NANO LETTERS
Strong Polarization Dependence of Double-Resonant Raman Intensities in Graphene
2008 Vol. 8, No. 12 4270-4274
Duhee Yoon,† Hyerim Moon, Young-Woo Son,*,‡,§ G. Samsonidze,|,⊥ Bae Ho Park,‡ Jin Bae Kim,# YoungPak Lee,# and Hyeonsik Cheong*,† Department of Physics, Sogang UniVersity, Seoul 121-742, Korea, Department of Physics, Konkuk UniVersity, Seoul 143-701, Korea, Korea Institute for AdVanced Study, Seoul 130-722, Korea, Department of Physics, UniVersity of California at Berkeley, Berkeley, California 94720, Materials Sciences DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720, Quantum Photonic Science Research Center, Hanyang UniVersity, Seoul 133-791, Korea Received July 7, 2008; Revised Manuscript Received September 23, 2008
ABSTRACT Spatially resolved and polarized micro-Raman spectroscopy on microcrystalline graphene shows strong polarization dependences of doubleresonance Raman intensities. The Raman intensity of the double-resonant 2D band is maximum when the excitation and detection polarizations are parallel and minimum when they are orthogonal, whereas that of the G band is isotropic. A calculation shows that this strong polarization dependence is a direct consequence of inhomogeneous optical absorption and emission mediated by electron-phonon interactions involved in the second-order Stokes-Stokes Raman scattering process.
Raman spectroscopy has been an important tool to study physical properties of graphitic materials for the last four decades.1 After the successful isolation of graphene,2-4 whose electronic states have a relativistic energy dispersion,2-5 Raman-based information has proven to be vital in characterizing the material.6-12 In this work, we report spatially resolved and polarized micro-Raman spectroscopy on microcrystalline graphene and, for the first time, show strong polarization dependences of double-resonance Raman intensities measured at the center of two-dimensional crystalline graphene. Our study reveals that this strong polarization dependence is a direct consequence of inhomogeneous optical absorption and emission mediated by electron-phonon interactions in the double-resonant Raman scattering processes. Such a peculiar nodal structure of optical absorption and emission originating from the relativistic energy dispersion9 has not been observed directly so far in clean twodimensional graphene due to the difficulty in selecting specific momentums of electrons.9-11
Electronic states in graphene have a relativistic energy dispersion described by the (2 + 1)-dimensional massless Dirac equation.2-5 Such a characteristic is of central importance to the novel physical properties predicted in graphene, some of which already have been observed in a number of different experimental setups.13-19 Notably, for a given polarization vector P of a linearly polarized incident (scattered) light, the light absorption or emission probability per unit time is known to be proportional to | ( (P × kˆ)·nˆ|2 where kˆ is a unit vector along the momentum of electron (hole) measured from the Dirac point, and nˆ is a unit vector orthogonal to the plane of graphene.9 It implies that the absorption (emission) of polarized light is absent when the direction of momentum of electron is parallel to the polarization vector while it is maximized when they are perpendicular to each other. Such a nodal structure in the absorption (emission) coefficient is peculiar to graphene unlike other materials having quadratic low energy dispersion relations for electrons.
* To whom correspondence should be addressed. E-mail:
[email protected] (Y.-W.S.);
[email protected] (H.C.). † Department of Physics, Sogang University. ‡ Department of Physics, Konkuk University. § Korea Institute for Advanced Study. | Department of Physics, University of California at Berkeley. ⊥ Materials Sciences Division, Lawrence Berkeley National Laboratory. # Quantum Photonic Science Research Center, Hanyang University.
