Calculated spectra : the Raman fingerprint of rhombohedral graphite 

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Carbon-based Materials

Carbon-based materials is a family of compounds formed by any stable arrange-ment of carbon atoms at different levels of hybridization (sp, sp2 or sp3). Diamond is an example of a stable sp3 arrangement where the rigidity is simply due to the pres-ence of the σ bonds that exist as a consequence of the sp3 hybridization. Graphite (3 dimensional), graphene (2 dimensional), carbon nanotubes (1 dimensional ), and fullerene (0 dimensional) are well known examples of stable sp2 arrangements. The sp arrangements are mostly found within the vast hydrocarbon family of carbyne. The physical and chemical properties of carbon-based materials are of great interest to the physical and industrial communities. In this chapter, we will be interested in sp2 bonded carbon crystals (see Fig. 1.1) which have attracted a lot of attention within the condensed matter community. This interest was started by Bernal’s discovery of the hexagonal symmetry of natural graphite and its layered structure (Bernal, 1924 [14]) during x-ray diffraction experiments (the studied graphite has since been named Bernal graphite). This family includes, in particular, graphite, graphene, few-layer (b) A piece of natural graphite. (c) Atomic structure of Bernal graphite showing the ABA stacking sequence and the hexagonal symmetry. (d) and (e) Typical STM images of graphene and few-layer graphene respectively (adapted from Ref. [21]). (f) High resolution STM image of one single carbon nanotube where the structure of the sp2 rings can be seen (adapted from Ref. [22]). (g) Atomic structure of carbon nanotubes with the three possible arrangements of the carbon rings (armchair, zigzag and chiral).
graphene and carbon nanotubes. In the next paragraphs we will discuss some proper-ties of these materials, before looking at how these properties can be studied by Raman spectroscopy.
Graphite is the hexagonal close-packed arrangement of carbon atoms. It belongs to the P 63/mmc space group with four atoms per unit cell and a six-fold rotation axis along the stacking direction (see Figs. 1.1(b) and (c)). The cohesion of the whole crystal is ensured by the sp2 bonds in the plane directions and by Van Der Waals interaction [14, 23] in the out-of-plane direction (c-axis direction). This gives rise to the so-called layered structure which is at the origin of the exceptional electronic and mechanical properties of graphite. From a thermal point of view, graphite has a high melting points (due to σ bonding) and a good thermal conductivity, making it suitable for heat transfer applications (e.g. nuclear reactors [24]). From a structural point of view, the possibility of intercalation in bulk graphite opened a vast field of research and gave rise to an independent class of materials called graphite intercalation compounds. The physical and chemical properties of these materials are tuned by controlling the nature and the amount of the intercalated element [25, 26, 27].
Another interesting aspect of graphite is the possibility to exfoliate a single or few-layer from the graphite bulk material. Over the last ten years, and following the discovery of graphene in 2004 [28] (Nobel prize in 2010), this property has opened a vast field of research on the so-called 2-dimensional (2D) materials. The discovery of graphene has shown that 2D materials can exist under normal conditions of temper-ature and pressure. Since then, new types of 2D materials, such as phosphorene and transition metal dichalcogenides, have been isolated and studied.

