J
ournal of Statistical Mechanics: Theory and Experiment
An IOP and SISSA journal
The Coulomb gap and low energy
statistics for Coulomb glasses
1
Materials Science Division, Argonne National Laboratory, Argonne,
IL 60439, USA
2 Department of Physics and Center of Advanced Materials and
Nanotechnology, University of Oslo, PO Box 1048 Blindern, 0316 Oslo, Norway
3 A F Ioffe Physico-Technical Institute of Russian Academy of Sciences,
194021 St Petersburg, Russia
E-mail: glatz@anl.gov, vinokour@anl.gov, joakim.bergli@fys.uio.no,
martin.kirkengen@fys.uio.no and iouri.galperine@fys.uio.no
Received 27 March 2008
Accepted 13 May 2008
Published 12 June 2008
Online at stacks.iop.org/JSTAT/2008/P06006
doi:10.1088/1742-5468/2008/06/P06006
Abstract. We study the statistics of local energy minima in the configuration
space of two-dimensional lattice Coulomb glasses with site disorder and the
behavior of the Coulomb gap depending on the strength of random site energies.
At intermediate disorder, i.e., when the typical strength of the disorder is of
the same order as the nearest-neighbor Coulomb energy, the high energy tail of
the distribution of the local minima is exponential. We furthermore analyze the
structure of the local minima and show that most sites of the system have the
same occupation numbers in all of these states. The density of states (DOS)
shows a transition from the crystalline state at zero disorder (with a hard gap)
to an intermediate, probably glassy state with a Coulomb gap. We analyze this
Coulomb gap in some detail and show that the DOS deviates slightly from the
traditional linear behavior in 2D. For finite systems these intermediate Coulomb
gap states disappear for large disorder strengths and only a random localized
state in which all electrons are in the minima of the random potential exists.
Dedication: This paper is dedicated to Thomas Nattermann, our dearest friend,
brilliant colleague, and outstanding teacher.
Keywords: disordered systems (theory), slow relaxation and glassy dynamics,
energy landscapes (theory)
c
2008
IOP Publishing Ltd and SISSA
1742-5468/08/P06006+12$30.00
J. Stat. Mech. (2008) P06006
Andreas Glatz1, Valerii M Vinokur1, Joakim Bergli2,
Martin Kirkengen2 and Yuri M Galperin2,3
The Coulomb gap and low energy statistics for Coulomb glasses
Contents
1. Introduction
2
2. Model and simulation
3
3. Statistics of the local minimal states
5
4. Coulomb gap
8
11
Acknowledgments
12
References
12
1. Introduction
Possible phases and the corresponding dynamic behaviors of strongly correlated disordered
systems remain among the central unresolved issues. Doped semiconductors in the
insulating state are exemplary systems, endowed with strong long range Coulomb
interactions and strong disorder. It was hypothesized that their combined action imposes a
glassy phase, resulting in the term electron or Coulomb glass [1]. Ever since, the Coulomb
glass has been a subject of intensive research; see [2]. However, despite impressive
developments, a consistent picture of its nature is still not available and consensus on
the interrelation between its inherent properties, like the Coulomb gap, slow dynamics,
and ageing, and the nature of the low lying metastable states and the mechanism of
hopping transport is not yet achieved. A classic qualitative derivation [3] of the Coulomb
gap induced in the single-particle density of states due to pair correlations remains the
major result in the field.
One of the main characteristics of the glassy state of an infinite system is the existence
of an infinite number of low lying states (valleys in the rugged free energy relief) separated
by barriers growing infinitely in the thermodynamic limit. This picture emerged from
the study of the infinite range interaction model for spin glasses (Sherrington–Kirkpatrick
spin glass) [4] and was later shown to imply an ultrametric structure of the configurational
space [5].
There was important recent progress in relating the appearance of the Coulomb gap
in electron systems to the generic glassy behavior [6], where a lattice model of spinless
interacting electrons in the presence of randomness was considered and a non-linear
screening theory was formulated to tackle Coulomb gap formation and its relation to
glassy freezing. The formation of the glass phase with a large number of metastable
states was identified via the emergence of a replica symmetry breaking instability.
In [7, 8], the connection between the glass phase characteristics and a Coulomb
gap was argued for, noting that within the locator approximation the correlated
electron system shares several key properties with the Sherrington–Kirkpatrick spin glass.
