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Resonance Raman scattering in graphene: Probing phonons and electrons
Solid State Communications 149 (2009) 1136–1139
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Resonance Raman scattering in graphene: Probing phonons and electrons L.M. Malard a,∗ , D.L. Mafra a , S.K. Doorn b , M.A. Pimenta a a
Departamento de Física, Universidade Federal de Minas Gerais, 30123-970, Belo Horizonte, Brazil
b
Chemistry Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
article
info
Article history: Received 7 November 2008 Received in revised form 6 February 2009 Accepted 6 February 2009 by the Guest Editors Available online 20 March 2009 PACS: 63.20.Kr 73.21.-b 78.30.-j 81.05.Uw
abstract In this work, by using different laser excitation energies, we obtain important electronic and vibrational properties of mono- and bi-layer graphene. For monolayer graphene, we determine the phonon dispersion near the Dirac point for the in-plane transverse optical (iTO) mode. This result is compared with recent calculations that take into account electron–electron correlations for the phonon dispersion around the K point. For bilayer graphene we extract the Slonczewski–Weiss–McClure band parameters and compare them with recent infrared measurements. We also analyze the second-order feature in the Raman spectrum for trilayer graphene. © 2009 Elsevier Ltd. All rights reserved.
Keywords: A. Nanostructures D. Electronic band structure D. Optical properties
1. Introduction Since the identification of mono and few graphene layers on a substrate [1], intensive work has been devoted to characterize this new material. New physics arose from this research such as the unconventional Hall effect [2–6]. In particular, Raman spectroscopy played an important role to characterize the number of layers [7–11], doping [12–14] and strain [15]. Moreover resonance Raman scattering (RRS) in graphene systems was shown to be an important tool to probe phonons and electrons [16,17]. The number of graphene layers can be obtained by Raman spectroscopy [7–9] and, in particular, to clearly distinguish a monolayer graphene from a double or few layer graphene sample. The two most prominent Raman bands arising from the double Resonance Raman (DRR) [18–20] process are the socalled induced-disorder D band around 1350 cm−1 (for a 2.41 eV laser excitation energy), and its associated second-order feature, that appears around 2700 cm−1 and that is historically called the G0 band. The denomination of this second-order feature is controversial in the literature: some authors prefer to call it the 2D band [7] or the D0 band [9], since it involves the same phonon mode that gives rise to the D-band. However, the observation
∗
Corresponding author. E-mail address: [email protected] (L.M. Malard).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.02.045
of the D-band requires the existence of a defect (or disorder) in the sample, justifying its denomination as D band, whereas the second-order feature appears in the spectrum of pure crystalline graphite, without any kind of defects or disorder. So, we will keep the historical denomination and call here this second-order feature as G0 band. The DRR process considers the scattering of electrons by phonons between points inside the first Brillouin zone. By this mechanism, it is possible to probe phonons that are not only at the zone center 0 point as for the usual first order Raman scattering. By combining RRS and the DRR model, one can infer information about the electronic and phonon dispersions of mono and few graphene layers. In this work, we studied the dependence of the G0 Raman band of mono-, bi- and tri-layer graphene as a function of the laser energy. We obtained the phonon dispersion of the inplane transverse optical (iTO) mode near the Dirac point for monolayer graphene and compared it with phonon dispersion calculations found in literature [21,22]. The electronic structure of bilayer graphene can be described in terms of the standard Slonczewski–Weiss–McClure (SWM) model for graphite [23,24]. By analyzing the dispersion of the G0 band of bilayer graphene when the laser energy is changed, it is possible to probe the splitting of the electronic dispersion branches and to obtain experimental values for the tight-binding parameters γ0 , γ1 , γ3 and γ4 . We compare our results with recent infrared measurements (IR) [25,26]. We also present a group theory analysis for
L.M. Malard et al. / Solid State Communications 149 (2009) 1136–1139
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the ABA trilayer graphene. With the selection rules for electronradiation interaction within the dipole approximation and for electron scattering by phonons, we discuss the DRR scattering showing how many processes can contribute for the G0 band of trilayer graphene. 2. Experimental details The graphene samples used is this experiment were obtained by a micro-mechanical cleavage of graphite on the surface of a Si sample with a 300 nm layer of SiO2 on the top. The bilayer flakes were identified in an optical microscope by the slight color change cause by white light intereference in the silicon oxide [27]. The number of layers was characterized by Raman spectroscopy using the procedure described by Ferrari et al. [7], where it is possible to distinguish between mono, bi and few layers graphene. For the Raman experiments in the visible range (Ar–Kr and dye lasers in the range 1.91–2.71 eV), we used a Dilor XY triple monochromator in the back scattering configuration. The spot size of the laser was ∼1 µm using a 100 × objective and the laser power was kept at 1.2 mW in order to avoid sample heating. The measurements in the near infrared range (Ti:Saphire laser 1.57–1.69 eV) were done using a triple monochromator (300/300/600 groves/mm grating set) with a 100 × objective and laser power of 1.0 mW. 3. Results and discussion 3.1. Phonons in monolayer graphene Phonon dispersion in graphene has been calculated by a number of different models in the literature [21,22]. Many of them contradict each other mainly near the 0 and K points where the Kohn anomaly occurs [28]. The electron–phonon coupling renormalizes the phonon frequency, causing a steep slope for phonon wavevectors near these two high symmetry points. More recently the iTO phonon dispersion near the K point was calculated also considering electron–electron interactions, predicting