Properties of graphene and few-layer graphene systems 

Graphene is a two-dimensional arrangement of sp2 bonded carbon atoms that follow a honeycomb pattern (see Fig. 1.1 (d)). It belongs to the P 6/mmm space group with two atoms per unit cell and a six-fold rotation axis in the out-of-plane direction. Fig. 1.1 (d) is a high resolution TEM (Transmission Electron Microscopy) image of one-atom-thick exfoliated sample, in which the hexagonal arrangement of carbon atoms is clearly observable. Graphene has two important properties that make it one of the most interesting materials nowadays. First, its high electronic mobility (15000-200000 cm2/V.s) exceeds any mobility of all the known semi-conductors. Second, its rigidity (that is due to the sp2 bonding) makes it one of the hardest materials. These two properties are relevant features in a wide domain of applications such as optoelectronics, transistors, sensors, solar cells and many other areas [29, 28, 30, 31].
Few-layer graphene systems are systematically found during the exfoliation pro-cedure from bulk graphite. Exfoliated samples may exhibit different thicknesses that are multiples of the inter-plane distance in graphite, as shown in Atomic Force Mi-croscopy (AFM) measurements [32]. These systems are called multi-layer graphene or few-layer graphene. The electronic and optical properties of FLG have been shown to be different from those of graphene and depend of the number of layers (composing the FLG sample) and the stacking mode sequence [33]. Fig. 1.1 (e) shows a typical high resolution Scanning Tunneling Microscope (STM) image of a few-layer graphene sample.
Another type of sp2 bonded carbon crystals is a one dimensional arrangement of carbon atoms to form the so-called carbon nanotubes. These systems are one di-mensional materials in the sense that their wave function decays exponentially in the directions perpendicular to the nanotube. Thus, the charge carriers are confined to just one dimension. This suggests that 1D materials have many interesting features one can only find in one dimension (e.g. Van Hove singularities). A SWNT (single wall carbon nanotube) can be described as a single layer of a graphene crystal that is rolled up into a seamless cylinder. Fig. 1.1 (f) is typical STM image which reveals the atomic arrangement of the sp2 rings within one single carbon nanotube. SWNTs are predicted to be metallic or semiconductors [34, 35] depending on their diameter and the helicity of the sp2 rings arrangement in their walls (zigzag, armchair or chiral, see Fig. 1.1 (g)). This dependence on the diameter and the helicity, which was confirmed in many experimental works [22], has opened a large area of possible applications in electronics and optics (for further reading, see the book of R. Saito, G. Dresselhaus and M.S. Dresselhaus [36] on carbon nanotubes).
In the rest of this chapter, we will focus on graphene and multilayer graphene systems, presenting firstly their electronic and vibrational properties.

Properties of graphene and few-layer graphene systems

In this section, the electronic and vibrational properties of graphene and FLG systems are discussed using Density Functional Theory (DFT) and Density Functional perturbation Theory (DFPT). For the sake of simplicity, only fundamental properties are presented in this chapter, leaving more detailed study (such as the effect of electrons correlation) to be presented in the subsequent chapters. The understanding of a key property, called Kohn anomaly, is essential to the broader understanding of the Raman spectra in FLG systems and will be presented in an independent section.

The adiabatic approximation

Let us consider a crystal formed by electrons and nuclei interacting via the Coulomb law. The time scale associated with the motion of nuclei is usually much smaller than the time scale associated with electrons due to the high mass ratio Mproton/me ∼ 1836. This difference in time scale suggests that the quantum treatments of electrons and nuclei are different. Indeed, this observation is taken into account within the adiabatic approximation [37], where we assume that the nuclei motion is decoupled from the electrons system. In the adiabatic approximation, the electrons are assumed to be capable of instantaneously adjusting their motion to the relatively slow motion of the ions. This can be cast in a formal mathematical way by assuming that the total wave function Ψ of the system can be written as a product of two wave functions containing the electronic and nuclear degrees of freedom separately [38] :
This approximation was successfully applied to the majority of materials including insulators metals and semimetals. However, the full proof does not give any guarantee of the validity of the adiabatic approximation on metallic systems. In a metal (or a semimetal), the energy difference between excited electronic states in Eq. 1.4 is zero and the electrons may, in principle, jump between excited states while being perturbed by the nuclear motion. This means that the two systems can no longer be considered independent. Nowadays, the adiabatic approximation is always used to study the electronic and vibrational properties of metallic and semimetallic materials with an accuracy of a few percent and it will be used in our work to study graphene, few-layer graphene and graphite.

