Calculation of Raman intensities using the Wannier interpolation scheme and the BZ reduction method 

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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 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.
The D peak (respectively D0 peak, see Fig. 1.7 (b)) at around 1350 cm-1 (1600 cm-1 resp.) (Fig. 1.7 (a) and (c)), characterizes the presence of defects. These peaks are absent in pristine graphene. The Raman process responsible for the D peak is the same as the one responsible for the 2D peak except that the electron (or the hole) is scattered by a defect and by a phonon (Fig. 1.7 (c)). Following the same argument as the one presented in the previous paragraph, one finds that the D peak is also double resonant and is assisted by phonons around the K and K0 .

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Uniaxial strain

Uniaxial strain in graphene consists of compressing or stretching the whole crystal out of its equilibrium shape along a given direction. The modification of the carbon-carbon bond due to strain has a direct consequence on the G Raman peak, as this latter is related to the E2g phonon mode at . While in unstrained graphene at the -point, the E2g phonon (1581 cm-1) is two-fold degenerate, strain introduces a lifting of this degeneracy, giving rise to two independent phonon modes [57]. The Raman signature of this splitting is a splitting of the G peak in graphene into two sub-peaks called G+ and G-. Each sub-peak is related to one of the two E2g strain-induced modes. This was observed and confirmed through the experiments reported in Refs. [55, 56]. Fig. 1.8 (a) shows the red-shift and the splitting of the G mode under uniaxial strain in graphene. An important observation was made by authors of Refs. [55, 56] concerning the polarization behavior of the G+ and G- sub-peaks. As these two peaks are assisted by two orthogonal phonon modes in graphene, their polarization behavior is predicted to be different. Polarization study 5 confirms this prediction. Furthermore, the angle between the strain direction and the phonon polarization vectors can be easily ex-tracted from the polarization behavior of the G+ and G- peaks. On the other hand, the two phonon polarization vectors arising from the splitting of the E2g mode are di-rectly related to the crystallographic orientation of the sample. Therefore, the relative position between the strain direction and the phonon polarization vectors of the two strain-induced E2g modes determine completely the orientation of the sample. Thus, Raman spectroscopy offers a simple and efficient way to determine the crystallographic orientation of the graphene sample which turns out to be crucial in a wide domain of applications such as in nanoribbons [58] and quantum dots [59] technologies.

Edges, defects and disorder

The study of defects in graphene was simplified enormously by the use of Raman spectroscopy. there exist Raman peaks characteristic of defects exist (they are present only if the sample has defects) : the D peak around 1350 cm-1 and the D0 peak around 1600 cm-1. This allows for the characterization of defected graphene through the use of Raman spectroscopy.
Several type of defects can be found in graphene. They can be regrouped in two big families, namely, point defects and extended defects. Examples of point defects include : vacancies, divacancies, Stones-Wales or charged impurities. On the other hand, extended defects are delocalized along a 1D region within the graphene sheet. Edges and grain boundaries are the most studied extended defects.
Point defects have been studied since the early days of Raman spectroscopy in graphite. Tuinstra and Koening [66] proposed a formula that gives the dependence of I(D), the intensity of the D peak, on the defect concentration : I(D) ∼ 1/L2D where LD is the average defect-defect distance [66]. This relation was further studied and confirmed in the work of Cançado et. al. [65]. However, in Ref. [65], the authors show the existence of two distinct regimes. For low defect concentration (LD > 3 nm) the Tuinstra and Koening formula holds. This is completely understandable within an independent defect picture where the contributions arising from all the defects add up constructively and I(D) is simply proportional to defect concentration (1/L2D). For high defect concentration (LD < 3 nm, see Fig. 1.10 (a)) the intensity of the D peak is shown to be inversely proportional to defect concentration and the Tuinstra and Koening formula no longer holds. This can be understood in a simple way. When the defect-defect distance reaches the limit of the electron-hole free path length before a phonon scattering (which is around ωD ∼ 3 nm), the Raman process that gives rise to the D peak will be less probable with an increasing defect concentration. While the scattering with a defect becomes more probable, the scattering with a phonon (which is essential for the appearance of the D peak) becomes less probable. This is due to the fact that, at high defect concentration, the electron-hole pair will generally recombine before the scattering with a phonon takes place [53]. The ratio I(D)/I(G) was proposed as a good estimator for defect concentration in defected graphene [65]. The existence of the previously discussed two regimes seems, at first, to be a limitation for the use of Raman spectroscopy, since for each I(D)/I(G) ratio, two values for defect concentration exist (see Fig. 1.10 (a)). However, the width dependence of the G peak with respect to defect concentration (see Fig. 1.10 (b)) makes it to possible to distinguish between the two regimes. More precisely, the G peak width is a monotone function of the defect concentration. It becomes larger for higher defect concentration.
Edges can be seen as extended defects because they break the translation symmetry (at least perpendicular to the edges orientation). Perfect edges can exist within two configurations: either armchair or zigzag. While the armchair edges activate the D and the D0 peak in the Raman spectrum of graphene, the zigzag edges have no D peak but always a D0 peak. Thus, Raman spectroscopy (more precisely the I(D)/I(G) ratio) can be used to determine the type of perfect edges.