a steeper slope for the phonon dispersion [22]. In this section, we focus on the experimental data for the iTO phonon mode for monolayer graphene and compare them with theoretical calculations. Fig. 1(a) shows the Raman spectrum of a monolayer graphene, where it is possible to observe the G band near 1580 cm−1 and the G0 band near 2700 cm−1 for a 1.72 eV laser excitation energy. The G0 -band position as a function of the laser excitation energy is shown in Fig. 1(b). The G0 band arises from the DRR scattering process mechanism [18–20], which consists of three steps. First, an electron with wavevector k measured from the K point in the valence band is excited to the conduction band by a photon with energy EL . Then, the electron is resonantly scattered by emitting a phonon, going to the vicinity of the K0 point with wavevector k0 and energy EL − Eph . The electron is scattered back to the K point and recombines back with a hole emitting a scattered photon with energy EL −2Eph . A schematic picture of the DRR process can be seen in Fig. 1(c), where the initial k and the intermediate k0 wavevectors connected by a phonon with wavevector q are shown. In graphene, the electronic structure near the Dirac point can be described by the linear expression E = h¯ vF k, where vF is the Fermi velocity. The DRR equations that relates the wavevectors of the electrons (k and k0 ) and the phonon (q) with the laser energy (EL ) and phonon energy (Eph ) can be written as [17]: EL = 2h¯ vF k
(1a)
Eph = h¯ vF (k − k )
(1b)
q=k+k.
(1c)
0
0
It has been shown that the main contribution to the G0 band comes when the electron is scattered along the high symmetry
Fig. 1. (a) Raman spectrum of a monolayer graphene performed with a 1.72 eV laser energy, showing the two most prominent G and G0 Raman bands. (b) The G0 Raman band position as a function of laser energy. (c) Schematic view of the DRR process, where the electron is scattered from k to k0 by emitting a phonon with wavevector q. The circles around K and K0 points represent the electronic equi-energy contours.
line 0 –K–M–K 0 –0 where its initial wavevector k is along the 0 K line [17]. It is possible to see in Fig. 1(c) that the initial (k) and the scattered (k0 ) electron wavevectors are along the K0 direction. Moreover, the phonon wavevector (q) translated to the 0 point has a direction along the high symmetry KM line, with modulus q = k + k0 measured from the K point. From the experimental data shown in Fig. 1(b) and using Eqs. (1a)–(1c) it is possible to convert the horizontal axes (EL ) to the phonon wavevector (q) obtaining the experimental values for the frequency of the iTO phonon as a function of the wavevector q. The iTO phonon frequency is obtained by dividing the frequency of the G0 band by two, since the DRR scattering for this process involves two iTO phonons. Fig. 2 shows the experimental phonon dispersion along the KM −1
direction with wavevector q given in Å as measured from the K point. We have considered three different values of vF : 0.84 × 106 m/s (black circles), 1.0 × 106 m/s (gray circles), and 1.1 × 106 m/s (white circles), which corresponds to the values of the
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L.M. Malard et al. / Solid State Communications 149 (2009) 1136–1139 Table 1 Experimental SWM parameters (in eV) for the band structure of bilayer graphene.
Raman IR (Zhang et al. [25]) IR (Kuzmenko et al. [26])
Fig. 2. Phonon dispersion of the iTO phonon mode along the KM direction in monolayer graphene. The experimental data are presented by considering three different values for vF : 0.84 × 106 m/s (black circles), 1.0 × 106 m/s (gray circles), and 1.1 × 106 m/s (white circles). The theoretical models from Popov et al. [21] (blue dotted curve), DFT (black dashed curve) and GW (red full curve) by Lazzeri et al. [22] are also depicted. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
nearest-neighbor tight-binding hopping energy γ0 of 2.6, 3.1 and 3.4 eV, respectively. Three different phonon models for the iTO mode are also shown in Fig. 2. They correspond to the force constant calculation by Popov et al. [21] (blue dotted curve), and the recent DFT (black dashed curve) and GW (red full curve) calculations by Lazzeri et al. [22]. Since we were able to measure the G0 band with laser lines in the near infrared range, it was possible to probe the phonon dispersion more closer to the K point. The best agreement is observed between the GW [22] calculation and the experimental data for vF = 1 × 106 m/s. This result reflects the importance of considering electron–electron interactions for the description of phonon dispersion near the K point, and shows that other theoretical models such as DFT underestimated the phonon slope near the K point. 3.2. Electrons in bilayer graphene The SWM [23,24] model can be used to describe the electronic dispersion in bilayer graphene. It has been shown that the parameters that give rise to electron–hole asymmetry play an important role in describing the electrons in bilayer graphene [16]. Resonance Raman scattering applied together with the DRR process can evaluate the SWM band parameters, giving insight into the bilayer graphene electronic structure. Differently from recent infrared measurements where the electronic structure is probed in the low energy spectrum, the RRS can probe electron states connected by photons from the visible to the near infrared range. Here, we review the combination of RRS and DRR models to evaluate the SWM parameters [16], and we compare them with values obtained by infrared measurements. Fig. 3(a) shows the Raman spectrum of a bilayer graphene taken at 2.41 eV laser energy showing the G and G0 bands. Fig. 3(b) and (c) show the G0 band acquired with laser energies 2.71 and 1.91 eV respectively and fitted with four Lorentzians, as discussed in Ref. [16]. The dispersion of the four peaks of the G0 band as a function of laser energy is shown in Fig. 3(d). We labeled each of the four peaks that comprise the G0 band as Pij , where i, j = 1, 2. Each one of the Pij processes is related with a specific DRR process (see Fig. 4 from Ref. [16]). The slopes of the four Pij processes are different, the larger slope (120 cm−1 /eV) being associated with the smallest frequency peak, the smaller slope (84 cm−1 /eV) associated with the higher frequency peak, and the two intermediate peaks having approximately the same slope (95 cm−1 /eV).
