Two-phonon resonant Raman scattering in graphene in the ultraviolet region
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Fig. 1. Left: Electronic band structure of graphene,
calculated within the NTB model. The red rectangle shows the relevant energy region
for Raman processes. Right: Phonon dispersion of graphene, calculated within
the NTB model. The red rectangle shows the phonon branches with largest
contribution to the two-phonon Raman spectra. |
Due to its characteristic electronic band structure with cone-like electronic bands (Dirac cones) close to the Fermi energy and strong electron-phonon coupling for certain phonons, the two-phonon Raman spectra of graphene show a few intense bands. 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, We derive the electronic band
structure and phonon dispersion (Fig. 1), matrix elements In Ref. [6], we extended the calculation of the two-phonon bands to the ultraviolet region. Figure 2 shows the calculated Raman spectra of graphene at laser excitation EL = 4.0, 5.0, and 6.0 eV. The intense feature marked by 2TO is observed as a symmetric band at lower excitations and is referred to as the 2D band. At the studied high-energy excitations, this band widens and becomes asymmetric. The two end points of it, I and II, come from phonons along ΓK and KM, the corresponding processes being termed inner and outer. The feature II is enhanced for phonons close to the M point because of the high phonon density of states of the TO branch there. For this reason, it is denoted here by 2TO@M. Apart from the broad band 2TO, two other features can also be identified. The 2LO band comes from scattering by two LO phonons close to the Γ point. This peak is enhanced for phonons, close to the overbending of the LO band (panel at EL = 6.0 eV). Another feature, 2LOTO, can be seen as a sharp peak. It is a combination band, originating from the LO and TO branches, close to the Γ point. At lower Raman shifts, there are several other less intense peaks, denoted by LOLA@M, 2LO@M, and LOTO@M, which are also enhanced by the high phonon density of states at M. Figure 3 shows parts of the conduction (π*) and valence (π) bands in the vicinity of the M point of the Brillouin zone of graphene. Four different scattering paths of the electrons, marked by Latin numbers, can be identified. For laser excitation EL, close to 5.0 eV, the scattering of electrons takes place around the inflexion point of the electronic bands at the M point. The high electronic density of states there results in enhancement of the Raman intensity of the bands. Figure 4 shows that the calculated Raman spectrum at EL = 5.0 eV is in excellent agreement with the measured one (Tyborski, 2015). Additionally, the calculations allowed to assign precisely the features, marked in red. 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. 6. V. N. Popov and Ph. Lambin, 2D Materials 3 (2016) 025014. |
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Valentin Popov