Two-dimensional analysis of the double-resonant 2D Raman mode in bilayer graphene
Felix Herziger, Matteo Calandra, Paola Gava, Patrick May, Michele Lazzeri, Francesco Mauri, Janina Maultzsch
TTwo-dimensional analysis of the double-resonant D Raman mode in bilayer graphene
Felix Herziger, ∗ Matteo Calandra, Paola Gava, Patrick May, Michele Lazzeri, Francesco Mauri, and Janina Maultzsch Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany Institut de Minéralogie, de Physique des Matériaux, et de Cosmochimie,UMR CNRS 7590, Sorbonne Universités - UPMC Univ Paris 06,MNHN, IRD, 4 Place Jussieu, F-75005 Paris, France
By computing the double-resonant Raman scattering cross-section completely from first principlesand including electron-electron interaction at the GW level, we unravel the dominant contributionsfor the double-resonant D -mode in bilayer graphene. We show that, in contrast to previous works,the so-called inner processes are dominant and that the D -mode lineshape is described by threedominant resonances around the K point. We show that the splitting of the TO phonon branch in Γ − K direction, as large as 12 cm − in GW approximation, is of great importance for a thoroughdescription of the D -mode lineshape. Finally, we present a method to extract the TO phononsplitting and the splitting of the electronic bands from experimental data. PACS numbers: 78.30.-j, 78.67.Wj, 81.05.ue, 63.22.Rc
Double-resonance Raman spectroscopy provides a ver-satile tool for investigating the electronic structure andphonon dispersion of graphitic systems by tuning thelaser energy [1]. In particular, the D and D Ramanmodes allow to investigate structural changes, such asthe number of layers, disorder, strain and doping in thesample [2–8].Especially the distinction between single, bi-, and few-layer graphene via measuring the D mode attractedgreat attention due to its simplicity [2]. In single-layergraphene, the double-resonance is often simplified to onesingle scattering process, well describing the experimen-tal peak shape of the D mode. Up to now, the D mode in bilayer graphene is described and interpretedwithin the framework of four scattering processes. Eachprocess was assigned to a different spectral feature inthe D -mode lineshape, phenomenologically explainingthe observed peakshape [2]. All successive studies on the D mode in bilayer graphene relied on this assignment[9–14]. Furthermore, the D mode in bilayer graphenehas been mainly discussed in terms of outer processes[2, 9–11, 14]. However, the importance of inner processeswas shown both theoretically and experimentally for the D mode in single-layer graphene [5, 15, 16]. In bilayergraphene, only very few works considered the possibilityof contributions from inner processes, but were still ne-glecting the splitting of the two TO (transversal optical)phonon branches [12, 13]. Hence, the role of different con-tributions to the double-resonance in bilayer graphene isstill under discussion and needs final clarification.In this letter, by completely calculating the double-resonant Raman cross-section from first principles and bycomparing with experimental spectra for different laserenergies, we unravel the dominant scattering processesin bilayer graphene. In contrast to previous works thatexplained the D -mode lineshape with four independentscattering processes [2, 9–14], we show that the D mode is described by three dominant resonances around the K point from inner processes plus a weaker contributionfrom outer processes. We show that the GW correctionto the TO phonon branch leads to a much larger TO split-ting than in LDA approximation. This splitting cannotbe neglected; we present an analysis to directly derive theTO phonon and electronic splitting in bilayer graphenewith high accuracy.Experimental Raman spectra were obtained from free-standing bilayer graphene in back-scattering geometryunder ambient conditions using a Horiba HR800 spec-trometer with a 1800 lines/mm grating with spectral res-olution of 1 cm − . During all measurements the laserpower was kept below 0.5 mW to avoid sample damagingor heating. Spectra were calibrated by standard neonlines. The freestanding bilayer graphene enables us toprobe the intrinsic D -mode lineshape, ensuring an ac-curate extraction of the fitting parameters [16], followingthe model of Basko [17].In Bernal-stacked bilayer graphene the π orbitals giverise to two valence and two conduction bands, denotedas π , π and π ∗ , π ∗ . Bilayer graphene possesses two TOphonon branches, each one degenerate with an LO (lon-gitudinal optical) branch at Γ . At Γ the TO branches aresplit into a symmetric and an anti-symmetric vibration.The symmetric TO phonon is an in-phase vibration be-tween the lower and upper layer and exhibits E g symme-try in the point group D d , whereas the anti-symmetricvibration is out-of-phase ( E u symmetry). Our calculatedfrequency splitting at Γ is approx. 