γ0
γ1
γ3
γ4
∆
2.9 – –
0.30 0.4 0.378
0.10 – –
0.12 0.15 –
0.018 0.015
By describing the electronic structure of bilayer graphene with the SWM model, and applying the DRR conditions for each one of the four Pij , it is possible to extract the SWM band parameters that gives the best fit of the experimental data shown in Fig. 3(d). The fitting procedure is described in details in Ref. [16]. The solid lines in Fig. 3(d) show the best fit considering a linear phonon dispersion in the measured energy range. In Table 1 we show the best set of parameters and we compare them with the same parameters obtained from IR measurements. Zhang et al. [25] measured the parameters γ1 , ∆ and γ4 , and used the values of γ0 and γ3 from the literature [25] in order to fit their data. Kuzmenko et al. [26] also used the values of γ0 , γ3 and γ4 from the literature [29] and obtained the parameters of γ1 and ∆. Notice that the values of γ4 obtained from Raman and IR [25] measurements are in good agreement, revealing a strong electron–hole asymmetry for bilayer graphene. This result is essential to explain the four peaks of the G0 band of bilayer graphene and, in particular, the splitting of the two intermediate peaks [16]. The parameter ∆, which is not included in our present analysis, is related with the energy difference between the A and B atomic sites and also contributes to electron–hole asymmetry. A more complete Raman experiment with laser energies extending from the near infrared to the visible region can give more information about the discrepancies in the literature concerning the values of the parameter γ3 . Moreover, this experiment will allow an accurate determination of the parameter γ1 , which has an important effect on the low energy range of the electronic dispersion. 4. G0 band in trilayer graphene The space group of ABA trilayer graphene is isomorphic to the D3h point group at the 0 point, with six atoms in the unit cell. Therefore, along the T line, the electronic representation is given by 2T + ⊕ 4T − in the C1h point group. Since the light polarization in the basal plane belongs to the totally symmetric representation T + , five possible electron–hole creation processes connecting the valence and conduction bands with same symmetry are possible. Fig. 4(a) shows the electronic structure around the K point of trilayer graphene. The electronic structure consists of three conduction and three valence bands [30,29,31], where the symmetry and respective allowed light absorption are shown in Fig. 4(a). The electron–phonon scattering from K to K0 can occur involving the three different iTO phonon branches, one with T + and two of them with T − symmetries. Each of the five possible excited electrons can be scattered by phonons to the three different conduction bands near the K0 point. Therefore, there are in principle fifteen different DRR processes associated with the G0 band for trilayer graphene. Fig. 4(b) shows the Raman spectrum of a trilayer graphene measured with 2.41 eV laser energy. We found that we need at least six peaks with a FWHM of ∼24 cm−1 necessary to correctly fit the G0 band. The dependence of the frequency of the six peaks on the laser energy in the visible range is shown in Fig. 4(c). The dispersion slopes from the lower to the highest frequency peaks are 140 cm−1 /eV, 126 cm−1 /eV, 94 cm−1 /eV, 87 cm−1 /eV, 82 cm−1 /eV and 72 cm−1 /eV. The two lowest frequency modes have a steeper slope when compared to mono or bilayer graphene, mainly due to a steeper electronic or phonon dispersion. The other
L.M. Malard et al. / Solid State Communications 149 (2009) 1136–1139
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Fig. 3. (a) Raman spectrum of bilayer graphene taken at 2.41 eV laser energy showing the G and G0 features. (b) G0 band fitted with four Lorenztians for 2.71 eV laser energy and (c) for 1.91 eV laser energy. (d) The dispersion of the four peaks of the G0 band as a function of the various laser energies used.
a
b
Acknowledgements We are grateful to D.C. Elias, J.C. Brant, F. Plentz and E.S. Alves for their help in preparing the graphene samples and useful discussions. We also thank Han Htoon for technical assistance and access to the LANL CINT facilities. This work was supported by Rede Nacional de Pesquisa em Nanotubos de Carbono - MCT, Brasil and FAPEMIG. L.M.M. and D.L.M acknowledge the support from the Brazilian Agency CNPq. References
c
Fig. 4. (a) The electronic structure of trilayer graphene around the K point, showing the symmetry of the valence and conduction bands and respective allowed light absorption. (b) The G0 band fitted with six Lorenztians for 2.41 eV laser energy. (c) The dispersion of the six G0 peaks as a function of the laser energy.
