Energy relaxation via confined and interface phonons in quantum-wire systems
C. R. Bennett
Department of Physics, University of Essex, Colchester, CO4 3SQ, England
B. Tanatar
Department of Physics, Bilkent University, Bilkent, 06533 Ankara, Turkey
Received 13 November 1996
We present a fully dynamical and finite temperature study of the hot-electron momentum relaxation rate and
the power loss in a coupled system of electrons and confined and interface phonons in a quantum-wire
structure. Renormalization effects due to electron-phonon interactions lead to an enhancement in the power
loss similar to the bulk phonon case. S0163-18299705112-6
I. INTRODUCTION
In recent years, the hot-electron energy relaxation phe-
nomenon in low-dimensional semiconductors attracted con-
siderable interest both experimentally
1
and theoretically.
2
The energy-loss mechanism is very important because of its
technological relevance, as most semiconductor-based de-
vices operate under high-field, hot-electron conditions. In
particular, hot-electron transistors with a base region made of
high-mobility semiconducting material like GaAs offer a
high-speed device. When a strong electric field is applied,
the electron gas attains a temperature higher than that of the
surrounding lattice. Equilibrium is reached by the emission
of different types of phonons depending on the temperature
regime. The advances in growth technology made it possible
to study quasi-one-dimensional Q1D electronic structures,
thereby improving our knowledge on low-dimensional sys-
tems. Theoretical work on the energy-loss rate in quantum
wires has started to appear.
3–5
In low-dimensional semiconductor structures phonon con-
finement becomes an essential part of the description of
electron-phonon interactions. Since the early observation of
confined phonons in GaAs/AlAs superlattices,
6
the phonon
modes in microstructures have attracted a great deal of
attention.
7,8
Among the various macroscopic pictures, the di-
electric continuum DC model
9
offers a simple framework
with which to address the phonon confinement effects. The
phonon modes in the DC model are i an infinite set of
confined modes with vanishing electrostatic potentials at the
interfaces that oscillate at the bulk LO-phonon frequency of
GaAs, and ii a set of modes with electrostatic potentials
attaining maxima at the interfaces. The interface modes lie
within the reststrahl band of GaAs and AlAs, and in quasi-
two-dimensional Q2D systems, it is found
10,11
that the
AlAs interface modes dominate the interaction. The situation
is similar in Q1D systems, as demonstrated in the confined
and interface polaron problem in cylindrical quantum
wires.
12
The purpose of this paper is to study the energy relaxation
via confined and interface phonons of an excited Q1D elec-
tron gas in a GaAs quantum wire embedded in AlAs mate-
rial. We employ the dielectric continuum model to describe
the phonon confinement effects and take the many-body
renormalization effects due to electron-phonon interactions
into account. Thus our work complements the recent study
by Zheng and Das Sarma
5
who considered the energy relax-
ation by bulk LO phonons. Earlier works taking phonon con-
finement effects into consideration in quantum wires have
neglected the many-body renormalization.
13
Hot-electron
experiments
14
to date are performed on wide quantum wires
with multisubband occupation, but it is conceivable that in
the near future quantum-wire structures with only the lowest
subband occupied
15
will be amenable to measurements di-
rectly relevant to calculations presented here. The many-
body effects change the phonon self-energy due to electron-
phonon interactions, renormalizing the phonon propagator
significantly at low temperatures.
We use the theory advanced by Das Sarma and
co-workers
2,5,16
to calculate the hot-electron power loss due
to confined and interface phonons. A test electron is assumed
to be injected into the quantum wire without modifying the
properties of the coupled electron-phonon system. The stan-
dard electron-scattering theory
17
is used, treating the system
to be not completely isolated. The coupled system is then in
quasiequilibrium and interacts with an external heat bath. We
note that the above viewpoint was challenged
18
predicting
differing results. Nevertheless, the electron-temperature
model
2,5,16
provides a suitable scheme to describe the energy
relaxation processes especially when hot-phonon effects are
not important.
The rest of this paper is organized as follows. In the next
section, we introduce the expressions for momentum relax-
ation rate and power loss in Q1D wires. Our numerical re-
sults for GaAs quantum wires embedded in AlAs material
are presented in Sec. III. We conclude with a brief summary
in Sec. IV.
II. THEORY
The quantum-wire model we use consists of an infinitely
long cylinder of radius R with hard walls.
19
Such a model
leads to an analytic expression
19
for the effective Coulomb
potential V ( q ) between the electrons within certain approxi-
mations. We assume that the linear electron density is such
that only the lowest subband is populated. To describe the
phonon confinement, we consider a GaAs quantum wire em-
PHYSICAL REVIEW B 15 MARCH 1997-I VOLUME 55, NUMBER 11
55 0163-1829/97/5511/71655/$10.00 7165 © 1997 The American Physical Society