5 cm − , compara-ble with the experimentally observed 6 cm − splittingin graphite [18]. Along the Γ − K direction, the GW -calculated TO splitting increases to values as large as12 cm − , whereas the splitting in LDA is approximatelytwo times smaller. The displacement patterns of the TOvibrations change away from Γ , we however extend thelabeling of the phonon branches throughout the BZ. a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t π π π ∗ π ∗ Γ Γ
K K M P P symmetric(a) π π π ∗ π ∗ Γ Γ
K K M P P anti-symmetric(b) inner outerTO LO LA ← Γ K M → (cid:16) ˚A − (cid:17) P h o n o n f r e qu e n c y (cid:0) c m − (cid:1) (d) q ν − q µ | k, l i| k, m i | k + q, j i| k + q, i i (c) Figure 1. Schematized illustration of the D -mode scattering processes along the Γ − K − M − K (cid:48) − Γ high-symmetry direction for(a) symmetric and (b) anti-symmetric processes. Inner and outer processes are marked with red and blue traces, respectively. (c)Goldstone diagram for a double-resonant e-h scattering process P ljmi . (d) GW -corrected phonon dispersion of bilayer grapheneclose to K , showing the TO splitting in Γ − K direction. The double-resonant D mode is a second-order Ra-man process, involving two TO phonons with wave vec-tor q (cid:54) = 0 . The process can be divided into four vir-tual transitions, (i) creation of an electron-hole pair bya photon with energy (cid:126) ω L , (ii) scattering of an elec-tron/hole state by a phonon with wave vector q , (iii)scattering of an electron/hole state by a phonon withwave vector − q and, (iv) recombination of the electron-hole pair. The observed frequency of this process is twicethe phonon frequency at q . As we have explicitly veri-fied, the processes where one phonon is scattered by anelectron and one phonon is scattered by a hole (diagramsof the kind shown in Fig. 1), are, by far, the most dom-inant contribution to the Raman cross-section [15]. Wewill refer to these processes as electron-hole scattering( e-h scattering). The scattering processes can be furtherdivided into symmetric/anti-symmetric and inner/outerprocesses. Symmetric processes are scattering events be-tween equivalent electronic bands at K and K (cid:48) , whereasfor an anti-symmetric process the band index is chang-ing. We refer to the term inner process, if the resonantphonon wave-vector stems from a sector of ± ◦ next tothe K − Γ direction with respect to K . Conversely, outerprocesses have phonon wave-vectors from ± ◦ next tothe K − M directions [Fig. 2(a)]. To simplify the labelingof the scattering processes, we enumerate the electronicbands starting from the energetically lowest band near K . Every scattering process P ljmi is then uniquely de-fined by four indices that are given by the band indicesof the initial electron m , of the excited electron l , of thescattered electron j , and of the scattered hole i . Sincethe incoming light couples mostly to only two ( → and → ) of the four possible optical transitions, four dif-ferent combinations of e-h scattering are allowed [2, 19].These are the symmetric processes P and P and theanti-symmetric processes P and P .Following Ref. [15], the two-phonon ( pp ) double-resonant Raman intensity is I ( ω ) = 1 N q (cid:88) q ,ν,µ I pp q νµ δ ( ω L − ω − ω ν − q − ω µ q )[ n ( ω ν − q ) + 1][ n ( ω µ q ) + 1] , (1) where ω µ q and n ( ω µ q ) are the phonon frequencies andthe Bose distributions for mode µ , respectively [20].The probability of exciting two phonons is I pp q νµ = (cid:12)(cid:12)(cid:12) N k (cid:80) k ,β K ppβ ( k , q , ν, µ ) (cid:12)(cid:12)(cid:12) , where the matrix elements K ppβ ( k , q , ν, µ ) are defined by expressions involving theelectron and phonon band dispersion, the electron-phonon coupling g µ k n, k + q m and the electron-light D k n, k m matrix-elements throughout the full BZ (see appendix Aof Ref. [15]). Here, k refers to the electron