modes have the same slopes as those of the two intermediate and the highest frequency modes of bilayer graphene. 5. Conclusions In this work we have shown how RRS and DRR scattering applied to graphene systems can provide useful information about its electronic and vibrational properties. We have presented the iTO phonon dispersion for monolayer graphene and compared it to different theoretical phonon models in the literature. The phonon dispersion GW calculation [22] agrees very well with our experimental data, showing that the electron–electron interactions play an important role in describing phonon dispersion near the K point. We also shown that the dispersion of the G0 band of bilayer graphene can be used to extract SWM parameters, which were compared with values obtained by infrared measurements. Finally, we presented the G0 band of trilayer graphene. Although fifteen DRR processes are possible, only six peaks are seen in the Raman spectrum. We also obtained the experimental dispersion of the six G0 band components with their respective slopes.
[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197. [3] Y. Zhang, Y. Tan, H. Stormer, P. Kim, Nature 438 (2005) 201. [4] E. McCann, V.I. Fal’ko, Phys. Rev. Lett. 96 (2006) 086805. [5] K.S. Novoselov, E. McCann, S.V. Morozov, V.I. Fal’ko, M.I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, A.K. Geim, Nature Phys. 2 (2006) 177. [6] F. Guinea, A.H. Castro Neto, N.M.R. Peres, Phys. Rev. B 73 (2006) 245426. [7] A.C. Ferrari, J.C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K.S. Novoselov, S. Roth, A.K. Geim, Phys. Rev. Lett. 97 (2006) 187401. [8] A. Gupta, G. Chen, P. Joshi, S. Tadigadapa, P.C. Eklund, Nanoletters 6 (2006) 2667. [9] D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, L. Wirtz, Nano Lett. 7 (2007) 238. [10] A.C. Ferrari, Solid State Commun. 143 (2007) 47. [11] D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, L. Wirtz, Solid State Commun. 143 (2007) 44. [12] S. Pisana, M. Lazzeri, C. Casiraghi, K.S. Novoselov, A.K. Geim, A.C. Ferrari, F. Mauri, Nature Mater. 6 (2007) 198. [13] J. Yan, Y. Zhang, P. Kim, A. Pinczuk, Phys. Rev. Lett. 98 (2007) 166802. [14] C. Casiraghi, S. Pisana, K.S. Novoselov, A.K. Geim, A.C. Ferrari, Appl. Phys. Lett. 91 (2007) 233108. [15] Zhen Hua Ni, Ting Yu, Yun Hao Lu, Ying Ying Wang, Yuan Ping Feng, Ze Xiang Shen, ACS Nano 2 (2008) 2301. [16] L.M. Malard, J. Nilsson, D.C. Elias, J.C. Brant, F. Plentz, E.S. Alves, A.H. Castro Neto, M.A. Pimenta, Phys. Rev. B 76 (2007) 201401. [17] D.L. Mafra, G. Samsonidze, L.M. Malard, D.C. Elias, J.C. Brant, F. Plentz, E.S. Alves, M.A. Pimenta, Phys. Rev. B 76 (2007) 233407. [18] A.V. Baranov, A.N. Bekhterev, Y.S. Bobovich, V.I. Petrov, Opt. Spectrosk. 62 (1987) 1036. [19] C. Thomsen, S. Reich, Phys. Rev. Lett. 85 (2000) 5214. [20] R. Saito, A. Jorio, A.G. Souza Filho, G. Dresselhaus, M.S. Dresselhaus, M.A. Pimenta, Phys. Rev. Lett. 88 (2002) 027401. [21] N.V. Popov, P. Lambin, Phys. Rev. B 73 (2006) 085407. [22] M. Lazzeri, C. Attaccalite, L. Wirtz, F. Mauri, Phys. Rev. B 78 (2008) 081406. [23] J.W. McClure, Phys. Rev. 108 (1957) 612. [24] J.C. Slonczewski, P.R. Weiss, Phys. Rev. 109 (1958) 272. [25] L.M. Zhang, Z.Q. Li, D.N. Basov, M.M. Fogler, Z. Hao, M.C. Martin, Phys. Rev. B 78 (2008) 235408. [26] A.B. Kuzmenko, E. van Heumen, D. van der Marel, P. Lerch, P. Blake, K.S. Novoselov, A.K. Geim, arXiv:0810.2400v1. [27] P. Blake, E.W. Hill, A.H. Castro Neto, K.S. Novoselov, D. Jiang, R. Yang, T.J. Booth, A.K. Geim, Appl. Phys. Lett. 91 (2007) 063124. [28] S. Piscanec, M. Lazzeri, F. Mauri, A.C. Ferrari, J. Robertson, Phys. Rev. Lett 93 (2004) 185503. [29] B. Partoens, F.M. Peeters, Phys. Rev. B 74 (2006) 075404. [30] S. Latil, L. Henrard, Phys. Rev. Lett. 97 (2006) 036803. [31] T. Ohta, A. Bostwick, J.L. McChesney, T. Seyller, K. Horn, Eli Rotenberg, Phys. Rev. Lett. 98 (2007) 206802.
