ e i >< v f |v i > (3.7)
where <Ψ f e |µ e |Ψ i e > is the electronic transition dipole moment; v i and v f are the nuclear vibrational wave functions of the initial and final states, respectively; and <v f |v i > is the Franck-Condon factor.

The simulation of the absorption spectra of nanoclusters usually involves two steps:

• Perform a structural optimization to obtain the minimum energy geometry of the system at a given level of theory.

• Calculate the vertical electronic excitation energies and the corresponding oscillator strengths using the TD-DFT method at the same level of theory .

The B3LYP functional was used with the same basis set for TD-DFT calculations for the absorption spectra. [32] The reported optical gaps correspond to the excitation energies for the first allowed transition.

Chapter 4

Theoretical Results and Discussions • Severely electron-deficient bonds (down to 1.46 electrons per bond)

• Different hybridization states (between sp 2 and sp 28 )

• Strong stabilization energies (up to 62 kcal mol -1 )

Magnetically induced ring current

Isomer D shows a high net diamagnetic aromatic ring current of 41.6 nAT -1 (see Figure 4.2)

which is quite comparable to the one of the symmetric nanocrystal (isomer A) of 42.0 nAT -1 , while the corresponding value for benzene is only 12.9 nAT -1 ,Figure4.3. It is worth noting that relatively strong electric currents of isomer D can be induced for any arbitrary orientation of the nanocluster relative to the magnetic field; i.e.,c o n t r a r yt ob e n z e n e ,i ti sn ol o n g e rn e c e s s a r yt o have a magnetic field perpendicular to a given molecular plane. Chapter 5

Applied Experimental Techniques:

An Overview

Experimental studies were performed to measure the optical absorption of plasma-born aromatic silicon nanoclusters in situ,toin v estigatetheirpermanen tdipolemomen tc haracterandtostudy the structure of deposited nanoclusters. We start the experimental section by providing a short overview of the capacitively coupled discharge and cavity enhanced absorption spectroscopy in order to guide the reader through the different topics treated in the following chapters.

Capacitively coupled discharge

A plasma is a partially ionized gas with a combination of free electrons, ions, radicals, and neutral species. Plasmas are often called the fourth state of matter. In nature, a plasma is formed when gas is exposed to high temperatures or subjected to high energy radiation such that the atoms decompose into freely moving charged particles. [START_REF] Lieberman | Plasma discharges and materials processing[END_REF]H o w e v e r ,t oc r e a t ea n ds u s t a i nap l a s m ai n the laboratory, energy input is required. Generally, this energy is transferred via coupling of an external electromagnetic field to the gas. Different coupling methods generate plasmas with different characteristics.

One of the widely used coupling methods is the one for capacitively coupled plasmas (CCP). A CCP system consists of two parallel plate electrodes separated by a small distance and embedded in a vacuum chamber as illustrated in Figure 5.1. The gap between the electrodes is filled with gas. The vacuum chamber and one of the electrodes are electrically grounded while the other electrode is powered via an impedance matching network. The energy is supplied as av o l t a g ed i fference between the cathode and the grounded plate in a time-varying manner.

Most commonly, a RF voltage is applied to the plates with an operation frequency of 13.56

MHz. 1 Electrons play a significant role in RF discharges. The electrical energy supplied by the 1 13.56 MHz is a band reserved for the Industrial Scientific and Medical Frequencies (ISM) which does not interfere with radio and television broadcasting bands. The authorities in charge of the management frequency ranges are the "Authority of Regulation of Telecommunications"and "European Radiocommunications Office".

The selection of the gas temperature is, thus, crucial in our experiments for the following reasons:

(i)w ew a n tt og e n e r a t es i l i c o nn a n o c l u s t e r so f⇠1nm only, and (ii)w ew a n tt oe n s u r et h a ta l l the optical absorption measurements are done before the agglomeration phase begins.

5.1.4

After-glow forces acting on particles When a plasma is switched off, particles will no longer remain trapped in the discharge by the confining electric fields. Different forces will act on the particles affecting their dynamics: [START_REF] Mikikian | Dusty plasma: synthesis, structure and dynamics of a dust cloud in a plasma[END_REF] • Gravitational force

F g = 4 3 ⇡r 3 p ⇢g (5.1)
where ⇢ and r p are the mass density and radius of a particle, respectively.

• Neutral drag force (Gas flow)

F nd = - 4 3 ⇡r 2 p m n n n v th ⇣ 1+↵ ⇡ 8
⌘ (v p -v n ) (5.2)
where m n , n n , v th = p 8k B T n /⇡m n are the mass, density and thermal speed of neutrals (T n being the gas temperature), respectively; v p is the particle velocity; v n is the neutral mean speed; and ↵ is the accommodation coefficient.

• Thermophoresis forces pushing particles away from hot surfaces

F T = - 32 15 r 2 p k tr v th rT (5.3)
where k tr is the translational part of the thermal conductivity and rT is the temperature gradient.

Optical Absorption Spectroscopy

When light is transmitted through a medium (solid, liquid, or gas), it experiences attenuation due to absorption and scattering; the former being the dominant loss contribution in many cases. This is the basic principle behind most absorption spectroscopy techniques: illuminate a sample with a light source of known intensity and measure the amount of light transmitted through the sample to a detector on the far side. The difference in intensity between the incident light intensity, I 0 ,a n dt r a n s m i t t e dl i g h ti n t e n s i t y ,I( x, λ), yields the absorbed light, ∆I =I 0 -I( x, λ), which Beer's law relates to the absorbance of the sample, ↵ (λ), as: [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF] I(x, λ)=I 0 e -↵(λ)x (5.4)

where λ is the wavelength and x is the path length. The absorption coefficient ↵ (λ)isrelatedto the molecular concentration of the target sample, C,andtheextinctioncoefficient, ✏ (λ), namely,

↵ (λ)=C ✏ (λ).
When employing absorption spectroscopy, it is important to ensure sensitivity.S e n s i t i v i t yc a nb e quantified as a minimum detectable absorption loss, i.e. the normalized standard deviation of the smallest detectable change in absorption. The minimum detectable absorption is reached when the transmitted light intensity change due to absorption is smaller than the intensity fluctuations inherent to the light source. Equation (5.4) shows that the absorption is intensity dependent and its sensitivity linearly depends on the optical path length. This is often the reason to use optical cavities with high reflective mirrors inside them; i.e. the cavity enhanced absorption method. This method enlarges the path length of the light, increasing the possibility of matterlight interaction and thus the measured signal. A variety of cavity enhanced absorption methods are used in absorption spectroscopy, as for instance, integrated cavity output spectroscopy, incoherent broadband cavity enhanced absorption spectroscopy (IBBCEAS), continuous wave cavity ring-down spectroscopy, and so on. All of the above mentioned methods utilize light reflection by mirrors to enhance the optical path length and increase the sensitivity while maintaining a small resonator size.

Incoherent broadband cavity enhanced absorption spectroscopy

A basic IBBCEAS setup consists of an incoherent light source (e.g. aX e n o na r cl a m p ,al i g h t emitting diode, or a supercontinuum source2 ), collimation optics, an optical cavity, the absorber of interest, and a detector.

IBBCEAS measurement principle

Consider an optically stable cavity continuously excited with incoherent light of intensity I in (Figure 5.5). Apart from the losses (1-R)duetoimperfectmirrorreflectivities,itisassumedthat the cavity is subject to additional losses (1-L) per pass. The intensity of the light transmitted by the cavity, I,c a nt h e nb ed e s c r i b e db yt h es u p e r p o s i t i o no ft h el i g h ta f t e ra nodd number of passes, n, leading to the following infinite sum [START_REF] Fiedler | Incoherent broad-band cavity-enhanced absorption spectroscopy[END_REF]:

I = I in (1 -R) 2 (1 -L) 1 X n=0 R 2n (1 -L) 2n (5.5)
Since R<1a n dL<1, the geometric series converges and Eq. ( 5.5) can be written in the form

I = I in (1 -R) 2 (1 -L) 1 -R 2 (1 -L) 2 (5.6)
For an empty resonator with L=0, the time integrated transmitted intensity I 0 is given by

I 0 = I in 1 -R 1+R (5.7)
Assuming the losses per pass to be solely due to Lambert-Beer attenuation and after applying some approximations 4 ,t h ea b s o r p t i o nc o e fficient, ✏, can finally be expressed by the following equation:

✏(λ)= (1 -R(λ)) d  I 0 (λ) I(λ) -1 (5.8)
where the absorption coefficient ✏(λ)[ c m -1 ]i ss o m e t i m e sr e f e r r e dt oa se x t i n c t i o nc oe fficient as one cannot distinguish between absorption and scattering losses; I 0 (λ) and I (λ)aretheintensity of light transmitted by the cavity without and with as a m p l ei nt h ec a v i t y ,r e s p e c t i v e l y .

The reflectivity of the mirrors, R(λ), are accurately determined. This is usually accomplished by measuring the reflectivity as a function of wavelength using known concentrations of a calibration sample.

Chapter 6

Performed Experiments and

Diagnostics

Due to the expectedly broad absorption features of silicon nanoclusters in the visible wavelength region of interest, and the potentially low concentration of particles in the sample region of the reactor, IBBCEAS is an optimum technique to detect the extinction in the chamber.

Experimental setup

In our experiment, a CCP reactor for the production of hydrogenated silicon nanoclusters was combined with a cavity enhanced absorption setup as a diagnostic tool. The components of the setup are shown schematically in Figure 6.1. The general experimental conditions and procedures are described below.

Plasma reactor

Inside a ⇠ 0.02 m 3 vacuum chamber, an RF (13.56 MHz) CCP discharge was generated (Cesar 133 RF power supply) at low pressure between a grounded and a powered electrode. The circular electrode surfaces had a diameter of 7 cm and were separated by 4 cm. The shower-head shape of the powered electrode allowed the injection of a gas mixture of silane (2%)i na r g o n( 9 8 %)

directly between the electrodes, in order to produce hydrogenated silicon nanoclusters in a pulsed discharge. The flow rate of the SiH 4 /Ar mixture was controlled by a mass flow controller (MFC-Bronkhorst F-201DV, Low-∆P-Flow), which was operated at a typical flow rate of 10 sccm.

A primary pump, which was connected to the chamber through an electronically controlled butterfly valve, was used to continuously remove the gas mixture from the chamber and maintain of 6 seconds between plasma discharge trigger pulses, which were generated by a specifically designed trigger board. A more detailed description of the trigger system is given below (also cf. Figure 6.2).

We heated the gas to typically 150 • C. At such a gas temperature, we ensure that the formation and accumulation of "first-generation" nanoparticles takes about 6 seconds before they start agglomerating. The discharge on-time was generally set to 2 s1 .B e f o r ee x p e r i m e n t sw e r ep e rformed, the chamber was fully pumped by a turbo pump overnight to a base pressure of ⇠ 10 -6 mbar while maintaining the chamber at 150 • C. The secondary pumping was done to clean the chamber and remove potential impurities.

Cavity-enhanced absorption setup

The light source used in the absorption measurements was a spatially coherent supercontinuum light (Fianium SC450) with a total output power of 5 W. It delivered pulses with a duration of ⇠5 ps and was operated at a repetition rate of 50 MHz. Its emission covered a broad spectral range from ⇠450 to 1800 nm; therefore appropriate combinations of long-and short-pass filters and/or bandpass filters were used to select an appropriate spectral region where the cavity mirrors are highly reflecting (∆λ ⇡ ±30 nm around the central wavelength). The light was coupled on-axis into the optical cavity for which the mirrors were connected to the reactor by vacuum flanges (see Figure 6.1). The cavity was formed by two plano-concave high-reflectivity (HR) mirrors (diameter 25 mm, radius of curvature -2 m) with a manufacturer-specified reflectivity of R > 0.9999. The separation of the cavity mirrors was 152 cm. This cavity length was chosen to maximize the distance between the discharge region and the mirrors in order to protect the mirrors from degradation due to interaction with highly reactive species formed in the plasma.

Additionally, the mirrors were purged with a small argon flow at a rate of typically 10 sccm. Whenever the mirrors were not in use, they were isolated from the plasma reactor through large diameter gate valves.

Two different sets of mirrors were used, with HR regions centred at 610 and 700 nm, respectively.

The light transmitted through the cavity was focused with an achromatic lens (f = 3 cm) onto the 1 mm entrance aperture of a circular-to-rectangular multi-core fiber bundle. The fiber bundle was connected to the entrance slit of a monochromator (Shamrock 303i), which was used to disperse the light with a blazed diffraction grating of 600 lines/mm (corresponding to a spectral resolution of 0.65 nm). The light exiting the monochromator was detected with a gated iCCD (Andor iStar, DH740-18 mm). An optical filter after the cavity was used to protect the iCCD from exposure to excessive near infrared radiation.

Mirror reflectivity calibration

The wavelength-dependence of the reflectivity of the mirrors, R(λ)i nE q . ( 5.8), was established independently in the University College Cork laboratory using the method described in Ref [START_REF] Washenfelder | Measurement of glyoxal using an incoherent broadband cavity enhanced absorption spectrometer[END_REF]. R(λ)wasdeterminedbasedontheRayleighscatteringcross-sections,↵ Ray ,ofheliumandcarbon dioxide gases as described in Eq. (6.1):

1 -R(λ) d = ( ICO 2
IHe )↵ CO2 Ray (λ) -↵ He Ray (λ) 1 -ICO 2 IHe (6.1) I CO2 and I He represent the light intensities through a cavity filled to 1 atmosphere with CO 2 and filled to the same pressure with helium gas, respectively. The main issue with the reflectivity measurements is that they are alignment-sensitive. It was very challenging to replicate the exact alignment used during the experiments of the extinction measurements carried out in the LPICM.

Experiment synchronization

A trigger board synchronized the plasma discharge with the gated detector. As shown schematically in Figure 6.2, the trigger board generated three trigger pulses. The initial trigger pulse determined the start and the duration of the plasma discharge (indicated by blue arrows in Figure 6.2). The falling edge of the initial pulse triggered two subsequent trigger pulses for the iCCD (indicated by green arrows in Figure 6.2). The first of these two pulses is generated immediately on the falling edge of the "plasma pulse" and was used to record the cavity transmission with the sample (I (λ) in Eq. (5.8)) shortly after the end of the discharge plasma, the second one was always generated 3.5 s later and triggered the measurement of the cavity transmission without the sample (I 0 (λ) in Eq. (5.8)), i.e. without silicon nanoclusters. The two pulses for the detector triggered a digital delay and a gate generator of the iCCD. The delay time and gate width (=integration time) could be freely chosen (vertical dashed lines in Figure 6.2), but are the same for the measurements of I and I 0 . The gate width was 10 ms in all cases. The gate delay, however, was varied for different experiments between ⇠ 1and⇠ 200 ms. The time difference of 3.5 s between the two pulses for the iCCD (green pulses) was sufficiently long to ensure that any residual nanoclusters were pumped out of the reactor between recording the sample (I)a n dt h e background (I 0 )t r a n s m i s s i o n . T y p i c a ls p e c t r aw e r ea v e r a g e do v e r1 0 0t o1 0 0 0p l a s m ap u l s e s . We stress here that our technique in measuring I and I 0 alternatingly between plasma pulses, rather than subsequently at the end of the total number of pulses, has the advantage that systematic errors, such as light source fluctuations, variations in residual optical losses due to scattering, very gradual misalignment effects or mirror drifts, and slow and small pressure variations, are effectively reduced due to the short time separation between the respective measurements of I and I 0 .

Plasma Plasma

Chapter 7

Experimental Results and Discussions

This chapter covers our experimental results concerning the aromatic silicon nanoclusters. In the first place, we present the extinction or absorption spectra of plasma-born aromatic silicon nanoclusters in the visible region using the experimental setup and synchronization presented in Chapter 6. Second, we exploit the absorption measurements performed in conjunction with an applied DC electric field (to be described in details hereafter) and provide a proof that those nanoclusters possess a permanent dipole moment. An estimation of the dipole moment is made using the absorption measurements combined with a theoretical study. Finally, we present a structural characterization of deposited aromatic silicon nanoclusters based on high-resolution transmission electron microscopy measurements.

Extinction measurements

Investigation of the spectral range 575-665 nm

The following measurements were carried out using a set of mirrors centered at 610 nm covering a spectral range of 575 -665 nm. In 4.0x10 -4

6.0x10 -4

8.0x10 -4

1.0x10 -3

1.2x10 -3 1.4x10 -3 1.6x10 -3 1.8x10 -3 2.0x10 -3 ε (cm -1
) wavelength (nm)

Investigation of the spectral range 655-745 nm

In Figure 7.2,w es h o wt h em e a s u r e de x t i n c t i o nu s i n gt h es e to fm i r r o r sc e n t e r e da t7 0 0n m covering a spectral range of 655-745 nm. The measuring conditions were similar to the measurements done for the previous spectral range. Although the measured extinction is essentially flat in the high reflectivity range of the mirrors, the spectrum in Figure 7.2 looks coherent with respect to Figure 7.1,t h a ti s ,t h es i g n a li nt h el a t t e rh a sam a x i m u ma r o u n d6 4 5n ma n dt h e n decreases until 670 nm, while the signal in the former spectrum decreases from its beginning at about 655 nm until 700 nm and then increases again.

Taking a closer look at the calculated absorption spectra of Si 19 H 12 nanoclusters at 0K, Figure 4.8, we notice a non-zero oscillator strength at about 650 nm, which is very close to the measured absorption peak at about 645 nm (Figure 7.1). Moreover, we find non-zero oscillator strengths at about 674 nm and 722 nm (see Figure 4.8-b). At these wavelengths, the measured spectrum using the 700 nm mirrors (Figure 7.2) shows some absorption features.

We would like to note that unlike the absorption measurements of single molecules where the absorption peaks always look sharp and well-defined, we do expect that the absorption peak of as i n g l eo fn a n oc l u s t e rf o r m e di nap l a s m ar e a c t o rt ob eb r o a dd u et ot h et h e r m a lm o v e m e n to f the cluster atoms under the experimental conditions. In addition, considering that our plasma conditions will not lead to the creation of one single isomer of Si 19 H 12 , but rather to a wide variety of similar silicon nanoclusters, the effective absorption over a relatively broad spectral region can readily be understood. 2.0x10 -6

3.0x10 -6

4.0x10 -6

5.0x10 -6

ε (cm -1
) wavelength (nm)

Effect of the gate delay time

In Figure 7.3, we show a set of extinction measurements done with a range of gate delay times, between 1 ms and 250 ms, using the 610 nm mirrors. The measurements were performed with ap l a s m ao n -t i m eo f2s ,g a sp r e s s u r ea n dt e m p e r a t u r eo f0 . 1 6 5m b a ra n d1 5 0 • C, respectively, a gate width of 10 ms, and a gas mixture flow of 10 sccm. We notice from Figure 7.3 that the intensity of the measured extinction varies with the delay time. The extinction increases with the increase in delay time until 150 ms and then rapidly decreases. This can be explained as follows: when we turn off the plasma, the absorbing nanoclusters, mainly located above the detection beam with the highest density near the shower head, move down swept by the gas flow.