To observe the unique nodal structure of the absorption or emission of light in graphene, it is important to select the direction of momentum k of electrons. Usually, the optical absorption (emission) does not choose a special momentum of electrons but rather average all momenta of electrons contributing to the process.9 A lower dimensional materials derived from graphene, for example, graphene nanoribbons
10.1021/nl8017498 CCC: $40.75 Published on Web 10/29/2008
2008 American Chemical Society
with a nanoscale confinement in one spatial dimension10 and edge states in a well defined step of graphite11 would be one possibility in selecting the direction of momentum along the ribbon or the edges and indeed show the aforementioned inhomogeneous optical process.10,11 Besides anistropies of electrons’ momentum in terminated or confined geometries of graphene, there is another process that can select electrons’ momenta in graphene. In the second-order Stokes-Stokes Raman scattering processes,20-22 the electron-phonon interactions in the 2D band are strong functions of the relative angle between the momenta of electron and the phonon involved.22,23 If one uses a specific polarization of the incident laser, electrons with a specific momentum will excite first. Then, after aforementioned electron-phonon scattering processes (twophonon scattering), electrons with another specific momentum scatter off and emit photons with specific polarization due to the nodal structures of optical transition matrix of graphene. Hence, the Raman intensity at the 2D band frequency will change significantly when the angle between the polarizations of incoming and outgoing light are varied. Motivated by the discussion, we have measured spatially resolved and polarized micro-Raman spectra on graphene. Graphene samples used in this work were prepared by micromechanical cleavage of graphite flakes and then put on silicon wafers with a 300 nm silicon oxide layer.2,3 We first searched possible single-layer graphene areas with the optical microscope that was coupled with a micro spectroscopy setup (Figure 1a,b), and then the single-layer graphene was exactly identified with the micro-Raman imaging technique and atomic force microscopy. The 514.5 nm (2.41 eV) line of an Ar ion laser, with a power of ∼1.2 mW, was used as the excitation source. The power was chosen to avoid heating of the sample.6 A 40× microscope objective (0.6 N.A.) was used to focus the laser beam onto the sample and collect the scattered light in the backscattering geometry (Z[Pi,Ps]Z). The scattered signal was dispersed with a JobinYvon Triax 550 spectrometer (1200 grooves/mm) and detected with a liquid-nitrogen-cooled CCD detector. The spatial resolution was less than 1 µm, and the spectral resolution was ∼1 cm-1. For polarized Raman measurements, a half-wave plate was used to rotate the polarization of the incident laser beam. For the scattered light, a linear polarizer was used as an analyzer and another half-wave plate was used to align the polarization of the light going into the spectrometer perpendicular to the groove direction of the grating. This ensures that the polarization dependence of the grating efficiency does not affect the result. The polarization dependence of the transmittance of the beamsplitter was separately measured and compensated for in the analysis of the data. Figure 1e shows typical Raman spectra from the graphene sample used in this work. There are two prominent spectral features, called G (ωG ∼ 1586 cm-1) and 2D (ω2D ∼ 2686 cm-1) bands, respectively. Also, there is the G* band (ωG* ∼2457 cm-1) with a relatively low intensity.21,22,24,25 On the basis of the intensity and shape of G and 2D bands, our spatially resolved Raman scattering measurement can deNano Lett., Vol. 8, No. 12, 2008
Figure 1. (a) Schematic diagram of the experimental set up. A polarizer is used to align the polarization of the incident laser beam to the s-polarization direction relative to the beam splitter. A halfwave plate is used to rotate the polarization of the incident laser beam on the sample and that of the scattered light. An analyzer picks up a specific polarization direction of the scattered light, and another half-wave plate aligns the polarization of the light going into the spectrometer perpendicular to the groove direction of the grating. (b) Optical microscope image of the graphene sample. (c) Image of the intensity of the Raman G band. The central red area corresponds to the single-layer graphene. (d) Image of the intensity of the Raman 2D band. The image is fully complementary to the G-band image. (e) Raman spectra obtained from the single-layer (top) and bilayer (bottom) areas of the sample.
termine the number of layers locally. Figure 1c is a local intensity map at ωG, and Figure 1d is one at ω2D. There is a clear complementarity between the two images such that a large central (red color in Figure 1c) area of the sample is identified to be the single layer while boundaries to be the 4271
bilayer (yellow color in Figure 1c). The top spectrum in Figure 1e is obtained from the single-layer region of the sample and the bottom spectrum from the bilayer region, which are consistent with the data in the literature.6-8,12 This assignment was also confirmed by an AFM measurement (not shown here).
