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Energy Loss Rates of Hot Electrons in Semiconducting Carbon Nanotubes
Margarita Tsaousidou *
Materials Science Department, University of Patras, Patras 26 504, Greece
*Email: rtsaous@upatras.gr
The study of the energy relaxation provides significant information about the carrier-phonon coupling in low-dimensional
semiconductors [1]. In the present paper we calculate the energy loss rate, P, of hot 1D electrons in semiconducting singlewall
carbon nanotubes (SWCNTs) due to their coupling to stretching and breathing phonons. A brief description of the
theoretical formalism is given and numerical calculations of P are performed for electron temperatures in the range 4-300 K
and for various values of the Fermi energy E F . These results can be particularly useful for steady-state electric field heating
or time-dependent experiments [2].
The SWCNT can be regarded as a long and infinitesimally thin cylinder of radius R. The electrons are free to move along
the direction of the nanotube axis (z-direction) while their motion in the xy-plane is quantized. The electron spectrum consists
of several 1D subbands associated with the quantized electron motion. We assume that the electrons are in internal thermal
equilibrium at a temperature T e higher than the lattice temperature T lat and their distribution is described by the Fermi-Dirac
distribution function. This approximation, the so-called electron temperature model, is valid when the electron-electron
relaxation time is much less than the energy loss time of the electrons. For simplicity reasons we consider only electronphonon
(e-p) scattering within the lowest 1D electron subband (quantum limit). The Fermi level is assumed to be close to the
ground subband minimum in order to eliminate non-parabolicity effects. The e-p interaction is described via a deformation
potential. It can be shown that for intrasubband scattering within the ground subband only the stretching and the breathing
phonon modes contribute to the e-p coupling. In the long-wavelength limit the stretching mode has a linear dispersion while
the breathing mode is dispersionless [3].
The average energy-loss rate due to e-p scattering is given by [4]
P =
1
N
∑
q
ω
where, N is the total number of electrons, ω q is the frequency of a phonon with wave vector q (q is along the z-axis), and
dN q /dt is the rate of change of the phonon distribution due to phonon emission and absorption processes. The above equation
can be written in the following convenient form:
q
dΝ
dt
1
ep 2
P = ∑ ω
| H ( q) | [ N q ( Te
) − N q ( Tlat
)] Im χ(q,
ω
N
q
q q )
where, |H ep (q)| is the e-p coupling matrix element, N q (T) is the Bose-Einstein distribution at temperature T and Im χ(q,ω q ) is
the imaginary part of the electronic polarizability function [5]. For qR