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Resonance Raman scattering in graphene: Probing phonons and electrons L.M. Malard a,∗ , D.L. Mafra a , S.K. Doorn b , M.A. Pimenta a a
Departamento de Física, Universidade Federal de Minas Gerais, 30123-970, Belo Horizonte, Brazil
b
Chemistry Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
article
info
Article history: Received 7 November 2008 Received in revised form 6 February 2009 Accepted 6 February 2009 by the Guest Editors Available online 20 March 2009 PACS: 63.20.Kr 73.21.-b 78.30.-j 81.05.Uw
abstract In this work, by using different laser excitation energies, we obtain important electronic and vibrational properties of mono- and bi-layer graphene. For monolayer graphene, we determine the phonon dispersion near the Dirac point for the in-plane transverse optical (iTO) mode. This result is compared with recent calculations that take into account electron–electron correlations for the phonon dispersion around the K point. For bilayer graphene we extract the Slonczewski–Weiss–McClure band parameters and compare them with recent infrared measurements. We also analyze the second-order feature in the Raman spectrum for trilayer graphene. © 2009 Elsevier Ltd. All rights reserved.
Keywords: A. Nanostructures D. Electronic band structure D. Optical properties
1. Introduction Since the identification of mono and few graphene layers on a substrate [1], intensive work has been devoted to characterize this new material. New physics arose from this research such as the unconventional Hall effect [2–6]. In particular, Raman spectroscopy played an important role to characterize the number of layers [7–11], doping [12–14] and strain [15]. Moreover resonance Raman scattering (RRS) in graphene systems was shown to be an important tool to probe phonons and electrons [16,17]. The number of graphene layers can be obtained by Raman spectroscopy [7–9] and, in particular, to clearly distinguish a monolayer graphene from a double or few layer graphene sample. The two most prominent Raman bands arising from the double Resonance Raman (DRR) [18–20] process are the socalled induced-disorder D band around 1350 cm−1 (for a 2.41 eV laser excitation energy), and its associated second-order feature, that appears around 2700 cm−1 and that is historically called the G0 band. The denomination of this second-order feature is controversial in the literature: some authors prefer to call it the 2D band [7] or the D0 band [9], since it involves the same phonon mode that gives rise to the D-band. However, the observation
∗
Corresponding author. E-mail address: [email protected] (L.M. Malard).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.02.045
of the D-band requires the existence of a defect (or disorder) in the sample, justifying its denomination as D band, whereas the second-order feature appears in the spectrum of pure crystalline graphite, without any kind of defects or disorder. So, we will keep the historical denomination and call here this second-order feature as G0 band. The DRR process considers the scattering of electrons by phonons between points inside the first Brillouin zone. By this mechanism, it is possible to probe phonons that are not only at the zone center 0 point as for the usual first order Raman scattering. By combining RRS and the DRR model, one can infer information about the electronic and phonon dispersions of mono and few graphene layers. In this work, we studied the dependence of the G0 Raman band of mono-, bi- and tri-layer graphene as a function of the laser energy. We obtained the phonon dispersion of the inplane transverse optical (iTO) mode near the Dirac point for monolayer graphene and compared it with phonon dispersion calculations found in literature [21,22]. The electronic structure of bilayer graphene can be described in terms of the standard Slonczewski–Weiss–McClure (SWM) model for graphite [23,24]. By analyzing the dispersion of the G0 band of bilayer graphene when the laser energy is changed, it is possible to probe the splitting of the electronic dispersion branches and to obtain experimental values for the tight-binding parameters γ0 , γ1 , γ3 and γ4 . We compare our results with recent infrared measurements (IR) [25,26]. We also present a group theory analysis for
L.M. Malard et al. / Solid State Communications 149 (2009) 1136–1139
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the ABA trilayer graphene. With the selection rules for electronradiation interaction within the dipole approximation and for electron scattering by phonons, we discuss the DRR scattering showing how many processes can contribute for the G0 band of trilayer graphene. 2. Experimental details The graphene samples used is this experiment were obtained by a micro-mechanical cleavage of graphite on the surface of a Si sample with a 300 nm layer of SiO2 on the top. The bilayer flakes were identified in an optical microscope by the slight color change cause by white light intereference in the silicon oxide [27]. The number of layers was characterized by Raman spectroscopy using the procedure described by Ferrari et al. [7], where it is possible to distinguish between mono, bi and few layers graphene. For the Raman experiments in the visible range (Ar–Kr and dye lasers in the range 1.91–2.71 eV), we used a Dilor XY triple monochromator in the back scattering configuration. The spot size of the laser was ∼1 µm using a 100 × objective and the laser power was kept at 1.2 mW in order to avoid sample heating. The measurements in the near infrared range (Ti:Saphire laser 1.57–1.69 eV) were done using a triple monochromator (300/300/600 groves/mm grating set) with a 100 × objective and laser power of 1.0 mW. 3. Results and discussion 3.1. Phonons in monolayer graphene Phonon dispersion in graphene has been calculated by a number of different models in the literature [21,22]. Many of them contradict each other mainly near the 0 and K points where the Kohn anomaly occurs [28]. The electron–phonon coupling renormalizes the phonon frequency, causing a steep slope for phonon wavevectors near these two high symmetry points. More recently the iTO phonon dispersion near the K point was calculated also considering electron–electron interactions, predicting a steeper slope for the phonon dispersion [22]. In this section, we focus on the