wave-vectorand β labels the different possibilities of electron and holescattering. We want to remind the reader of the impor-tance of quantum interference in the double-resonanceprocess. Scattering processes with the same final state ( q , µ, ν ) but different intermediate states can observe in-terference. Consequently, scattering processes at differ-ent q do not interfere. In most previous works on the D mode in bilayer graphene, the interference between dif-ferent processes was completely neglected. However, aswill be shown later, quantum interference has remarkableimpact on the D -mode lineshape in bilayer graphene.Due to the difficulties in obtaining g µ k n, k + q m and D k n, k m directly from first principles, previous publica-tions used matrix elements derived from tight-bindingmodels [15, 21, 22]. Here, we overcome this difficulty byusing Wannier interpolation [23] of the electron-phononand the electron-light matrix elements, as developed inRef. [24]. We first calculate from first principles in LDAapproximation [25] the unscreened electric dipole and thescreened electron-phonon matrix elements on a × electron-momentum grid and a × phonon momen-tum grid. We then interpolate them to denser × electron-momentum grid randomly shifted from the ori-gin and a -points phonon momentum grid, cover-ing a sufficiently large region around the K points. Thephonon bands were Fourier interpolated from a × phonon momentum grid. The electronic, the TO phononbands, and g µ k n, k + q m were GW -corrected, following theapproach given in Ref. [15] (see SI [30]). The electronbroadening γ was chosen to be twice as large as that inRef. [15] to account for additional electron-electron in-teraction [26], namely γ = 0 . × ( (cid:126) ω L / − . K ← M Γ → − . − . − . . . . . q y (cid:16) ˚A − (cid:17) − . − . − . . . . . P P + P P . . . . . . − . − . − . . . . . q x (cid:0) / ˚A (cid:1) q y (cid:16) ˚A − (cid:17) . . . . . . − . − . − . . . . . q x (cid:0) / ˚A (cid:1) . . . . . . N o r m . R a m a n c r o ss - s ec t i o n ( a r b . un i t s ) (a) (b) 1.96 eV(c) 2.33 eV (d) 2.54 eV Figure 2. (a) Illustration of the phonon wave-vector sectorsfor inner (red) and outer (blue) processes around the K point.The solid and dashed white lines denote the K − M and K − Γ high-symmetry lines, respectively. (b)-(d) Plots of the nor-malized D -mode scattering cross-section I q around the K point for different laser energies. eV. This choice gives better agreement with experiments(SI). Finally, the δ -function in Eq. (1) is broadened withan 8 cm − Lorentzian [27].Fig. 2 presents calculated contour plots of I q = (cid:80) ν,µ I pp q νµ for the double-resonant D mode in bilayergraphene for various (cid:126) ω L . Three resonances around the K point contribute to the D mode. These regions areattributed to, from inside to outside, the P , the anti-symmetric P and P , and the P processes. Asthe resonant phonon wave-vectors of the anti-symmetricprocesses are nearly degenerated, the resulting phononfrequencies are very similar, disproving previous assign-ments of anti-symmetric processes to different spectralfeatures of the D mode [2, 9–14]. Furthermore, thedominant contributions to the D -mode scattering cross-section stem from the K − Γ direction which can be iden-tified with inner processes.We will now turn our discussion to the calculated Ra-man spectra of the D mode in bilayer graphene. Fig.3 compares the calculated Raman spectra with spectrafrom freestanding bilayer graphene at different (cid:126) ω L . Theoverall agreement between calculation and experimentaldata is very good, although there is a slight mismatch infrequency. The calculated frequencies are approximately10 cm − too high, yet our calculations reproduce the line-shape of the D mode, i.e. , the relative intensities of thedifferent contributions, very well.Fig. 4 shows the decomposition of the calculated D -mode spectrum at 1.96 eV excitation energy into its dif-ferent contributions. The decomposition for other (cid:126) ω L isaccordingly (SI). As in single-layer graphene, we confirmthat in bilayer graphene the e-h scattering processes dom- (cid:0) cm − (cid:1) I n t e n s i t y ( a r b . un i t s ) (cid:0) cm − (cid:1) Figure 3. Comparison of calculated D -mode spectra withRaman spectra from freestanding bilayer graphene at different (cid:126) ω L . Calculations and experimental data are shown as redand black curves, respectively. Spectra are normalized andvertically offset. inate compared to all other scattering paths. Further-more, inner processes dominate over outer ones. By ex-plicitly decomposing the D mode into the four differentprocesses in Fig. 4(b), we find that the symmetric P and P processes are on the low- and high-frequency sideof the D mode, respectively. The frequencies of the anti-symmetric processes are in between the symmetric contri-butions and nearly degenerate, as already inferred fromFig. 