Lattice-gas model for electron-hole coupling in disordered media
V. I. Yudson
Center for Chemical Physics, Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada N6A 3K7
and Institute of Spectroscopy, Russian Academy of Sciences, Troitzk, Moscow region 142092, Russia
M. R. Singh
Center for Chemical Physics, Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada N6A 3K7
Received 23 February 1998
We study an effective lattice-gas model for electron-hole coupling in disordered semiconductor structures.
Despite its simplicity, the model turns out to be quite rich. It possesses several crossover regimes between
phases of bound electron-hole pairs excitons and unbound electrons and holes. It has been shown that a
sufficiently strong disorder promotes dissociation of bound electron-hole pairs and may decrease considerably
the range of existence of exciton gas. S0163-18299805948-7
I. INTRODUCTION
There is considerable research interest in systems of ex-
citons in semiconductor quantum wells QW’s. This interest
is connected with the important role excitons play in light-
matter interaction processes in light-emitting devices and
semiconductor microcavities. Another reason for the re-
search interest is related to a possibility of Bose-Einstein
condensation of excitons in QW’s.
1
In this respect, special
attention is attracted to spatially indirect excitons formed by
spatially separated electrons e and holes ( h ).
2
Due to a
strongly enhanced annihilation time of spatially indirect ex-
citons, these systems are especially attractive with respect to
the search for collective phenomena. There is a rapidly in-
creasing amount of publications devoted to both
theoretical
3–6
and experimental
7,8
investigations of indirect
e -h coupling; see also references 4–6 and 8. Some experi-
mental evidence has been reported
8
for a stable excitonic
ground state in a strong magnetic field, which favors the
stability of the excitonic phase.
9
One of the intrinsic physical problems one meets when
dealing with excitons in semiconductor nanostructures is the
presence of a disorder that hinders manifestation of collec-
tive or coherent effects of exciton-exciton or exciton-light
interaction. The disorder corresponds to structure imperfec-
tions unavoidable in the course of fabrication. For thin QW’s
and, especially, in the case of indirect excitons, the disorder
is mainly determined by an interface roughness and a thick-
ness variation of QW’s. The presence of a moderate disorder
results in scattering of excitons and is responsible for rela-
tively small values of the exciton diffusion coefficient. With
an increase of the disorder excitons may become localized.
With respect to two-dimensional 2D systems a more cor-
rect statement is that the exciton localization length is be-
coming small as compared to the system size. However,
even in this ‘‘strong-localization’’ regime, typical variations
V
0
of the random potential V ( r) may be still small with
respect to the exciton binding energy E
0
, so that the internal
structure of excitons is weakly affected by the disorder.
The question addressed in this paper relates to the oppo-
site case when the random potential variation V
0
is greater
than the exciton binding energy E
0
. This situation may oc-
cur in sufficiently thin QW’s. For instance, as small as 0.1
nm thickness variation L in a GaAs QW of an average
thickness L =3 nm this corresponds to the experiment
8
would result in the variation of the electron confinement en-
ergy
2
2
L /( m
e
L
3
) 30 meV, which is considerably
greater than the exciton binding energy. In general, electrons
and holes in semiconductors are subject to different varia-
tions of their potential energies, V
e
( r) and V
h
( r), respec-
tively. Moreover, the correlation between V
e
( r) and V
h
( r)
may be rather weak. The difference in the electron and hole
random potentials influences the internal degrees of freedom
of an e -h pair. Our task is to investigate under which cir-
cumstances this may lead to the exciton dissociation.
An exciton created at an arbitrary place is not in the most
favorable—lowest potential energy—state of the e -h pair. At
large as compared to the exciton binding energy and non-
correlated random-potential variations there is a good chance
for the electron and the hole to find lower energy positions
that may be quite far from each other, so that at first glance,
the e -h pair should dissociate see Fig. 1. On the other hand,
for finite and finite-size correlated potentials, there is always
a possibility for the e -h pair to find a position that corre-
sponds to the minimum of both V
e
( r) and V
h
( r). Electron-
hole coupling in the vicinity of this point and the formation
of the exciton state leads to an additional lowering of the
energy. Thus, speaking about thermodynamically equilib-
rium states one might come to the opposite conclusion that
the disorder may have only a little influence on the existence
of excitons. However, the latter reasoning relates to a single
exciton. But a single exciton cannot survive even in an or-
dered macroscopic sample at thermal equilibrium with a
nonzero temperature T, it will unavoidably dissociate into an
electron and a hole. At T 0, only a finite-density gas of
bound e -h pairs coexisting with a gas of free electrons and
holes may be in the ‘‘ionization equilibrium.’’
10
The subject
of the present study is the influence of a random potential on
the ionization equilibrium condition for e -h pairs. We shall
restrict our consideration to the case of low electron and hole
densities when we may neglect screening effects.
The quantum-mechanical problem of a particle subjected
PHYSICAL REVIEW B 15 DECEMBER 1998-II VOLUME 58, NUMBER 24
PRB 58 0163-1829/98/5824/162027/$15.00 16 202 ©1998 The American Physical Society