Double 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. Some of the special points of the Brillouin zone are also shown.	INTRODUCTION 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 bands originating from double resonant scattering processes (Thomsen and Reich, 2000). The double resonant processes are favoured by the specific cone-like electronic bands (Dirac cones) close to the Fermi energy (Dirac points) (Fig. 1). A double resonant process comprises 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 (Fig. 2). The processes with one "ep" process and one "hp" process give major contribution to the Raman intensity. We calculate the electronic band structure, phonon dispersion, electronic lifetime and double resonant Raman intensity within the non-orthogonal tight-binding (NTB) model [1,2,3]. This model uses ab-initio derived matrix elements and has no adjustable parameters except for the downscaling parameter of 0.9 for the phonon frequencies. The dynamical matrix uses electronic response to the ionic displacement, derived in first-order perturbation theory. The Raman intensity is derived in fourth-order perturbation theory. The electron-photon and electron-phonon matrix elements are calculated within the NTB model [3].
Fig.2. Schematic representation of the double resonant 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.	Overtone double resonant Raman spectrum The overtone spectrum is due to two phonons from the same branch, labeled by 1,2,…,6 (Fig. 4) [3]. 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. 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, arXiv).
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.	COMBINATION double resonant Raman spectrum The combination spectrum is due to two phonons from two different branches, labeled by 1,2,…,6 (Fig. 6) [3]. 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).
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.	TOTAL DOUBLE RESONANT RAMAN SPECTRUM The total double resonant Raman spectrum exhibits mainly three measurable bands: D+D", 2D, and 2D'. The 2D band is the most intense among the three and is always observed. Two of our results have been confirmed recently (Berciaud, 2013; May, 2013) in support of the use of our NTB model for Raman spectra calculations. References: 1. V. N. Popov, New J. Phys. 6 (2004) 1-17. 2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407. 3. V. N. Popov and Ph. Lambin, Eur. Phys. J. B 85 (2012) 418/1-8.

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Valentin Popov

July 25, 2012