2. This disagrees with all previous works [2, 9–14],attributing substantially different phonon frequencies tothe anti-symmetric processes. As seen in Fig. 4(b), thedecomposition of the single processes is not additive, i.e. ,the sum of the four processes does not yield the totalspectrum. This can be directly attributed to quantum in-terference effects between the anti-symmetric processes.By decomposing the spectrum into the single processes asin Fig. 4(b), interference between the P ljmi is prohibited.However, P and P exhibit a large overlap in recipro-cal space and interfere constructively. Decomposing thetotal spectrum into symmetric and anti-symmetric con-tributions and thus allowing interference between thoseprocesses yields the spectrum in Fig. 4(c). This decom-position is additive. The constructive interference hasremarkable impact on the D -mode lineshape, i.e. , theintensity of the anti-symmetric processes is drasticallyenhanced, highlighting the importance of quantum inter-ference effects in the double-resonance process.Up to now, we described the D mode in terms ofthree dominant resonances that split up into inner andouter contributions. Thus, one might expect six sepa-rate peaks in the D -mode spectrum in total. This is incontrast to the experimentally observed lineshape, whereusually three or four peaks can be distinguished. How-ever, the decomposition in Fig. 4(a) and (b) shows thatinner and outer contributions for the P , P , and P processes are nearly degenerate in frequency, thus reduc-ing the number of observable D -mode peaks for these all contributionse-h scatteringinner processesouter processes (a)2600 2700 2800Raman shift (cid:0) cm − (cid:1) I n t e n s i t y ( a r b . un i t s ) e-h scattering P P P P (b)2600 2700 2800Raman shift (cid:0) cm − (cid:1) e-h scatteringanti-symmetricprocesses symmetric processes (c)2600 2700 2800Raman shift (cid:0) cm − (cid:1) outerinner00 . E l ec t r o n i c s p li tt i n g ( e V ) outer inner0 0 . . | K − q | (cid:0) / ˚A (cid:1) P h o n o n s p li tt i n g (cid:0) c m − (cid:1) (d)(e) Figure 4. Calculated D -mode spectra at 1.96 eV excitation energy. Decomposition of the calculated spectra into (a) e-h scattering processes, as well as inner and outer contributions, (b) the four different scattering processes P ljmi (without interferencebetween the different processes), and (c) into symmetric and anti-symmetric processes (including interference). (d) and (e)show the experimental values for the electronic and TO phonon splitting, respectively. The solid (red) and dashed (blue) linesdenote the DFT calculated splittings in inner and outer direction, respectively. Open, green circles are data points from Ref.[11]. Filled, black circles represent data points from this work. processes to two. Only the P process exhibits a split-ting between inner and outer contributions that is largeenough to be detected in experiments; it is responsiblefor the third and fourth peak in the D -mode lineshape.Therefore, in experiments the D mode should be fittedwith four peaks, where the assignment of the peaks, fromlowest to highest frequency, is P , P /P , inner P ,and outer P . In previous works, the inner P con-tribution was erroneously assigned to an anti-symmetricscattering process, whereas the outer contribution, i.e. ,the small high-frequency shoulder of the D mode, wasattributed to a symmetric process. Here, we showed thatthese two peaks result from the same scattering process( P ). Our assignment of the third and fourth D -modepeaks to inner and outer P contributions is supportedby recent experiments on strained bilayer graphene [28].Due to different dispersions of inner and outer processes,both contributions to P merge with increasing laser en-ergy. Therefore, at higher laser energies the fourth peakvanishes. This can be seen in the spectrum of the free-standing bilayer graphene at 2.54 eV excitation energy inFig. 