They cross the detection beam after a few milliseconds, with a maximum density of nanoclusters reaching the light beam at 150 ms. Waiting for a longer time (200 ms), we find a drop in the extinction indicating that most of the absorbing nanoclusters had already crossed.

Effect of an applied DC electric field

Previous theoretical studies have shown that aromatic silicon nanoclusters possess a strong permanent dipole moment (see Section 2.1). In order to confirm and to explore such a characteristic quantitatively, we will examine the effect of an applied DC electric field on the measured absorption spectra. But first, what role does an electric field play on polar molecules? The next two subsections will clarify it.

Permanent dipole moment

In general, molecules are electrically neutral, but they can have an asymmetric charge distribution, as for example the water molecule. In other words, the centers of positive charge density and negative charge density do not coincide. [72] This charge separation leads to an electrical dipole moment and the molecule, then, is called a polar molecule. Classically, the dipole of a molecule can be viewed as one point charge +q and one negative point charge -q separated by ad i s t a n c er (see Figure 7. 4.0x10 -4

6.0x10 -4

8.0x10 -4

1.0x10 -3

1.2x10 -3 1.4x10 -3 1.6x10 -3 1.8x10 -3 2.0x10 -3 0V 120V ε (cm -1
) wavelength (nm) The extinction is measured using the 610 nm mirrors and an applied DC bias of 120V. The plasma duration is 2 s and the gate width and delay are 10 ms and 100 ms, respectively. The gas mixture flow is 10 sccm and the pressure and temperature of the gas are 0.115 mbar and 150 Fractional Difference wavelength (nm)

How does the extinction change as a function of the applied DC bias?

We have performed a set of extinction measurements with and without an applied electric field for a range of DC biases (25V, 100V, 160V, and 240V) under the same experimental conditions indicated above. Figure 7.9 shows the extinctions measured with an applied bias compared to that without any bias for each experiment. In all our measurements, we find a decrease in the absorption signal in the presence of an applied electric field. However, the decrease of the absorption depends on the strength of the applied field and follows a certain trend. In order to clarify this dependence, we plot in Figure 7.10 the percentage of the reduction in the absorption

( ✏0V -✏DC ✏0V ⇥100
) as a function of the DC bias. Here, we notice that at 25V and 100V, there is only a slight decrease in the absorption. However, we find that the amount of reduction starts to increase linearly with higher DC bias to become about 10.7% and 19% at 160V and 240V, respectively. How can we explain such a behavior? Can it be helpful in providing a quantitative estimation of the permanent dipole moment of aromatic silicon nanoclusters?

DC electric field simulations

In order to better understand the inhomogeneity of the electric field in the reactor chamber, simulations were the best way to give us a full demonstration. The simulations were done, thanks to François Silva, using the finite element methods with a 3D electrostatic solver. First, we designed our reactor chamber (see Figure 7.11 for a cross-sectional view) taking into account its exact configuration and dimensions. Thick blue lines marked in In the following, we track the behavior of the nanoclusters, under the influence of the dipole forces after we switch-off the plasma. In order to simplify our analysis, we separate between the dipole forces acting on the nanoclusters in the z-direction and those in the plane of the light beam. Since the reactor chamber has a perfect cylindrical configuration, we perform our analysis

Dipole forces acting on the nanoclusters

Along the z-axes:

The gradient of the z-component of the electric field (E z ), along say L 1 in Figure (7.11), gives the dipole force acting on the nanocluster in the z-direction. force and the thermophoretic force, can be neglected in our case due to the very small size of the silicon nanoclusters and the absence of a significant temperature gradient in our reactor, respectively. Therefore, we conclude that the movement of the nanoclusters in the z-direction is mostly determined by the neutral drag force. Consequently, nanoclusters, initially located in the same horizontal plane, will reach the light beam at the same time. Along the polar axes:

At this point, we would like to demonstrate how the inhomogeneity of the electric field influences the movement of the nanoclusters and, hence, their distribution in the plane of the light beam.

In Figure 7.15,w ep l o tt h eg r a d i e n to fE r along the polar axis with ✓=0 for a range of potentials (25V, 100V, 160V and 240V), which is similar to all rays with any theta in the polar system due to cylindrical symmetry (see Figure 7.16 for a 3D representation). One can notice from Figure 7.15 that the gradients of E r ,t h a tc o r r e s p o n dt ot h ed i p o l ef o r c e s ,a r ep o s i t i v ei nt h ec e n t e r of the reactor and progressively increase until about 24 mm from the center. After that, the forces show a sudden drop and become negative around 30 mm. Then, they start to increase (with a negative sign) until around 36 mm (i.e. parallel to the edge of the electrode). Finally, the forces decrease again, keeping their negative sign, reaching the walls of the reactor. This means that the dipole forces act on the nanoclusters in such a way that, at some instants, they push them toward the rector walls (positive forces) and, at other instants, toward the center of the reactor (negative forces). Generally speaking, while the nanoclusters are being drifted from the top electrode toward the detection beam by the neutral drag force, they also spread out under the influence of those dipole forces. Consequently, the nanoclusters will change their planar distribution with respect to their initial distribution depending on the applied bias.

In our experiments, we have measured the extinction (or the absorption), as follows:

✏(λ)= X i σ i Z d 0 n i (x)dx = 1 -R d ✓ I 0 I -1 ◆ (7.3)
where the sum i is over all relevant species, σ (cm 2 particles -1 )isthecross-section,andn is the number density (particles cm -3 ). Therefore, equation 7.3 indicates that the measured absorption is proportional to the density of the nanoclusters in the volume of the light beam.

Distribution of the nanoclusters from trajectory simulations

While the nanoclusters are moving toward the detection beam in the presence of the applied inhomogeneous electric field, two phenomena are happening at the same time:

• The nanoclusters spread out under the influence of the dipole forces establishing a new distribution in the plane.

• This plane is shifted down to the plane of the detection beam.

In order to gain a deeper insight into the densities of the nanoclusters in the plane of the detection beam, we follow their trajectories from their initial positions in a region below the top electrode until they reach the light beam, where we obtain their new distribution.

Trajectories simulations

The simulations of the nanocluster trajectories were performed as follows:

• We start with an equal distribution (or density) of particles in each 1 mm ring section in a plane that is 5 mm below the top electrode (see Figure 7.17 for an illustration). The mass of the particle corresponds to the mass of a Si 19 H 12 nanocluster (546 amu).

• Using Newton's equations of motion, we follow the trajectory of the particles under the influence of the dipole forces (calculated from the gradient of E r ) with a time step of 1 ms until they reach the plane of the detection beam. Thus, we obtain the final position of each particle on that plane.

• From the numb er of particles in each ring section, we find the new densities and, hence, the new distribution of particles.

proof that aromatic silicon nanoclusters with a size of ⇠1nm possess a permanent dipole moment between 2.0 and 2.5 Debye. This result is in perfect agreement with previous estimations of the dipole moment of the Si 19 H 12 isomers, based on ab initio calculations.[47] Interestingly, silicon nanoparticles (< 5 nm) with ordered structures, formed in silane-based RF plasmas, having crystalline lattices other than the diamond-cubic are reported in the literature, as for instance the cubic (bcc and bc8) structures [START_REF] Viera | Atomic structure of the nanocrystalline si particles appearing in nanostructured si thin films produced in low-temperature radiofrequency plasmas[END_REF]orthehexagonal-close-packedstructure. [START_REF] Fontcuberta I Morral | In situ investigation of polymorphous silicon deposition[END_REF] Although those phases have mainly been reported as possible high-pressure phases of silicon, they have also shown to exist in nanostructured silicon obtained from a RF plasma. There can be several explanations for the formation of different phases, although it is not well understood yet: first, it is accepted that the diamond structure is not the most stable structure for small nanoparticles. Veprek et al. [START_REF] Veprek | A thermodynamic criterion of the crystalline-to-amorphous transition in silicon[END_REF]reportedondiamond-structurednanocrystalsformedindifferent chemistries and claimed that 3 nm represents the lower limit size for their stability. Several factors related to the process of particle formation by RF plasma such as the process of temperature, pressure, and the inert gas dilution, may play an important role in the construction of the particle structure. [START_REF] Viera | Atomic structure of the nanocrystalline si particles appearing in nanostructured si thin films produced in low-temperature radiofrequency plasmas[END_REF]O nt h eo t h e rh a n d ,t h ef o r m a t i o no fa nu n c o m m o np h a s eo fs i l i c o nc a n be attributed to the "chemistry with a hammer" mechanism. This phenomenon explains why reactions that are normally observed only at extremely high reactant pressures, can be carried out in low-pressure, ultra-high vacuum conditions where the collision energy of the incident particle on a surface is a key factor in determining the observed chemistry of unexpected reactions. [START_REF] Ceyer | New mechanisms for chemistry at surfaces[END_REF] If a charged nanocluster escapes from our plasma bulk, it will collide with high impact energy on the hard surface, such that extreme conditions of pressure and temperature are experienced by the atoms inside the cluster. At the same time, the atoms cannot easily "escape" since they are sterically confined by the surrounding cluster atoms. Consequently, unusual chemical reactions between the concerned atoms can take place that otherwise would only become possible at extremely high pressures and temperatures. [START_REF] Vach | Nonadiabatic ladder climbing during molecular collisions[END_REF][START_REF] Nguyen | Mechanically-induced generation of highly reactive excited-state oxygen molecules in clustersurface scattering[END_REF] d= 0.20 nm d= 0.21 nm d= 0.21 nm

Conclusion

An experiment was designed, built and performed to study the absorption of aromatic silicon nanoclusters in the visible region, in situ, in a plasma reactor using IBBCEAS. A plasma reactor for the generation of Si m H n nanoclusters was combined with a cavity enhanced absorption setup as a diagnostic tool. The plasma parameters were optimized such that only ⇠1nm particles were formed. We synchronized the absorption measurements with the nanocluster production and we measured the transmitted light intensities, with and without the absorbing species, alternatingly between plasma pulses. Using a spatially coherent supercontinuum light source, several filters, and two sets of highly reflecting mirrors, covering two spectral ranges between 575-665 nm and between 655-745 nm, respectively, we found an extinction peak in the visible around 645 nm while standard tetrahedral silicon nanoclusters of this size can only absorb in the ultraviolet.

The experiments have also demonstrated that our aromatic silicon nanoclusters possess a permanent dipole moment. This property was deduced from the decrease in the measured extinction in the presence of an applied DC bias. The decrease in the absorption was a consequence of the inhomogeneity of the electric field in the reactor chamber, so that the dipole forces influenced the movement of nanoclusters. Moreover, measurements have shown that the change in the absorption signal has a unique dependence on the strength of the applied electric field. Simulations of the electric field helped us to understand the behavior of the nanoclusters under the influence of the dipole forces. We obtained a coherence between the simulations of the density of nanoclusters in the plane of the detection beam and the measured extinction for different applied bias voltages. Now, we have a compelling evidence that aromatic silicon nanoclusters possess a permanent dipole moment between 2.0 and 2.5 Debye!

We were eager to p erform additional studies on the nano clusters but, after their dep osition. To this end, we performed HRTEM measurements on samples with different deposition times, but with similar optimized plasma conditions. Our HRTEM analysis on selected areas have shown the presence of some crystalline nanopartcles along with amorphous ones. We found that some of the 2.0 to 3.5 nm crystalline nanoparticles attain a primitive hexagonal structure. This indicates that exotic forms of silicon can be produced by a RF plasma and not only under extreme conditions.

Chapter 8

Introduction

In 1959, Richard P. Feynman gave a lecture titled "There's plenty of room at the bottom.

An invitation to enter a new field of physics," at the American Physical Society meeting in Caltech. [START_REF] Feynman | Plenty of Room at the Bottom[END_REF] There, Feynman raised several questions, such as What could we do with layered structures with just the right layers? What would the properties of materials be if we could really arrange the atoms the way we want them? His inspiring talk motivated generations of scientists to discover new low-dimensional materials. In 1966, Frindt isolated a single crystal of MoS 2 ,only several molecular layers thick, by using the scotch tape method. [START_REF] Frindt | Single crystals of MoS2 several molecular layers thick[END_REF]Materialssciencehadamajor scientific breakthrough in 2004 when Konstantin Novoselov, Andre Geim, and their colleagues isolated a single layer of carbon, graphene, again through the scotch tape exfoliation of graphite and measured its electronic transport properties. [START_REF] Novoselov | Electric field effect in atomically thin carbon films[END_REF] The importance of this achievement was sealed in 2010, when the Nobel prize was awarded to both researchers. The freestanding 2D nature of graphene gave rise to some of its exotic and favorable aspects that are lacking in its bulk counterpart, graphite. Since then, the 2D materials library has been greatly expanding to include transition metal chalcogenides (MoS

Classification of materials based on dimensions

The shrinkage of the dimensions of materials to the nanometer range has an enormous impact and industrial potential deriving from the fact that new and fascinating properties emerge at the nanoscale due to quantum effects. Nanostructured materials are low-dimensional materials comprising of building units of a nanoscale size (<100 nm) at least in one direction. Hundreds of novel nanostructured materials have been obtained in the past two decades, and the need in their classification is ripened. [START_REF] Gleiter | Nanostructured materials: Basic concepts and microstructure[END_REF][START_REF] Pokropivny | Classification of nanostructures by dimensionality and concept of surface forms engineering in nanomaterial science[END_REF] The classification is based on the number of dimensions which are not confined to the nanoscale range, such as zero-dimensional (0D) quantum dots, one-dimensional (1D) nanowires, and 2D layers or sheets. Those different nanostructured materials exhibit dimension-tunable electronic and optical properties, which are important attributes in nano-device applications. Therefore, dimensionality is a major feature that discriminates various types of nanostructures; not only it defines the atomic structure of a material, but it also determines its properties to a significant degree. Consequently, the same chemical element or compound can exhibit different properties in different dimensions.

Graphene: The father of 2D materials

During the first decades of the twentieth century, the existence of 2D materials was a highly debated issue in the physics community. According to Landau and Pierls, 2D materials are thermodynamically unstable at any finite temperature due to thermal lattice fluctuations which would most likely decompose the crystal. [START_REF] Landau | Zur theorie der phasenumwandlungen II[END_REF][START_REF] Peierls | Quelques proprietes typiques des corpses solides[END_REF]L a t e ro n ,t h i sa r g u m e n tw a se x p a n d e db y

Mermin [START_REF] Mermin | Crystalline order in two dimensions[END_REF]a n di tw a ss u p p o r t e db ye x p e r i m e n t a lo b s e r v a t i o n st h a tt h em e l t i n gt e m p e r a t u r e of thin films rapidly reduces with decreasing film thickness. [START_REF] Venables | Nucleation and growth of thin films[END_REF]O n l yt h es u c c e s s f u li s o l a t i o no f graphene from its mother crystal graphite in 2004 ruled out the doubts on the possible stability of 2D materials.

Graphene is a single, tightly packed layer of carbon atoms that are bonded together in a hexagonal honeycomb lattice, to lead to electron mobilities a hundred times larger for graphene [START_REF] Bolotin | Ultrahigh electron mobility in suspended graphene[END_REF] with a potential for much faster electronics than with conventional silicon. However, the main drawback in graphene is its lack of a controllable band gap which makes it not suitable for the use as an electronic switch. Therefore, to use graphene in future nanoelectronic devices, a band gap will need to be engineered into it without compromising any of its other properties. Doping, functionalization, and substrate-induced band gap opening are among the most common techniques and methods reported. [START_REF] Zhang | Opening the band gap of graphene through silicon doping for the improved performance of graphene GaAs heterojunction solar cells[END_REF][START_REF] Zhang | Aryl functionalization as a route to band gap engineering in single layer graphene devices[END_REF][START_REF] Zhou | Substrate-induced bandgap opening in epitaxial graphene[END_REF] For hybridization to take place, the 2s orbital must combine with the 2p x ,t h e2 p y ,a n dt h e 2p z orbitals to form four identical sp 3 hybrid orbitals (Figure 8.2a), having 25% of s-character and 75% of p-character. Since the carbon atom has 4 valence electrons, each electron fills one sp 3 hybrid orbital. The carbon which has undergone hybridization is ready to interact with four orbitals and adopt a tetrahedral geometry with bond angles of 109.5 • . This is the case of diamond which makes it extremely rigid and a strong crystal. In the case that a carbon atom forms three bonds, one 2s orbital and only two 2p orbitals take part in the hybridization process. This hybridization forms three energetically equivalent sp 2 hybrid orbitals, which adopt a trigonal planar geometry with bond angles of 120 Therefore, the key to making graphene-like silicon or germanium materials is to take a broader view of two-dimensionality, that is to say, one is not strictly looking for the elusive sp 2 bonding in silicon or germanium. [START_REF] Lew Yan Voon | Is silicene the next graphene?[END_REF] 8.3 Silicene and germanene: novel 2D materials

(E 3p -E 3s =5 . 6 6e V )a n dC( E 2p -E 2s =1 0 . 6 0e V ) ) .

The crystal structure

Silicene and germanene are novel 2D allotropes of silicon and germanium, respectively. They have a honeycomb structure that resembles that of graphene. However, the resemblance is not perfect; silicene and germanene are buckled while graphene is perfectly planar. The buckling parameter is defined as the vertical distance between the two planes of Si or Ge atoms in the same layer (see bottom panel in Figure 8.

Ab r i e fh i s t o r y

Research on silicene from the experimental and theoretical communities got a real boost after its experimental synthesis on substrates only in 2012, [START_REF] Vogt | Silicene: Compelling experimental evidence for graphenelike two-dimensional silicon[END_REF] leaving silicene with a relatively short history. Despite that, both silicon and germanium materials were first investigated in 1994 as an infinite 2D Si and Ge corrogated stage, a decade before the emergence of graphene, by Takeda and Shiraishi using first-principles total-energy calculations. [START_REF] Takeda | Theoretical possibility of stage corrugation in Si and Ge analogs of graphite[END_REF] In their work, the authors have shown the preference of Si and Ge to form a buckled structure, unlike C atoms in graphene which prefer to remain flat. However, there work was ignored for almost a decade because there was a long standing belief that materials with a monolayer thickness were thermodynamically unstable and could not exist. [START_REF] Landau | Zur theorie der phasenumwandlungen II[END_REF][START_REF] Peierls | Quelques proprietes typiques des corpses solides[END_REF].