Electronic properties

Fig. 1.2 (a) shows a schematic for the honeycomb structure of graphene in which the unit cell has two atoms labeled A and B. The whole lattice can be regarded as the superposition of two sub-lattices (A and B sub-lattices) arising from translation symmetry operations acting on the A and B atoms in the unit cell. The overlap between the hybridized orbitals of the carbon atoms during the formation of graphene gives rise to a σ bonding in the graphene plane while the remaining π orbitals overlap gives rise to a π bonding. The σ bonds are mainly responsible for the mechanical properties of graphene and they ensure the in-plane cohesion of the whole crystal, while the π bonds are mostly responsible for optical and electronic properties of graphene.
Fig. 1.2 (c) shows the electronic structure of graphene calculated with density functional theory (DFT) in the LDA approximation 1. The Fermi surface in graphene is reduced to two points : K and K0, edges of the first Brillouin zone (BZ) (see Fig. 1.2 (b)). Due to the hexagonal symmetry of the graphene honeycomb lattice with two atoms per unit cell, the dispersion of the electronic bands in graphene near the Fermi level is found to be linear. This implies that the charge carriers in graphene are described by a Dirac-like Hamiltonian, with a Fermi velocity of vf = 106 m/s ∼ 0.003 c, where c is the light velocity. This characteristic makes graphene very interesting from a fundamental point of view as it offers a solid bridge between theoretical electrodynam-ics of massless fermions in high-energy physics and the practical low-energy condensed matter physics.
In few-layer graphene systems, in addition to the in-plan σ and π-bondings, we have the interaction between the π orbitals from different graphene sheets. This interaction makes the electronic bands dispersion in FLG different from that in graphene, as there are more conduction and valence bands in the vicinity of the Fermi level (see the case of bilayer in Fig. 1.2 (d)). Although the linear dispersion is lost in FLG, the semimetallic character is preserved and the Fermi surface is always reduced to the points K and K0 (See Fig. 1.2 (d) for the bilayer case). After the isolation of FLG (2004) it was soon realized that new optical and electronic features can be found within FLG. For two layers, gated bilayer graphene was shown to possess a widely tunable band gap (up to 250 meV) [41]. When there is more layers (Nlayers > 2), the possibility of having different stacking sequences is shown to be of crucial importance [42, 42]. We will study the stacking order in FLG more rigorously when we deal with the application of Raman spectroscopy on stacking order determination.

Vibrational properties

The motion of the nuclei of given system determines its vibrational properties. The quantum treatment of nuclei is carried out, in general, within the adiabatic approxi-mation. Let us consider a crystal structure belonging to a given Bravais lattice with Na atoms per unit cell. The positions of the atoms may be labeled by an index I; this contains the unit cell index l to which a given atom belongs and the position of the atom within that unit cell indexed by s; I ≡ {l, s}. The position of the Ith atom is thus :
where Rl is the position of the lth unit cell in the Bravais lattice; τs is the equilibrium position of the atom in the unit cell, and uI = ul,s indicates a deviation from the equilibrium position. In order to access the nuclei eigenstates Ψi(R), one has to solve where uαI is the αth Cartesian component of uI . Usually, the thermal wavelength of nuclei in solids is small compared to their distance. This allows us, in general, to treat nuclei as classical particles which results in the so-called classical nuclei approximation. In this approximation, one solves the Newtonian equations of motion for the ions and not the full Schrödinger equation in Eq. 1.7. To do that, one shall use the classical version of the previous Hamiltonian, which describes a set of decoupled harmonic oscillators. The vibrational frequencies ω of these oscillators are determined as the solution of the following linear system :
where 1is the identity matrix. The derivative of the energy with respect to atomic displacement in the previous equation is called the inter-atomic force constants (IFC), usually denoted by Cs,tα,β(Rl, Rm). Due to translation invariance, the IFC is a function of Rn = Rl − Rm. The linear system in Eq. 1.10 is infinite and its resolution is usually carried out by the Fourier transform technique [6]. Let us define the mass scaled Fourier transform of the IFC :
where q is a vector belonging the first BZ in reciprocal space. The quantities ω2 are now defined for each wave vector q as the solution of the following linear system :
The subscripts s, t in Eq. 1.12 run over the atoms within the unit cell. The quan-tity Dstαβ(q) is called the dynamical matrix. It is a second rank tensor of dimension (3 × Na) × (3 × Na) (Na is the number of atoms per unit cell) with eigenvalues ωq2ν and eigenvector eqν representing the squared-frequency and the polarization of the decoupled harmonic oscillators of the system in Eq. 1.9 now called phonons. The displacement uI of an atom in the unit cell Rl at a position τs is now given by :
Within DFT, one can show that the dynamical matrix (or the IFC) is a functional of the first derivative of the electronic density at the equilibrium nuclear positions (see Ref. [6]). This fundamental result, first stated by De Cicco and Johnson (1969) and by Pick, Cohen, and Martin (1970) [6], transform the nuclear problem in Eq. 1.10 into a simpler one where we just need to calculate (self-consistently) the change of the electronic density with respect to a phonon perturbation. This type of calculation, is usually carried out using density functional perturbation theory (DFPT) [6] which constitutes our method to study vibrational properties.
An atomic displacement generated by a given phonon mode may or not be invariant under the action of a crystal symmetry. The set of symmetry operations under which an atomic displacement is invariant, is called the symmetry group of the phonon mode that generate this atomic displacement. Clearly, this symmetry group is a sub-group of the crystal space group. It is important to determine the symmetry properties of phonons in a crystal, since they determine whether or not a phonon mode is Raman active. This kind of analysis is carried out using group theory. A full discussion of phonon symmetries is beyond the scope of the present work, but the reader can find good references in the domain (see for example Ref. [43]). Table 1.1 recapitulates commonly used notations in the literature to label the phonon modes in crystals with respect to their symmetry properties.