3. Here, the small high-frequency shoulder cannotbe identified any more, giving further evidence to our as-signment of the three dominant contributions to the D mode in bilayer graphene.In previous works, the TO splitting in bilayer graphenehas always been neglected in the double-resonance, asonly outer processes were considered and the TO splittingalong K − M is of the order of 1.5 cm − [2]. However, weproved that inner processes are dominant. In fact, along K − Γ the TO splitting is as large as 12 cm − in GW approximation. We observe that the dominant contribu-tions to symmetric processes stem from scattering withsymmetric TO phonons, whereas the dominant contribu-tions to anti-symmetric processes result from scatteringwith anti-symmetric TO phonons (see SI).The fact that symmetric and anti-symmetric processes couple to different phonon branches has remarkable im-pact on the D -mode lineshape. If all scattering pro-cesses would couple to the same phonon branch, all con-tributions would be equidistantly spaced in frequency.This is true for outer contributions [Fig. 4(a)], since theTO splitting along K − M is neglibile. However, the dom-inant contributions stem from inner processes and there-fore, the TO splitting must be taken into account. Sincethe inner anti-symmetric processes couple to the ener-getically higher TO branch along K − Γ , their frequencyis upshifted with respect to the center between the sym-metric processes. This upshift is a direct measure of theTO splitting and can be easily accessed experimentally.Furthermore, one can also extract the splitting of theelectronic bands from the D -mode spectrum, as this pa-rameter is directly connected to the frequency differencebetween the symmetric processes and the laser-energy de-pendent shift rate of the D mode (see SI). Figures 4(d)and (e) present the measured TO phonon and electronicsplitting in comparison with data from DFT+ GW cal-culations. As can be seen, the experimental values arein good agreement with the calculated curve along theinner direction. However, a discrepancy in the TO split-ting between theory and experiment can be observed for q vectors close to K . The TO phonon splitting is largestalong Γ − K and decreases away from this high-symmetryline. Since also phonons away from the high-symmetrydirection contribute to the double resonance [29], the ex-perimentally measured TO splitting is expected to besmaller than theoretically predicted along Γ − K . Thus,the theoretical curves should represent lower and upperlimits for the experimental values. The fact that the ex-perimental data is also outside those boundaries indicatesthat the commonly assumed GW correction might stillunderestimate the TO splitting, which is probably largerthan 15 cm − close to K . Finally, we should note thatall results for the D mode in bilayer graphene are alsovalid for the D mode.In conclusion, we demonstrated that the double-resonant D Raman mode in bilayer graphene is de-scribed by three dominant contributions, contradictingall previous works on this topic. We showed that in-ner processes contribute most to the Raman scatteringcross-section, as in single-layer graphene. Moreover, wedemonstrated that the TO phonon splitting is of greatimportance for a correct analysis of the D -mode line-shape. The TO phonon and electronic splitting can bedirectly extracted from experimental Raman spectra us-ing the presented analysis. Our results highlight the keyrole of inner processes and finally clarify the origin of thecomplex D -mode lineshape in bilayer graphene.FH, PM, and JM acknowledge financial support fromthe DFG under Grant no. MA 4079/3-1 and the Euro-pean Research Council (ERC) Grant no. 259286. MCand FM acknowledge support from the Graphene Flag-ship and from the French state funds (reference ANR-11-IDEX-0004-02, ANR-11-BS04-0019 and ANR-13-IS10-0003-01). Computer facilities were provided by CINES,CCRT and IDRIS (project no. x2014091202). ∗ [email protected][1] C. Thomsen and S. Reich, Phys. Rev. Lett. , 5214(2000).[2] A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi,M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S.Novoselov, S. Roth, and A. K. 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B , 045419 (2014). upplementary material for the paper:Two-dimensional analysis of the double-resonant D Raman mode in bilayer graphene