Even before the term "silicene" was given, 1D structures of silicon were first synthesized on a Ag ies appeared: investigations on the properties of hydrogenated silicene and germanene, [START_REF] Houssa | Electronic properties of hydrogenated silicene and germanene[END_REF]o n the possibility to tune the band gap of silicene and germanene by the application of an electric field perpendicular to the sheet, [START_REF] Ni | Tunable bandgap in silicene and germanene[END_REF]a n do nt h eq u a n t u ms p i nH a l le ffect in silicene and germanene. [START_REF] Liu | Quantum spin hall effect in silicene and two-dimensional germanium[END_REF] The first compelling experimental evidence for the existence of a silicene monolayer was reported by Vogt et al in 2012 by depositing silicon atoms on a Ag (111) substrate. [START_REF] Vogt | Silicene: Compelling experimental evidence for graphenelike two-dimensional silicon[END_REF] In this work, a 3x3 supercell of silicene perfectly matched with the 4x4 supercell of the Ag [START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF] surface and agreed with DFT geometry optimizations. Even though there was a previous attempt to grow silicene on Ag [START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF] of the variation of the experimental conditions. [START_REF] Feng | Evidence of silicene in honeycomb structures of silicon on Ag(111)[END_REF][START_REF] Lin | Structure of silicene grown on Ag(111)[END_REF]N o to n l yo ns i l v e rs u r f a c e s ,s i l i c e n e was also experimentally synthesized on other metallic surfaces such as zirconium diboride thin films [START_REF] Fleurence | Experimental evidence for epitaxial silicene on diboride thin films[END_REF] and on an Ir(111) substrate. [START_REF] Meng | Buckled silicene formation on Ir(111)[END_REF] On the other hand, the first reports on the synthesis of germanene date back to 2014. The first paper, by Li et al., reported the growth of germanene on a Pt (111) substrate, [START_REF] Li | Buckled germanene formation on Pt(111)[END_REF]a n da second one, by Davila et al., [START_REF] Dávila | Germanene: a novel two-dimensional germanium allotrope akin to graphene and silicene[END_REF]s t a t e di t ss y n t h e s i so nA u( 1 1 1 ) . A f t e rs e v e r a ld a y s ,a n o t h e r paper appeared, by Bampoulis et al.,reportingtheformationofgermanenehoneycom bstructure terminating the Ge-Pt clusters. [START_REF] Bamp | Germanene termination of Ge2Pt crystals on Ge(110)[END_REF]F i n a l l y ,i n2 0 1 5 ,D e v i r a zet al. synthesized a continuous germanene layer on an Al [START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF] surface. [START_REF] Derivaz | Continuous germanene layer on Al(111)[END_REF] Silicene and germanene have been almost exclusively grown on metallic substrates. On metals, their 2D Dirac character is, in most cases, either altered or completely destroyed because of the hybridization of the relevant electronic states near the Fermi level with electronic states of the metallic substrates. [START_REF] Cahangirov | Electronic structure of silicene on Ag(111): Strong hybridization effects[END_REF][START_REF] Gori | Origin of Dirac-cone-like features in silicon structures on Ag(111) and Ag(110)[END_REF][START_REF] Guo | Absence and presence of Dirac electrons in silicene on substrates[END_REF] Besides, the epitaxial growth of silicene on Ag substrates has been recently questioned. [START_REF] Prévot | Monitoring Si growth on Ag(111) with scanning tunneling microscopy reveals that silicene structure involves silver atoms[END_REF][START_REF] Satta | Silicon reactivity at the Ag(111) surface[END_REF][START_REF] Ronci | Silicon-induced faceting at the Ag(110) surface[END_REF] It was reported that Si atoms strongly interact with the Ag substrate and penetrate into the first layer of the Ag surface, expelling Ag atoms and inducing reconstructions, faceting, or growing reconstructed islands inside the first layer rather than remaining on top of the substrate. These observations imply that the growth dynamics of an ultrathin silicon layer on any metal substrate is extremely complicated and does not necessarily result into genuine silicene.

Our attempts to grow genuine silicene and germanene

To date, however, reports on the successful growth of germanene and silicene on nonmetallic substrates are rather scarce. A single report on the growth of germanene on MoS 2 [START_REF] Zhang | Structural and electronic properties of germanene on MoS2[END_REF]a ppeared while our work on germanene was still under review. In view of a viable solution to the above-mentioned issues associated with metallic substrates, we use in our work a highly oriented pyrolytic graphite (HOPG) substrate that, due to its sp 2 configuration, approximates well the electronic and structural properties of a fully honeycombed structure of silicon and germanium. Moreover, HOPG is chemically inert, and this is expected to avoid alloy formation, i.e., intermixing of Ge or Si and C atoms which can easily occur on metallic substrates.

Outline of Part II

Chapters 9 and 10 are devoted to our theoretical and experimental studies of silicene and germanene, respectively. The following two chapters are presented in a different way; i.e.,f o re a c h study performed, the method is described at the beginning of the section followed by the corresponding results and discussions. Moreover, we refer the interested reader to Appendices A and B for a theoretical background on molecular dynamics simulations and DFT methods, respectively, and to Appendices E and F for basic concepts in solid state systems and for an overview of the experimental techniques used in the present work, respectively.

Chapter 9

Formation of Silicene Nanosheets on

Graphite

In this chapter, we present and discuss our results on the growth of silicene nanosheets on top of HOPG surfaces based on a combined experimental and theoretical work. We provide strong evidence that 2D alloy-free silicene is formed. However, before we go ahead and discuss the above-mentioned study, let us present our theoretical results we have obtained, in the first place, on the interaction of silicon atoms with one particular metallic substrate: the silver surface.

Theoretical results

Si atoms on a Ag(111) surface

This study aims at the investigation of the interaction of Si atoms with a Ag surface. It was performed in the midst of all the experimental work published on the claimed growth of a silicene sheet on Ag [START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF] surfaces. During this period, no research work argued the growth of silicene on Ag [START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF], except for the debate on its electronic properties. 1 From a theoretical approach, we decided to take a closer look at the interaction of Si atoms on Ag [START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF], but from a dynamical one. This study was crucial due to 1) the lack of molecular dynamics simulations of the silicene growth on metallic surfaces, and 2) the prominence of dynamical studies, not to mention ab initio ones, where one is able to "visualize" the motion of atoms at the atomic scale as chemical reactions occur.

AIMD simulations were performed, at quasi realistic experimental conditions, as follows. We have started with a (8⇥4) four-layer slab2 Ag (111) surface previously heated to 500 K using at h e r m o s t a t . Finally, after 3.6 ps, we clearly see how Ag 1 is released from the surface, Ag 2 sits in its place, and one Si atom quickly "sneaks in" and substitutes Ag 2 . This Si atom binds to the Ag atoms in the top layer and the second layer, marked by a green arrow in Figure 9.1-d. Here we would like to note that such behavior is well reproduced in our simulations. We have modelled another Ag [START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF] surface, a (8⇥8) Ag( 111), and we have obtained exactly the same representative behavior. We continued our AIMD investigations and added 6 more Si atoms to the structure obtained after 4 ps. In addition, we have compared the effect of the interaction of Si monomers, dimers, and trimers on the Ag surface. First, we heated a (8⇥8) three-layer slab Ag (111) surface to 500 K. After several ps of equilibration at 500 K, we placed Si atoms and followed their trajectories for several ps. The mostly reported structure in the literature was the (4⇥4) superstucture, which is also known as (3⇥3) silicene on top of (4⇥4) Ag. This superstructure consists of 18 Si atoms and 16

Ag atoms per Ag layer. In our simulations, we took twice the size of the Ag superstructure. To consider a full coverage or 1 monolayer (ML) of silicene on top of a (8⇥8) Ag surface, we need 36 Si atoms. Here, we studied the case of 2, 4, 6 and 12 Si atoms that correspond to a coverage of ⇠ 0.05 ML, 0.11 ML, 0.2 ML, and 0.3 ML, respectively. We took into account Si monomers, dimers and trimers interacting with the surface. Figure 9.3 shows a plot of the mean square displacement of the top Ag atoms averaged over several MD steps, and calculated for different Si coverages as well as for the different cases of Si. We noticed that a strong displacement of the Ag atoms takes place due to Si monomers and this displacement increases with the increase of the number of Si atoms added. For the dimer and trimer cases, we found a weaker effect on the displacement of the Ag atoms relative to the monomer case. This finding shows that mostly Si monomers are responsible for the modification of the Ag surface and the expelling of Ag atoms.

Our findings demonstrate that the Ag [START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF] surface is strongly modified upon the interaction with Si atoms. Our simulations reveal that Ag atoms are released from the surface and Si atoms become members of the top substrate layer. Those results have contradicted all previously reported ones which always describe the formation of a silicene sheet on top of an unreconstructed Ag surface. Our MD simulations suggest that the formed silicene sheet corresponds to an alloyed structure, that is, a mixture of Si-Ag atoms, and not to a pristine silicene sheet.

Silicene nanosheets on HOPG

Ab initio calculations were performed by using DFT within the generalized-gradient approximation (GGA) as implemented in VASP-5.3.5. [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using aplane-w a v ebasisset[END_REF] The PBE [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] with the van der Waals-D2 [START_REF] Grimme | Semiempirical gga-type density functional constructed with a long-range dispersion correction[END_REF] functional 3 was used to describe the exchange-correlation. The core electrons were described by the projector-augmented wave (PAW) method. [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] The plane-wave basis set was restricted to a cutoff energy of 400 eV.

Silicene/HOPG structure optimizations

The HOPG surface was modelled by a four-layer slab in ABAB stacking of graphene layers within periodic boundary conditions (PBC), keeping the bottom two layers fixed. The vacuum region is as thick as 2.0 nm. The entire system, silicene on top of HOPG, was fully relaxed by a conjugate-gradient method until the forces acting on each atom were less than 0.1 eV/nm. The convergence criterion of total energy for self-consistent field calculations was chosen to be 10 -4

eV. We have used 3⇥3⇥1a n d9 ⇥9⇥1k -p o i n tm e s h e si n c l u d i n gt h eΓ point in order to sample The thermal stability of the silicene monolayer even above RT underlines the essential contribution of the van der Waals forces binding the silicon atoms in a honeycomb pattern to the graphite surface. From our simulations of a perfect silicene sheet on top of HOPG, we did not perceive any 3D silicon cluster formation even at 350 • C. Therefore, we will take a closer look at the details of the growth dynamics below. At this point, however, we can already conclude that a perfect silicene sheet remains stable at such a high temperature of the substrate, supporting the STM measurements (see Figure 9.10 below) where silicene zones were observed in many regions.

Growth of a silicene sheet on HOPG

In order to go deeper in the atomic-scale view of the growth process of the silicene layer on HOPG, we have carried out AIMD simulations under quasi realistic experimental conditions. In the experiments, Si atoms were evaporated at an extremely small deposition rate (0.01 nm/min); that is, a deposited Si atom will have lost all of its initial impact information a long time before the next Si atom will arrive. Consequently, we have started the simulation of the growth mechanism with Si atoms randomly positioned on top of the graphite surface and not by explicitly simulating the deposition process atom by atom. It is necessary to introduce such a model because it is impossible to follow the growth with AIMD methods on a time scale of several minutes. To comply with the experimental conditions, we have started the simulations with a coverage of Si atoms above the graphite surface, which was less than 1 ML, followed the system evolution at RT, and then added Si atoms successively until we reached a 1 ML coverage as in the experiment.

In Figure 9.7, we present the general trends observed during the growth. The initial structure consisted of a 0.75 ML of Si atoms, that is, 18 Si atoms, randomly distributed at about 0.35 nm above the graphite surface with relatively large lateral distances between them, such that the initial interaction between those Si atoms before moving was negligible, as depicted in Figure 9.7-a. Following the trajectories for 900 MD steps (0.9 ps evolution time), we can notice from Figure 9.7-b how Si atoms bind together in a long chain (black ellipse), while other Si atoms move closer to each other to form the first hexagon (red circle). After 1 ps of system evolution from the starting configuration, three Si atoms were added on the bare regions of the graphite surface to have a 0.88 ML coverage. We notice that two hexagons are formed (red circles) along with small polygonal rings after 900 MD steps (Figure 9.7-c). After another 1 ps, three more Si atoms were added to get a 0.96 ML coverage. Figure 9.7-d is a snapshot of the structure resulting from its self-organization after about 1300 MD steps. Nearly all Si atoms in the system now bind in a hexagonal network with one large polygon (black ellipse). Finally, after around 1.4 ps of system evolution, one more Si atom was added in a random location to reach a full coverage (1 ML). Figure 9.7-e is a snapshot of the system trajectory after an additional 633 MD steps showing the honeycomb network of the silicene monolayer with only little defects. The formed silicene sheet lies about 0.4 nm above the HOPG surface, which agrees very well with our STM measurements.

for the formation of a default-free honeycomb structure.

Interestingly, we have observed at an atomistic scale how Si hexagons spontaneously form on top of the HOPG substrate at RT. This growth mechanism only becomes possible due to the van der Waals interactions between silicon and carbon atoms which are strong enough to define a flat growth plane necessary for the silicene sheet formation, but weak enough to leave the silicene properties largely unchanged from the free-standing silicene case; that is, in the present system, one can create a stable silicene monolayer without any covalent bonds to its supporting substrate underneath.

Experimental results

• Preparation of HOPG surface: A HOPG (from GE Advanced Ceramics, USA, 12 mm⇥12 mm⇥1m m )s a m p l ew a su s e da sas u b s t r a t e . Af r e s hs u r f a c eo fg r a p h i t ew a so b t a i n e db y peeling the HOPG substrate with a scotch tape in a nitrogen atmosphere and was then transferred into a UHV chamber.

• Deposition of silicon: High-purity silicon (Sil'tronix ST, ⇢ =1 -1 0Ω•cm) was evaporated from a tungsten basket-shaped crucible located at 200 mm from the substrate. The deposition was done in UHV conditions (base pressure low 10 -10 Torr) at a constant rate of 0.01 nm/min (0.04 ML/ min) monitored by an Inficon quartz balance (properly calibrated for the Si density and carefully positioned close to the sample holder). Deposition was carried out keeping the substrate at RT.

STM measurements

STM imaging was performed using an Omicron-STM system with electrochemically etched tungsten tips. The STM was calibrated by acquiring atomically resolved images of the bare HOPG substrate. All images were acquired in the constant current mode and were unfiltered apart from the rigid plane subtraction.

We present, in 2 a, where a is the hexagonal lattice parameter, by measuring the distance between the spots and the center, we obtain a value of a =0 . 2 4± 0.01 nm that corresponds well to the lattice constant of carbon atoms arranged in a hexagonal pattern for a bare HOPG surface. The inner spots correspond to the atoms arranged with a wider lattice parameter. The ratio between a couple of outer and inner spots is 1.71± 0.01, which is a factor of p 3 of the HOPG hexagonal lattice parameter and corresponds to a lattice parameter of 0.41±0.02 nm. This is strong evidence in favor of the formation of a ( p 3 ⇥ p 3)R30 • Si adlayer. To support this finding, we show in Figure 9.9-a the STM image and in Figure 9.9-b the corresponding Fourier transform of the bare HOPG substrate before Si deposition. It is worth noting that the Fourier transform presents only a single hexagonal pattern, giving a lattice parameter of 0.24 ± 0.01 nm. Interestingly, the Si adlayer hexagonal pattern of Figure 9.8 is different from that of the HOPG (see also the discussion of Figure 9.10 below), in which only one atom out of two of the hexagonal honeycomb structure is observed (note for comparison Figure 9.9-b and the HOPG atomic model in Figure 9.9-c). For HOPG, this is due to the characteristic alternation between filled and empty p z orbitals normal to the surface, while for the Si adlayer, all Si atoms can be simultaneously observed, but at different heights, suggesting that the adlayer is not completely flat as for HOPG and presents a small buckling. ) is perfectly comparable with that obtained with our ab initio calculations (see above). Moreover, following the black horizontal line profile drawn in Figure 9.10-a, we identify the presence of the buckling of the silicon nearest neighbors, which results in about 0.05± 0.01 nm. This accounts well for the theoretical model of silicene in the low buckling configuration (as discussed in the theoretical part) and reported in the literature for free-standing silicene [START_REF] Cahangirov | Two-and one-dimensional honeycomb structures of silicon and germanium[END_REF]and for silicene on graphene. [START_REF] Cai | Stability and electronic properties of two-dimensional silicene and germanene on graphene[END_REF] Besides the well-known growth of Si clusters [START_REF] Scheier | Growth of silicon nanoclusters on graphite[END_REF]onHOPGandon epitaxial graphene on 6HSiC(0001), [START_REF] Sone | Si growth on graphenen surfaces on 6H-SiC(0001) substrates[END_REF]w eha v eobserv edgen uinesiliceneregionsinthepresen t study. The most striking difference with the existing literature on silicene growth on metallic substrates is the absence of any atomic surface reconstruction and/or superstructure. [START_REF] Vogt | Silicene: Compelling experimental evidence for graphenelike two-dimensional silicon[END_REF] Finally, we would like to point out an inherent problem with our recording of the STM images: at present, we have to operate our STM at RT. From our AIMD simulations (see above), we know that Si atoms that are bonded with van der Waals forces to the HOPG substrate experience much larger displacements around their equilibrium position than Si atoms that are bonded covalently to a silver substrate. In order to achieve the same image quality as those previously published on silver, we estimate that we have to cool down our samples to about liquid nitrogen temperatures to reduce the large root-mean-square displacement.

STS measurements

The scanning tunneling spectroscopy (STS) measurements helped us to investigate the local electron density of states near the Fermi level. During the acquisition, the tunneling current was recorded as a function of the applied bias voltage (I vs V), the feedback loop was disabled, and the set-point current, which regulates the tip-sample distance, remained unchanged during the voltage scan. Current-voltage curves were collected over grids of points equally spaced on the scanned sample area, and the I-V spectra were averaged over a set of several curves.

In Figure 9.11, we show the current versus voltage curve obtained on the silicene region. The experimental curve in Figure 9.11-a is an average of many acquisitions, where several currents a quasi-continuous layer with large patchworks (clearest regions indicated by light blue arrows) extending over tens of nanometers, but not completely covering the HOPG substrate (darkest zones as indicated by white arrows) probably because of the large mismatch between the silicon and graphite lattices. In these conditions, we estimate that the percentage of coverage of HOPG is about 80%. Due to the exposure to air, we cannot exclude the formation of a silicon oxide layer, although a low reactivity of the silicene surface with O 2 has recently been reported. [START_REF] Liu | Quantum spin hall effect in silicene and two-dimensional germanium[END_REF] Moreover, we note from the AFM image that the silicene areas form flat surfaces (with a roughness of 0.1-0.2 nm measured by the line profile of the clearest zones of Figure 9.12-d). The white spots are due to the formation of small Si clusters (1 nm high) which decorate the HOPG monatomic steps, as indicated in Figure 9.12-c by white arrows.

