Two-phonon resonant Raman scattering in graphene

Fig. 1. Schematic representation of the electronic band structure close to the Fermi energy: the conduction and valence bands are cone-like (Dirac cones) with apices (Dirac points) at the K and K' points of the hexagonal Brillouin zone of graphene.	Graphene has a single Raman-active phonon of E2g symmetry, observed as an intense band in the Raman spectra. Apart from this band, the spectra exhibit a few other intense bands due to two-phonon Raman scattering. It has been argued that the enhancement of the two-phonon Raman bands of graphene is due to the specific electronic band structure of graphene with cone-like electronic bands (Dirac cones) close to the Fermi energy (Dirac points) (Fig. 1). The origin of the enhancement of the two-phonon Raman processes can be revealed by considering the Raman scattering process from a quantum-mechanical point of view. In the quantum-mechanical description of the Raman process, one takes into account the electrons, photons and phonons of the system, and their interactions. The resonant Raman intensity for Stokes processes is derived in fourth-order quantum-mechanical perturbation theory (Martin & Falicov, 1983) Here, , ; is the energy of the initial state, is the incident photon energy (laser excitation); , , are the energies of the intermediate (a, b, c) and final (f) states of the system;are the matrix elements between initial, intermediate, and final states; and are momentum matrix elements; and are electron-phonon matrix elements; γ is the broadening parameter, equal to the sum of the halfwidths of conduction and valence states. Here, the electronic band structure, phonon dispersion, matrix elements and broadening parameter γ are entirely derived within a non-orthogonal tight-binding (NTB) model [1-4]. The Raman scattering process can be represented as several virtual ones: absorption of an incident photon with creation of an electron-hole pair, double scattering of the created electron/hole by phonons, and recombination of the electron-hole pair with emission of a photon. There are altogether eight such processes. For four of the processes, an electron is scattered twice by two phonons with opposite momenta or a hole is scattered twice by two phonons with opposite momenta. It can be shown that, in the denominator of the intensity, the real part of two can be zero, hence, the Raman intensity is enhanced. Such resonant processes are termed double resonance processes (Thomsen). For the other four processes, an electron and a hole are scattered by two phonons with opposite momenta (Fig. 2). It can be shown that for such processes, the real part of all three can be zero and the intensity is enhanced. Such processes are termed triple resonance processes (Venezuela). The triple resonance processes give major contribution to the two-phonon Raman bands. Figure 4 shows the overtone two-phonon Raman spectrum of graphene, which is due to two phonons from the same branch, labeled by 1,2,…,6 [5]. The major contributions to the spectrum come from the 5th branch close to the K point (2TO@K and 2TO'@K phonons) and the 6th branch close to the Γ point (2LO@Γ phonons). The former give rise to the 2D band, and the latter give rise to the 2D' band. Figure 5 shows the dependence of the Raman shift and the Raman intensity of the overtone bands on the laser excitation. The calculated values are in excellent agreement with the available experimental data. The double scattering processes by 2TO@K phonons are usually called inner processes and those by 2TO'@K phonons are called outer processes. The contribution from inner processes is about an order of magnitude larger than that from outer processes. Recently, this has been confirmed experimentally (Berciaud, 2013). Figure 6 shows the combination two-phonon Raman spectrum of graphene, which is due to two phonons from two different branches, labeled by 1,2,…,6 [5]. The largest contributions to the spectrum come from TO'LA'@K and TOLA@K phonons. The former is not observed as a separate band because it overlaps with the much more intense 2D band. The TOLA@K and TO'LO'@K give rise to the D+D" band - a result, which has recently been confirmed theoretically (May, 2013). The remaining contributions are at least one order of magnitude smaller and are rarely observed (Sato, 2011). Figure 7 shows the dependence of the Raman shift and Raman intensity of the combination bands on the laser excitation. The calculated values are in excellent agreement with the existing experimental data. Figure 8 shows the total two-phonon Raman spectrum, which exhibits mainly three measurable bands: D+D", 2D, and 2D'. The 2D band is the most intense among the three and is often observed. References: 1. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407. 2. V. N. Popov, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436. 3. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407 4. V. N. Popov and Ph. Lambin, Phys. Rev. B 74 (2006) 075415 5. V. N. Popov and Ph. Lambin, Eur. Phys. J. B 85 (2012) 418.
Fig.2. Schematic representation of the resonant Raman scattering processes. The solid lines are cross-sections of the Dirac cones at the K and K' points. The dashed lines are virtual processes: "eh^+" and "eh^−" are electron-hole creation and destruction processes, resp., "ep" and "hp" are electron and hole scattering processes by a phonon
Fig. 3. Phonon dispersion of graphene. The six phonon branches are marked by the acronyms LO, TO, ZO, LA, TA, and ZA. The letters O and A stand for “optical” and “acoustic”, respectively; L, T, and Z denote in-plane longitudinal, in-plane transverse, and out-of-plane atomic displacement, respectively. The crosses mark the phonons, which play major role in the double resonant processes
Fig. 4. Calculated overtone Raman spectra at laser photon energy of 2.0 eV. The graphs show the contributions to the spectra from two phonons from the same branch labeled by 1,2,…,6.
Fig. 5. Calculated Raman shift (left) and Raman intensity (right) of the overtone double resonant Raman bands. The symbols are experimental data (Mafra, 2007).
Fig. 6. Calculated combination Raman spectra at laser photon energy of 2.0 eV. The graphs show the contributions to the spectra from two phonons from two different branches labeled by 1,2,…,6.
Fig. 7. Calculated Raman shift (left) and Raman intensity (right) of the combination double resonant Raman bands. The symbols are experimental data (squares, Mafra, 2007; circles, Sato, 2011).
Fig. 8. Calculated total Raman spectra at laser photon energy of 2.0 eV. The graphs show the contributions to the spectra from the overtone and combination spectra.

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Valentin Popov

July 25, 2012