Counting the number of layers

In this section, we will show how Raman spectroscopy can be used to determine the number of layers in few-layer graphene. The study presented here is relatively close to the main subject of the present thesis which concerns stacking order determination of graphene layers by mean of Raman spectroscopy.
In the early days of graphene, the number of layers in isolated graphene flakes was determined by either Atomic Force Microscopy (AFM) or optical absorption in selected substrates [67]. Until 2006, these two techniques (accompanied with STM and HRSTM) have been the only techniques used to identify monolayer and FLG. The complexity of the equipment required for these techniques was a major limitation in the research filed of FLG. On the other hand, Raman spectroscopy was already known to be one of the most available non-destructive characterization tools and it was worthy to investigate the Raman response from graphene samples with different number of layers. Indeed, this investigation was carried out by Ferrari et. al. who showed that the shape and width of the 2D Raman peak in graphene and FLG drastically change according to the number of layers [13]. In the case of graphene, the simple structure of the electronic bands (two Dirac cones at K and K0) implies the existence of one single resonant phonon mode according to the double resonant model [54] (see Fig. 1.11 (c)) and consequently gives rise to a single Lorentzian 2D peak (Fig. 1.10 (a)). In bilayer graphene the electronic structure is more complicated, with two valence and two conductions bands near the Fermi level. This « double bands » structure gives rise to four resonant phonon modes and, as shown in Ref. [68], to a more complicated 2D Raman peak with four sub-peaks (see Fig. 1.11 (a) and (d)). For Nlayers > 3, the situation is more or less the same, with more complicated electronic bands near the Fermi level and more resonant phonons modes. The resulting 2D peaks are larger with more sub-peaks contributions (see Fig. 1.11(a))[13].
Other Raman peaks can be used to probe the number of layers in FLG : the C shear mode peaks around 40 cm-1, as shown in Fig. 1.11 (b), and the layer breathing mode peaks (LBM) (1700-1800 cm-1). However, these peaks are less intense than the 2D peak and hard to detect. The shear mode peak (see Sec. 1.2.3) is related to layer displacements parallel to each other. A direct evidence for single layer graphene would be the absence of the shear mode in the Raman spectrum. Though, we should not use the absence of a peak as a characterization tool (because one can never be sure why a peak is absent [53]). For Nlayers > 2, the frequency of this mode evolves with the number of layers and can also be used to probe Nlayers. The LBM (see Sec. 1.2.3) corresponds to the displacement of layers along the stacking direction. These phonon vibrations are detected in the Raman spectrum between 1650 and 1800 cm-1. Contrary to the 2D peak which is due to in plane vibrations and sensitive to the electronic structure change with the number of layers, these last two signatures (the C and the LBM peaks) are related to phonons that are sensitive to interlayer coupling; thus, they are a direct probe of stacking in FLG.

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Stacking order determination in few-layer graphene