Properties of graphene and few-layer graphene systems 

Figure 1.3 – (a) and (b) Ab-initio phonon dispersion of graphene and bilayer graphene respectively (technical details are given in Chap. 3) . The « i » and « o » prefixes in labeling the phonon modes stand for « in-plane » and « out-of-plane » vibrations respectively. (c) Schematics for the -point phonon displacement pattern for graphene and bilayer graphene, where each phonon mode in graphene gives to two phonon modes in bilayer. The Raman (R) and infrared (IR) activity is also indicated. Figure (c) is adapted from Ref. [44].
We will now introduce now some useful definitions regarding the type of phonon looking at it independently from its symmetry properties. A phonon is called lon-gitudinal (transverse) if the corresponding displacement of the atoms uI is parallel (perpendicular) to the phonon wave vector q. Acoustic (optical) phonons are those for which the displacement of atoms in the unit cell is in-phase (out of phase). If the atomic displacement of the atoms following a given phonon mode is in the plane (out of plane) in a layered material, the phonon is called an in-plane (out of plane) vibration.
Applying the above theory to the case of graphene gives the phonon bands disper-sion presented in Fig. 1.3 (a). Graphene has two atoms per unit cell, so we have in total 6 phonon modes, namely 3 optical and 3 acoustic modes. At the point, they are reduced to 4 independent modes due to symmetry. These modes are depicted in Usually, in crystals with a center of symmetry (e.g. graphene), phonons that have the index « g » (from the German « gerade ») are symmetric with respect to inversion and are Raman active (see Fig.1.3 (c)). In graphene, at the point, there are only two phonon modes that are Raman active. The first Raman mode is the E2g (see Fig. 1.3 (c)) mode which is a doubly degenerate in-plane optical vibration. The second mode is the B mode (Fig. 1.3 (c)). Although the B mode is Raman active, this mode is undetected in Raman scattering experiments because of its weak electron-phonon coupling which can be put down to its out-of-plane vibration character (see Sec. 1.3).
Few-layer graphene have almost the same phonon band dispersion as graphene (see the case of bilayer in Fig. 1.3). Although phonons symmetries are different, we still have the E2g vibration at the point in the BZ which is Raman active. Furthermore, in FLG, a set of new phonon modes which are not present in the case of single layer graphene can be found. These modes are related to layers displacement parallel or perpendicular to the stacking direction. Namely, the Layer Breathing Modes (LBM) and the shear mode (see Fig. 1.3 (c)). These phonons have weak vibrational frequencies because of the weak Van Der Waals interaction between the graphene plan and this makes them hard to detect in Raman or neutron scattering experiments.