( ) ( ) f 0 0 0 f 0 0 -f 0 and f 0 0 0 0 0 0 0 0
(2)
and eˆL (eˆS) is the unit polarization vector of the incident (scattered) light.26 If we define the unit polarization vectors
Figure 2a is a collection of Raman spectra measured for different polarization configurations. First, we focused the incident laser beam on the center of a single-layer graphene sample and aligned its polarization (eˆL) along one of the edges (R ) 0). Then, the Raman spectra were taken as a function of the polarization of the analyzer (eˆA) (see the inset of Figure 2a) with eˆL fixed. After the spectra were normalized for the polarization dependent transmittance of the beam splitter, the intensity at ωG is found to be constant for all polarization directions but that at ω2D shows clear polarization dependence. As will be shown later, the intensity of the G band must be independent of the polarizations of the incident or scattered photons. Therefore, when comparing the intensities of the 2D bands from different spectra, one can eliminate experimental fluctuations between different measurements by normalizing the intensity of the 2D band to that of the G band. Figure 2b is a plot of the intensity of the 2D band normalized to that of the G band. The normalized intensity is maximum when the scattered polarization is parallel to the incident polarization and minimum when the two polarizations are perpendicular to each other. The largest normalized intensity of the 2D band is about 3.3 times the smallest. The normalized intensity of the G* band also shows the same strong polarization dependence, and the ratio between the maximum and minimum intensity is similar to the case of 2D band. We also find that the observed variation of the intensity of Raman 2D band does not change at all if the polarization of the incident laser beam is altered. After rotating the incident polarization (R) in 30° steps, we repeated the measurements. The same oscillating behaviors of the 2D and G* band were observed for all incident polarization directions, i.e., the variation of the 2D band intensity only depends on the relative angle (β) between polarizations of the analyzer and the incident laser (Figure 2c). It indicates that the observed behavior does not result from some anisotropy in the experimental setup or structural anisotropy of the sample such as defects or edges. Hence, we conclude that the observed variation of intensities of the double-resonance Raman processes (2D and G* bands) indeed originate from the intrinsic material properties of the present two-dimensional microcrystalline graphene sample. The invariant intensity of the G mode with respect to polarizations can be understood using group theory. The G mode in graphene exhibits E2g symmetry (space group P63/ mmc).25,26 The Raman intensity (IG) for the mode can be calculated according to IG∝|
∑
2 eˆ · Ri · eˆS|2 i)1 L
(1)
where R1 and R2 are the doubly degenerate Raman polarizability tensors for the E2g symmetry25,26 4272
Figure 2. (a) Polarized Raman spectra obtained for the incident laser beam with a fixed polarization (eˆL) while the polarization of the analyzer (eˆA) is varied with respect to eˆL with an angle β. The spectra were taken with R ) 0. The intensity of the G band does not change with β. (b) The variation of the intensities of the 2D and G* bands normalized to that of the G band as a function of the polarization of the scattered light. (c) Normalized intensity of the 2D band as a function of the relative angle β measured for different incident polarization R. Regardless of R, the normalized intensity shows the same dependence on β. Nano Lett., Vol. 8, No. 12, 2008
Figure 3. (a) Schematic for the process of double-resonance Stokes-Stokes Raman scattering (2D band). An incident laser beam excites an electron near K-point of which momentum (k) lie on a contour for Ec - Ev ) 2.41 eV (left cone) and is scattered to the state near K′-point of which momentum (k′) lies on a contour for Ec - Ev ) 2.09 eV (right cone) with an emission of a zone boundary transverse optical phonon (iTO) with a momentum of k-k′. The momenta of electrons (k and k′) and the polarization of incident laser beam are parametrized by angles θ, θ′, and R with respect to the KΓ direction, respectively. (b) Contour plot (unit of 10-4 eV) for the absolute value of the optical dipole transition matrix elements for the light absorption process as functions of the direction of electron momentum (θ) and the polarization direction of incoming photon (R). (c) Contour plot (unit of 10-4 eV) for the absolute value of electron-phonon scattering matrix element for the Stokes process involving the iTO phonon mode as functions of directions of electrons (θ and θ′). (d) Polar plot for the normalized intensity ratio of the 2D peak to the G peak as a function of the relative angle (β). The curve is from the calculations and dots from the experiment.
