The Raman fingerprint of rhombohedral graphite

TL;DR

This paper provides a first-principles analysis of the 2D Raman spectrum in various multilayer graphene stackings, enabling non-destructive identification of rhombohedral graphite, which is crucial for exploring its potential correlated electronic states.

Contribution

It offers the first theoretical description of the 2D Raman fingerprint in rhombohedral multilayer graphene and practical methods for its identification.

Findings

First-principles Raman spectra for 3 and 4 layer graphene

Identification protocols for ABC stacking sequences

Enhanced understanding of Raman features in Bernal and rhombohedral graphite

Abstract

Multi-layer graphene with rhombohedral stacking is a promising carbon phase possibly displaying correlated states like magnetism or superconductivity due to the occurrence of a flat surface band at the Fermi level. Recently, flakes of thickness up to 17 layers were tentatively attributed ABC sequences although the Raman fingerprint of rhombohedral multilayer graphene is currently unknown and the 2D resonant Raman spectrum of Bernal graphite not understood. We provide a first principles description of the 2D Raman peak in three and four layers graphene (all stackings) as well as in Bernal, rhombohedral and an alternation of Bernal and rhombohedral graphite. We give practical prescriptions to identify long range sequences of ABC multi-layer graphene. Our work is a prerequisite to experimental non-destructive identification and synthesis of rhombohedral graphite.

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Full text

Bernal graphite Bernal (1924) with AB stacked graphene is the most stable form of graphite. Recently, however, rhombohedral stacked multi-layers graphene (RMG) with ABC stacking, see Fig. 1 (a), attracted an increasing attention as theoretical calculations suggest the occurrence of a dispersionless electronic band (bandwith smaller than 2 meV) at the Fermi level Kopnina and Heikkiläa (2014); Xiao et al. (2011). This flat band with extremely large effective mass, constitutes a very promising candidate for highly correlated states of matter such as magnetism Pamuk et al. (2017) or room-temperature superconductivity Precker et al. (2016).

As ABC-stacked graphite is metastable Charlier et al. (1994), the synthesis of long sequences of ABC graphene layers is a real challenge. For a random sequence of $N$ graphene layers stacked along the $c$ axis, a purely statistical argument states that the probability to obtain $N$ layers with ABC order is $1/2^{N-1}$. In reality the probability is even lower as all stackings are not equally probable as energetics favour the Bernal one with respect to the others. This explains why three and four layer graphene flakes with ABC-stacking are systematically found Mak et al. (2010); Lui et al. (2010); Cong et al. (2011), while it is highly unprobable to obtain long range ABC-stacking order. Recently, it has been suggested that pentalayers graphene with rhombohedral stacking can be grown epitaxially on 3C-SiC(111) Pierucci et al. (2015). Finally, Henni et al. Henni et al. (2016) were able to isolate multilayer graphene flakes with ABC sequences exceeding 17 graphene sheets. However, while for three and four ABC stacked graphene layers an optical signature exists Mak et al. (2010); Lui et al. (2010), a clear fingerprint of long-range rombohedral order is lacking.

Raman spectroscopy, and in particular the 2D double resonant Raman peak, has proven to be a very powerful technique to investigate structural and physical properties of graphene flakes. It can be used to count the number of layers Ferrari et al. (2006), detect charged impurities Casiraghi et al. (2007), measure the strain-induced deformation of the electronic structure Huang et al. (2010); Mohr et al. , measure the phonon dispersion May et al. (2013); Herziger et al. (2014) and many other properties (for a review see Ferrari and Basko (2013)). However, despite its crucial importance, the theoretical understanding of the 2D double resonant Raman spectrum has been obtained only for graphene Venezuela et al. (2011); Basko (2008); Popov and Lambin (2013) and bilayer graphene Herziger et al. (2014). Even the basic case of bulk Bernal graphite is not completely understood.

In this work we provide a complete first principles description of the 2D Raman peak in three and four layer graphene for all possible stackings, as well as for bulk AB, bulk ABC and a periodic mixing of the two (the so-called ABCB graphite). We present calculations for several laser energies and we give practical prescriptions to identify long sequences of ABC stacked multilayer graphene.