Following the discovery of graphene, one of the most exciting fields of research has been to explore the electronic and optical properties of graphene layers. It was soon realized that FLG systems possess completely different electronic structures and new features can be found. For more than two layers, three possible stacking orders can occur in FLG : the Bernal, rhombohedral and mixed stacking (see Fig. 1.12 (a)). In the Bernal stacking mode (also called AB stacking), half of the atoms in a given layer are positioned perpendicularly between neighboring atoms from the plans on top and below, while the other half are between the centers of the hexagons. On the contrary, in rhombohedral stacking (also called ABC stacking), half of the atoms in each layer are positioned directly above of their neighbors on the layer below, and below the centers of the hexagons from the layer above, the opposite occurs for the other half. Mixed stacking is a mixture of rhombohedral and Bernal sequences.
In the case of trilayer graphene, while Bernal stacked trilayer is always a semimetal, transport measurements on suspended ABC-trilayer show the occurrence of a 42 meV gap at the Fermi level [70] that has been attributed to the occurrence of a magnetic ground states (non vanishing spin density) [70, 71]. This finding was the first indication of the importance of stacking in few-layer graphene. Another aspect of rhombohedral-stacked multilayer graphene (RMG) is the existence of an extremely flat band in the electronic structure. This feature implies the existence of a highly correlated electrons system and the possible appearance of exotic states of matter such as magnetic order or room temperature superconductivity. The flat band in RMG is a surface state with an increasing extent in reciprocal space with an increasing number of layers [16, 15]. Evidences for this flat band were reported in flakes with 5 layers in Ref. [72] through scanning tunneling spectroscopy and angle-resolved photoemission spectroscopy. More recently, Henni et. al. [73] were able to isolate multilayer graphene flakes with ABC sequences exceeding 17 graphene sheets. These samples were tentatively attributed to ABC stacking via magneto-Raman measurements.
In view of the previous findings, stacking in FLG is now understood to be of crucial importance. In order to identify the stacking mode in FLG, the method of reference, has been, so far, infrared absorption [74, 75, 76] 9. Raman spectroscopy has been also reported to be a reliable method for distinguishing layer stacking in trilayer and tetralayer graphene [69]. However, the reported study of Ref. [69] were first based on infrared absorption as a reference method. This means that, in a first stage, light absorption experiments were performed in order to identify the stacking order by comparing the obtained absorption spectra to simulation results of the op-tical conductivity. Raman spectra were later attributed to Bernal and rhombohedral stackings accordingly.
Infrared absorption technique has several limitations such as the low spatial reso-lution and the high sensitive instrumentation that is required for the detection of the infrared (IR) signal [69]. Moreover, for stacking determination, it is reliable only in the case of trilayer graphene where there are exactly two possible stacking modes : the Bernal and rhombohedral stackings (see Fig. 1.12 (b)). In this case, the light absorp-tion can be identified easily as originating from of a Bernal or a rhombohedral stacking by a simple comparison to optical conductivity simulations. All other possible spectra can be seen as arising from large domains with a mixed stacking. This is possible since the infrared signal is not very localized in space (compared to laser sources used in Raman experiments). For tetralayer graphene the situation is different. Three possible stackings can be found for tetralayer graphene, namely, the ABAB, ABCA and the ABCB (which is equivalent to the ABAC stacking). The ABCB stacking is predicted by to be less stable [77]. However, it can exist under certain (T ,P ) conditions, or produced by means of other technique then exfoliation which was used in Ref. [77]. In this case one can not be sure, for example, whether the absorption spectra of the infrared light arise from an ABCB stacking or from a mixture of ABAB and ABCA stacking signals.
In view of the previous discussions, we understand that a powerful technique to determine the stacking in FLG is highly required. In the next section, we will show how Raman spectroscopy can be used to fulfill this requirement.

The role of Raman simulations in stacking order determi-nation

A Raman spectroscopy simulation consists of providing the theoretical Raman spec-tra for a given system based on the knowledge of its electron and phonon properties.
It can be carried out within either parametric models (e.g. tight-binding) or from first-principles. The theoretical frame work for this type of computation is perturba-tion theory. Raman spectroscopy, contrary to the IR absorption technique, is a simple and powerful tool as its spectral sensitivity reaches the theoretical Density Functional Theory (DFT) limit on the phonon frequencies (few cm-1). Furthermore, it can reach high spatial resolution nowadays and, as discussed in previous sections, many authors have reported its reliability in characterizing graphene and few-layer graphene system.
The purpose of the present thesis is to investigate theoretically the use Raman spectroscopy to determine the type of stacking in FLG and graphite systems. Ra-man spectra will be calculated from first principles for all possible stacking and then compared to experimental results in order to identify the stacking sequence. We ex-pect this contribution to widen the use of Raman spectroscopy to include stacking determination in few-layer graphene and bulk graphites.
Of all the peaks present in the Raman spectrum of FLG (and also graphite), the 2D peak is expected to be more sensitive to stacking. This is the case at least for trilayer graphene, where the 2D peak has been shown to be useful in distinguishing between ABA and ABC stacking orders while the G peak is almost the same for the two stackings [69]. The sensitivity of the double resonant 2D peak to stacking is analogous to its sensitivity to the number of layers. This similarity, which justify our use of the 2D peak to differentiate between different stackings, can be understood as follows. The double resonant condition verified by the 2D peak in FLG and graphite imposes a selection rule on the phonon wave vector [54] and only phonons around K and K0 contribute to the Raman cross section. From a geometrical point of view, as shown in Fig. 1.13, if one tries to connect points in the Dirac cones near then K and K0 points using a given phonon wave vector, only few electronic states will be connected (these states are indicated by red lines in Fig. 1.13). This means that the electronic region contributing to the 2D peak is very narrow. Thus, it is possible, in principle, to probe tiny modifications in the electronic structure of few-layer graphene such as the ones induced by a change in the number of layers or the stacking order. If the electronic regions contributing to the Raman cross section were large, the 2D peak would not be In view of the previous discussions, simulation of the Raman 2D peak will be at the heart of this work. We will aim to give the theoretical Raman signature for all possible stacking in FLG and graphite.

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
Conclusions and perspectives
Appendices 
A Inclusion of electron correlations effects. 
B Goldstone diagrams for double resonant Raman scattering 
C TO-character calculation 
D Spectra analysis at higher excitation energies 
Résumé en français 
Bibliography

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