Felix Herziger, ∗ Matteo Calandra, Paola Gava, Patrick May, Michele Lazzeri, Francesco Mauri, and Janina Maultzsch Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany IMPMC, Université Pierre et Marie Curie, CNRS, 4 place Jussieu, 75252 Paris, France GW correction to the TO phonon modes in bilayer graphene Phonon dispersions are obtained from ab initio
DFT calculations in the linear-response scheme and corrected with GW , similar to what was done in Refs. [1–3] for single-layer graphene. In more detail, we correct the dispersion ofthe two TO branches near K , namely we define ( ω GW q ν ) = 0 . × erfc (cid:18) | q − K α | a / π − . . (cid:19) × (cid:2) ( α K − ω DF T q ν ) + ∆ (cid:3) (1)where erfc( ) is the error function, ν = 1 , is a label for the two TO branches, a is the graphene lattice constant,and K α is the closest vector to q among those equivalent to K . The constants α K and ∆ are defined as α K = 1 . and ∆ = 42 . Ryd and are identical for both TO phonon modes.In order to apply the GW correction to all phonon wave vectors in the Brillouin Zone, we use the phonon frequencyWannier interpolation method developed in Ref. [4] section III D. The comparison between the DFT and DFT+ GW phonon branches is shown in Fig. 1. In previous calculations at the GW level, the splitting of the two TO branchesalong Γ − K was found to be negligible and similar to that along K − M [5], in contradiction to our results. Thisdiscrepancy is explained by the coarse × phonon-momentum mesh used in Ref. [5] and the use of Fourierinterpolation. In our case, this error is not present due to the use (i) of a larger phonon momentum grid ( × )and (ii) of a Wannier interpolation scheme to obtain phonon frequencies at any electron and phonon momentum inthe Brillouin zone with high accuracy [4]. TO GW TO LDA
LO LA ← Γ K M → (cid:16) ˚A − (cid:17) P h o n o n f r e qu e n c y (cid:0) c m − (cid:1) (a) ∆ TO GW ∆ TO LDA ← Γ K M → (cid:16) ˚A − (cid:17) T O ph o n o n s p li tt i n g (cid:0) c m − (cid:1) (b) Figure 1. (a) Comparison of TO phonon frequencies in bilayer graphene calculated in LDA approximation (blue curve) andwith GW correction (red curve). LO and LA phonon branches were not GW corrected. The numbers along the LO and TObranches refer to their branch indices at this specific wave vector. (b) Splitting of the two TO phonon branches along the Γ − K − M high-symmetry direction in bilayer graphene. Colors are chosen as in (a). upplementary material: Two-dimensional analysis of the double-resonant D Raman mode in bilayer graphene (a) @1.96 eVexp. 2600 2700 2800Raman shift (cid:0) cm − (cid:1) I n t e n s i t y ( a r b . un i t s ) phonons with GW correction, . × γ phonons from LDA calculation, . × γ phonons with GW correction, . × γ (b) @2.33 eVexp. 2600 2700 2800Raman shift (cid:0) cm − (cid:1) phonons with GW correction, . × γ phonons from LDA calculation, . × γ phonons with GW correction, . × γ (c) @2.41 eVexp. 2600 2700 2800Raman shift (cid:0) cm − (cid:1) phonons with GW correction, . × γ phonons from LDA calculation, . × γ phonons with GW correction, . × γ Figure 2. Calculated Raman spectra of the D mode for different broadenings γ and with/without GW correction for the TOphonons in comparison with experimental spectra from freestanding bilayer graphene. Influence of electronic broadening and GW correction on the calculated Raman spectra The electronic broadening γ in our calculations was calculated according to γ = 0 . × (cid:18) ~ ω L − . (cid:19) eV , (2)where ~ ω L represents the laser excitation energy. Using this formula, we found very good agreement between thecalculated D -mode spectra and the experimentally observed lineshape. Using Eq. (2) for calculation of the electronicbroadening yields values twice as large as in Ref. [3]. However, Ref. [3] only considered electron-phonon interactionsfor the derivation of the broadening and neglected electron-electron interaction. However, it was shown recently byMak et al. that electron-electron interaction can contribute as much as electron-phonon interactions to the decayrate of excited charge carriers [6]. In fact, the decay rates nearly double by including electron-electron interaction.Another recent work on the optical properties of bilayer graphene used electronic broadenings as large as 125 meV,but argued that the real value must be even larger to reproduce the broad excitonic M -point absorption peak [7].Figure 2 compares