One-phonon resonant Raman
scattering in single-walled carbon nanotubes
The Raman scattering in single-walled carbon nanotubes (SWNT) is
essentially resonant as Raman signal is observed only for laser excitations
close to an optical transition Eii. For the calculation of the
resonant Raman intensity of the Raman bands of any SWNT, one can use a model of
the electronic band structure of the SWNT [1,2]. The quantum-mechanical description of the Raman
scattering process can be done considering the system of electrons, photons and
phonons, and their interactions [3,4]. 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 [5]; γ
is the broadening parameter, equal to
the sum of the halfwidths of conduction and valence states [6]. 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 one-phonon Raman bands of SWNTs can have quite complicated shapes,
determined by three important parameters: the
separation of the optical transition energies, the optical phonon frequency, and the excited electron state lifetime. The presence of the three
parameters can make difficult the approximate description of the intensity but
nevertheless such a description can effectively be realized [5].
Considering a particular optical transition Eii, only a pair of conduction and valence bands can be
taken into account in the calculation of the intensity. Therefore, the matrix
elements are scalars, which will denoted with
and
. For a Raman band, observed at the transition Eii, it is advantageous to introduce the peak Raman intensity defined
as
,
where mii* is the effective mass at Eii. The parameters Jii
can be pulled out of
the sum in the expression for I and
the remaining summation over the Brillouin zone can be carried out as an
integration. The result is the shape function for the Raman band:
,
where 
is the phonon energy. The Raman intensity can then be
described approximately by the expression [5]
.
Depending on the signs of Jii
and Fii, strong constructive or destructive interference
is expected to take place for close optical transitions, as it is the case with
metallic nanotubes.
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, L. Henrard, and Ph. Lambin, Nano Letters 4 (2004) 1795-1799.
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 73 (2006) 165425.
6. V. N.
Popov and Ph. Lambin, Phys. Rev. B 74
(2006) 075415.
Valentin Popov
September 21, 2006