Conclusion

This work is devoted to the investigation of a new allotropic form of silicon, namely silicene.

A new form of silicon whose lattice structure resembles that of graphene, yet includes a small buckling of the Si atoms, and whose synthesis on supporting substrates remains a challenging issue.

Our preliminary investigations on the interactions of Si atoms with a silver surface, based on AIMD simulations, have shown a non-negligible interaction between Si and Ag atoms. Our results have shown that Si atoms penetrate into the first layer of the silver surface. They can expel Ag atoms and form bonds with other Si atoms inside the first layer rather than remaining on top of the substrate.

To solve the growth problems, we have performed a combined experimental and theoretical study to investigate the growth of silicene on HOPG. In this study, we directly deposited Si on a chemically inert graphite substrate at RT. Based on AFM, STM, and AIMD simulations, we revealed the growth of silicon nanosheets where the substrate-silicon interaction is minimized.

STM measurements clearly displayed the atomically resolved unit cell and the small buckling of the silicene honeycomb structure. Similar to the C atoms in graphene, each of the Si atoms has three nearest and six second nearest neighbors, thus demonstrating its dominant sp 2 configuration. Our STS investigations confirmed the metallic character of the deposited silicene, in excellent agreement with our band structure calculations that have also exhibited the presence of a Dirac cone.

Chapter 10

Van der Waals Hetero epitaxy of

Germanene Islands on Graphite

The success in synthesizing silicene nanosheets on HOPG surface have spurred our interest to carry out new investigations on another group IV element, germanium. This chapter presents our results on germanene/HOPG heterostructures based on a combined theoretical and experimental study.

10.1 Theoretical results

Germanene/HOPG structure optimisations

Ab initio electronic structure and total energy calculations were performed using VASP-5.3.5. [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using aplane-w a v ebasisset[END_REF] The calculations were based on DFT and the PAW representations. [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] The exchange-correlation interaction was treated with the GGA parametrized by the PBE formula. [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] The van der Waals correction to the GGA calculations was included by using the D2 method of Grimme. [START_REF] Grimme | Semiempirical gga-type density functional constructed with a long-range dispersion correction[END_REF] The wave functions were expanded in plane waves up to a kinetic energy cutoff of 400 eV. For BZ integrations, a Γ-centered k-point mesh of size 4⇥4⇥1w a su s e df o rt h eg e o m e t r yo p t i m i z a t i o n s and of size 11⇥11⇥1 for the density of states computations. The optimization of the atomic positions was performed via the conjugate-gradient method with the maximal force equal to 0.01 eV Å -1 . The convergence criterion of the total energy for SCF calculations was chosen to be 10 -4 eV.

The graphite substrate was modelled by a four-layer slab in ABAB stacking of graphene layers with PBC in three dimensions and keeping the bottom two layers fixed at the optimized bulk values. Two neighboring slabs were separated by a vacuum space of about 15 Å along the zdirection that is perpendicular to the surface to avoid interactions between the different slabs.

For establishing the calculation mo dels for the germanene/HOPG structure and to minimize the lattice mismatch between the two lattices, different configurations have been investigated.

Several models have been created by combining different supercells of germanene and graphite with a rotation angle between the two surfaces.

Our first principles calculations show that the van der Waals interaction is responsible for the bonding of the germanene layer to the graphite substrate. Eventual covalent bonding between the Ge and C atoms is ruled out by the search for the minimum-energy structure. The weak bonding to the substrate allows for an out-of-plane buckling and in-plane misorientation of the germanene layer with respect to the graphite substrate, enabling a partial relief of the misfit strain of the system. As a consequence of the weak interaction, our simulations show multiple configurations having similar stability which could coexist at finite temperatures. The binding energy per Ge atom, defined as

E b = (E Ge/HOP G -N Ge ⇥ E Ge -N C ⇥ E C ) n Ge (10.1)
where E Ge/HOP G is the total energy in the supercell germanene/HOPG; N Ge xE Ge and N C xE C are the total energies in the monolayer germanene and graphite unit cells, respectively; and n Ge is the number of Ge atoms in the germanene/HOPG supercell. Equation (10.1) shows a maximum difference in E b for the different investigated configurations to be about 40 meV. The structural parameters are then defined by averaging over the configurations with the uncertainty being the statistical dispersion.

In Figure 10.1,w ed i s p l a yab a l l -a n d -s t i c km od e lo ft h eg e r m a n e n eo ng r a p h i t eh e t e r o s t r u c t u r e .

Despite sharing a honeycomb structure, the lattice constant of graphite is markedly smaller (about 38%) than that expected for germanene; consequently, the two lattices can be unequivocally distinguished by STM (see below). The difference in the relative size is evident from the schematic illustration of the germanene/HOPG system reported in Figure10.1-a. Such a lattice mismatch, which would be giant for classical epitaxial growth based on covalent bonding, is easy to handle in van der Waals heteroepitaxy because of the weakly bound heterojunction. [START_REF] Koma | Van der waals epitaxy-anew growth method for a highly lattice-mismatched system[END_REF][START_REF] Hong | Controlled van der Waals heteroepitaxy of InAs nanowires on carbon honeycomb lattices[END_REF] From DFT, the relaxed germanene structure has a Ge bond length of (2.45 ± 0.07) Å with a buckling of (0.69 ± 0.07) Å, and its distance to the top layer of graphite is (3.28 ± 0.04) Å.

Electronic band structure calculations

In Figure 10.2-a, we show the calculated electronic band structure of the germanene/HOPG heterosystem along the line paths M-K, K-Γ,a n dΓ-M of the BZ. An enlarged picture in the vicinity of the Fermi level is displayed in Figure 10.2-b. To facilitate the comprehension of the band structure of the heterosystem, the contributions of germanene (Ge) and graphite (C) to the bands are shown in green and red, respectively. We find that the Dirac cone of germanene supported on graphene is mapped to the Γ point. Thin dashed lines are also added as a guide to the eye to highlight the linearity of the electronic bands of germanene at the Γ point. When germanene is paired up to an inert substrate like graphite, the van der Waals interaction at the interface breaks the sublattice symmetry in germanene and opens a small band gap. We find that the band gap of germanene supported on HOPG is about 80 meV, larger than the

STS measurements

In order to better understand the electronic properties of the deposited germanene layer, we have measured the STS spectra at different areas on the surface at RT with the Z-feedback off.

First, let us compare the STS spectra taken on top of a germanene island, Figure 10.5-b, with that on the bare HOPG substrate, Figure 10.5-c. As expected, both curves reveal a metallic character with a finite DOS close to the Fermi level. However, while the STS curve for HOPG has a minimum at the Fermi level, the germanene curve is shifted toward the empty states, wellmatching the results of our DOS calculations (see Figure 10.5-a). Actually, it has recently been shown that the metallic behavior of the "silicene" I/V curve has been exploited for realizing a field effect transistor based on ambipolar charge transport effects. [START_REF] Tao | Silicene field-effect transistors operating at room temperature[END_REF]B e s i d e st o po g r a p h y ,t h i s electronic fingerprint clearly distinguishes germanene from the graphite substrate. Conversely, STS curves taken on 3D clusters (inset of Figure 10.5-c) show the absence of electronic states at the Fermi level, confirming that 3D Ge nanoislands are semiconducting. [START_REF] Pacchioni | Silicon and germanium clusters. a theoretical study of their electronic structures and properties[END_REF] 10.

Conclusion

Flat 2D germanium sheets, showing a honeycomb lattice, grown on top of a HOPG surface were the result of a successful collaboration between theory and experiments. Germanene islands were synthesized on HOPG by depositing submonolayers of Ge on graphite at RT and subsequent annealing to 350 • C. STM showed that the germanene islands have a small buckling with no atomic reconstruction and did not give any hints for alloy formation and hybridization with the substrate. Our DFT calculations of the structural properties agreed well with our experimental findings and indicated that the germanene sheet interacts only weakly with the substrate underneath. Our band structure calculations confirm that the Dirac cone of free-standing germanene is preserved for layers supported on graphite. The germanene islands show a small, but characteristic charge transfer with the graphite substrate which is predicted by our ab initio simulations in excellent quantitative agreement with our STS measurements. Chapter 11

Conclusions and Perspectives

Our economy consistently pushes for the development of revolutionary materials to meet our ever-increasing technological demands. The search for novel silicon allotropes and corresponding synthesis methods represents the beginning of a wide-open field which is expected to yield further exotic structures with superlative properties. In this context, we focused our studies on new allotropes of silicon and explored their miscellaneous and fascinating properties. Not only silicon, but also its brother, germanium, is part of our research interests.

Two topics were the subject of this thesis:

1. Plasma-born aromatic silicon nanoclusters.

2. Two-dimensional silicon and germanium materials.

In the first topic, we focused our theoretical study mainly on the isomer scaffold of hydrogenated silicon clusters with a Si 19 H 12 stoichiometry, as one specific example, in order to demonstrate that a cluster of a given stoichiometry can possess a huge variety of physical and chemical properties resulting from apparently minor differences in its structure. Looking at different previously reported stoichiometries, we notice how the properties can further dramatically be changed by modifying the cluster size by just a few Si atoms and/or by the different degrees of hydrogenation of those clusters. This result can fully be exploited for the fabrication of cheaper and less toxic nanotechnological components and devices since it is a powerful alternative to the more traditional strategy where different properties are obtained by changing the chemical composition often involving the incorporation of chemicals as toxic as PbS, PbSe, CdS, and CdSe or as expensive as Ag and Au.

Si 19 H 12 nanoclusters exhibit high electron delocalization as can be concluded from their magnetic shielding values and their magnetically induced aromatic ring currents that largely exceed the ones of benzene. The ELF calculations demonstrate that the electrons around the overcoordinated Si atoms behave similar to the ones in a homogeneous electron gas showing some clear resemblance with small metal clusters, resulting in an extended range of absorption wavelengths from the ultraviolet to the infrared. Using NEB calculations of the Si 19 H 12 scaffold, we established how many isomers may be yet more stable than the symmetric nanocrystal, but that all members of this scaffold manifest very comparable physical, optical, and chemical properties despite their asymmetric structures. This last finding has demonstrated the potential wealth of non-tetrahedral hydrogenated silicon nanoclusters that all derive their ultrastability and their special features from extensive electron delocalization that is induced by their electron deficient bonds. All isomers of the presently investigated scaffold turned out actually to be so much more stable than their tetrahedral counterparts that it appeared quite unlikely that the latter ones are formed at all by self-assembly. This conclusion is supported by the fact that we have never observed the spontaneous formation of tetrahedral clusters in the presently investigated cluster size range during the AIMD simulation of the cluster growth.

For a given stoichiometry, the degree of electron delo calization dep ends on the details of the cluster structure. Since delocalized electrons are less tightly bonded, the resulting calculated absorption spectra shift toward longer wavelengths in comparison with perfectly tetrahedral silicon clusters of the same size. In order to prove this, we performed several experiments to investigate the absorption spectra of plasma-born hydrogenated silicon clusters with a size of less than 2 nm in the visible spectral region. To this end, we combined a cavity enhanced absorption setup as a diagnostic tool with a capacitively-coupled plasma reactor. We optimized the plasma parameters, such as the gas temperature and pressure, the plasma duration, pulse cycle, etc, in such a way that we only create "first-generation" nanoclusters that, based on our AIMD simulations, would spontaneously form with a variety of non-tetrahedral structures. We synchronized the incoherent broadband cavity enhanced absorption measurements with the nanoclusters production and we measured the transmitted light intensities "simultaneously" with and without the absorbing species between two successive plasma pulses. Using a spatially coherent supercontinuum light source, a bunch of filters, and two sets of highly reflecting mirrors, covering two spectral ranges between 575-665 nm and between 655-745nm, respectively, we found an obvious extinction peak around 645 nm. The size of this peak have also shown to vary with the density of absorbing nanoclusters crossing the detection beam by varying the time between the end of the discharge and the measurement of the cavity transmission.

Besides, we performed numerous extinction measurements varying the discharge time (from a few milliseconds to a few seconds), the reactor pressure and temperature, and the gas flow rate.

The results will be analysed, in a future study, for an understanding of the nucleation and growth of silicon nanoclusters in a plasma reactor.

Moreover, we performed extensive extinction measurements with-and without an electric field applied during the cavity transmission measurements. Here, we have always obtained a decrease in the extinction peak in the presence of an applied DC bias which depends on the strength of the electric field. This effect can only be attributed to the presence of a permanent dipole moment in plasma-born silicon nanoclusters, where the inhomogeneity of the electric field in the reactor chamber plays an important role. Simulations of the electric field have helped us to analyse and understand the behavior of aromatic silicon nanoclusters under the influence of the dipole forces. From the coherence between our simulations of the density or distribution of nanoclusters in the plane of the detection beam and the extinction measurements, we found the permanent dipole moment of aromatic silicon nanoclusters to be between 2.0 and 2.5 Debye in excellent agreement with our prior ab initio calculations. Such a property is crucial for many applications that rely on the use of polar molecules to enhance the properties of materials, as for instance the modification of the work function of electrodes of an organic field effect transistor.

Furthermore, the dep osition of such silicon nano clusters under optimized plasma conditions have shown the formation of crystalline nanoparticles along with amorphous ones. From our HRTEM characterization, we found that the crystalline nanoparticles, with a size of 2.0 to 3.5 nm, attain a primitive hexagonal structure. Such a crystalline structure of silicon obtained from a RF plasma along with other structures reported in the literature, such as the bc8 and the bcc, usually form after exposing diamond silicon to extremely high pressures. Since the appearance of exotic forms of silicon by a RF plasma is not well understood yet, we propose an experiment to examine where such a crystallization takes place. For instance, we can deposit the nanoclusters (under different optimized plasma conditions) on a soft surface (gel), instead of a hard one, placed under a grid to control the impact velocity. Then, we can collect the nanoclusters and perform HRTEM measurements. From the obtained results, we can deduce if the crystallization occurs in the plasma or if a high impact energy is needed. If the latter is true, we will have direct experimental evidence for the theoretically predicted "chemistry with a hammer" mechanism.

Finally, a plasma-born mixture of different structures deposited within a thin film might be advantageous; e.g. for efficient large-band light absorption as needed, for instance, in photovoltaic cells and protective coatings. Indeed, combining slightly different stoichiometries and structures, light of the entire solar spectrum between 1.0 eV and 5.2 eV can be absorbed using only nanoclusters having a size of about twenty silicon atoms. Therefore, this new allotropic form of nanosilicon may pave the way for new competitive solar materials and architectures. We predict that this unique combination of metal and semiconductor properties within the same pure, but nontetrahedral hydrogenated silicon nanoclusters will be an extremely rich source for future bandgap engineering employing nothing else but silicon and hydrogen atoms.

In the second topic, we focused on a combined theoretical and experimental study of the growth of a new allotropic form of silicon and germanium, silicene and germanene, on top of a graphite surface. We have obtained, and for the first time, several silicene areas with a structure very similar to that of graphene by the deposition of one ML of Si on an inert HOPG substrate at RT and under UHV conditions. The AFM images supported the formation of a quasi-continuous 2D silicon layer leaving some areas of HOPG uncovered together with the formation of small 3D Si clusters. The STM data supported the existence of a graphene-like structure with a very small buckling of the silicon nearest neighbors. The atomic resolution images obtained by STM did not show any surface reconstruction, as found when silicene is deposited on metal surfaces, suggesting a negligible electronic interaction/hybridization with the substrate underneath. Our observation suggested that a growth mode with a full release of strain has occured. Our AIMD simulations predicted the thermal stability of a perfect silicene monolayer on a HOPG surface at RT and 350 • C, highlighting the essential role of van der Waals forces bonding silicene to the surface. The simulations of the growth mechanism of the silicene sheet on HOPG at RT showed how the silicon hexagons spontaneously form and remain at a vertical distance corresponding to the van der Waals distance above the surface.

Similarly, flat 2D sheets showing a honeycomb lattice matching that of germanene were synthesized for the first time by depositing submonolayers of Ge on HOPG at RT followed by annealing to 350 • C. The overlayers also showed a small buckling and no atomic reconstruction suggesting the absence of any alloy formation and hybridization with the substrate. Ab initio calculations indicated that the Dirac cone of free-standing germanene is preserved for layers deposited on HOPG. Compared to the substrate, the germanene islands showed a characteristic charge transfer which was predicted by simulations and observed with STS spectroscopy. The two different approaches lead to an astonishing quantitative agreement.

HOPG has been found to be a good candidate to form silicene and germanene layers with remarkable electronic properties very different from any other known silicon and germanium-based nanostructures. The growth of silicene and germanene on a nonmetallic and inert substrate is an important step toward the realization of alloy-free layers with properties that approach the free-standing ones. We believe that we have succeeded in observing these small, but true, silicene and germanene areas because of a rather slow deposition rate of Si and Ge, and we are confident that in the future by using more sophisticated growth methods (as molecular beam epitaxy) larger areas can be obtained. Only in the case of a continuous silicene and germanene layer formation, we will be able to detach and transfer these ultrathin films for device applications.

In future works, the ideal deposition temperature has, for instance, to be determined more carefully: hot enough to overcome all energy barriers to silicene and germanene growth, but low enough to avoid 3D cluster formation. Optimized experimental deposition conditions will favor the extent of the pure silicene and germanene domains, eventually pushing them to the size limits imposed by the terrace size of the graphite substrate possibly making the use of graphene instead of graphite necessary for extremely large 2D layers.

The ultimate silicene/graphene heterostructure can then be lifted off from the underlying substrate and be deposited on SiO2 as already reported in the literature. The silicene/graphene/-SiO2 system can then be integrated in functional devices. Such systems can be grown on a very large scale corresponding to the requirements of the electronic industry. Besides this immediate compatibility with powerful silicon technologies and the absence of any alloying problems, the silicene on graphite system presents a much less expensive solution than strategies based on silver substrates. The present research is in the forefront of 2D materials with revolutionary future applications for nanotechnological devices as, for example, ultra-high-speed electronic devices and for functional electronic devices based on silicene and germanene.

• Compute the forces acting on all particles according to Eq. (A.2). This is the most timeconsuming calculation.