Remarkably, recent experimental studies of low frequency resistance noise in silicon
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5. Conclusions
The Coulomb gap and low energy statistics for Coulomb glasses
2. Model and simulation
In the present work we study the low energy properties of the lattice Coulomb model
with on-site disorder potential (no positional disorder). The Hamiltonian for this system
is defined as
e2 (ni − ν)(nj − ν)
εi ,
εi = Uαi ni +
,
(1)
H=
2
r
ij
i
j=i
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inversion layers indicated glassy freezing and the presence of long range correlations,
consistent with the hierarchical picture of glassy dynamics [9].
In this paper we re-examine in the first part the distribution of local minimal states
(which we define as states that are energetically stable against all single-particle jumps)
of the lattice model for a disordered Coulomb system (cf [10]), using an improved
minimization algorithm which is explained in detail. We show that the high energy tail
of the distribution of the numerically found local minimal states of the system can be
fitted with an exponential distribution and study the structure of these local minima to
some detail. The overlap between all local minima obtained is calculated and shows that
the number of difference sites of two arbitrarily chosen states is almost invariant (7.4%
for a 1002 -site system). A spatial analysis of the overlap reveals that the occupation
numbers of most sites (about 75% for a 1002 -site system) are frozen in all local minima.
It has been suggested [11] that flips of local, multi-site aggregates are responsible for the
low frequency noise of such systems. We tried to identify such aggregates by identifying
localized clusters of sites that appear in two different occupation states in our set of
local minima. However, we were not able to identify such aggregates from analyzing
the overlaps of our local minimal states. The numerically obtained overlap map of the
calculated minima rather suggests that a transition between two minima has to occur
by a global rearrangement of electrons on all ‘active’ sites (sites which have a non-frozen
occupation number throughout all minima). Since the total number of local minima for
the 1002 -site system studied is probably very large (the number of all possible states is of
order 103008 ), we cannot prove that the idea of such aggregates is wrong; we could have
missed identifying them in the set of 20 000 analyzed minima. More work is needed to
reveal the mechanisms of the low temperature dynamics of Coulomb glass systems.
In the second part, the distribution of addition or single-particle energies of these low
energy local minima for a half-filled Coulomb glass is analyzed numerically for differing
disorder strength. For small disorder strength the systems exhibits a hard Coulomb gap in
this density of states (DOS) indicating a global crystalline electron structure, whereas at
intermediate values a linear Coulomb gap forms (glassy/microcrystalline structure) which
vanishes for strong disorder (random localized state) at finite disorder strength in finite
systems. We analyzes the ‘linear’ Coulomb gap for a large set of single-particle energies
(of order 108 ) and found that it follows a power law with exponent close to 1.23 rather
than the classical linear behavior.
In the last part of the paper we study the width of the hard gap Δh with increasing
disorder U, starting with an initial Wigner crystal configuration. This crystalline structure
is stable for U < Uc , which manifests itself in a linear decrease of Δh and breaks down at
Uc where Δh drops to zero.
The Coulomb gap and low energy statistics for Coulomb glasses
where i and j label the lattice sites, rij = |ri − rj |, Uαi with αi ∈ [−1; 1] define the
quenched uniformly distributed random site energies, and ni are the occupation numbers
(values 0 or 1) with average ν.
Apart from the site energies εi, another energy associated with a single site is useful
to introduce: the energy difference needed to add an additional electron to an empty site
of the system or remove one from an occupied site: the addition energy. Assume that
the system is in a state with energy Hm and that at an unoccupied site i (i.e., ni = 0)
an additional electron is inserted into the system. The energy needed for this process is
given by
(2)
If we neglect the change of the filling factor ν in H̃m this energy can be written as
Δεi =
2εi − Uαi (ni + ν)
.
ni − ν
(3)
This expression is equivalent to the ‘single-particle energy’ proposed by Efros and
Shklovskii [3, 12, 13]:
nj − ν
.
(4)
Δεi = Uαi + e2
r
ij
j=i
For an occupied site this expression gives the energy needed to remove the electron from
this site.
In this paper, we are interested in the static low energy properties of model (1).
Therefore, the central numerical method which is used here is the minimization algorithm
which we describe in some detail in the following. First, the system is initialized with a
state according to the filling factor which is either a random distribution of electrons or,
in the second part of the paper, with an ordered crystalline configuration.