of incident and scattered lights as (cos R, sin R, 0) and (cos γ, sin γ, 0), respectively, the Raman intensity can be expressed as I|[cos2(R + γ) + sin2(R + γ)] ) I| where I| is the intensity when both polarizations are parallel to each other. So, the Raman intensity of the G band does not depend on the polarization direction of either incident or scattered light. This explains our observation on the constant Raman intensity at ωG regardless of variation of the polarizations of the incident laser and the analyzer. A peculiar variation of intensities of the 2D band originates from interplay between the nodal structures of electron-photon interactions and the momentum dependent anisotropic electron-phonon interactions in graphene. The differential cross-section for the second-order Stokes-Stokes Raman 2 scattering process in graphene is proportional to |Kλ′λ ea (Ee,Ea)| , where Ea and Ee are the energies of the incident and scattered photons of polarizations λ and λ′, respectively, and Kλ′λ ea is an appropriate high-order matrix element of interaction between the electromagnetic field and graphene.20-23,27 We calculate the Kλ′λ ea within a tight-binding approximation with a nonorthogonal single-orbital basis set for carbon π-orbitals22,23,28 (For the detailed form of the matrix element, see Supporting Information). Figure 3a depicts the energy conservation in the intervalley second-order Stokes-Stokes resonance Raman (SRR) spectrum for the 2D band, which can be expressed in terms of (Ea - Ee)/p ) 2ωiTO ) ω2D, where p is the Planck’s constant and ωiTO is the frequency of the zone boundary in-plane transverse optical phonon Nano Lett., Vol. 8, No. 12, 2008
(iTO) mode.22,23,27,29 The G* band comes from the combination of the zone boundary in-plane longitudinal acoustic (iLA) and iTO modes.21-23,27,29 The light absorption (emission) involves electrons of which wave vectors are on the contour lines for Ec - Ev ) 2.41 (2.09) eV corresponding to the wavelength of the 514.5 nm Ar ion laser used in the experiment and the measured Raman frequency shift (ω2D), where Ec(v) is the conduction (valence) band of graphene. Figure 3b is the calculated optical dipole transition matrix element for the light absorption process. It indeed demonstrates the nodal structures, which is zero when the momentum of an electron is parallel to the polarization direction of incident photon. The electron-phonon scattering for the 2D band involves two phonons with opposite momentum. Figure 3c is the calculated matrix element between two electron states of which the momenta lying on two contours show strong anisotropy: the scattering is maximized when two electrons’ momenta point to opposite directions and zero when the two are parallel to each other, in good agreement with previous studies.22,23 Combining the matrix elements of each process for the SRR intensity together with a projection of the scattered light to the analyzer (eˆA), the resulting Raman intensity at ω2D shows the strong polarization dependence and agrees well with the experiment results (Figure 3d). In the calculation, the maximum intensity occurs when eˆL is parallel to eˆA while the minimum occurs when 4273