experimental data for the iTO phonon mode for monolayer graphene and compare them with theoretical calculations. Fig. 1(a) shows the Raman spectrum of a monolayer graphene, where it is possible to observe the G band near 1580 cm−1 and the G0 band near 2700 cm−1 for a 1.72 eV laser excitation energy. The G0 -band position as a function of the laser excitation energy is shown in Fig. 1(b). The G0 band arises from the DRR scattering process mechanism [18–20], which consists of three steps. First, an electron with wavevector k measured from the K point in the valence band is excited to the conduction band by a photon with energy EL . Then, the electron is resonantly scattered by emitting a phonon, going to the vicinity of the K0 point with wavevector k0 and energy EL − Eph . The electron is scattered back to the K point and recombines back with a hole emitting a scattered photon with energy EL −2Eph . A schematic picture of the DRR process can be seen in Fig. 1(c), where the initial k and the intermediate k0 wavevectors connected by a phonon with wavevector q are shown. In graphene, the electronic structure near the Dirac point can be described by the linear expression E = h¯ vF k, where vF is the Fermi velocity. The DRR equations that relates the wavevectors of the electrons (k and k0 ) and the phonon (q) with the laser energy (EL ) and phonon energy (Eph ) can be written as [17]: EL = 2h¯ vF k
(1a)
Eph = h¯ vF (k − k )
(1b)
q=k+k.
(1c)
0
0
It has been shown that the main contribution to the G0 band comes when the electron is scattered along the high symmetry
Fig. 1. (a) Raman spectrum of a monolayer graphene performed with a 1.72 eV laser energy, showing the two most prominent G and G0 Raman bands. (b) The G0 Raman band position as a function of laser energy. (c) Schematic view of the DRR process, where the electron is scattered from k to k0 by emitting a phonon with wavevector q. The circles around K and K0 points represent the electronic equi-energy contours.
line 0 –K–M–K 0 –0 where its initial wavevector k is along the 0 K line [17]. It is possible to see in Fig. 1(c) that the initial (k) and the scattered (k0 ) electron wavevectors are along the K0 direction. Moreover, the phonon wavevector (q) translated to the 0 point has a direction along the high symmetry KM line, with modulus q = k + k0 measured from the K point. From the experimental data shown in Fig. 1(b) and using Eqs. (1a)–(1c) it is possible to convert the horizontal axes (EL ) to the phonon wavevector (q) obtaining the experimental values for the frequency of the iTO phonon as a function of the wavevector q. The iTO phonon frequency is obtained by dividing the frequency of the G0 band by two, since the DRR scattering for this process involves two iTO phonons. Fig. 2 shows the experimental phonon dispersion along the KM −1
direction with wavevector q given in Å as measured from the K point. We have considered three different values of vF : 0.84 × 106 m/s (black circles), 1.0 × 106 m/s (gray circles), and 1.1 × 106 m/s (white circles), which corresponds to the values of the
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L.M. Malard et al. / Solid State Communications 149 (2009) 1136–1139 Table 1 Experimental SWM parameters (in eV) for the band structure of bilayer graphene.
Raman IR (Zhang et al. [25]) IR (Kuzmenko et al. [26])
Fig. 2. Phonon dispersion of the iTO phonon mode along the KM direction in monolayer graphene. The experimental data are presented by considering three different values for vF : 0.84 × 106 m/s (black circles), 1.0 × 106 m/s (gray circles), and 1.1 × 106 m/s (white circles). The theoretical models from Popov et al. [21] (blue dotted curve), DFT (black dashed curve) and GW (red full curve) by Lazzeri et al. [22] are also depicted. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
nearest-neighbor tight-binding hopping energy γ0 of 2.6, 3.1 and 3.4 eV, respectively. Three different phonon models for the iTO mode are also shown in Fig. 2. They correspond to the force constant calculation by Popov et al. [21] (blue dotted curve), and the recent DFT (black dashed curve) and GW (red full curve) calculations by Lazzeri et al. [22]. Since we were able to measure the G0 band with laser lines in the near infrared range, it was possible to probe the phonon dispersion more closer to the K point. The best agreement is observed between the GW [22] calculation and the experimental data for vF = 1 × 106 m/s. This result reflects the importance of considering electron–electron interactions for the description of phonon dispersion near the K point, and shows that other theoretical models such as DFT underestimated the phonon slope near the K point. 3.2. Electrons in bilayer graphene The SWM [23,24] model can be used to describe the electronic dispersion in bilayer graphene. It has been shown that the parameters that give rise to electron–hole asymmetry play an important role in describing the electrons in bilayer graphene [16]. Resonance Raman scattering applied together with the DRR process can evaluate the SWM band parameters, giving insight into the bilayer graphene electronic structure. Differently from recent infrared measurements where the electronic structure is probed in the low energy spectrum, the RRS can probe electron states connected by photons from the visible to the near infrared range. Here, we review the combination of RRS and DRR models to evaluate the SWM parameters [16], and we compare them with values obtained by infrared measurements. Fig. 3(a) shows the Raman spectrum of a bilayer graphene taken at 2.41 eV laser energy showing the G and G0 bands. Fig. 3(b) and (c) show the G0 band acquired with laser energies 2.71 and 1.91 eV respectively and fitted with four Lorentzians, as discussed in Ref. [16]. The dispersion of the four peaks of the G0 band as a function of laser energy is shown in Fig. 3(d). We labeled each of the four peaks that comprise the G0 band as Pij , where i, j = 1, 2. Each one of the Pij processes is related with a specific DRR process (see Fig. 4 from Ref. [16]). The slopes of the four Pij processes are different, the larger slope (120 cm−1 /eV) being associated with the smallest frequency peak, the smaller slope (84 cm−1 /eV) associated with the higher frequency peak, and the two intermediate peaks having approximately the same slope (95 cm−1 /eV).