The most important elementary process which is needed for the minimization
algorithm is the pair exchange. It is necessary to have an efficient way to calculate the
total system energy difference of the states before and after an exchange of particles/holes.
We denote the initial energy as H and the energy of the system after an electron has
been moved between sites k and l as H̃. The occupation numbers are ni and ñi with
ñi = ni ∀i = k, l and ñk = nl , ñl = nk . The energy difference is then
Δεkl ≡ H̃ − H = Uαk (ñk − nk ) + Uαl (ñl − nl )
rM (ñk − ν)(ñj − ν)
rM (ñl − ν)(ñj − ν)
+ e2
+ e2
rkj
rlj
j=k
j=l
(ñk − ν)(ñl − ν)
rkl
rM (nk − ν)(nj − ν)
rM (nl − ν)(nj − ν)
− e2
− e2
rkj
rlj
j=k
j=l
− e2 γlk
+ e2 γlk
(nk − ν)(nl − ν)
,
rkl
with γlk = Θ(rM − rkl ) where Θ(x) is the Heaviside unit step function.
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J. Stat. Mech. (2008) P06006
Δεi ≡ H̃m |ni →1 − Hm .
The Coulomb gap and low energy statistics for Coulomb glasses
or, in terms of the addition energy,
2 γkl
(nk − nl ) .
Δεkl = (nk − nl ) Δεl − Δεk − e
rkl
(6)
The minimization algorithm consists of several steps. In the initial step all nearestneighbor (NN) pair exchange energies for one site are calculated and the exchange which
lowers the total energy most is executed, resulting in a complete recalculation of H. All
sites in the system are addressed by a random permutation of the site indices. If no NN
pair exchange in the whole system can lower the system energy further, an addition energy
optimization is performed, meaning that first the electron with the maximal addition
(1)
energy of all occupied sites Δεmax hops to the hole with the minimal addition energy
(0)
(1)
(0)
Δεmin as long as Δεmax > Δεmax which according to (6) lowers the system energy. Second,
(1)
we calculate the differences δ of Δεmax and the addition energies of all occupied sites. If
δ < e2 /a all possible pair exchanges within a radius of e2 /δ are checked (see (6)) and the
one minimizing the energy most is executed. After this step we start over again until no
further pair exchange is possible, i.e., all possible pair exchanges in this final state lead
to an increase of the system energy. We define these final states as local minima in the
present work. Additionally, multi-site exchanges can be checked using expressions similar
to (5) or (6). Additionally to pair exchanges, we checked compact four-site aggregates, and
found that those cannot lower the system energy significantly and can be neglected in the
following considerations. The simulations were performed for two-dimensional systems of
size (number of lattice sites) N between 502 and 10002 with periodic boundary conditions
and filling factor ν = 0.5. In the following we use dimensionless units: distances are
measured in units of the lattice constant a and the disorder strength U and Coulomb
energy in units of e2 /a.
3. Statistics of the local minimal states
In the first part we study the distribution of the local minima states of the lattice model (1)
at intermediate disorder strengths. We consider the system in which the disorder spread,
U, is equal to the Coulomb interaction at the distance of the lattice constant, i.e., U = 1.
In general, the behavior of the systems depends on the interplay between disorder and
Coulomb interaction: for vanishing disorder the system has a crystalline structure which
goes over to glassy behavior upon increasing U. With U = 1 we believe that we are well
into the glassy regime.
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J. Stat. Mech. (2008) P06006
rM denotes the maximal interaction radius in the simulation and therefore restricts
the sums for the Coulomb interaction parts. Since we use sufficiently large systems here,
we set rM = L/2 and therefore avoid Ewald summation for the Coulomb interaction.
Using the occupation number relations and the site energies of H the above expression
can be written as
nk + ν
nl + ν
Δεkl = U(nk − nl ) αk
− αl
nk − ν
nl − ν
εl
(nk − nl )2
εk
2
,
(5)
−
− e γkl
+ 2(nk − nl )
nl − ν nk − ν
rkl
The Coulomb gap and low energy statistics for Coulomb glasses
Starting from many initial random distributions of electrons we use the minimization
algorithm to calculate the local minimal states. The distribution of the energies for those
states is shown in figure 1 (right) for 20 000 local minimum states of the N = 1002 system
used for the analysis of overlaps (see below). It seems that the high energy tail can be
fitted with an exponential law, nmin (E) ∼ e−E/E0 . In order to further substantiate this
claim we repeated the procedure for a smaller system (N = 502 ) which allows us to get
better statistics (8 × 106 minima). The results are shown in figure 1 (left) and confirm
that the high energy tail is better fitted with an exponential than a Gaussian.