those two are perpendicular to each other. The calculated maximum intensity is 3.13 times higher than the minimum. With the help of the calculations, we can understand the polarization dependent SRR intensity by introducing a simple heuristic model. When an incident (scattered) light has a polarization eˆL ) (cos R, sin R, 0) [eˆS ) (cos γ, sin γ, 0)], the absorption (emission) matrix element can be expressed as sin(R-θ)[sin(γ-θ′)], where θ(θ′) is an angle between the momenta of electron and the KΓ direction of the first Brillouin zone of graphene (Figure 3a). From the calculation (Figure 3c), the electron-phonon scattering matrix element between two electrons can be approximated to be proportional to sin2[(θ – θ′)/2]. The scattering intensity at an angle γ is given as I(R, γ) )
|∫ ∫
2π
0
sin(R - θ) sin4
( θ -2 θ′ ) sin(γ - θ ′ ) dθdθ ′ |
2
(3)
Considering a projection of the scattered light to the analyzer with a polarization eˆA ) (cos(R + β), sin(R + β), 0) (Figure 2a), the resulting intensity after integrating out all relevant angles can be expressed as
∫
2π
0
I| I(R, γ) cos2(φ - γ) dγ ) (2cos2 β + 1) ) I(β) 3
(4)
where I| is a maximum Raman intensity for the 2D band when eˆA is parallel to eˆL (β ) 0 or π) and φ ) R + β. From this simpler model, the maximum intensity is three time the minimum. Because the bilayer graphene also exhibits its characteristic double-resonance Raman spectrum,12 the present simple model will also play an important role in understanding its polarization dependence of Raman spectrum. In summary, we have presented the first experimental confirmation of the nodal structure of the electron-photon interaction in two-dimensional microcrystalline graphene through spatially resolved and polarized Raman spectroscopy. Combined with theoretical calculations, we clearly demonstrate that the observed polarization dependence of higherorder Raman scattering processes indeed originate from interplay between the special properties of graphene-nodal electron-photon interaction and anisotropic electron-phonon interactions involved. Our present experimental and theoretical work suggest that, in addition to the present extensive usage of Raman spectroscopy to characterize the nanocarbon materials, the polarized Raman spectroscopy is also an invaluable tool to study the peculiar electron-photon and electron-phonon interactions in them. Acknowledgment. This work was supported in part by Korea Research Foundation (no. KRF-2007-314-C00093). Y.-W.S. is supported in part by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MEST) no. R01-2007-000-10654-0. J.B.K., Y.P.L., and H.C. are supported in part by KOSEF grant funded by MEST (Quantum Photonic Science Research Center). D.Y. and H.C. are supported in part by Nano R&D program through KOSEF funded by MEST (2008-03717). B.H.P. was supported by the KOSEF NRL Program grant funded by MEST (No. R0A-2008-000-20052-0). 4274