γ0
γ1
γ3
γ4
∆
2.9 – –
0.30 0.4 0.378
0.10 – –
0.12 0.15 –
0.018 0.015
By describing the electronic structure of bilayer graphene with the SWM model, and applying the DRR conditions for each one of the four Pij , it is possible to extract the SWM band parameters that gives the best fit of the experimental data shown in Fig. 3(d). The fitting procedure is described in details in Ref. [16]. The solid lines in Fig. 3(d) show the best fit considering a linear phonon dispersion in the measured energy range. In Table 1 we show the best set of parameters and we compare them with the same parameters obtained from IR measurements. Zhang et al. [25] measured the parameters γ1 , ∆ and γ4 , and used the values of γ0 and γ3 from the literature [25] in order to fit their data. Kuzmenko et al. [26] also used the values of γ0 , γ3 and γ4 from the literature [29] and obtained the parameters of γ1 and ∆. Notice that the values of γ4 obtained from Raman and IR [25] measurements are in good agreement, revealing a strong electron–hole asymmetry for bilayer graphene. This result is essential to explain the four peaks of the G0 band of bilayer graphene and, in particular, the splitting of the two intermediate peaks [16]. The parameter ∆, which is not included in our present analysis, is related with the energy difference between the A and B atomic sites and also contributes to electron–hole asymmetry. A more complete Raman experiment with laser energies extending from the near infrared to the visible region can give more information about the discrepancies in the literature concerning the values of the parameter γ3 . Moreover, this experiment will allow an accurate determination of the parameter γ1 , which has an important effect on the low energy range of the electronic dispersion. 4. G0 band in trilayer graphene The space group of ABA trilayer graphene is isomorphic to the D3h point group at the 0 point, with six atoms in the unit cell. Therefore, along the T line, the electronic representation is given by 2T + ⊕ 4T − in the C1h point group. Since the light polarization in the basal plane belongs to the totally symmetric representation T + , five possible electron–hole creation processes connecting the valence and conduction bands with same symmetry are possible. Fig. 4(a) shows the electronic structure around the K point of trilayer graphene. The electronic structure consists of three conduction and three valence bands [30,29,31], where the symmetry and respective allowed light absorption are shown in Fig. 4(a). The electron–phonon scattering from K to K0 can occur involving the three different iTO phonon branches, one with T + and two of them with T − symmetries. Each of the five possible excited electrons can be scattered by phonons to the three different conduction bands near the K0 point. Therefore, there are in principle fifteen different DRR processes associated with the G0 band for trilayer graphene. Fig. 4(b) shows the Raman spectrum of a trilayer graphene measured with 2.41 eV laser energy. We found that we need at least six peaks with a FWHM of ∼24 cm−1 necessary to correctly fit the G0 band. The dependence of the frequency of the six peaks on the laser energy in the visible range is shown in Fig. 4(c). The dispersion slopes from the lower to the highest frequency peaks are 140 cm−1 /eV, 126 cm−1 /eV, 94 cm−1 /eV, 87 cm−1 /eV, 82 cm−1 /eV and 72 cm−1 /eV. The two lowest frequency modes have a steeper slope when compared to mono or bilayer graphene, mainly due to a steeper electronic or phonon dispersion. The other
L.M. Malard et al. / Solid State Communications 149 (2009) 1136–1139
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Fig. 3. (a) Raman spectrum of bilayer graphene taken at 2.41 eV laser energy showing the G and G0 features. (b) G0 band fitted with four Lorenztians for 2.71 eV laser energy and (c) for 1.91 eV laser energy. (d) The dispersion of the four peaks of the G0 band as a function of the various laser energies used.
a
b
Acknowledgements We are grateful to D.C. Elias, J.C. Brant, F. Plentz and E.S. Alves for their help in preparing the graphene samples and useful discussions. We also thank Han Htoon for technical assistance and access to the LANL CINT facilities. This work was supported by Rede Nacional de Pesquisa em Nanotubos de Carbono - MCT, Brasil and FAPEMIG. L.M.M. and D.L.M acknowledge the support from the Brazilian Agency CNPq. References
c
Fig. 4. (a) The electronic structure of trilayer graphene around the K point, showing the symmetry of the valence and conduction bands and respective allowed light absorption. (b) The G0 band fitted with six Lorenztians for 2.41 eV laser energy. (c) The dispersion of the six G0 peaks as a function of the laser energy.