Additionally to the two distributions discussed, with good statistics, we calculated
the distributions of many more system sizes N between 502 and 1002 as well as for 1502
and 2002 systems. For all these distributions we calculated the mean energy Em and
standard deviation (figure 2 (left)) as well as the collapsed distribution, where we shifted
the distribution by Em and rescaled the axis by the standard deviation σ (figure 2 (right)).
Em shows clearly a linear dependence on the system size N = L2 , whereas σ shows a
square root dependence on N. The linear behavior of Em on N shows that the energy
is an extensive variable, as would be expected if the system could be considered as a
sum of individual subsystems weakly interacting with each other. In the presence of long
range Coulomb interactions this is not obviously the case, and our results show that for the
local minimal states screening is sufficient for this to happen. Using this picture, it follows
directly from probability theory that the square of the standard deviation would be the
sum of all variances of the subsystems and therefore lead to the square root dependence on
N. At this point we remark that the number of calculated local minimum states represents
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J. Stat. Mech. (2008) P06006
Figure 1. Energy distribution of local minimum states of systems of size N in
half-log representation with a linear fit (i.e., an exponential distribution) at the
high energy tail. Left: for 8 × 106 local minima of a N = 502 system. The black
diamonds represent the ‘square root’ of the distribution (in arbitrary units), i.e.,
a straight line here would indicate a Gaussian distribution. While at the central
peak it is close to that, the right tail shows a clear curvature, and we conclude
that the tail is not Gaussian. Right: for 2 × 104 local minima of a N = 1002
system. As a guide to the eye a Gaussian distribution at the maximum is shown
(dashed red curve). For the low energy tail not enough data points are available
for a reasonable analysis.
The Coulomb gap and low energy statistics for Coulomb glasses
only a small fraction of all possible metastable states of the systems considered. However,
in the absence of correlation between the energy of a minimum and the probability of
reaching this minimum using our algorithm, our histograms will still be representative.
At the present time we are checking this correlation, and our preliminary results for small
systems indicate some correlation. More work is required to verify this and to understand
how the correlation scales with the system size. We also plan to understand the relation
between our algorithm for finding local minima and a physical relaxation process (modeled
by a kinetic Monte Carlo algorithm).
For all computed minima of the N = 1002 system, we also calculated the normalized
site occupation number difference
|nαi − nβi |
Δαβ = N −1
i
corresponding to all pairs of minima {α, β}. We call the quantity Δαβ (i) ≡ |nαi − nβi | the
local occupation number difference of minima α and β.
The distribution of Δαβ , i.e., the fraction of the system which have different
occupation numbers for two different minima, is shown in figure 3 (left). For the filling
factor used, 0.5, about 7.4% of all sites (not only the occupied ones, as shown in [10])
have different occupation numbers when comparing different minima. This can be seen in
the spatially resolved ‘activity’ map in figure 3 (right): the color coded regions show the
places in the system where electrons/holes differ between different local minima—the color
is determined by the numbers of differences between all calculated low energy states at a
particular site. One sees that the occupation numbers of most sites are frozen in all local
minima (about 75% of all sites have the same occupation in all local minima). In order to
understand the structure of the local minima, we compared the
occupation numbers of all
sites i which have the same local occupation difference counts α,β Δαβ (i). There are only
a few of these collections of sites which have more than four sites, but even these do not
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J. Stat. Mech. (2008) P06006
Figure 2. Left: dependence of the mean value Em and standard deviation σ(n)
of the distribution of the local minimum states for 2D systems on the system
size. Em depends clearly linearly on the system size N = L2 and σ[n(E)] as the
square root of the system size. Right: rescaling all distributions according to
x = (E − Em )/σ and y = n(E)/n0 σ collapses them onto one distribution.