• Integrate the above second order differential equations, Eq. (A.1). Due to the many-body nature of the problem, the equations of motion are discretized and solved numerically.

• As the particles "move", their trajectories are analysed, providing instantaneous and averaged properties.

The second and third steps are repeated for a chosen number of time-steps (∆t)i naM D simulation.

A.1.1 Numerical integration of the equations of motion

The aim of the numerical integration of Newton's equations of motion is to find positions r i (t+ ∆t)andvelocitiesv i (t+ ∆t)attimet+∆t in terms of the already known positions and velocities at time t.T ot h i se n d ,t h eV e r l e ta l g o r i t h mi su s e di no u rM Ds i m u l a t i o n sb e c a u s eo fi t ss i m p l i c i t y and its stability. The basic formula for this algorithm is derived from the second-order Taylor expansions for the positions of a particle at (t+∆t):

r i (t + ∆t)=r i (t)+v i (t)∆t + 1 2 F i (t) m i (∆t) 2 + O((∆t) 3 ) (A.3)
similarly,

r i (t -∆t)=r i (t) -v i (t)∆t + 1 2
F i (t) m i (∆t) 2 -O((∆t) 3 ) (A.4)
By adding Eqs. (A.3)a n d( A.4), we obtain:

r i (t + ∆t)=2r i (t) -r i (t -∆t)+ F i (t) m i (∆t) 2 + O((∆t) 4 ) (A.5)
Equation (A.5)i sa c c u r a t eu pt ot e r m so ft h ef o u r t hp o w e ri n∆t. The velocity is computed by subtracting Eq. (A.4)f r o mE q . ( A.3):

v i (t)= r i (t + ∆t) -r i (t -∆t) 2∆t + O((∆t) 2 ) (A.6)
The choice of ∆t is an important consideration in any MD simulation. Exact trajectories correspond to the limit of an infinitesimally small integration step in order to conserve the total energy of the system. For a too large ∆t,ap a r t i c l ec a nfi n di t s e l funphysically close to another particle. This will be reflected in a "jump" in the potential energy and the total energy of the system. Moreover, the period of the bond vibration must also be considered. For instance, bonds involving light atoms vibrate with periods of several femtoseconds, implying that ∆t should be on a subfemtosecond scale to ensure stability of the integration and to follow adequately the movement of all atoms.

Moreover,

VNN = 1 2 N -1 X I=1 N X J=I+1 Z I Z J |R I -R J | (B.5) Vee = 1 2 n-1 X i=1 n X j=i+1 1 |r i -r j | (B.6) VNe = - N X I=1 n X i=1 Z I |R I -r i | (B.7)
here Z I and Z J are the atomic numbers of nuclei I and J.

Due to the complexity of the many-body problem, Eq. (B.1)b e c o m e sd i fficult to be solved and an approximation is needed.

B.1.1 Born-Oppenheimer approximation

In 1927, Born and Oppenheimer [START_REF] Born | Zur quantentheorie der molekeln[END_REF]s h o w e dt h a tt oav e r yg o o da p p r o x i m a t i o n ,b a s e do nt h e huge difference in mass between nuclei and electrons, the electronic and nuclear problems can be separated. Hence, it is assumed that the electron cloud "equilibrates" quickly for each instantaneous configuration of the heavy nuclei. In other words, the electronic wave functions can be found by assuming that the nuclei are fixed. The nuclei, in turn, move in the field of the averaged electron densities. Consequently, the electronic and nuclear motions are said to be decoupled. Therefore, the total wave function of a given electronic state splits into two components: one which describes the nuclear motion, χ(R), and one which describes the electronic motion, (R,r), for a fixed nuclear coordinate, R. Therefore, the total wave function has the form:

Ψ(R, r)=χ(R) (R, r) (B.8)
Then, the electronic wave function (R,r)isobtainedfromthesolutionoftheelectronicSchrödinger equation:

- 1 2 n X i=1 r 2 i + Vee (r)+ VNe (R, r) ! (R, r)=E e (R) (R, r) (B.9)
Equation (B.9)i ss o l v e df o re a c hs e to ffi x e dn u c l e a rp o s i t i o n s ,R. Consequently, the electronic eigenvalue, E e ,e x p l i c i t l yd e pe n d so nR. The nuclei, in turn, are assumed to move according the to nuclear Schrödinger equation:

-

N X I ✓ 1 2M I ◆ r 2 I + VNN (R)+E e (R) ! χ(R)=E n (R) (B.10)
With the Born-Oppenheimer (BO) approximation, the many-body problem is reduced to the solution of the nuclear motion on a single potential energy surface (PES), where the electronic potential energy, E e ,p l a y sp a r ti nt h ep o t e n t i a le n e r g yo ft h en u c l e i :

V (R)= 1 2 N -1 X I=1 N X J=I+1 Z I Z J |R I -R J | + E e (

B.2 Kohn-Sham density functional theory

In modern AIMD, the electronic structure method most commonly used is the Kohn-Sham formulation of density functional theory (DFT). The basis for DFT is the proof by Hohenberg and Kohn [START_REF] Hohenb | Inhomogeneous electron gas[END_REF]t h a tt h eg r o u n ds t a t ee l e c t r o n i ce n e r g yi sd e t e r m i n e dc o m p l e t e l yb yt h ee l e c t r o n density, ⇢. The electron density is the square of the wave function, integrated over the electron coordinates. While the complexity of a wave function increases exponentially with the number of electrons, the electron density has the same number of variables, independent of the system size. [START_REF] Jensen | Introduction to Computational Chemistry[END_REF] The goal of DFT methods is to design functionals 1 connecting the electron density with the energy.

Early attempts at designing DFT models tried to express all the energy components as a functional of the electron density, but these methods had poor performance. The success of modern DFT methods is based on the suggestion by Kohn and Sham [START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF]thattheelectronkineticenergy should be calculated from an auxiliary set of one-electron orbitals, "Kohn-Sham orbitals", used for representing the electron density. The Kohn-Sham method is based on paramaterization of the density ⇢(r )i nt e r m so fK o h n -S h a mo r b i t a l sφ i (r ):

⇢(r)= N X i=1 |φ i (r)| 2 (B.12)
and a decomposition of the total energy functional according to:

E[⇢]=T [⇢]+E H [⇢]+E xc [⇢]+E ext [⇢] (B. 13
)
where

T [⇢]= X i Z φ ⇤ i (r)(- 1 2 r 2 )φ i (r)d 3 r (B.14)
1 A function is a prescription for producing a number from a set of variables (coordinates). A functional is a prescription for producing a number from a function, which in turn depends on variables. The wave function and the electron density are thus functions, while the energy depending on a wave function or an an electron density is a functional.

is the contribution of the kinetic energy approximated by that of non-interacting electrons in states φ i (r ).

E H = ZZ ⇢(r)⇢(r 0 ) |r -r 0 | drdr 0 (B.15)
is the Coulomb electron-electron repulsion in the Hartree approximation, while

E ext = Z V ext (r)⇢(r) (B.16)
describes the interaction with the external field, with

V ext = - X I Z I |R I -r| (B.17)
the Coulomb interaction between electrons and nuclei.

E xc [⇢]i st h eexchange-correlation energy. It accounts for "exchange" and "correlation" effects.

The key to Kohn-Sham theory is to calculate the kinetic energy under the assumption of noninteracting electrons from Eq. (B.14). However, the electrons, in reality, are interacting, and Eq. (B.14) does not provide the total kinetic energy. Therefore, the remaining kinetic energy is absorbed into the E xc [⇢] term. This complicated functional has no explicit form. It has to be approximated, as will be discussed shortly.

Variation of the total energy functional, Eq. (B.13), with respect to φ i ⇤ (r)l e a d st oa ne ffective one-electron equation for the determination of φ i (r), the Kohn-Sham equations:

✓ - 1 2 r 2 + Z ⇢(r 0 ) |r -r 0 | d 3 r 0 + V ext (r)+ δE xc [⇢(r)] δ⇢(r) ◆ φ i (r)=✏ i φ i (r) (B.18)
The Kohn-Sham equations describe electrons moving in one-electron potential given by

V ef f (r)=V ext (r)+ Z ⇢(r 0 ) |r -r 0 | d 3 r 0 + V xc [⇢(r)] (B.19)
where the exchange correlation potential is given by the variational derivation of the exchangecorrelation energy, proposed by Hohenberg and Kohn. [START_REF] Hohenb | Inhomogeneous electron gas[END_REF] The LDA consists of locally approximating the true exchange-correlation energy of a system by the exchange-correlation energy associated with a homogeneous electron gas of the same density. The homogeneous gas is the only system for which the form of the exchange-correlation energy is known precisely.

V xc (r)= δE xc [⇢(r)] δ⇢(r) (B.
The LDA is only dependent on local density, and the total energy is commonly written as

E LDA xc [⇢(r)] = Z d 3 (r)⇢(r)✏ xc [⇢(r)] (B.21)
where ✏ xc [⇢(r)] is the exchange-correlation energy density corresponding to a homogeneous electron gas of density ⇢(r).

• Generalized gradient approximation

A major drawback in LDA is its incapability of describing non-homogeneous electron densities. One way to improve this approximation is to consider, not only the electron density, but also its gradient. The electron density gradient describes the non-homogeneity of the true electron density. As a result of this procedure, the generalized gradient approximation (GGA) can be conveniently written in terms of an analytic function known as the enhancement factor, F xc [⇢(r),r⇢(r)], that directly modifies the LDA energy density,

E GGA xc [⇢(r)] = Z d 3 (r)⇢(r)✏ hom xc (⇢(r))F xc [⇢(r), r⇢(r)] (B.22)
In recent years, several exchange-correlation GGA implementations have become a kind of standard choice, namely: Perdew-Wang (PW86), [START_REF] Perdew | Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation[END_REF] Perdew-Wang (PW91), [START_REF] Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF]a n d Perdew-Burke-Ernzerhof (PBE). [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]Moreover,severalGGAfunctionalsweredevelopedin the form of an exchange functional (B) [START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF]oracorrelationfunctional((LYP), [START_REF] Lee | Development of the colle-salvetti correlation-energy formula into a functional of the electron density[END_REF](B95), and (P86) [START_REF] Perdew | Density-functional approximation for the correlation energy of the inhomogeneous electron gas[END_REF]). A combination between different exchange and correlation functionals is made to describe completely the systems under study. Some of the most common combinations are: BP86 and BPW91.

• Hybrid Functionals

Hybrid functionals were developed to provide a better description to the exchange part of the GGA functional. Since the exchange part in Hartree-Fock (HF) is defined exactly, the hybrid functional uses a combination of HF, GGA, and LDA functionals to describe the exact exchange and correlation part. One of the most commonly used hybrid functional is B3LYP (Becke, three-parameter, Lee-Yang-Parr),

E B3LY P xc = E LDA x + a 0 (E HF x -E LDA x )+a x (E GGA x -E LDA x )+E LDA c + a c (E GGA c -E LDA c ) (B.

B.2.2 Basis sets

To solve the Kohn-Sham equations, Eq. (B.18), the single particle wave function can be mathematically expressed as a linear combination of a set of given functions. This set of functions is called a basis set and is formed in different ways depending on the systems under study and the required accuracy. Localized basis sets are one type and are based on atom-centred basis functions, while plane wave basis sets are another type and are used for calculations involving periodic systems.

B.2.2.1 Localized basis sets

Localized basis sets can be further classified into two main types, Slater type orbitals (STO) [START_REF] Slater | Atomic shielding constants[END_REF] and Gaussian type orbitals(GTO). [START_REF] Boys | Electronic wave functions. i. a general method of calculation for the stationary states of any molecular system[END_REF] They both have the same form where N nl is a normalization constant; ⇣,i nE q . ( B.25), is a constant related to the nucleus effective charge; and ↵,i nE q . ( B.26), is the orbital exponent. Although the STO gives a more accurate description of the atomic orbital, the GTO can be numerically solved more efficiently.

χ nlm (r, ✓, φ)=R nl (r)Y lm (✓, φ) (B.
However, a linear combination of several GTOs, called contractions, is able to mimic the STO accuracy. The more of GTOs included in the contraction, the better they are able to model the accuracy of STO functions. As a result, the Gaussian type basis functions are the most popular localized basis sets used today.

When using GTOs to mimic STOs, the new basis sets are given a new name. They may be in the format of a minimal basis sets or a split-valence basis sets. The conventional nomenclature for the minimal basis sets is STO-x G, where x is a constant and represents the number of Gaussian primitive functions used to comprise a single basis function. One way to improve the STO-x G is to use two or three functions for each of the minimal basis functions, i.e. double zeta and triple zata basis set,r e s p e c t i v e l y . A n o t h e rc o n v e n i e n ta n de fficient way to build a basis set is to let the valence functions doubled (or tripled) while each inner shell orbital use a single function, known as split valence shell basis sets. One typical example is the 6-311G basis set. It implies that each of the inner core shell orbitals consists of one basis function that is a contraction of 6p r i m i t i v eG a u s s i a n s ;w h i l ee a c ho ft h ev a l e n c eo r b i t a l si sc o m p o s e do ft h r e eb a s i sf u n c t i o n s , the first one composed of a contraction of 3 primitive Gaussians, the other two composed of 1 primitive Gaussian respectively. Moreover, in order to increase the accuracy of the calculations, additional functions, such as polarization functions (⇤)a n ddiffusion(+) functions need to be added to the basis.

B.2.2.2 Planewaves and pseudopotentials

The electron orbitals, used to express the single particle density, defined in the Kohn-Sham equations, Eq. (B.18), may be expanded in terms of a planewave basis set. Planewave basis sets are the natural choice in the case of solids that exhibit periodicity. Therefore, the electron wave functions can be written as

nk (r)= X K c nk (K)e i(k+K)•r (B.27)
where K are reciprocal lattice vectors. The number of planewaves used in the basis can be controlled by imposing an energy cut-off. Thus, for a given energy cut-off (E cut )a l lp l a n e w a v e s with a kinetic-energy smaller than (E cut )a r ei n c l u d e di nt h eb a s i ss e t .

Although the use of a planewave basis set is advantageous in periodic systems, the computational cost is prohibitive if we treat all electrons in a system explicitly. This problem can be circumvented by the implementation of pseudopotentials. [START_REF] Phillips | New method for calculating wave functions in crystals and molecules[END_REF] This approximation is based on the assumption that only valence electrons strongly participate in the bond formation. Therefore, in the pseudopotential approach, the nuclei and core electrons are described by an effective, smooth potential which is constructed in such a way to reproduce the correct effect on the remaining valence electrons. The Kohn-Sham equations are then solved only for valence electrons, thus reducing the computational cost. On the other hand, all information about the real wave function close to the nucleus is lost, making it hard to compute properties which rely on the core region.

The projector-augmented wave (PAW) method, [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF]originallydevelopedbyBlöchl, [START_REF] Blöchl | Pro jector augmented-wave metho d[END_REF]

Ψ n (r)= Ψn (r)+ X a X i (φ a i (r) -φa i (r)) < pa i | Ψn > (B.28)
where φ a i , φa i are atom-dependent partial waves, and pa i are some fixed projector functions. The partial waves φ a i and φa i are the basis sets used to expand Ψ n (r) and Ψn (r),r e s pe c t i v e l y . φ a i are usually chosen to be the solutions of the Kohn-Sham equations for the isolated atom, and φa i are their smooth continuation inside the augmentation spheres. The projectors, pa i ,a r ef u n c t i o n s chosen to obtain a rapid convergence of the expansion of the wave functions.

B.2.3 Dispersion interactions in DFT

Dispersion interactions are weak van der Waals interactions. This type of interaction is due to long range dynamical correlation effects. It arises from the instantaneous fluctuation in the charge distribution which creates instantaneous dipole moments even though the system does not possess a permanent dipole moment. Traditional DFT functionals, such as LDA, GGA, and hybrids, are incapable of modelling dispersion interactions because of their nonlocal character.

Therefore, dispersion corrected DFT functionals are good practical alternatives to conventional wave function methods. One approach is the DFT-dispersion (DFT-D) method. It is based on adding an explicit dispersion term to the conventional DFT energy.

E (DF T -disp) = E KS-DF T + E disp (B.29)
In the present work, we have used the DFT-D2 method of Grimme. [START_REF] Grimme | Semiempirical gga-type density functional constructed with a long-range dispersion correction[END_REF] The dispersion correction is given by an attractive semi-empirical pair potential:

E (disp) = s 6 X µ<µ 0 V PP (Z µ ,Z µ 0 ,R µ,µ 0 ) (B.30)
where s 6 is a global scaling factor that is specific for each exchange-correlation functional. The pair potential between two atoms (Z 1 and Z 2 )s e p a r a t e db yr 12 is

V PP (Z 1 ,Z 2 ,r 12 )=- p C 6 (Z 1 )C 6 (Z 2 ) r 6 12 f r 12 R 0 (Z 1 )+R 0 (Z 2 ) (B.31)
where C 6 and R 0 are element-specific dispersion coefficients and van der Waals radii, respectively, and

f (x)= 1 1+e -dx (B.32)
is a cut-off function.

Empirical DFT-D methods have been implemented in many computational packages because they are easy to implement and the forces required e.g. for geometry optimizations are easily computed.

B.3 Structure optimization

The principle of ab initio calculations is to find the electronic ground state of the system, i.e.

a particular arrangement of atoms that corresponds to the lowest potential energy. In order to search for the optimal geometry, an iterative procedure is followed. The quantum-mechanical code performs several ionc relaxation steps; i.e.,alln ucleiaredisplacedinappropriatedirections (according to energy gradients). After each ionic step, the ground state energy of the particular arrangement is evaluated by means of self-consistent cycles of electronic relaxation steps. Once the difference between two consecutive energy values is lower than a specified threshold value,

C.1.1 Linear response TD-DFT

The time-dependent Kohn-Sham equation, Eq. (C.1), can be applied to essentially any timedependent situation and can be solved in two ways: real-time propagation and linear response theory. If the time-dependent external potential is weak, as in normal spectroscopic experiments, perturbation theory can be applied and it is thus sufficient to use the linear-response theory through which the electronic excitation can be extracted (thus the optical absorption spectra).

On the other hand, if the time-dpendent potential is strong (e.g.,stronglaserfield),thereal-time propagation [START_REF] Castro | Propagators for the time-dependent kohn-sham equations[END_REF]o ft h es y s t e ms h o u l db eu s e dt od e s c r i b en o n -l i n e a rp h e n o m e n a .

Linear response of a Kohn-Sham system: In linear response TD-DFT, the linear response of the density to the applied field is a crucial quantity. The linear response of an interacting (real )s y s t e m( χ)c a nb ed e r i v e df r o mt h a to fi t sc o r r e s p o n d i n gn o n -i n t e r a c t i n g( Kohn-Sham) system (χ s ), whose form can be known from the general linear-response theory. [START_REF] Nogueira | Time-dependent density-functional theory[END_REF] Therefore, the density response related properties can be extracted.