experimental and theoretical spectra calculated with different electronic broadenings γ for differentlaser excitation energies. For comparison, the spectra are normalized to the contribution of the anti-symmetric pro-cesses. As can be seen, the spectra calculated with electronic broadenings . × γ do not reproduce the experimentallyobserved lineshape. In contrast, spectra that were calculated using electronic broadening from Eq. (2) show goodagreement with experimentally obtained spectra.Figure 2 also shows the influence of the GW correction on the calculated spectra. As already inferred in Fig. 1, thecalculated Raman spectra, where DFT-calculated phonon frequencies were used, exhibit frequencies that are clearly toohigh. Furthermore, the overall linewidth of the calculated spectra does not fit the experimentally observed lineshape, i.e. , the calculated lineshape is too narrow (the same phonon broadening of 8 cm − was used for all calculations). Incontrast to the DFT+ GW calculated phonon dispersion, the TO phonon branch exhibits a reduced slope in LDAapproximation. Thus, the frequency separation of the different scattering processes in LDA approximation is smallerand, therefore, the linewidth is reduced. Spectra, where the DFT+ GW calculated TO frequencies were used, showgood agreement in both the absolute frequencies and the overall linewidth of the D mode with measured spectra offreestanding bilayer graphene. 2upplementary material: Two-dimensional analysis of the double-resonant D Raman mode in bilayer graphene e-h scattering,only Γ -K-M line e-h scattering,only Γ -K-M line, ν = sym, µ = sym e-h scattering,only Γ -K-M line, ν = asym, µ = asym e-h scattering,only Γ -K-M line, ν = sym, µ = asym(a) 1.96eV2500 2600 2700 2800Raman shift (cid:0) cm − (cid:1) I n t e n s i t y ( a r b . un i t s ) (b) 2.33eV2500 2600 2700 2800Raman shift (cid:0) cm − (cid:1) (c) 2.41eV2500 2600 2700 2800Raman shift (cid:0) cm − (cid:1) (d) 2.54eV2500 2600 2700 2800Raman shift (cid:0) cm − (cid:1) Figure 3. Calculated spectra along the Γ − K − M high-symmetry direction. The black curve represents the calculated spectrawithout any restrictions on the phonon indices in Eq. (3). The contributions from scattering with symmetric phonons ( ν = sym, µ = sym) is shown in red, contributions from scattering with anti-symmetric phonons ( ν = asym, µ = asym) is shown as a bluecurve. Contributions to the mode that result from scattering with both symmetric and anti-symmetric phonons are depictedwith a green curve. Contributions to the mode from scattering with different TO phonons The coupling of different scattering processes to different TO branches in the -mode double-resonance process ofbilayer graphene has never been considered so far. In order to separate the different contributions from the two TOphonon branches to the -mode Raman scattering cross-section we proceeded as follows. We calculated the overlapbetween the eigenvectors of the six highest phonon branches with the symmetric and anti-symmetric TO vibrationat a q vector along the K − Γ direction very close to K . Since outside the high-symmetry directions the symmetricand anti-symmetric TO branches cannot be separated univocally by their vibration pattern, we only considered theTO branches along the Γ − K − M high-symmetry direction in this analysis. Next, we restricted the summation ofthe two-phonon double-resonant Raman intensity [3] I ( ω ) = 1 N q X q ,ν,µ I pp q νµ δ ( ω L − ω − ω ν − q − ω µ q )[ n ( ω ν − q ) + 1][ n ( ω µ q ) + 1] (3)to the indices ν , µ of the symmetric (sym) or anti-symmetric (asym) TO branches along Γ − K − M . The calculatedspectra for this separation are shown in Fig. 3 for different laser excitation energies. As can be seen, the dominantcontribution to the symmetric processes stems from scattering with symmetric TO phonons. Vice versa, the dominantcontribution to the anti-symmetric processes stems from scattering with anti-symmetric TO phonons. Scattering withthe combination of symmetric and anti-symmetric TO phonons gives only minor contribution to the -mode intensity. Calculation of the TO phonon and electronic splitting