Kohn anomalies

Kohn anomalies (KA) [45] in metallic and semimetallic systems are abrupt soft-ening of the phonon frequencies for phonons with wave vector q which connect two points on the Fermi surface. To explain why these anomalies occur at these particular phonon wave vectors, let us first remind the expression of the dressed phonon fre-quency according to perturbation theory when electron-phonon 3 interaction is taken into account. We have [46] :
Figure 1.4 – (Color online) Upper panel: Lines are the phonon dispersion of graphene (GE), calculated at the experimental and equilibrium lattice spacings (aexp = 2.46Å and ath = 2.248Å). Experimental data from Ref. [47]. The red straight lines at and K are obtained from evaluating the phonon self-energy within DFT. The two lower panels correspond to the dotted window in the upper panel. Here, graphite (GI) computed frequencies are also shown. The points are theoretical frequencies obtained by direct calculation. A single GE band corresponds to two almost-degenerate GI-bands. Adapted from Ref. [48].
If the phonon frequencies are neglected in the denominators in Eq. 1.15, the effect of the electron-phonon interaction on the phonon frequencies is said to be static. This approach is called static in the sense that the obtained dressed frequencies can be recast in a perturbative approach where the phonon is treated as a time independent perturbation. In this case, the energy denominators go to zero for wave vector q such that k,n − k +q,m = 0 on the Fermi surface. This singular behavior gives rise to KA and a softening of the phonon frequency at wave vector q that connect two points on the Fermi surface.
In graphene, graphite and FLG, this condition is satisfied for q = and q = K. The phonon dispersions of graphene and graphite, as measured experimentally, exhibit an anomalous behavior at these particular wave vectors (see Fig. 1.4). This anomalous behavior has been explained by the presence of KA in Ref. [48]. In Fig. 1.4 (upper panel), the two sharp kinks in the phonon dispersion at and K points indicate the two Kohn anomalies at K and . Due to the presence of Kohn anomalies, the phonon frequencies in graphene, graphite and FLG undergo a softening as high as 80 cm-1 [48].
Experimentally, as shown in Fig. 1.4, we observe that the Kohn anomaly in graphene and graphite is visible only for the E2g phonon mode at and the A01 phonon mode at K. The other branches do not soften, eventhough, their phonon wave vector satisfies the Kohn anomaly condition. In fact, in Eq. 1.15 we see that the electron-phonon coupling enhances the KA. Since the E2g mode at or the A01 mode at K possess the highest electron-phonon couplings, Kohn anomalies are visible only for these two branches. The other branches have an electron-phonon coupling which is one order of magnitude smaller than the A01 phonon at K [48] and do not show any softening.
Taking into account the existence of Kohn anomalies while studying the Raman spectrum of graphene, FLG, carbon nanotubes and graphite is of great importance. In fact, a redshift of the peaks that arise from phonons affected by the Kohn anomaly, is expected. In carbon nanotubes (see Sec. 1.1), while metallic nanotubes possess Kohn anomalies, semiconducting nanotubes do not. Thus, Raman spectroscopy can be used to distinguish between metallic and semiconducting carbon nanotubes [48] simply by observing the occurrence the peak’s redshift.
Dynamical case If phonon frequencies are not neglected in the denominators of Eq. 1.15, the treat-ment of the electron-phonon coupling is said to be dynamic. We notice that dynamical effects move the Kohn anomaly with respect to the static case since the singularity in the denominators is now shifted due to the presence of the ~ω term.
Doped graphene, is a spectacular system where the static approximation fails mis-erably [49] 4. In fact, the static phonon frequency of the E2g mode in graphene at the BZ center does not show any dependence on the doping, in contradiction with experi-mental measurements (see Ref. [49] and references therein). As revealed in Ref. [49], the observed dependence on doping as well as the stiffening of the G Raman peak in doped graphene is a consequence of moving the KA away from the q = 0 case due to dynamical effects. Indeed, by evaluating the integral (or the sum over k, n, m) in the expression given the self-energy with linearized electronic bands (graphene) we find the following expression for the dressed E2g phonon frequency at the -point [49]:
where α is a constant : α = 4.39 × 10−3 and f is the Fermi level shift due to doping. Thus, while the static approximation works very well for most metallic systems, dy-namical effects are very important while studying doped graphene. We will return to discuss these effects and their implications on the Raman spectrum of doped graphene in Sec. 1.4.2.