If a time-dependent field (v appl (t)) is applied to an unperturbed interacting system, this will cause a change in the time-dependent density. The linear changes of the density can be found from the density-density response function, χ (the so-called susceptibility):

δ⇢(r,t)= ZZ χ(r, r 0 ,t-t 0 )δv appl (r 0 ,t)dt 0 d 3 r 0 (C.4) Equation (C.4
)m e a n st h a ti fas m a l lc h a n g ei nt h ee x t e r n a lp o t e n t i a li sm a d ea tp o i n tr 0 and time t 0 ,t h ed e n s i t y -d e n s i t yr e s po n s ef u n c t i o n( χ(r, r 0 ,t -t 0 )) tells us how the density will change at point r and later time t.

When the field v appl (t) is applied to the unperturbed Kohn-Sham system which corresponds to the interacting system, it will cause a variation in the Kohn-Sham potential. Thus the linearized total variation of the Kohn-Sham effective potential is

δv ef f (r,t)=δv appl (r,t)+δv H (r,t)+δv xc (r,t) (C.5)
where

δv H (r,t)= Z δ⇢(r 0 ,t) |r -r 0 | d 3 r 0 (C.6)
and

δv xc (r,t)= Z f xc (r, r 0 ,t-t 0 )δ⇢(r 0 ,t 0 )dt 0 d 3 r 0 (C.7)
The time-dependent exchange-correlation kernel, f xc , is a key quantity of TD-DFT in the linear response regime. It is a functional of the corresponding ground state density,

f xc [⇢](r, r 0 ,t-t 0 )= δv xc (r,t) δ⇢(r 0 ,t 0 ) (C.8)
Then the linearized density variation of the Kohn-Sham system to the applied field can be calculated via the non-interacting Kohn-Sham response function (χ s ):

δ⇢(r,t)= ZZ χ s (r, r 0 ,t-t 0 )δv ef f (r 0 ,t)dt 0 d 3 r 0 (C.9)

According to the Kohn-Sham assumption, the density response in Eq. (C.4)a n dE q . ( C.9) should be equal, resulting in an expression connecting χ and χ s in real space. For convenience, the equation is transformed to the frequency representation via Fourier transformation:

χ(r, r 0 ,w)=χ s (r, r 0 ,w)+ ZZ χ s (r, r 1 ,w) ✓ 1 (|r 1 -r 2 |) + f xc (r 1 , r 2 .w) ◆ χ(r 2 , r 0 ,w)d 3 r 1 d 3 r 2 (C.10)
Now all the terms are functionals of the ground state density. Eq. (C.10)c o n t a i n st h ek e yt o electronic excitations via TD-DFT. When the frequency of the applied field, w,m a t c h e sat r u e transition frequency of the system, the response function χ has a pole as a function of w.

The frequency-dependent, non interacting Kohn-Sham density response function is known as:

χ s (r, r 0 ,w)= X q ✓ ⇠ q (r)⇠ ⇤ q (r 0 ) w -w q + i⌘ - ⇠ ⇤ q (r)⇠ q (r 0 ) w + ww q -i⌘ ◆ (C.11)
where q is a double index (a, i )r e p r e s e n t i n gat r a n s i t i o nf r o ma no c c u p i e dK o h n -S h a mo r b i t a l (φ i )t oa nu n o c c u p i e dK o h n -S h a mo r b i t a l( φ a ); w q is a transition energy which is equal to the difference of the Kohn-Sham orbital energies

w q = ✏ a -✏ i (C.12)
and ⇠ q (r)d e n o t e s

⇠ q (r)=φ ⇤ i (r)φ a (r) (C.13)
In 1995, Casida [START_REF] Casida | Time-dependent density functional response theory of molecular systems: Theory, computational methods, and functionals[END_REF]derivedtheessentialequationstoextracttheexactenergiesoftheinteracting system from the density-density response function. His approach is nowadays implemented in many electronic-structure codes and is popular for calculating the optical properties of a system.

Casida reformulated the calculation of the poles (the transition energies) of χ into a generalized Hermitian eigenvalue problem:

Ω qq 0 F I = w 2 I F I (C.14)
where the Ω qq 0 matrix is defined as Ω qq 0 (w)=w 2 q δ q,q 0 +4 q w q w 0 q K q,q 0 (w) (C. [START_REF] Hückel | Quantum theoretical contributions to the problem of aromatic and non-saturated compounds[END_REF] where q(a, i )andq 0 (b, j )aredoubleindexes: q denotes an electronic transition from an occupied Kohn-Sham orbital, φ i ,t oa nu n o c c u p i e dK o h n -S h a mo r b i t a l ,φ a ,a n dq 0 denotes an electronic transition from an occupied Kohn-Sham orbital, φ j ,t oa nu n o c c u p i e dK o h n -S h a mo r b i t a l ,φ b ; K q,q 0 is the frequency-dependent coupling matrix K q,q 0 (w)= ZZ ⇠ ⇤ q (r)f HXC (r, r 0 ,w)⇠ q (r 0 )d 3 rd 3 r 0 (C. [START_REF] Iwamoto | Persilastaffanes: Design, synthesis, structure, and conjugation between silicon cages[END_REF] here, f Hxc (r, r 0 , w) is the response kernel including the Hartree kernel and exchange-correlation kernel,

f Hxc (r, r 0 ,w)= 1 |rr 0 | + f xc (r, r 0 ,w) (C.17)

and the frequency-dependent xc kernel is given by f xc (r, r 0 ,w)= Z e iw(t-t 0 ) f xc (r, r 0 ,t-t 0 )d(tt 0 ) (C.18)

The eigenvalues of Eq. (C.15)p r o v i d et h ee x c i t a t i o ne n e r g i e sw I and the eigenvectors can be used to obtain the oscillator strengths, f,d e fi n e di nE q . ( 3.6), and to assign the symmetry of each transition. Now, if the intensity is increased and E gets closer to the Coulomb field strength, the induced electric dipoles in the medium cannot respond linearly anymore; i.e.,t h ed i s p l a c e m e n to ft h e electrons becomes so large that the restoring forces on the electrons are no longer proportional to their displacements, thus, higher order terms contribute to the polarization: [START_REF] Heicht | Optics.A d d i s o n -W i s e l e y[END_REF] P = P (1) + P where χ (n) is the n-th order susceptibility of the material. In its most general form, all susceptibilities are tensors that determine the spatial direction of the polarization vector as induced by multiple field vectors applied simultaneously to the material.

The usual linear susceptibility, χ,i sm u c hg r e a t e rt h a nt h eo t h e rc o e fficients of the nonlinear terms, χ (2) , χ (3) ,e t c . a n dh e n c et h el a t t e rc o n t r i b u t en o t i c e a b l yo n l ya th i g h -a m p l i t u d efi e l d s .

If we suppose that a light wave of the form As the harmonic light wave sweeps through the medium, it creates a polarization wave, i.e. an undulating redistribution of charge within the material in response to the field. If only the linear term (i.e. P (1) )w e r ee ffective, the electric polarization wave would correspond to an oscillatory current following along with the incident light. The light thereafter reradiated in such a process is the usual refracted wave generally propagating with a reduced speed and having the same frequency as the incident light.

E = E o sin !t (D.
[170]P (1) can give the Pockels effect (the linear electro-optic effect), i.e. the linear electro-optic effect, where the refractive index of a medium is modified in proportion to the applied electric field strength. The Pockels effect exists only in certain crystals which lack center symmetry.

The second-order nonlinear polarization, P (2) , i.e. the second term in Eq. (D.6), arises from a χ (2) nonlinearity. It is composed of two components that are of great interest: the first corresponds to a constant bias polarization varying as E 2 0 . If an intense plane-polarized beam traverses an appropriate crystal, the presence of quadratic linearity will, in part, be manifested by a constant electric polarization of the medium. A voltage difference, proportional to the beam flux density, will accordingly appear across the crystal. This effect is known as optical rectification. [START_REF] Heicht | Optics.A d d i s o n -W i s e l e y[END_REF]A n o t h e ri m p o r t a n te ffect that takes place when a material is subjected to an intense electric field is the Kerr effect. The material becomes birefringent, i.e. ac h a n g ei n the refractive index of the material takes place that is directly proportional to the square of the electric field, i.e. the light intensity. This refractive index variation is responsible for the nonlinear optical effect of self-focusing of light, for example. The second component (according to the cos 2!t term) corresponds to a variation in electric polarization at twice the fundamental frequency. The reradiated light which arises from the driven oscillators also has a component at this same frequency, 2!, and the process is termed second-harmonic generation. Now, if two or more primary beams of different frequencies propagate through the material, other effects can take place, such as sum and difference frequency conversion (frequency mixing).

In order to have a non-vanishing χ (2) component, the material must not exhibit inversion symmetry.

The third-order nonlinear polarization, P (3) ,a r i s e sf r o maχ (3) nonlinearity that quantifies the strength of various related nonlinear processes such as, the nonlinear contribution to the refractive index, four-wave mixing, self-phase modulation, and third harmonic generation, depending on the involved light frequencies and phase matching conditions.

Higher orders of nonlinearity can be neglected, since their strength of susceptibility decreases as

χ n /χ n+1 ⇡ 10 12 .
The supercontinuum light is generally generated in photonic crystal fibers that possess centrosymmetric structures, therefore the χ (2) term vanishes and Eq. D.6 becomes: P = P (1) + P (3) + ... = ✏ 0 χE 0 sin !t + 3 4 ✏ 0 χ (3) E 3 0 sin !t (D.7)

We note that due to variations in the refractive index of the fib er, a lack of phase b etween frequencies ! and 3! takes place and, hence, the third-harmonic of polarization term (sin 3!t) can be neglected in Eq. (D.7).

Equation D.7 can then be reformulated as: Since the strong field of a laser beam is capable of inducing an appreciable intensity-dependent refractive index change in the medium, the medium then reacts back and inflicts a phase change on the incoming wave, resulting in self-phase modulation (SPM).

P = ✏ 0  χ + 3 4 χ (

SPM frequency shift of a laser pulse

For an ultrashort pulse with a Gaussian shape and constant phase, the intensity at time t is given by

I(t)=I 0 exp( -t 2 τ 2 ) (D.11)
where I 0 is the peak intensity and τ is half the pulse duration.

The electric field of the incident laser pulse propagating in z direction is expressed as:

E in (z 0 ,t)=E 0 (z, t)e i(knz-w0t) + c.c. = E 0 (z, t)e iφ0 (D.12)

where k is the wavenumber and n is the refractive index of the material. As the pulse propagates in a non-linear optical medium of length L,t h ei n t e n s i t ya ta n yo n ep o i n ti nt h em e d i u mr i s e s and then falls as the pulse goes past. This will produce a time-varying refractive index:

dn(I) dt = n 2 dI dt = n 2 I 0 -2t τ 2 exp( -t 2 τ 2 ) (D.13)
This variation in the refractive index produces a shift in the instantaneous phase of the pulse: φ(t)=ω 0 (t) -2π λ 0 n(I)L (D. [START_REF] Hückel | Quantum-theoretical contributions to the benzene problem. I. the electron configuration of benzene and related compounds[END_REF] where ω 0 and λ 0 are the carrier frequency and (vacuum) wavelength of the pulse, and L is the distance the pulse has propagated. This phase shift results in a frequency shift of the pulse.

Since the time derivative of the phase of a wave, φ(t), is simply the angular frequency of the wave, the instantaneous frequency ω(t) will be: where c n,k+K are now coefficients that can be varied to determine the lowest energy solution.

ω(t)= dφ (
This also converts the Kohn-Sham equations, Eq. (B.18), from an integral to a set of algebraic equations that can readily be solved using matrix algebra.

Bloch functions, nk ,a n dt h e i re i g e n v a l u e sa r ep e r i o d i ci nk-space, i.e. for a reciprocal lattice vector that does not lie in the first BZ, K,w eh a v e n,k+K (r)= nk (r) ; ✏ n,k+K = ✏ nk (E.8) Hence, a full set of levels can be described with k restricted to the first BZ. For each n,t h e set of electronic levels specified by ✏ nk ,i sc a l l e da nenergy band. This leads to a description of energy levels of an electron in a periodic potential in terms of a family of continuous functions, ✏ nk or (✏ n (k)), each with the periodicity of the reciprocal lattice. The information contained in these functions is referred to as the band structure of the solid. [START_REF] Ashcroft | Solid State Physics.S a u n d e r sC o l l e g e ,P h i l a d e l p h i a[END_REF]M o r e o v e r ,t h es y m m e t r y of the lattice allows one to calculate all the possible bands by selecting a line path in the BZ (irreducible BZ). This line path includes all the important high symmetry points.

E.3 Density of states

Calculating the density of states (DOS) is another way to analyse the band structure. The DOS corresponds to the number of electronic states or unit energy within a given energy interval. It is expressed as

g(E)= X i δ(E -✏ i ) (E.9)
where ✏ i denote the one-electron energies. So the integral of (E) over an energy interval, E 1 to E 2 ,giv esthen um berofone-electronstatesinthatin terv al. Whentheδ-functionsaremultiplied by a weight factor that describes some property of the one-electron state φ i at energy ✏ i ,v a r i o u s types of DOS are obtained. One example is the projected DOS (PDOS) for molecules

µ (E)= X i | < χ µ |φ i > | 2 δ(E -✏ i ) (E.10)
The PDOS provides the projection of a function, χ µ ,o n t ot h eo r b i t a lφ i as a weight factor for the importance of χ µ in the orbital. In other words, with PDOS, information about the different contributions of the different orbitals is computed. One, then, knows which states (s, p, d ) contribute at which energy.

E.4 Kpoint sampling

Bloch theorem eliminates the need to calculate an infinite number of wave functions, because there is only a finite number of electrons in the unit (super) cell. However, there is still an infinite number of discrete k points that must be considered, and the energy of the unit cell is calculated as an integral over these points. It turns out that wave functions at k points that are close to each other are similar, thus an interpolation scheme can be used with a finite number of k points. The present work employs the Monkhorst-Pack scheme, [START_REF] Monkhorst | Special points for brillouin-zone integrations[END_REF]

1. 1

 1 Silicon phases obtained from high pressure treatments. ............... 1.2 Silicene: A 2D allotrope of silicon. .......................... 1.3 Structure of a tricyclic aromatic isomer of hexasilabenzene. ............ 2.1 The geometrical structure of Si 19 H 12 (isomerA). ................... 2.2 Calculated absorption spectra of a tetrahedral (Si 20 H 20 ) vs an o n -t e t r a h e d r a l (Si 19 H 12 )n a n o c l u s t e r . ................................. 2.3 Time evolution of the amplitude of the third harmonic of the discharge current in as i l a n e -b a s e dd u s tf o r m i n gp l a s m a . ......................... 2.4 Different stages of the growth of dust particles in a plasma reactor ........ 3.1 Schematic illustration of induced magnetic field lines and induced ring currents . 3.2 Schematic illustration of the NEB method ...................... 4.1 The geometrical structure of different isomers of Si 19 H 12 .............. 4.2 Signed modulus of the magnetically induced current densities of isomer D of the Si 19 H 12 . ......................................... 4.3 Magnetically induced current lines for benzene C 6 H 6 in comparison with those for the highly-symmetric Si 19 H 12 isomer A. ....................... 4.4 ELF plots in different planes for the Si 19 H 12 nanocrystal (isomer A) and for C 6 H 6 . 4.5 ELF plots in different planes for the presently most stable isomer D. ....... 4.6 ELF plots for three small metal clusters in comparison with isomer D. ...... 4.7 Si 19 H 12 scaffold resulting from a Nudged Elastic Band calculation between the initial Si 19 H 12 nanocrystal, isomer A, and isomer D. ................ 4.8 Absorption spectra calculated for two isomers of the Si 19 H 12 nanocluster. ....4.9 Absorption spectrum calculated for the active solar cell layer. ........... 5.1 A typical capacitively coupled plasma setup. ..................... 5.2 Scheme of an RF discharge in a parallel-plate reactor with capacitive coupling of the RF generator. .................................... 5.3 Pulsed excitation parameters. ............................. 5.4 Time evolution of the amplitude of the third harmonics for different gas temperatures. .......................................... 5.5 Schematic illustration of light passing through an optical cavity in IBBCEAS .. 6.1 Schematics of the experimental setup. ........................ 6.2 Typical trigger sequence to synchronize the iCCD with the RF power supply. .. 7.1 Extinction measured using the 610 nm mirrors. ................... 7.2 Extinction measured using the 700 nm mirrors. ................... 7.3 Extinction measured using the 610 nm mirrors for a range of gate delay times. . 7.4 Electric dipole moment ................................. 7.5 Polar molecule in an electric field. .......................... 7.6 Trigger sequence with an applied DC bias. ...................... xi List of Figures xii 7.7 Extinction measurements with and without an applied bias of 120V. ....... 7.8 Fractional difference between the measured extinction with and without an applied DC bias of 120V. .................................... 7.9 A set of extinction measurements with and without applied DC bias. ....... 7.10 Reduction in the measured absorption as function of the applied DC bias. .... 7.11 A schematic representation of the reactor chamber. ................. 7.12 Cross-sectional view of the electric field simulations. ................ 7.13 Dipole forces acting on the nanoclusters in the z-direction. ............. 7.14 A comparison of the different forces acting on a spherical nanoparticle as a function of its diameter. ..................................... 7.15 The gradient of E r along the polar axis with ✓=0 for the different applied biases. 7.16 Three-dimensional representation of the gradient of E r calculated for a 240V bias. 7.17 Schematic illustration of the distribution of particles on a plane. ......... 7.18 Reduction in the measured absorption and calculated particle density as a function of the applied DC bias. ................................. 7.19 HRTEM images of silicon nanoparticles obtained after different deposition times. 7.20 HRTEM images of crystalline silicon nanoparticles. ................. 8.1 Band structure and ball-and-stick model of graphene. ............... 8.2 Hybridization of atomic orbitals. ........................... 8.3 Free-standing silicene and germanene. ........................ 9.1 AIMD snapshots of 6 Si atoms on Ag(111). ..................... 9.2 Typical AIMD snapshots of 12 Si atoms on Ag(111). ................ 9.3 Mean square displacement of the top Ag atoms averaged over several MD steps as function of Si coverage investigated separately for Si monomers, dimers and trimers. 9.4 Atomic structure model of the silicene on HOPG system. ............. 9.5 The electronic band structure calculated from DFT. ................ 9.6 AIMD snapshots of a silicene sheet on HOPG. ................... 9.7 Typical snapshots from AIMD simulations of the epitaxial growth of a silicene monolayer on HOPG at RT. .............................. 9.8 Scanning tunneling microscopy images of one monolayer silicon deposited on HOPG at room temperature. ............................. 9.9 Scanning tunneling microscopy image of the bare HOPG substrate. ........ 9.10 Scanning tunneling microscopy image of one monolayer of Si deposited on HOPG at room temperature. ................................. 9.11 Scanning tunneling spectroscopy measurement. ................... 9.12 Xray photoemission spectra and atomic force microscopy images. ......... 10.1 Atomic structure model of the germanene on HOPG system. ........... 10.2 Electronic band structure of germanene/HOPG by DFT calculation. ....... 10.3 Atomic-resolution STM images of the bare graphite surface used as a template for the germanene growth. ................................. 10.4 STM images showing 2D islands formed after Ge deposition at room temperature and subsequent annealing to 350 • C. ......................... 10.5 Scanning tunneling spectroscopy measurements and density of states calculations. D.1 Demonstration of self-phase modulation in a laser pulse .............. E.1 The unit cell and Wigner-Seitz cell. .......................... E.2 Real space lattice and Brillouin zone of graphene. .................. E.3 Brillouin zone sampling. ................................ E.4 2D representation of PBC. ............................... E.5 Schematic representation of a slab model. ...................... Chapter 1