Double resonant spectra are calculated from first principles using the method developed in Ref. Herziger et al. (2014). The electrons and phonons bands where first calculated by using the Quantum ESPRESSO  Giannozzi et al. (2009) code in the local density approximation with norm-conserving pseudopotentials and an energy cutoff of 70 Ry. Electronic integration was performed on k-point grids of $64\times 64$ for three and four layer systems and $64\times 64\times 4$ for Bernal graphite. For rhombohedral graphite we use the hexagonal unit cell containing three layers (6 atoms/unit cell) and a $64\times 64\times 4$ k-point grid. The dynamical matrices and the electron-phonon coupling were first calculated in linear response on sparse phonon momentum grids ($6\times 6$ for three and four layer graphene and $6\times 6\times 3$ for bulk graphites) and then both were Wannier interpolated throghout the Brillouin zone (BZ) using the method of Ref. Calandra et al. (2010). The electronic bands and phonon frequencies were corrected for the electron-electron interaction as performed in Ref. Herziger et al. (2014). The double resonant Raman cross section was calculated on ultra-dense phonon grids for reciprocal space integration, namely, grids of of $300\times 300$ for 3 and 4 layers and up to $300\times 300\times 16$ for bulk graphites and electron grids as large as $256\times 256$ for few layers and $128\times 128\times 16$ for bulk graphites. As only a small percentage of the phonon and electron momenta in the grids actually contributes to the cross section, we develop an authomatic technique to identify the subset of relevant points (see Ref. sup ). The electron lifetime was chosen as in Ref. Herziger et al. (2014) and it was kept the same for all calculations.

We first calculate the double resonant Raman spectra for three and four layers, where a determination of the stacking sequence has been obtained by optical measurements Lui et al. (2010). The results are shown in Fig. 2 for three layers and several laser energies and in Fig. 3 for four layers. Additional results for the four-layers case are presented in the supplemental material sup . Overall we find an excellent agreement between our parameter-free ab initio calculation and experimental data. We reproduce all spectral features in position, width and intensities as well as the laser energy dependence of the spectra. Our results clearly demonstrate that the spectra substantially differ from one stacking to the other. This is mostly due to the difference in electronic structure between different stackings and to the dominance of symmetric inner processes (see Fig. 1 (b) and supplemental materials sup ). Indeed, while along the $\bf\Gamma-K$ direction, the electronic structure is stacking independent for both three and four layers, it differs along the $\bf K-M$ high symmetry line. As a consequence, electron-hole pairs are created/destroyed at slightly different points in the BZ for the same incident laser energy. As shown in Fig. 1 (b), inner processes imply electron-hole pairs creation and distraction in the BZ region close to $\bf K-M-K$ and thus the resulting spectrum mostly feel the difference in electronic structure close to this high symmetry direction. A similar effect occurs in four layer graphene (see Fig. 3) Having validated our calculation against experimental data on three and four layers graphene, we switch to the case of bulk graphite. We first consider bulk AB graphite (Bernal graphite) for which several experimental data are available. We calculate the spectra for different laser energies finding an excellent agreement with experimental data (see Fig. 4). We then consider in more details the spectra at $\omega_{L}=1.96$ eV. The 2D peak is composed of a main peak at $\approx 2683$ cm*-1* and a shoulder around $2640$ cm*-1*, as shown in Fig. 4. Both features are well described by the calculation. The shape and intensities of the D+D${}^{{}^{\prime\prime}}$ overtone structure at $\approx 2456$ cm*-1*, although at slightly lower energy in the calculation, are also very well reproduced.

