One-phonon
resonant Raman scattering in graphene
The Raman scattering in graphene is always
resonant because for all laser excitations in the visible range there are pairs
of valence and conduction states matching the laser photon energy. The relevant
states belong to the conduction and valence bands, crossing at the Fermi energy
and forming conic surfaces in space, so-called Dirac cones.
For calculation of the resonant Raman spectrum
of graphene, quantum-mechanical
description of the Raman scattering process is adopted, which considers the
electrons, photons and phonons of the system, and their interactions. The
resonant one-phonon Raman intensity for Stokes processes is derived in
third-order quantum-mechanical perturbation theory
.
Here, 
, 
; 
is the energy of the initial state, 
is the incident photon energy (laser excitation); 
, 
are the energies
of the intermediate (a, b) states of the system; f stands for final state; 
and 
are momentum matrix elements for incident and scattered
photons, resp.; 
is the
electron-phonon matrix element; γ
is the broadening parameter, equal to
the sum of the halfwidths of conduction and valence states. The one-phonon
Raman process can be represented by a sequence of virtual processes of
electron-hole creation and annihilation, and electron-phonon scattering.
The calculation of the one-phonon Raman
spectrum is performed within a non-orthogonal tight-binding (NTB) model applied
to the electronic band structure [1,2], phonon dispersion [3], electron-photon
and electron-phonon matrix elements [4], broadening parameter [5], and the
quantum-mechanical expression for the intensity [6]. This approach was
originally applied to single-walled carbon nanotubes.
Graphene has a single one-phonon Raman band,
called the G band. It originates from the Raman-active in-plane optical phonon
of symmetry E2g. The Raman shift does not change
with the laser excitation, while the intensity of the G band is an increasing
function of the laser excitation. The shift and intensity of the G band do not change
with the change of the angles of the incident and scattered light [7]. Indeed,
let 
and 
are the polarization vectors of the incident and scattered
light for backscattering geometry, and 
is the polarization angle of the Raman-active phonon. Then,
the light polarization vectors are
,
, and the polarization vectors of the two components of the E2g phonon are 
and 
. Substituting the polarization vectors in the intensity
and taking into account that for hexagonal symmetry the
non-zero components of the Raman tensor 
are 
, one gets 
.
The one-phonon Raman spectra were also
calculated for graphene with three types of point defects: mono-vacancy,
di-vacancy, and Stone-Wales defects [8] within the NTB model. The
characteristic Raman peaks for the different point defects were identified and
their behaviour with the laser excitation was investigated.
References:
1. V. N. Popov, New J. Phys. 6 (2004) 17.
2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407.
3. V. N. Popov and Ph. Lambin, Phys. Rev. B 73
(2006) 085407.
4. V. N. Popov, L. Henrard, and Ph. Lambin,
Phys. Rev. B 72 (2005) 035436.
5. V. N. Popov and Ph. Lambin, Phys. Rev. B 74 (2006) 075415.
6. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 165425.
7. V. N. Popov, L. Henrard, and Ph. Lambin,
Carbon 54 (2013) 86.
8. V. N. Popov, L. Henrard, and Ph. Lambin,
Carbon 47 (2009) 2455.
Valentin Popov
September 21, 2006