Raman scattering

Inelastic light scattering is one of the most powerful tools used to study fundamen-tal properties of materials. Raman scattering, a well known example, is associated with inelastic scattering of light by elementary excitations of the materials. Histori-cally, Raman scattering has been associated with phonons, although other elementary excitations can be involved (plasmons for example) which may give rise to interesting phenomena [2]. In this work, only scattering by electrons and phonons will be consid-ered. In Raman scattering, the energy difference between the incoming and outgoing light is equal to the sum of the absorbed or emitted phonon energies. This energy difference is usually called the Raman shift. A Raman experiment consists of measur-ing Raman spectra which give the intensity of the scattered light as a function of the Raman shift.
The electromagnetic field interacting with the material system, during the Raman scattering, can be seen as quanta of energy called photon. These photons are charac-terized by a wave vector k and frequency ωL as dictated by standard electrodynamics. The incoming radiation ωLi will scatter either elastically (ωLi = ωLo) or inelastically (ωLi 6= ωLo), where ωLo is the frequency of the scattered light. In the latter case, the material system is promoted to an excited state which, in our case, corresponds to a phonon. The phonon (of frequency ωqν ) may either be emitted (Stokes processes) or absorbed (anti-Stokes processes) by the material system. Energy conservation law implies : ωqν = ωLi − ωLo for Stokes processes and ωqν = ωLo − ωLi for anti-Stokes processes, see Fig. 1.5. If the laser energy ωLi is much smaller than the material band gap, Raman scattering is said to be non-resonant. In this regime Raman scattering can be formulated in terms of the polarizability tensor [5, 6, 4]. This is the so-called macroscopic view of Raman scattering (or equivalently, the dielectric approach to Ra-man scattering). If, on the other hand, the laser energy is comparable to the material band gap, Raman scattering is said to be resonant. In this regime, the macroscopic view is no longer adequate and Raman scattering has to be formulated from a micro-
scopic view in which Raman cross sections are calculated directly from perturbation theory [12]. In the next two paragraphs a brief discussion of the two formulations is given. Since graphene, FLG, and graphite are semimetallic systems, Raman scattering in these materials is always resonant with any external laser energy. Hence, this work will be based on a microscopic view of Raman scattering.

Macroscopic view

In this section, Raman scattering is formulated within a simple classical macro-scopic view which is relevant to introduce some important quantities and definitions such as the Raman tensor. Suppose a given dielectric material is interacting with an oscillating electromagnetic field : E(r, t) = E(ki, ωi) cos(ki.r − ωit) where ki and ωi are the momentum and frequency of the incident electromagnetic wave. This interac-tion induces a polarization P(r, t) = P(ki, ωi) cos(ki.r − ωi.t) in the dielectric with the same space and time periodicity. The relation between the amplitudes of the incident radiation and the induced polarization is by definition given through the polarizability tensor. Namely, in the linear regime we have :
where α and β indicate the Cartesian directions. If the material system is at a finite temperature, the polarizability χ is subject to thermal fluctuations. In the presence of a phonon with polarization vector eqν and frequency ωqν (one single monochromatic phonon perturbation), the collective atomic displacement Q(r, t) associated with this phonon can be expressed as a plane wave:
where the vector Q(q, ωqν ) is the amplitude of the atomic vibrations. The components of this vector are the vibrational amplitudes of each atom in the unit cell. Normally, the atomic displacements are small compared to the lattice spacing so one can develop the polarizability tensor as Taylor expansion in Q(r, t). At the first order of this expansion one has :
The first term between brackets in the previous equation corresponds to Stokes scat-tering, while the second corresponds to anti-Stokes scattering. Within this classical description, Raman scattering is simply related to the dependence of the polarizability tensor on the ionic degrees of freedom. As well known, the classical description of Raman scattering does not provide the correct ratio between Stokes and anti-Stokes processes. In order to describe correctly this ratio, one needs to pass to the quantized version of the previous theory where the atomic displacements are treated as quantum operators.