Supporting Information Available: Details on the matrix element in the second-order Stokes-Stokes Raman scattering process. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Dresselhaus, M. S.; Dresselhaus, G.; Saito, R.; Jorio, A. Phys. Rep. 2005, 409, 47. (2) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438, 197. (3) Zhang, Y.; Tan, Y.-W.; Stormer, H. L.; Kim, P. Nature 2005, 438, 201. (4) Berger, C.; Song, Z.; Li, X.; Wu, X.; Brown, N.; Naud, C.; Mayou, D.; Li, T.; Hass, J.; Marchenkov, A. N.; Conrad, E. H.; First, P. N.; de Heer, W. A. Science 2006, 312, 1191. (5) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. 2007, Preprint at http://arXiv.org/0709.1163. (6) Ferrari, A. C.; Meyer, J. C.; Scardaci, V.; Casiraghi, C.; Lazzeri, M.; Mauri, F.; Piscanec, S.; Jiang, D.; Novoselov, K. S.; Roth, S.; Geim, A. K. Phys. ReV. Lett. 2006, 97, 187401. (7) Gupta, A.; Chen, G.; Joshi, P.; Tadigadapa, S.; Eklund, P. C. Nano Lett. 2006, 6, 2667. (8) Graf, D.; Molitor, F.; Ensslin, K.; Stampfer, C.; Jungen, A.; Hierold, C.; Wirtz, L. Nano Lett. 2007, 7, 238. (9) Gru¨neis, A.; Saito, R.; Samsonidze, G.; Kimura, T.; Pimenta, M. A.; Jorio, A.; Souza Filho, A. G.; Dresselhaus, G.; Dresselhaus, M. S. Phys. ReV. B 2003, 67, 165402. (10) Canc¸ado, L. G.; Pimenta, M. A.; Neves, B. R. A.; Mederiros-Ribeiros, G.; Enoki, T.; Kobayashi, Y.; Takai, K.; Fukui, K.; Dresselhaus, M. S.; Saito, R.; Jorio, A. Phys. ReV. Lett. 2004, 93, 047403. (11) Canc¸ado, L. G.; Pimenta, M. A.; Neves, B. R. A.; Dantas, M. S. S.; Jorio, A. Phys. ReV. Lett. 2004, 93, 247401. (12) Malard, L. M.; Nilsson, J.; Elias, D. C.; Brant, J. C.; Plentz, F.; Alves, E. S.; Castro Neto, A. H.; Pimenta, M. A. Phys. ReV. B 2007, 76, 201401(R). (13) Zhou, S. Y.; Gweon, G. -H.; Graf, J.; Fedorov, A. V.; Spataru, C. D.; Diehl, R. D.; Kopelevich, Y.; Lee, D. -H.; Louie, S. G.; Lanzara, A. Nat. Phys 2006, 2, 595. (14) Bostwick, A.; Ohta, T.; Seyller, T.; Horn, K.; Rotenberg, E. Nat. Phys. 2007, 3, 36. (15) Pisana, S.; Lazzeri, M.; Casiraghi, C.; Novoselov, K. S.; Geim, A. K.; Ferrari, A. C.; Mauri, F. Nat. Mat. 2007, 6, 198. (16) Rutter, G. M.; Crain, J. N.; Guisinger, N. S.; Li, T.; First, P. N.; Stroscio, J. A. Science 2007, 317, 219. (17) Williams, J. R.; DiCarlo, L.; Marcus, C. M. Science 2007, 317, 638. (18) Huard, B.; Sulpizio, J. A.; Stander, N.; Todd, K.; Yang, B.; GoldhaberGordon, D. Phys. ReV. Lett. 2007, 98, 236803. ¨ zyilmaz, B.; Jarillo-Herrero, P.; Efetov, D.; Abanin, D. A.; Levitov, (19) O L. S.; Kim, P. Phys. ReV. Lett. 2007, 99, 166804. (20) Basko, D. M. Phys. ReV. B 2007, 76, 081405(R). (21) Maultzch, J.; Reich, S.; Thomsen, C. Phys. ReV. B 2004, 70, 155403. (22) Samsonidze, G. Ge. Photophysics of Carbon Nanotubes. Ph. D. Thesis,Massachusetts Institute of Technology, 2007. (23) Mafra, D. L.; Samsonidze, G.; Malard, L. M.; Elias, D. C.; Brant, J. C.; Plentz, F.; Alves, E. S.; Pimenta, M. A. Phys. ReV. B 2007, 76, 233407. (24) Shimada, T.; Sugai, T.; Fantimi, C.; Souza, M.; Canc¸ado, L. G.; Jorio, A.; Pimenta, M. A.; Saito, R.; Gru¨neis, A.; Dresselhaus, G.; Dresselhaus, M. S.; Ohno, Y.; Mizutani, T.; Shinohara, H. Carbon 2005, 43, 1049. (25) Tan, P.; Hu, C.; Dong, J.; Shen, W.; Zhang, B. Phys. ReV. B 2001, 64, 214301. (26) Loudon, R. AdV. Phys. 1964, 13, 423. (27) Thomsen, C.; Reich, S. Phys. ReV. Lett. 2000, 85, 5214. (28) Jiang, J.; Saito, R.; Samsonidze, G.; Chou, S. G.; Jorio, A.; Dresselhaus, G.; Dresselhaus, M. S. Phys. ReV. B 2005, 72, 235408. (29) Saito, R.; Gru¨neis, A.; Samsonidze, G.; Brar, V. W.; Dresselhaus, G.; Dresselhaus, M. S.; Jorio, A.; Canc¸ado, L. G.; Fantini, C.; Pimenta, M. A.; Souza Filho, A. G. New J. Phys. 2003, 5, 157, and references therein.
NL8017498
Nano Lett., Vol. 8, No. 12, 2008
