Thermodynamics and excitations of Coulomb glass
Vikas Malik and Deepak Kumar
School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India
Received 13 March 2007; revised manuscript received 7 June 2007; published 24 September 2007
We have calculated the phase diagram and density of states of single-particle excitations of a lattice model
of the Coulomb glass. We employ the replica method to average over disorder and then use linked-cluster
expansion to obtain the free energy in the random phase approximation. We find that the system has a transition
to an antiferromagnetic phase below a critical disorder. Above the critical disorder, the system has a glassy
“independent spin phase” at zero temperature which crosses over to paramagnetic phase as the temperature is
increased. In the antiferromagnetic phase, the single-particle excitation density density of states DOS has a
feature including the gap at the Fermi level due to long-range order. To obtain DOS in the glassy phase, we
analyze cavity-field equations for local magnetization occupation. We obtain DOS at nonzero temperatures
and find that with temperature the DOS at the Fermi level increases quadratically and the Coulomb gap
decreases linearly.
DOI: 10.1103/PhysRevB.76.125207 PACS numbers: 71.23.An, 73.20.Qt, 72.80.Ng
I. INTRODUCTION
The Coulomb glass has been a problem of long standing
interest as it involves a very interesting interplay of disorder
and Coulomb interactions. The initial interest in the problem
lies in describing the hopping transport in bands where the
electronic states are localized around Fermi level as is ob-
tained in doped, compensated semiconductors
1
at sufficiently
low temperatures. It was recognized early by Pollak
2,3
and
Srinivasan
4
that in such systems, Coulomb interactions play
a crucial role: they cause a depletion in the single-particle
density of states around the Fermi level. Efros and Shk-
lovskii ES made these arguments more precise by invoking
stability considerations of the ground state to show that this
depletion is actually a soft gap, now known as Coulomb gap,
with a density of states having the form g -
d-1
,
where is the chemical potential and d is the dimension of
the system.
5,6
They also pointed out its very significant effect
on conduction in the variable range hopping VRH regime.
The concept of VRH was introduced earlier by Mott,
7,8
who
argued that when the temperatures become too low to permit
nearest neighbor hopping, the electrons optimize between the
hop distance and the activation energy, which results in
the temperature dependence of conductivity of the form
exp-T
M
/ T
1/4
. Efros and Shklovskii
5
showed that interac-
tions between electrons would cause a qualitative change in
the temperature dependence of conductance to the form
exp-T
ES
/ T
1/2
due to the presence of the soft Coulomb
gap. A large number of experimental studies of conduction
have been made in the VRH regime examining the above
temperature dependences and the crossover between them.
1,9
Recently, experimental studies have focused on other as-
pects of the dynamical behavior of Coulomb glasses.
10–14
In
studies by Ovadyahu and co-workers,
10–12
the relaxation of
conductivity is measured after carrier concentration of the
sample has been changed by field effect or photoexcitation.
The decay of excess conductivity is found to be logarithmi-
cally slow and shows aging effects typical of glasses at low
temperatures. Another set of experiments which point toward
strong correlation effects in these system is due to Massey
and Lee,
13
who by combining tunneling conduction studies
with bulk conduction behavior argue that the Coulomb gap is
significantly lower than Efros-Shklovskii prediction, imply-
ing a role for multielectron excitations. Tunneling studies of
Sandow et al.
14
suggest strong correlation effects by showing
large departures from ES form of density of states.
In the last three decades or so, there have been numerous
numerical and analytical studies to explore various transport
and thermodynamic properties of Coulomb glasses.
15–30
Par-
ticular attention has been focused on the numerical compu-
tation of the density of states DOS for the single-particle
excitation. While all the studies find the gap, there are dif-
ferences regarding the form of the gap in various numerical
studies.
16,31–34
Many studies agree with the ES prediction,
but other studies find results which are either lower or higher
than these values. A main problem of the numerical calcula-
tions is that the system has a complex energy landscape with
innumerable metastable states, which makes it difficult to
reach the ground state. To relax from a metastable state to
another one typically needs multielectron excitations, for
which the phase space is too large to tackle numerically. The
metastable states also exhibit the Coulomb gap features,
33
but a clear power law over the full energy range is not seen
in these studies. The finite-size effects are particularly strong
for low energies.
33
Most of the theoretical studies have been done on a lattice
model, first discussed by Efros.
6
In this model, the electron
states are assumed to be localized around centers placed on a
regular lattice of N
s
sites. When the number of electrons N is
N
s
/2, the Hamiltonian of the system is taken to be
H =
i
i
n
i
+
1
2
ij
e
2
|r
i
- r
j
|
n
i
- 1/2n
j
- 1/2 , 1
where n
i
and
i
denote the electron occupation number and
the energy of the localized state at site i, respectively. The
variable n
i
takes values 0 and 1, as the typical on-site Cou-
lomb energy is too large to permit more than one electron per
site. The site energies are taken to be independent random
variables with a site-independent probability distribution
PHYSICAL REVIEW B 76, 125207 2007
1098-0121/2007/7612/1252079 ©2007 The American Physical Society 125207-1