In order to detect signatures of different kinds of stackings, we perform calculations for the case of ABC bulk graphite and ABCB bulk graphite. ABCB bulk graphite is interesting as it corresponds to a sequence …ABCCBA… that is an equal mixing of trilayers with rhombohedral (labeled [ABC or CBA]) and Bernal (labeled (BAB or BCB))) stackings. Thus, the differences between bulk AB and ABCB stackings can be seen as fingerprint of local rhombohedricity (i.e. few ABC layers) while the differences between bulk ABCB and bulk ABC are signatures of long range rhombohedral order. The results are depicted in Fig.5, where they are compared with the spectrum of Ref. Henni et al. (2016) that has been tentatively attributed to 17 layers ABC-stacked graphene. Both the 2D peaks spectra of bulk ABC and ABCB graphite are substantially broader than the one of Bernal graphite. Thus the increased width of the 2D peak at $\omega_{L}=1.96$ eV is a fingerprint of short range ABC sequences.

Even if both bulk ABC and bulk ABCB spectra seems similar, they differ for the presence of a feature at $\approx 2560$ cm*-1* in the ABC case that is completely missing in the ABCB stacking. As the bulk ABC structure differs from the bulk ABCB one for the occurrence of long range rhombohedral order, the feature at $\approx 2576$ cm*-1* at $\omega_{L}=1.96$ eV can be seen as a fingerprint of long range rhombohedral order. The good agreement between our theoretical calculation and the experimental spectrum in Ref. Henni et al. (2016), both from what concerns the 2D peak width and shape as well as the presence of the feature at $\approx 2576$ cm*-1* suggests that the samples in Ref. Henni et al. (2016) contains long range sequences of rhombohedral stacked multilayer graphene.

It is worthwhile to discuss a bit more the width of the 2D spectra for rhombohedral and Bernal graphite as the larger width of the 2D peak in the former with respect to the latter seems counterintuitive. Indeed, bulk AB-stacked graphite has two couples of $\pi$, $\pi^{*}$ electronic bands in the BZ, while ABC graphite only one in the rhombohedral cell. So one could naively think that AB-stacked graphite should have more allowed dipolar transitions. However, this is without taking into account the electronic $k_{z}$ dispersion, that, as shown in supplemental material sup , is substantially different along K-M-K in the two case. The different kz electronic band dispersion implies that different electron and phonon momenta contribute to the Raman cross section. This is clearly seen in the in Fig. 6 where the resonant phonon momenta contributing to the 2D peak cross sections are highlighted in a contour plot (see also Ref. Herziger et al. (2014) for more technical explanations). While the Bernal case includes very sharp resonances in phonon momenta, the resonance is much broader in the rhombohedral case due to the different band dispersion along $z$. This explain the larger width of the 2D peak in the rhombohedral case. More detailed analysis of electronic transitions contributing to the 2D two-phonon resonant cross section are given in the supplemental material sup .

In this work we performed parameter-free first principles calculation of the two-phonon resonant 2D and D+D${}^{{}^{\prime\prime}}$ peaks in three and four layer graphene for all possible stackings, as well as for bulk AB, ABC and ABCB graphite, that is a periodic arrangement of AB and ABC graphites. Our calculations carried out for several laser energies are in excellent agreement with experimental data available for three and four layers with AB and ABC stacking sequences as well as for Bernal graphite. In the case of ABC-stacked graphite, we compare our calculation with recently synthesized flakes from Ref. Henni et al. (2016) that were tentatively attributed to 17 layers ABC sequences. Our theoretical bulk ABC-stacked graphite spectra confirm this attribution. Furthermore, we have shown how to distinguish between short and long range rhombohedral order.

A. T. is indebted to the IDS-FunMat European network. M. C. and F. M. acknowledges support from the European Union Horizon 2020 research and innovation programme under Grant agreement No. 696656-GrapheneCore1, PRACE for awarding us access to resource on Marenostrum at BSC and the computer facilities provided by CINES, IDRIS, and CEA TGCC (Grant EDARI No. 2017091202). J.-C.C. acknowledges financial support from the Fédération Wallonie-Bruxelles through the ARC entitled 3D Nanoarchitecturing of 2D crystals (N$\degree$ 16/21-077), from the European Union’s Horizon 2020 researchers and innovation programme (N$\degree$ 696656), and from the Belgian FNRS.