The Coulomb gap and low energy statistics for Coulomb glasses
represent n-site aggregates, which are collections of n sites fixed throughout all states—
possibly spread over the whole system—having only two complementary states. This
observation suggests that transitions between different local minima occur through global
rearrangements of electrons on the active sites and in particular cannot be understood in
terms of flips or inversions of different n-site aggregates.
Due to the small—compared to the total—number of local minima, we cannot exclude
the possibility that n-site aggregates are important for the low temperature dynamics of
Coulomb glasses. However, the study of these transitions at finite temperatures is beyond
the scope of this work and will be the subject of further investigations.
4. Coulomb gap
In the second part of the paper we analyze the dependence of the density of states of
the local minima on the disorder strength U. It is clear that a crystalline configuration
defines the ground state of the system for U = 0, meaning that all occupied/free sites have
(1),(0)
(1)
(0)
the same (negative/positive) addition energy Δεi
with Δεi = −Δεi (see figure 4
(0)
(1)
(lower, left)). Here we can define a hard gap as Δh ≡ Δεmin − Δεmax . In the opposite
limit U → ∞ the disorder potential dominates the Coulomb interaction such that the
electrons are in the minima of the random potential and therefore the addition energies
are uniformly distributed (see figure 4 (lower, right)). Increasing the disorder strengths
also decrease Δh until the low temperature DOS approaches the linear/quadratic Coulomb
gap for intermediate disorder strength for 2D/3D systems, respectively [13]. In figure 4
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Figure 3. Left: distribution of the difference of the local minima Δαβ (0 for
identical states, 1 for complementary ones). The distribution is fitted with a
Gaussian curve, with mean 0.074 and width 0.020. 2 × 108 overlap integrals are
calculated. Right: the ‘active’ regions of the deep energy configurations, i.e.,
the placesin the system where electrons/holes typically differ for different local
minima,
α,β Δαβ (i). Data are shown for 20 000 different local minima of a
2
N = 100 system and the color code represents the number of local differences
in occupation numbers for all possible configuration overlaps. The background,
dark-gray color represents the frozen site at which the occupation number does
not change for all local minima—these are about 75% of all sites.
The Coulomb gap and low energy statistics for Coulomb glasses
(top) the DOS for a 2D N = 10002 is shown—the (apparently) linear dependence is stable
for a certain interval of U. As mentioned before, all simulations are done for half-filling
ν = 1/2 which results in the symmetrical structure of the DOS.
The above is the classical Efros–Shklovskii picture [3, 13] for states which are stable
against single-particle jumps or pair exchanges—which we can easily reproduce with our
algorithm. However, a closer look at the Coulomb gap at U = 1 reveals deviations
from this well known linear behavior of the Coulomb gap: we calculated the averaged
DOS for 56 local minima of a N = 10002 system in order to get a good statistical
foundation, and analyzed the gap in a log–log representation. Although not visible in the
linear representation, the gap shows a |Δε|α behavior with an exponent α = 1.23 ± 0.02
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Figure 4. Coulomb gap for differing disorder strength. Plotted are normalized
histograms for the addition/single-particle energies. The upper plot shows the
linear behavior of the DOS for a two-dimensional system in the ‘glassy’ region
(U = 1) with fitted linear graphs. The lower graphs show the DOS for a Wigner
crystal (left) and a strongly disordered system (right, U = 10) in which the
Coulomb gap has almost disappeared. All DOS are calculated for systems of size
N = 10002 . (d denotes the resolution of the histogram.)
The Coulomb gap and low energy statistics for Coulomb glasses
Uc
Figure 6. Disorder strength dependence of the hard Coulomb gap Δh for an
initial crystalline configuration. Simulations are done for N = 2562 systems.
rather than α = 1, see figure 5. First indications of this deviation were already reported
in [14, 15], however with rather low, non-conclusive accuracy. In order to understand this
discrepancy, further analytical studies are needed.
In the following we study the transition from the crystalline state to the Coulomb glass
state with increasing disorder strength in more detail. Therefore we start the minimization
algorithm with the two possible crystal configurations for half-filling at given U and
calculate the hard gap at the end. The average of these is plotted in figure 6. One
clearly sees a sharp transition of Δh at a critical disorder strength Uc to zero. For the
simulated system of size N = 2562 we have Uc ≈ 1/4. Below Uc , Δh decreases linearly since
(1)
Δεmax ∝ Uαmax + Ec with αmax ≈ 1 and Ec = −e2 /aUc,max = const being the Coulomb
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Figure 5. Averaged Coulomb gap at U = 1 for 56 local minima of a N = 10002
system. On a log–log scale one clearly sees that the Coulomb gap deviates
slightly from the classical |Δε| behavior. It rather follows a |Δε|α power law
with exponent α = 1.23 ± 0.02.