modes have the same slopes as those of the two intermediate and the highest frequency modes of bilayer graphene. 5. Conclusions In this work we have shown how RRS and DRR scattering applied to graphene systems can provide useful information about its electronic and vibrational properties. We have presented the iTO phonon dispersion for monolayer graphene and compared it to different theoretical phonon models in the literature. The phonon dispersion GW calculation [22] agrees very well with our experimental data, showing that the electron–electron interactions play an important role in describing phonon dispersion near the K point. We also shown that the dispersion of the G0 band of bilayer graphene can be used to extract SWM parameters, which were compared with values obtained by infrared measurements. Finally, we presented the G0 band of trilayer graphene. Although fifteen DRR processes are possible, only six peaks are seen in the Raman spectrum. We also obtained the experimental dispersion of the six G0 band components with their respective slopes.
[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197. [3] Y. Zhang, Y. Tan, H. Stormer, P. Kim, Nature 438 (2005) 201. [4] E. McCann, V.I. Fal’ko, Phys. Rev. Lett. 96 (2006) 086805. [5] K.S. Novoselov, E. McCann, S.V. Morozov, V.I. Fal’ko, M.I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, A.K. Geim, Nature Phys. 2 (2006) 177. [6] F. Guinea, A.H. Castro Neto, N.M.R. Peres, Phys. Rev. B 73 (2006) 245426. [7] A.C. Ferrari, J.C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K.S. Novoselov, S. Roth, A.K. Geim, Phys. Rev. Lett. 97 (2006) 187401. [8] A. Gupta, G. Chen, P. Joshi, S. Tadigadapa, P.C. Eklund, Nanoletters 6 (2006) 2667. [9] D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, L. Wirtz, Nano Lett. 7 (2007) 238. [10] A.C. Ferrari, Solid State Commun. 143 (2007) 47. [11] D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen, C. Hierold, L. Wirtz, Solid State Commun. 143 (2007) 44. [12] S. Pisana, M. Lazzeri, C. Casiraghi, K.S. Novoselov, A.K. Geim, A.C. Ferrari, F. Mauri, Nature Mater. 6 (2007) 198. [13] J. Yan, Y. Zhang, P. Kim, A. Pinczuk, Phys. Rev. Lett. 98 (2007) 166802. [14] C. Casiraghi, S. Pisana, K.S. Novoselov, A.K. Geim, A.C. Ferrari, Appl. Phys. Lett. 91 (2007) 233108. [15] Zhen Hua Ni, Ting Yu, Yun Hao Lu, Ying Ying Wang, Yuan Ping Feng, Ze Xiang Shen, ACS Nano 2 (2008) 2301. [16] L.M. Malard, J. Nilsson, D.C. Elias, J.C. Brant, F. Plentz, E.S. Alves, A.H. Castro Neto, M.A. Pimenta, Phys. Rev. B 76 (2007) 201401. [17] D.L. Mafra, G. Samsonidze, L.M. Malard, D.C. Elias, J.C. Brant, F. Plentz, E.S. Alves, M.A. Pimenta, Phys. Rev. B 76 (2007) 233407. [18] A.V. Baranov, A.N. Bekhterev, Y.S. Bobovich, V.I. Petrov, Opt. Spectrosk. 62 (1987) 1036. [19] C. Thomsen, S. Reich, Phys. Rev. Lett. 85 (2000) 5214. [20] R. Saito, A. Jorio, A.G. Souza Filho, G. Dresselhaus, M.S. Dresselhaus, M.A. Pimenta, Phys. Rev. Lett. 88 (2002) 027401. [21] N.V. Popov, P. Lambin, Phys. Rev. B 73 (2006) 085407. [22] M. Lazzeri, C. Attaccalite, L. Wirtz, F. Mauri, Phys. Rev. B 78 (2008) 081406. [23] J.W. McClure, Phys. Rev. 108 (1957) 612. [24] J.C. Slonczewski, P.R. Weiss, Phys. Rev. 109 (1958) 272. [25] L.M. Zhang, Z.Q. Li, D.N. Basov, M.M. Fogler, Z. Hao, M.C. Martin, Phys. Rev. B 78 (2008) 235408. [26] A.B. Kuzmenko, E. van Heumen, D. van der Marel, P. Lerch, P. Blake, K.S. Novoselov, A.K. Geim, arXiv:0810.2400v1. [27] P. Blake, E.W. Hill, A.H. Castro Neto, K.S. Novoselov, D. Jiang, R. Yang, T.J. Booth, A.K. Geim, Appl. Phys. Lett. 91 (2007) 063124. [28] S. Piscanec, M. Lazzeri, F. Mauri, A.C. Ferrari, J. Robertson, Phys. Rev. Lett 93 (2004) 185503. [29] B. Partoens, F.M. Peeters, Phys. Rev. B 74 (2006) 075404. [30] S. Latil, L. Henrard, Phys. Rev. Lett. 97 (2006) 036803. [31] T. Ohta, A. Bostwick, J.L. McChesney, T. Seyller, K. Horn, Eli Rotenberg, Phys. Rev. Lett. 98 (2007) 206802.