Momentum conservation implies ks ± q = ki where ks is the momentum of the scattered light and the ± sign is for Stokes and anti-Stokes processes. Since |ki,s| is much smaller than the typical size of the BZ in crystalline systems, the phonon wave vector involved in the one phonon processes described above is restricted to zone center phonons. This constitutes the fundamental selection rule for Raman scattering. If the Taylor expansion in Eq. 1.19 is pushed to higher orders, Raman scattering will involve multi-phonon emission or absorption and the fundamental selection will impose that the sum of all the emitted or absorbed phonon wave vectors is zero. Let us denote the incident (scattered) light polarization by ei (es). According to the dipole radiation formula at long distances, the scattered intensity is given by :
More selection rules can be derived from symmetry properties of the Raman tensor which can be deduced from the combined symmetry of the involved phonon and the po-larizability tensor. The simplest one would be for centrosymmetric systems where only symmetric phonons at BZ center are Raman active (the modes that have a subscript « g » as discussed in Sec. 1.2.3). The inversion symmetry in these systems implies that the Raman tensor must be invariant under inversion. Given that the tensor ∂χ/∂Q is symmetric under inversion, anti-symmetric phonons give rise to an anti-symmetric Raman tensor which is in contradiction with the inversion symmetry requirement.
Let us assume the total system formed by the material and the electromagnetic field to be described by the total Hamiltonian Htot = H + Hl + Hel, where H is the Hamiltonian describing the material system, Hl is the Hamiltonian describing the radiation field and the term Hel represents the interaction between the two systems. The Hamiltonian H can be decomposed, in a first approximation, into an ionic and an electronic part according to the adiabatic approximation (see Sec. 1.2.1), H = He + Hp. Furthermore, the interaction between electrons and phonons is described by the Hamiltonian Hep (see the next chapter for an exact description of the electron-phonon interaction). Under these conventions, Raman scattering with phonons can be see as a three steps process (see Fig. 1.6) : (a) first, an electron hole pair is created after the annihilation of an incoming photon via Hel (b) second, the electron or the hole is diffused by emitting or absorbing a certain number of phonons n with a momentum qi=1,n such that Pni=1 qi = 0 (c) the electron and hole meet again and recombine to emit the scattered photon via the Hamiltonian Hel. Perturbation theory up to the n + 2 order (where the Hamiltonians Hel and Hel are treated as perturbations) can be used to describe such a process. In this section, the formula that gives the scattered Raman intensity as a function of electron and phonon properties is presented. Its detailed derivation is provided in the next chapter. The scattered intensity of n-order Raman scattering is given by the generalized Fermi golden rule [2] :
where i and f stand for initial and final states and Ei, Ef the corresponding total energies (material system and the radiation field). The kets |sii represent the total system intermediate states with energies E0,..,n and γ is the sum of electron and hole lifetimes. In this work, we will be interested in Raman scattering such that the initial and final electronic states of the material system are assumed to be the electronic ground state. A more general Formula can be derived for Raman scattering with defects by including the defect perturbation matrix elements in Eq. 1.27 (See Ref.[19]).
In this thesis, we will focus on two-phonon Stokes Raman scattering where the ionic part of the material final state contains two phonons. The processes described by Eq. 1.27 are generally associated with lines which are much weaker than first-order Raman lines, since they arise from higher order processes in the perturbation series. Graphene, FLG and graphite are notable exceptions. During the intermediate virtual transition the energy is not necessarily conserved and the three denominators of Eq.1.27 are generally different from zero. However, in graphene, FLG and graphite two or more of the denominators of Eq. 1.27 can be equal to zero simultaneously and the intermediate electronic states are real excited states of the material. In the literature, this is called resonance condition and double-resonant in the case of two phonons Raman scattering. Raman peaks associated to resonant processes have an intensity comparable to that of lowest-order processes and are called resonant peaks and double resonant in the case of two phonons scattering.