Figure 1 . 2 :

 12 Figure 1.2: Silicene: A 2D allotrope of silicon.

Figure 1 . 3 :

 13 Figure 1.3: Structure of a tricyclic aromatic isomer of hexasilabenzene.T h eh e xasilabenzene isomer 3a crystallizes as a naphthalene solvate (3a•2C 10H8).The inset shows hexasilabenzene isomer 3a (Tip, 2,4,6-triisopropylphenyl) for clarity.[START_REF] Abersfelder | A tricyclic aromatic isomer of hexasilabenzene[END_REF]

  [START_REF] Vach | Controlled growth of silicon nanocrystals in a plasma reactor[END_REF]F i r s t ,fl u i dm o d e lc a l c u l a t i o n sp r o v i d e dar e a l i s t i cd e s c r i ption of the low-temperature silane plasma, such as the relative density of the different plasma species, their temperature, and their collision interval times.[START_REF] Vach | Growth dynamics of hydrogenated silicon nanoparticles under realistic conditions of a plasma reactor[END_REF]U s i n gt h eo b t a i n e dd a t aa si nput parameters for the molecular dynamics simulations, the dynamics of the cluster growth, as ar e s u l to ft h ec o n s e c u t i v ec a p t u r eo fp l a s m am o l e c u l e sa n dr a d i c a l s( S i H 4 ,S i H 3 ,S i H 2 ), was followed. The simulations have always shown the spontaneous organization of Si atoms around one central Si atom leading to overcoordinated amorphous clusters. Moreover, simulations have revealed the crucial role of atomic and molecular hydrogen fluxes. The interaction of the growing cluster with hydrogen atoms and molecules determines its final structure; i.e.,c hemicalreactions with hydrogen atoms heat the growing clusters permitting the atoms to overcome local energy barriers and attain new positions, while the collisions with hydrogen molecules cool the cluster down and fix the cluster atoms in their new positions. Therefore, the resulting overcoordinated cluster could either become crystalline or remain amorphous.[START_REF] Vach | Controlled growth of silicon nanocrystals in a plasma reactor[END_REF]

2. 1 .

 1 An a t u r a lb o n do r b i t a la n a l y s i sr e v e a l e dt h a tm o s tS i -S ib o n d sa r epoor in electrons: 1.64 electrons per bond in the Si-Si bonds of the central hexagon, 1.84 electrons per bond in the bonds between the next-neighbor Si atoms belonging to an external and the central hexagon, and 1.00 electron per bond in the six bonds between the central Si atom and its 6 nearestneighbors. Furthermore, the sum of all stabilization energies due to electron delocalization in Si 19 H 12 was found to be about 12 times higher than in benzene. The highly delocalized nature of the electrons provide the structure with a great stability. In contrast to the extrastability in convential aromatic structures, such as benzene, which originates from the excess number of electrons in a single bond, the ultrastability in the present case originates from the deficiency in the number of electrons to form single bonds between neighboring atoms. Consequently, electrons in both situations will be highly delocalized providing stability to the structures.[START_REF] Vach | Electron-deficiency aromaticity in silicon nanoclusters[END_REF]

Figure 2 . 1 :

 21 Figure 2.1: The geometrical structure of Si19H12 (isomer A). Interestingly, a detailed analysis of the electron density calculations have shown that the nontetrahedral nanoclusters possess a permanent dipole moment. Isomer A, for instance, has ap e r m a n e n td i p o l em o m e n to f⇠2 Debye. The dipole moment stems from the way the charges are distributed and from the shape of the nanocluster; the relative high density of charges in the vicinity of the inner overcoordinated Si atom and its non-central position, result in a perma-

Figure 4 .

 4 3 also demonstrates the similitude between the magnetically induced current lines for benzene and those for the Si 19 H 12 (isomer A). Moreover, isomer D shows high magnetic shielding values (NICS) up to -27 ppm similar to the NICS values of isomer A that exceed those of benzene by a factor of about three.[START_REF] Vach | Electron-deficiency aromaticity in silicon nanoclusters[END_REF]

Figure 5 . 5 :

 55 Figure 5.5: Schematic illustration of light passing through an optical cavity in IBBCEAS.O u rc a v i t yh a v i n gal e n g t hd of 1m approximately could create an effective path length of 10 km due to the high reflectivity mirrors.

Figure 6 . 1 :

 61 Figure 6.1: Schematics of the experimental setup. M1,M 2:h i g hr e fl e c t i v i t y( H R ) mirrors (R>0.9999). OF1 and OF2:c o m b i n a t i o no fo p t i c a lb a n d w i d t ha n dc u t -o ff/on filters. The HR mirrors were purged with high-grade argon (99.999%).

Figure 6 . 2 :

 62 Figure 6.2: Typical trigger sequence to synchronize the iCCD with the RF power supply. Blue: plasma triggers, green: iCCD triggers.

Figure 7 . 1 ,

 71 w eshowthemeasuredextinction(unitsofcm -1 ), according to Eq. (5.8). The light intensities transmitted through the cavity with and without plasma were measured for a gas pressure of 0.115 mbar and a temperature of 150 • C. The plasma on-time was 2 s and the gate delay time (the time between the end of the RF pulse and the beginning of the acquisition measurement) was 100 ms. The spectrum was averaged over long times (1000 acquisitions). As can be seen from Figure7.1, the spectrum shows an extinction peak in the visible around 645 nm and then, the signal decreases until around 670 nm.

Figure 7 . 1 :

 71 Figure 7.1: Extinction measured using the 610 nm mirrors. Plasma on-time is 2 s, gas mixture flow is 10 sccm, gate width and delay are 10 ms and 100 ms, respectively. Gas pressure and temperature are 0.115 mbar and 150 • C, respectively.

Figure 7 . 2 :

 72 Figure 7.2: Extinction spectrum measured using the 700 nm mirrors. Plasma on-time is 2 s, gas mixture flow is 10 sccm, gate width and delay are 10 ms and 100 ms, respectively. Gas pressure and temperature are 0.115 mbar and 150 • C, respectively.

Figure 7 . 3 :

 73 Figure 7.3: Extinction measured using the 610 nm mirrors for a range of gate delay times.P l a s m ao n -t i m ei s2s ,g a sm i x t u r efl o wi s1 0s c c m ,a n dg a t ew i d t hi s1 0m s . Gas pressure and temperature are 0.165 mbar and 150 • C, respectively.

  4). The electric dipole moment, p,d e fi n e da s p = qr (7.1)

Figure 7 . 7 :

 77 Figure 7.7: Extinction measurements with and without an applied bias of 120V.The extinction is measured using the 610 nm mirrors and an applied DC bias of 120V. The plasma duration is 2 s and the gate width and delay are 10 ms and 100 ms, respectively. The gas mixture flow is 10 sccm and the pressure and temperature of the gas are 0.115 mbar and 150 • C, respectively.

Figure 7 . 8 :

 78 Figure 7.8: Fractional difference between the measured extinction with and without an applied DC bias of 120V.

Figure 7 . 9 :Figure 7 . 10 :

 79710 Figure 7.9: A set of extinction measurements with and without applied DC bias. Applied bias of : (a) 25V, (b) 100V, (c) 160V, and (d) 240V. The extinction is measured using the 610 nm mirrors. The plasma duration is 2 s and the gate width and delay are 10 ms and 100 ms, respectively. The gas mixture flow is 10 sccm and the pressure and temperature of the gas are 0.115 mbar and 15o • C, respectively.

Figure 7 .Figure 7 .

 77 Figure 7.11: A schematic representation of the reactor chamber. La n dL n indicate the lines along which the simulations of the electric field were taken into consideration in our analysis: x-component of the electric field through the light beam along L, z-component of the electric field from the top to the bottom electrode along L1 until L12.

Figure 7 .

 7 Figure 7.12 shows the potential and the electric field vector map after applying a 240V bias. It is clear from Figure7.12-b that the the electric field is symmetric in the reactor, it is mostly homogeneous in the center (vertical arrows), and strongly inhomogeneous near the edges of the top electrode.

Figure 7 . 12 :

 712 Figure 7.12: Cross-sectional view of the electric field simulations.E l e c t r i cp o t e n t i a l between the two electrodes (a) and electric field vector map (b) after applying a 240V bias.

Figure 7 .

 7 Figure 7.14),[73]w efi n dt h a tt h ec a l c u l a t e dd i p o l ef o r c e sa r ea b o u tf o u ro r d e r so fm a g n i t u d e weaker. All other forces acting on the nanoclusters (see Section 5.1.4), such as the gravitational

Figure 7 . 13 :

 713 Figure 7.13: Dipole forces acting on the nanoclusters in the z-direction.T h ed i p o l e forces are calculated assuming that the nanoclusters possess a permanent dipole moment of 2 Debye and for an applied DC bias of 240 V.

Figure 7 .

 7 Figure 7.14: A comparison of the different forces acting on a spherical nanoparticle as a function of its diameter.[73]F e is the electrostatic force exerted by the electric field on the particle in the discharge, Fi is the ion drag force, Fth is the thermophoretic force, Fn is the neutral drag force, and Fg is the gravitational force.

Figure 7 . 18 :

 718 Figure 7.18: Reduction in the measured absorption and calculated particle density as a function of the applied DC bias.

Figure ( 7 .Figure 7 .

 77 Figure(7.19) shows TEM images for the three samples. After 20 min deposition, we obtained agglomerates with the smallest between 15 and 30 nm (see Figure7.19-a). When we decreased the deposition time to 5 min, we obtained smaller agglomerates, with sizes ranging between 10

Figure 7 . 20 :

 720 Figure 7.20: HRTEM images of crystalline silicon nanoparticles. Fourier transform analysis has been used for three different nanoparticles to explore their crystalline structure. "d" corresponds to the interplanar distance calculated from the different Fourier transforms of the HRTEM images.

Figure 8 . 1 .

 81 It is perhaps the most investigated material in the past decade due to its huge advantages over conventional materials in electronics industry. One reason is the extremely high mechanical strength of the graphene sheet; the C-C bonds in graphene are about 25% stronger than in diamond. Another reason is that graphene has a peculiar electronic structure. It is neither a semiconductor nor a metal, rather it is exactly in between, and can be called a semimetal. It does not show a band gap, but the density of states is zero at the Fermi level. The electronic bands cross the Fermi level near the six corners of the hexagonal Brillouin zone (BZ). At these points, the dispersion relation, i.e. the change of the electron energy of a band within the BZ, is linear (see Figure8.1). Such a feature leads to a zero effective mass for electrons and holes near these six points, i.e. the electrons and holes behave as if they are massless or relativistic particles, so-called Dirac fermions, and are described by the Dirac equation. The six corners of the BZ are called Dirac points. The cones at the Dirac points are located at the high symmetry K points in the BZ of the honeycomb lattice. This has shown

Figure 8 . 2 :

 82 Figure 8.2: Hybridization of atomic orbitals.( a )sp 3 hybridization. (b)sp 2 hybridization.

Figure 8 . 3 :

 83 Figure 8.3: Free-standing silicene and germanene.( a )T o pv i e wo ft h es i l i c e n eh e x a g onal structure. Red and yellow atoms represent the higher and lower atoms, respectively. The buckling is highlighted in one hexagon in the bottom panel. (b) Inclined view of the germanene lattice structure and its band structure. A zoom into the vicinity of the Dirac point shows the linear bands of germanene.

( 110 )

 110 surface by Leandri et al.[START_REF] Leandri | Self-aligned silicon quantum wires on Ag (110)[END_REF] In 2007, Si-based structures were investigated using Tight-Binding Hamiltonians by Guzmán-Verri and Lew Yan Voon.[START_REF] Guzmán-Verri | Electronic structure of silicon-based nanostructures[END_REF] They have shown that silicon graphene-like sheets exhibit linearly crossing bands at the Fermi level and the name "silicene" was used for the first time in this study. In 2009, Cahangirov et al.r e p o r t e das t u d yb a s e do n first-principles calculations on the stability of both planar and buckled silicene and germanene structures.[START_REF] Cahangirov | Two-and one-dimensional honeycomb structures of silicon and germanium[END_REF] Their calculations on the phonon dispersions showed imaginary frequencies for the planar structures clearly indicating their instability, while all phonon modes were positive for the buckled ones. This denoted that buckled silicene and germanene had restoring forces to any possible deformation contrary to the planar structures that were unstable. Inspired by the theoretical prediction of the stability of silicon structures with sp 2 -like hybridization (buckled structures) and its promising electronic properties, Le Lay et al.c o n t i n u e d t o s t u d y t h e d eposition of Si onto silver substrates to form silicene nanoribbons[START_REF] Kara | Physics of silicene stripes[END_REF][START_REF] Lay | Physics and chemistry of silicene nanoribbons[END_REF][START_REF] De Padova | Evidence of graphene-like electronic signature in silicene nanoribbons[END_REF]. In 2010 and after the Nobel Prize in Physics was awarded to Geim and Novoselov for their work on graphene, research on silicene and other 2D materials had received a strong boost. Many theoretical stud-

  in 2010 by Lalmi et al., their results, based on STM measurements, showed a silicene monolayer with a relatively short lattice parameter, [116]a n dt h e i rr e s u l t s were questioned by Vogt et al.M o r e o v e r ,m a n yg r o u p sr e p o r t e dt h ee x p e r i m e n t a lr e a l i z a t i o no f silicene on Ag (111) surfaces and showed the emergence of other phases of silicene as a result

  Figure 9.1-c, Ag 1 is pulled out of the surface and Ag 2 tries to fill the vacancy left behind Ag 1 .

Figure 9 . 1 :

 91 Figure 9.1: AIMD snapshots of 6 Si atoms on Ag(111). (a) After 0.217 ps. (b) After 1.4 ps, a strong interaction between Si and Ag1 causes Ag1 to slightly move out of the surface. (c) After 3 ps, Ag1 is pulled out and Ag2 fills the vacancy behind. (d) After 3.6 ps, a Si atom substitutes the Ag2 atom in the top surface layer. Si atoms are shown in yellow and Ag atoms in light green.

Figure 9 . 2 -

 92 a shows a snapshot of the trajectory taken after 2 ps from the addition of the additional 6 atoms. Here, we find that the Si atoms continue to interact strongly with the top Ag atoms expelling them out of the top surface layer. Now, we have two Si atoms (marked with blue arrows) filling sites of the Ag top layer. After 3.9 ps, Figure9.2-b, we find three Si atoms bonding together inside the first layer of the surface of the substrate.

Figure 9 . 2 :

 92 Figure 9.2: Typical AIMD snapshots of 12 Si atoms on Ag(111). (a) After 2 ps, two Si atoms (marked with blue arrows) filling the sites of the Ag top layer (b) After 3 ps, three Si atoms bond together with other Ag atoms in the top surface layer. (c) top view of (b). Si atoms are shown in yellow and Ag atoms in light green.

2 )Figure 9 . 3 :

 293 Figure 9.3: Mean square displacement of the top Ag atoms averaged over several MD steps as function of Si coverage investigated separately for Si monomers, dimers and trimers.

Figure 9 .

 9 8-a,b, typical STM images obtained after one ML deposition of Si on HOPG kept at RT. We observe the formation of some small Si islands (very bright regions) rising above flatter zones. These flatter areas appear like a patchwork of hexagonal patterned regions with different lattice spacings. Their extension and size prevent their interpretation in terms of Moiré interference or other artifacts induced on the HOPG surface by the Si deposition. White circles in Figure9.8-a evidence the areas with wider lattice spacing, while in the top-left corner, a hexagonal region with a lower lattice spacing is clearly visible. In Figure9.8-c,d, we report the bi-dimensional Fourier transform of the two STM images. Both of them confirm the STM image texture observation, exhibiting the presence of two hexagonal arrangements rotated by 30 • with respect to another. The observed external spots correspond to four {10} and two {-11} spots of the HOPG. Indeed, since the distance d 10 (or equivalently d -11 )i se q u a lt o p 3

Figure 9 . 8 :

 98 Figure 9.8: Scanning tunneling microscopy images of one monolayer silicon deposited on HOPG at room temperature.( a , b )S T Mi m a g e s( c o l l e c t e dw i t hV sample = +0.3 V, tunneling current, Itunn = 0.3 nA) after one silicon monolayer deposited on a HOPG substrate at RT. Note the coexistence of silicene areas and small, 1 nm high, silicon clusters (white regions). (c,d) Bidimensional Fourier transform of the two STM images: two hexagons (the external one due to HOPG and the internal one due to silicene) can be noted to be rotated by about 30 • with respect to each other.

Figure 9 . 9 :

 99 Figure 9.9: Scanning tunneling microscopy image of the bare HOPG substrate. (a) STM image of the bare HOPG before the deposition of the silicon adlayer exhibiting the characteristic hexagonal pattern in which only one atom of two of the hexagonal honeycomb structure is observed due to the characteristic alternation between filled and empty pz orbitals normal to the surface. (b): Bi-dimensional Fourier Transform of the STM image of the HOPG surface. (c) HOPG atomic model.