The TO phonon and electronic splitting in bilayer graphene can be easily obtained from the measured D -modespectra. First, we want to recall the assignment of the observed D -mode peaks, from lowest to highest frequency, tothe different scattering processes, which is as follows:peak 1 → P (sym.), peak 2 → P /P (anti-sym.), peak 3 → inner P (sym.), peak 4 → outer P (sym.)As discussed in the main text of this letter, the P peak splits up into an innner and outer contribution, which isclearly observable for smaller laser excitation energies. For the calculation of the TO phonon and electronic splittingwe used the inner component, i.e. , the third -mode peak, as we want to investigate the TO splitting along K − Γ and the electronic splitting along the K − M direction. Figure 4(a) shows an exemplary fit of a measured D -modespectrum at 2.33 eV laser energy using a fit of the form f ( ω ) = X i =1 a i · f i ( ω, ω i , Γ i ) , (4)3upplementary material: Two-dimensional analysis of the double-resonant D Raman mode in bilayer graphene ω ω ω ω ω × TO phonon splitting ∆ ω ¯ hω L = 2 . eV2580 2600 2620 2640 2660 2680 2700 2720 2740 2760 2780 2800Raman shift (cid:0) cm − (cid:1) I n t e n s i t y ( a r b . un i t s ) q q q higher TO branchlower TO branchTO splitting2 × ∆ ω ← Γ K → (cid:16) ˚A − (cid:17) P h o n o n f r e qu e n c y (cid:0) c m − (cid:1) (a) (b) Figure 4. (a) Exemplary fit of a experimental -mode spectrum from freestanding bilayer graphene at 2.33 eV laser excitationenergy with four peaks based on the model proposed by Basko [8]. The solid red curve represents the fit to the experimentalspectrum. The peak positions of the single peaks are marked with dashed lines, the center between the symmetric scatteringprocesses is shown by the dotted line. The upshift of the anti-symmetric processes with respect to the center between thesymmetric contributions is indicated. (b) Resonant phonon wave-vectors in the phonon dispersion of bilayer graphene for2.33 eV laser excitation energy. The dashed, gray arrow indicates the phonon frequency of the P process, if all processeswould couple to the same phonon branch. The upshift of the anti-symmetric processes due to coupling to the energeticallyhigher TO branch is indicated. where f i ( ω, ω i , Γ i ) = (cid:18) Γ i − / ) ( ω − ω i ) + Γ i (cid:19) / (5)is a normalized function following the model of Basko [8]. Here, a i reflects the peak amplitude, ω i represents thecentral peak frequency and Γ i is the full width at half maximum (FWHM). Fitting parameters for different laserexcitation energies are given in Tab. I. In Fig. 4(a) the peak positions of the P , P , and inner P processesare marked by vertical dashed lines. Furthermore, the center between the symmetric processes P and inner P is indicated by a dotted line. Since the main contribution to the anti-symmetric scattering processes stems fromthe energetically higher TO phonon branch along Γ − K , their frequency ω is upshifted with respect to the centerbetween the symmetric processes / × ( ω + ω ) . This upshift is directly related to the TO phonon splitting.In fact, the observed upshift is two times the TO splitting, since the D mode is a two-phonon process. Fig. 4(b)shows the situation in a different way. Here, the resonant phonon wave-vectors for the double-resonance process inbilayer graphene with 2.33 eV laser energy are shown. The dashed, gray arrow indicates the phonon frequency ofthe anti-symmetric process, if this process would result from scattering with symmetric TO phonons. As we alreadyknow, the main contribution stems from scattering with anti-symmetric TO phonons and therefore, the frequency ofthese processes is upshifted. The effect can be easily seen in this figure. For the calculation of the electronic splitting,the frequency difference between the symmetric processes is divided by the dispersion of the D -mode peaks withlaser excitation energy. Thus, the TO phonon and electronic splitting can be calculated using the following formulas ∆ ω = 12 (cid:18) ω − ω + ω (cid:19) and ∆ ε = ω − ω d ω/ d ~ ω L , where ω ljmi refers to the peak positions of processes P ljmi and d ω/ d ~ ω L is the laser-energy dependent shift rate of the -mode peaks. It is important to notice that the D -mode dispersion is non-linear, as can be seen from Ref. [9].We used a quadratic fit to the data points to obtain the dispersion. Decomposition of the D mode spectra for different laser energies Figure 5 presents the decomposition of the D mode into the different contributions at different laser excitation4upplementary material: Two-dimensional analysis of the double-resonant D Raman mode in bilayer graphene