Raman spectroscopy in graphene

The example of graphene is very suitable to introduce the reader to the field of Raman spectroscopy in carbon-based materials in general and FLG and graphite in particular. It is also a good example where Raman spectroscopy has been proven to be very useful. In this section we will discuss the most important Raman peaks in graphene and how they are used to probe its electronic and vibrational properties.
The G peak at 1580 cm-1 (Fig. 1.7 (a) and (b)) is a first order peak, and is related to the bond stretching phonon (E2g) at the points in the BZ. The Raman process giving rise to the G mode (see Fig. 1.7 (b)) can be described as follows. First, an electron is exited from the conduction band to an empty state in the valence band. Second, the exited electron will be scattered by a phonon from the BZ center within the same Dirac cone. Finally, the recombination process takes place with the emission of the scattered photon. Although it is a simple first order peak, it has been shown (see Ref. [51]) that the structure of the G peak involves electronic transitions from a wide region in the BZ. Thus, even if it is a first order peak, ab-initio simulations of this peak is quite challenging. Only recently, the G peak was finally reproduced from first-principles in Ref. [52].
The 2D Raman peak, which will be extensively studied in this work, is situated around 2700 cm-1 (Fig. 1.7 (a) and (c)). It is a two phonons Raman peak and it is double resonant. Now we will discuss the exact meaning of this double resonance character. For graphene (and graphite), if one tries to carry out a simple numerical evaluation of the Raman cross section in Eq. 1.27 by neglecting all the matrix elements in the numerators for a given laser energy, one finds that for a certain phonon wave vector q near the K point in the BZ, the scattered intensity is clearly enhanced. This idea was explored by C. Thomson and S. Reich in Ref. [54] where they showed that this enhancement is a result of the double resonant character of two phonons peaks in graphene as discussed in the next paragraph.
The Raman process responsible for the 2D peak consists of (see Fig. 1.7 (c)): (a) the creation of an electron-hole pair near the Dirac point K at wave vector K + ki
(b) the scattering by a phonon with momentum q of the electron or a the hole (c) the scattering by a phonon with momentum −q of the electron or a the hole (−q because of momentum conservation) (d) the recombination of the electron-hole pair  between 1000 cm-1 and 3300cm-1. (b) double-Resonant Raman processes responsible of the G, D and 2D peaks in graphene. The black lines are the Dirac cones, red and blue arrows are optical transitions and dashed lines represent the scattering by a defect or a phonon. Adapted from [53].
and the emission of the scattered photon. If the phonon that is involved in the previous process is around the K point in the BZ, namely q = K + d where d is a small vector, from momentum conservation, the scattered electron will have a momentum of K + ki + K + d ∼ 2K + ki ∼ K0 + ki. Let us neglect the phonon energy for simplicity. The intermediate electronic states before the scattering (at momentum K + ki) and after the scattering (at momentum K0 +ki) are real excited states of the material (Fig. 1.7 (c)). In Eq. 1.27, this means that, at least, two denominators in the expression of I are simultaneously equal to zero. This kind of process is called double resonant Raman scattering. Due to this resonance, the intensity of the 2D peak is enhanced and is comparable to that of the first order G peak. According to this view, the double resonant condition acts as a selection rule on the phonon wave vector and only phonons around K (and K0) contribute to the Raman cross section of the 2D peak.

Table of contents :

1 Raman spectroscopy in graphene and few-layer graphene 
1.1 Carbon-based Materials
1.2 Properties of graphene and few-layer graphene systems
1.2.1 The adiabatic approximation
1.2.2 Electronic properties
1.2.3 Vibrational properties
1.2.4 Kohn anomalies
1.3 Raman scattering
1.4 Raman spectroscopy in graphene
1.4.1 Uniaxial strain
1.4.2 Doping
1.4.3 Edges, defects and disorder
1.4.4 Counting the number of layers
1.5 Stacking order determination in few-layer graphene
1.6 The role of Raman simulations in stacking order determination
2 Raman simulations and challenges 
2.1 Theory of Raman scattering : An overview
2.1.1 Second quantization for electrons and phonons
2.1.2 Electron-phonon coupling
2.1.3 Electron-photon coupling
2.1.4 Second order Raman scattering
2.2 Electron correlations effects beyond LDA
2.2.1 Effects on electron energies
2.2.2 Effects on phonon frequencies and the electron-phonon coupling
2.3 Difficulties of ab-initio resonant Raman calculation
3 Calculation of Raman intensities using the Wannier interpolation scheme and the BZ reduction method
3.1 Wannier interpolation
3.1.1 Maximally localized Wannier functions
3.1.2 Wannier Interpolation schemes
3.1.3 Summary of Wannier interpolation
3.1.4 Case study : single and double layer graphene
3.2 Reduction method
3.2.1 General formulation
3.2.2 Case study : single and double layer graphene
3.3 General procedure for Raman simulations
3.4 Tests of Raman simulations in single and double layer graphene
4 Application for 2D systems : trilayer and tetralayer graphene
4.1 State-of-the-art of Raman simulation in few-layer graphene
4.2 Electron bands dispersion
4.3 Phonon frequencies
4.4 Calculated spectra
4.5 Analysis of trilayer graphene Raman spectra
4.6 TO branch dispersions from Raman spectra in trilayer graphene
4.7 The Difference between ABA and ABC trilayers 2D peaks
4.8 Conclusion
5 Application for 3D systems : Bernal and rhombohedral graphites 
5.1 State-of-the-art of Raman simulation in graphite
5.2 Electron bands and phonon frequencies
5.3 Calculated spectra : the Raman fingerprint of rhombohedral graphite
5.4 Width of the 2D peak in AB and ABC-stacked graphite
5.5 Spectra analysis of AB-stacked graphite
5.6 TO phonon bands dispersion from Raman spectra
5.7 Conclusion

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