Dielectric function of
single-walled carbon nanotubes (SWNTs)
The dielectric function of SWNTs can be calculated within the
independent-particle approximation [1,2]. The most important quantity describing
the optical absorption is the dielectric function. The imaginary part of the
dielectric function in the random-phase approximation is given by
where ħω is the
photon energy, e is the elementary
charge, and m is the electron mass.
The sum is over all occupied (v) and
unoccupied (c) states. pkl'cklv,μ is the
matrix element of the component of the momentum operator in the direction μ of the light polarization
Substituting the one-electron wavefunction 
expressed as a linear combination of atomic
orbitals 
in the expression above, one obtains the non-zero matrix elements
where
For z-axis along the tube
axis, the quantities
are given by
and 
. The latter two expressions express the selection rules for
allowed dipole optical transitions, namely, optical transitions are only
allowed between states with the same l
for parallel polarization and between states with l and l' differing by 1
for perpendicular polarization. Further on, from Maxwell’s relation 
(
is the complex refractive index), the refractive index 
and the extinction
coefficient 
are readily obtained.
The relations 
(c is the light
velocity in vacuum) and 
allow one to derive
the absorption coefficient α and
the reflection coefficient for normal incidence R.
Let us consider a single pair of valence and conduction bands with
maximum and minimum separated by a direct gap Ecv corresponding to an allowed optical transition v→c. In the effective mass
approximation, it is straightforward to show that the contribution to ε2 from these bands is
given by
Here mcv* is the reduced effective mass for a transition between the two bands and
pcv,μ is
the momentum matrix element at Ecv.
Alternatively, for a pair of valence and conduction bands with minimum and
maximum separated by energy Ecv
one obtains
In the general case, the graph ε2(ω) will consist of two types of
spikes close in form to those described by the two expressions above. From
their derivation it is clear that the electron density of states versus ω will have the same two types of
spikes.
The optical transition energies for parallel/perpendicular light
polarization can be derived from the positions of the spikes of ε2(ω). The resulting data can conveniently be arranged in an
energy-tube radius plot (the resonance chart) [1,2].
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.
Valentin Popov