Figure 9 . 10 :

 910 Figure 9.10: Scanning tunneling microscopy image of one monolayer of Si deposited on HOPG at room temperature.( a )S T Mi m a g e( c o l l e c t e dw i t hV sample =+0.3 V, Itunn =0 . 3n A )o b t a i n e dw i t ht h eH O P Gs u b s t r a t ek e p ta tR Td u r i n gs i l i c o ne v a po r a t i o n . Ball-and-stick models have been superimposed on the image representing Si atoms (blue and red) and C atoms (black) for better understanding. (b) Line profile obtained along the blue dashed line drawn in (a). The red arrow indicates the height of the silicene layer with respect to the HOPG surface. (c) The line profile obtained along the solid black line drawn in (a) determines the presence of a buckling of about 0.05 nm of the silicon nearest neighbors. (d) Ball-and-stick model drawn for silicene with a low buckling of the honeycomb structure. Red balls represent the highest Si atoms, while the yellow ones are slightly lower in height.

Figure 10 . 5 :

 105 Figure 10.5: Scanning tunneling spectroscopy measurements and density of states calculations. (a) Calculated total DOS for germanene on graphite. (b, c) Normalized differential tunneling conductance (V/I)dI/dV obtained by STS at room temperature: (b) on top of a germanene island; (c) on the HOPG substrate (main panel), on a Ge 3D cluster (inset). In contrast to clusters, germanene shows a finite DOS at the Fermi level. The axis labels of the inset are the same as for the main panel. Each experimental curve is an average of 10 acquisitions.

20 )

 20 The Kohn-Sham formalism solves the Schrödinger equation iteratively in a self-consistent field (SCF) fashion. SCF is needed because the Coulomb potential and the exchange-correlation potential, Eq. (B.20), are functionals of the density which itself depends on the unknown orbitals.B.2.1 Exchange-correlation functionals • Local density approximationThe difference between various DFT methods is the choice of the functional form for the exchange-correletion energy, E xc[⇢]. Several approximation methods to the E xc [⇢]a r e available. The oldest and simplest functional is the local density approximation (LDA)

  [START_REF] Vach | Growth dynamics of hydrogenated silicon nanoparticles under realistic conditions of a plasma reactor[END_REF] where a 0 , a x ,a n da c are three fitting parameters; E x GGA ,E c GGA ,a n dE c LDA are the B88 exchange functional,[START_REF] Becke | Density-functional exchange-energy approximation with correct asymptotic behavior[END_REF]L Y Pc o r r e l a t i o nf u n c t i o n a l ,[START_REF] Lee | Development of the colle-salvetti correlation-energy formula into a functional of the electron density[END_REF]a n dV o s k o -W i l k -N u s a i r( V W N )LDA[START_REF] Vosko | Accurate spindependent electron liquid correlation energies for local spin density calculations: a critical analysis[END_REF]c o r r e l a t i o nf u n c t i o n a l ,r e s p e c t i v e l y .

  [START_REF] Ro Ca I Cabarro Cas | Growth and optoelectronic properties of polymorphous silicon thin films[END_REF] where n, l,a n dm are the principle quantum numbers and Y l,m (✓,φ)i sas p h e r i c a lh a r m o n i c function. The main difference in both types lies in the radial part, R nl (r),STO : R nl (r)=N nl (⇣)r n-1 e -⇣r (B.25)GT O : R nl (r)=N nl (↵)r n-1 e -↵r 2 (B.26)

  isoften used as an improvement to the pseudopotentials. The PAW allows to use a computationally convenient representation of the wave function while maintaining the all-electron information available. This is achieved by describing the all-electron wave function in terms of a smooth pseudo-wave function ( Ψ)i nt h ev a l e n c er e g i o na n dal o c a lb a s i ss e te x p a n s i o ni nt h ec o r er egion, called the augmentation sphere, around each atom a.T h et r u e ,a l l -e l e c t r o n ,K o h n -S h a m wave function is, thus, defined as

( 2 )

 2 + P (3) + ... = ✏ o h χE + χ (2) E 2 + χ (3) E 3 + ... i (D.3)

  4) is incident on the medium. The resulting polarizationP = ✏ 0 χE 0 sin !t + ✏ 0 χ (2) E 2 0 sin 2 !t + ✏ 0 χ (3) E 3 0 sin 3 !t + ... (D.5)can be rewritten asP = ✏ 0 χE 0 sin !t + ✏ 0 χ (2) !t -sin 3!t)+... (D.6)

3 ) E 2 0

 32 E 0 sin(!t) (D.8)The expression in the square brackets is referred as the effective susceptibility χ eff = χ L + χ NL and contains both linear (L)a n dn o n l i n e a r( NL) terms. Therefore, a nonlinear contribution is added to the total refractive index of the material in Eq. (D.2)t ob e c o m e n(w, I)=n 0 (w)+n 2 I (D.9)where I is the intensity of the electric field, n 0 is the linear refractive index, and n 2 is the induced

-t 2 τ 2 ◆ (D. 16 )Figure D. 1

 2161 Figure D.1 shows how the instantaneous frequency is red-shifted to lower frequencies (i.e. longer wavelengths) in the leading edge (the left-hand wing) of a pulse and blue-shifted to higher frequencies (i.e. shorter wavelengths) in the trailing edge. The consequence is a broadening of the pulse spectrum resulting from the time-dependent intensity slope.[START_REF] Yefet | A review of cavity design for kerr lens mode-locked solid-state lasers[END_REF] For a large spectral broadening,o n en e e d sas t r o n gS P M( i.e.,al a r g et i m ed e r i v a t i v ei n the phase change). This obviously favors the use of short pump laser pulses. In practice, with sufficiently intense femtosecond laser pulses, a spectral broadening of 20,000 cm -1 (500 nm)

Figure F. 1 :

 1 Figure F.1: Schematic illustration of scanning tunneling microscopy.( a )T u n n e l i n g process between the tip and sample across a vacuum barrier. The electron wave functions Ψ decay exponentially into vacuum with a small overlap, allowing electrons to tunnel from one electrode to the other. With a positive bias voltage applied to the sample, electrons tunnel preferentially from the tip into unoccupied sample states. (b) Schematic view of the STM.[179] (c) Tunneling in the constant-current mode. The dotted line shows the path followed by the STM tip. Notice that the substituted atoms shown in red exhibit a different apparent height to the purple surface atoms due to their different local density of states. (d) STM tip operating in the constant-height mode. The tip maintains a constant height over the surface and records the current, shown by a yellow glow.

Table 4 . 1 :

 41 Summary of the skeleton energies for different hydrogenated silicon clusters.

	Cluster	E coh,sk [eV/atom]
	Si 19 H 12 Isomer A	2.9587[47]
	Si 19 H 12 Isomer B	3.0282[47]
	Si 19 H 12 Isomer C	3.1043[47]
	Si 19 H 12 Isomer D	3.1439
	Si 21 H 12	3.0742
	Si 29 H 24	3.0565
	Si 20 H 20	2.8481
	4.1.2 Electron-deficiency in silicon nanoclusters
	Ad e t a i l e dN B Oa n a l y s i so fi s o m e rDa tt h eM P 2 / 6 -3 1 1 + + g ( d , p )l e v e lo ft h e o r yr e v e a l s :

  • ,l e a v i n go n ep-orbital unchanged that sets at a right angle to the plane of the hybrid orbitals. Graphite undergoes the sp 2 hybridization and the carbon atoms form a 2D sheet of hexagons in which each atom has one unpaired electron left over. Those electrons will go freely across the matrix of atoms if one applies an electric current allowing it to readily conduct electricity.Silicon [[Ne]3s 2 3p 2 ]a n dg e r m a n i u m[ A r ]3 d 10 4s 2 4p 2 ,o nt h eo t h e rh a n d ,l i ebe l o wc a r bo ni nt h e same group in the periodic table. Therefore, they have the same number of valence electrons as carbon. However, the energy difference between the valence s and p orbitals is lower. For instance, the energy difference in silicon is only half of the corresponding value for carbon, i.e. (Si

Structure optimizations were also done with plane-wave basis sets using VASP leading to essentially identical results as with atom-centred basis sets. Quasi identical results based on different methods and functionals underlines the reliability of the calculations.

A brief introduction to the supercontinuum light source is presented in Appendix D.

A full description of our experimental setup is presented in Chapter[START_REF] Duclos | E x p e r imental study of the crystal stability and equation of state of Si to 248 GPa[END_REF].

The discharge on-time was also varied between lower and higher values in a number of experiments to measure the absorption of nanoclusters during their growth. Those experiments are not presented in this thesis, but will be the focus of future studies.

The debate on the electronic properties are briefly presented in the general introduction of part II,S e c t i o n 8.3.2.

A slab model is described in Section E.5 of Appendix E.

Acknowledgements

support team at IDRIS for all their great help. I would like to convey my hearty thanks to Hariri Foundation for Sustainable Human Development; Mrs. Bahia Hariri, the president, for the scholarship awarded to me and for opening a whole new world for me, Mrs. Dima El Hassan and Hiba Hneine for always being there for me.

Part I

Plasma-Born Aromatic Silicon

Nanoclusters

Part II

Two-Dimensional Materials

Appendix A

Molecular Dynamics Simulations

A.1 The basic idea

With the advent of high speed and massively parallel computing, simulations play a very important role in science today to the extent that their results can be compared directly with experimental results. Computer simulations have become an extremely powerful tool not only to understand and interpret experimental results at the microscopic level, but also to predict and study cases which are not accessible experimentally or which imply very expensive experiments.

One of the most commonly used theoretical approach is the molecular dynamics (MD) method, in which the time evolution of a set of interacting particles is followed by integrating their equations of motion. The particles interact by exerting forces on each other and they follow the laws of classical mechanics, notably Newton's second law of motion:

where, F i is the force acting upon the i th particle at time t; m i , a i ,a n dr i are the mass, acceleration, and position of particle i,r e s p e c t i v e l y .

With the interatomic potential, U (r 1 ,r 2 ,...,r N ), which represents the potential energy of N interacting particles in a system as a function of their positions r i , being known, the force acting upon each particle can be determined by the negative gradient with respect to the particle displacement as:

The solution of the set of equations (A.1)g i v e st h ep o s i t i o n sa n dv e l o c i t i e so fe a c hp a r t i c l ei n the system at any time; i.e.,t h ep h a s e -s p a c et r aj e c t o r i e so ft h es y s t e m .

Therefore, in order to calculate a MD trajectory of a system we need, in general, to perform the following:

• Specify the initial positions and velocities of all particles of the system of interest.

A.2 Classical vs ab initio molecular dynamics simulations

The quality of a MD calculation depends largely on the method by which the forces are calculated. In classical MD simulations, the forces are computed from "force field" models or empirical potentials with a specific functional form representing the physics and chemistry of the system of interest. The adjustable parameters are chosen such that the empirical potential represents a good fit to the results of ab initio calculations of small model systems, or they may be based on experimental data. Since most force fields are not able to model chemical reactions because of the requirement of breaking and forming bonds, it is often necessary to turn to the methodology of ab initio molecular dynamics (AIMD). In an AIMD calculation, finite-temperature dynamical trajectories are generated by using forces obtained directly from electronic structure calculations performed "on the fly" as the simulation proceeds. Thus, AIMD permits chemical bond breaking and forming events to occur. [START_REF] Iftimie | Ab initio molecular dynamics: Concepts, recent developments, and future trends[END_REF] Appendix B

Density Functional Theory

B.1 The Schrödinger equation

If we are interested in describing the electron distribution in detail, there is no substitute for quantum mechanics. Electrons are, to a certain extent, delocalized and "shared" by many nuclei and the resulting electronic cloud determines the chemical bonding. The Schrödinger equation is the fundamental equation of quantum mechanics. In principle, the solution for a system of N nuclei and n electrons can be found in the time-independent non-relativistic Schrödinger equation,

where R and r are collective variables of the nuclear and electronic coordinates, respectively. E is the total energy of the system and Ψ is the wavefunction of the 3n electronic coordinates and 3N nuclear coordinates. The many-body Hamiltonian operator, Ĥ,i sr e p r e s e n t e da s :

where the first two terms represent the nuclear and electronic kinetic energy operators, respectively. The third and fourth terms represent the nuclear-nuclear and electron-electron Coulomb repulsion, respectively, and the last term represents the nuclear-electron Coulomb attraction.

The above terms are expressed in atomic units as follows:

where r 2 I and r 2 i are the Laplacian operators involving the differentiation with respect to the I-th nucleus coordinate and the i-th electron coordinate at the R I and r i positions, respectively. the self-consistent cycle of ionic relaxations is stopped and the new positions of all atoms result in the relaxed (optimized) structure.

Several numerical algorithms are developed for finding the ground state according to the energy and its gradients. [START_REF] Wimmer | Fundamentals of the electronic structure of surfaces,v o l u m e2 . E l s e v i e r[END_REF]O n eo p t i m a ls e a r c hs t r a t e g yi st h eConjugate-Gradient method, which starts at an arbitrary point in a multidimensional space and evaluates the gradient of the function. The negative of the gradient vector gives the direction of the steepest descent which is followed until a minimum is reached along this line. In this method, the search direction contains information not only from the gradient at the current step, but also from the previous gradient.

The procedure is repeated until the value can no longer be lowered within a given convergence threshold of the residual forces on atoms.

The time dependent Kohn-Sham potential takes the following form

where the external potential, v ext (r,t), represents the potential the electrons experience due to the nuclear attraction and due to any field (perturbation) applied to the system, the second term is the Coulomb term, and the last term is the time-dependent exchange-correlation functional.

v xc is usually adopted within the adiabatic approximation. [START_REF] Bauernschmitt | Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory[END_REF] The time-dependent density is then given by the sum of the time-dependent Kohn-Sham orbitals,

What is a supercontinuum light source?

Asupercon tin uumligh tsource,firstin v en tedinthelate1960sb yR.AlfanoandS.Shapiro, [START_REF] Alfano | Observation of selfphase modulation and small-scale filaments in crystals and glasses[END_REF][START_REF] Alfano | Emission in the region 4000 to 7000 Å via four-photon coupling in glass[END_REF]c a nb e s tb ed e s c r i b e da s" b r o a da sal a m p ,b r i g h ta sal a s e r " . B r i d g i n gt h eg a pb e t w e e n incandescent lamps and lasers, a supercontinuum source combines the spatial coherence and brightness properties of a laser with the ultrabroadband spectrum of a white-light incandescent bulb. Supercontinuum light is generated by focusing a high-peak-intensity laser pulse into a dielectric material, which will strongly polarize the material in such a way that the induced polarization depends nonlinearly on the electric field of the incident light. This nonlinear lightmatter interaction is behind the transformation of the incident narrowband light into broadband radiation. Because it is produced in the guided modes of an optical fiber, the supercontinuum preserves the spatial coherence of the incident light. This results in an arbitrary-wavelength source of spatially coherent light. [START_REF] Dudley | Supercontinuum light[END_REF] When light propagates through a dielectric material, it induces a polarization, P, which is a charge separation that can be quantified as a dipole moment per volume. Therefore, the interaction between the electric field of a light wave and the material strongly depends on the strength of the electric field. For light with low intensity; i.e.,i ft h ea m p l i t u d eo ft h ee l e c t r i cfi e l di s much less than the Coulomb field of an atom, the displacement of the electrons is small and the restoring force can follow the electric-field linearly. Hence, the induced electric dipole moments are proportional to E, and the response is linear resulting in a polarization, P, of the medium described by

where χ is the linear susceptibility of the medium and ✏ 0 is the electric permittivity of free space.

In an isotropic material, χ is related to the refractive index n of the material: The lengths of the principle axes, or edges, of the unit cell (a, b, c)andtheanglesbet w een them (↵, β, γ)a r et h elattice constants,a l s oc a l l e dl a t t i c ep a r a m e t e r s( F i g u r eE.1-c).

• Lattice systems -the 14 Bravais Lattices: Lattices can be classified into "systems", each system being characterized by the shape of its associated unit cell. In three dimensions, the lattices are categorized into seven crystal systems: cubic, hexagonal, trigonal, tetragonal, orthorhombic, monoclinic, and triclinic. Each lattice is assigned to one or more of the following classifications (lattice types) based on the positions of the lattice points within the cell: primitive, body-centred, face-centred, base-centred, and rhombohedral.

Altogether, there are fourteen different ways of distributing lattice points to make space lattices in a three-dimensional lattice, called the Bravais lattices.

The structure of a crystal can, thus, be constructed by combining following elements: (i) the lattice type, which defines the location of the lattice points within the unit cell; (ii) the lattice parameters, that define the size and shape of the unit cell; (iii) the motif or the basis, which gives a list of atoms associated with each lattice point, along with their fractional coordinates relative to the lattice point.

Appendix F

Experimental Techniques

F.1 Scanning tunneling microscopy/spectroscopy

The field of nanotechnology was boosted in 1981 when the scanning tunneling microscope (STM)

was invented with which researchers are capable of observing atoms. The STM was developed by Binning and Rohrer [START_REF] Binning | Tunneling through a controllable vacuum gap[END_REF] at IBM laboratories in Zurich, for which they were awarded the Nobel Prize in Physics. The STM is a powerful tool for investigations not only of the topography of (semi) conducting surfaces with ultimately atomic resolution in real space, but also of the local electronic properties.

The STM is a non-optical microscope whose operation is based on quantum mechanical principles (see Figure F.1). A very sharp metallic tip is positioned over the specimen under study at a distance of the diameter of an atom (typically 5-10 ), and a bias voltage (V bias )i sa p p l i e d between them. Depending on the characteristics of that voltage, the electrons can jump from one side to the other by a tunneling effect, producing a weak current flow, known as "tunneling current", whose value is approximately a few picoAmperes. Therefore, when the tip scans the surface along the X and Y directions, raising and lowering it by a piezoelectric mechanism to control its level, the tunneling current is measured at each point of the digital XY matrix.

The tunneling current decays exponentially with the distance gap z and is strongly affected by the density of states of the sample at the Fermi level, ⇢ s (E F ):

where m is the mass of the electron and φ is the workfunction.

The STM can be operated in two different modes:

F.3 X-ray photoelectron spectroscopy

X-ray photoelectron spectroscopy (XPS) is a widely used quantitative spectroscopic technique to determine elemental composition, as well as chemical and electronic states of the elements in a sample. [START_REF] Crescenzi | Electron scattering and related spectroscopies[END_REF] The principle of XPS is based on the photoelectric effect. The XPS spectra are obtained by irradiating a sample surface with a beam of X-rays. The X-ray photon energy, produced either by an X-ray source such as Mg K↵ (1253.6 eV) and Al K↵ (1486.6 eV) or from a synchrotron radiation light source, is transferred to a core-level electron. If this energy is higher than the electron binding energy, the electron will be ejected from the atom (see Figure F.3).

Once ejected, the photoelectron will travel with a certain velocity, and therefore will have a certain kinetic energy which can be determined via the following equation:

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