The Coulomb gap and low energy statistics for Coulomb glasses
energy of the crystal configuration (at an occupied site). At Uc the global crystalline
structure breaks down and the hard gap disappears.
For an infinite system we can estimate Uc as follows. If we assume a crystalline
configuration, one can find two places where an electron occupies a site with αi = 1
and a hole a site with αj = −1. Additionally, we notice that if the addition energy (4)
of an electron/hole becomes larger/smaller than zero they can be exchanged, lowering
the system energy; see equation (6). Therefore the upper boundary for U at which the
crystalline structure is stable is given by
(3D)
Uc,max
∞ ∞
(−1)n+m
√
= −2
≈ 0.8078,
n2 + m2
n=1 m=0
∞
(−1)n1 +n2 +n3
= −1/2
≈ 0.8738.
2
2
2
n
+
n
+
n
1
2
3
n1,2,3 =−∞
n1 =n2 =n3 =0
These values are up to a factor 1/2 the 2D or 3D Madelung constants. Here we should
remark that the ‘delta’ peaks for the crystalline DOS shown in figure 4 (U = 0) are
(2D)
at addition energies ±0.78 which are close to ±Uc,max , where the peaks for the infinite
Coulomb interaction range system would be. Furthermore, if we assume that the electron
and hole with αi = 1, −1 are nearest neighbors and use (6), we find Uc = Uc,max − 1/2
(2D)
(3D)
and therefore Uc
≈ 0.3078 and Uc
≈ 0.3738. Both values are fairly close to the
numerically calculated one for 2D systems.
However, it should be remarked that Uc depends slightly on the screening radius of
the Coulomb interaction in finite systems, which we usually took as half the simulated
system size, but not on the disorder configuration. The behavior is, however, essentially
the same and especially for U ≥ O(1), Δh = 0 for all system sizes analyzed, meaning
there is no global crystalline order any longer.
A detailed study of the gap structure and the behavior at large disorder strength will
be subject of forthcoming works.
5. Conclusions
In conclusion, we have analyzed the statistical properties of local energy minima for a
disordered lattice Coulomb system. The lattice Coulomb model was studied numerically
before by many authors looking at various aspects. However, the statistical properties of
the local minima in the configurational space were studied either in too small systems,
or not in sufficient detail. In previous works [16]–[18] system sizes of the order 102 were
used. The optimized numerical procedures that we used in this paper and reference [10],
and latest computational equipment, made it possible to analyze systems 100 to 104 times
larger, which have an astronomically large number of different local energy minima states.
Typical times for the minimization algorithm range from O(1 s) for N = 1002 to O(1 h)
for N = 10002 .
We found that the high energy tail of the numerically calculated distribution of the
local minima can be fitted with an exponential function.
The study of the structure of the local minima suggests that transitions between
different local minima probably occur through global rearrangements of electrons on
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(2D)
Uc,max
The Coulomb gap and low energy statistics for Coulomb glasses
Acknowledgments
We are grateful to the referee of this paper for his/her detailed, insightful, and constructive
criticism, helping us to improve the presentation of this work substantially. This work was
supported by the US Department of Energy Office of Science under the Contract No DEAC02-06CH11357 and by the Norwegian Research Council under the STORFORSK
program, and the Norway–USA Bilateral collaboration program.
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doi:10.1088/1742-5468/2008/06/P06006
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J. Stat. Mech. (2008) P06006
certain active sites and in particular cannot be understood in terms of flips or inversions
of different, independent n-site aggregates. However, due to the large number of possible
local minima states, the possibility of a role of aggregates in the dynamics cannot be
excluded completely.
Finally, we studied the distribution of addition energies of these low energy minimal
states for a half-filled Coulomb glass at different disorder strengths. For small disorder
strength the systems exhibits a hard gap in the density of states (DOS) indicating a
global crystalline electron structure, whereas at intermediate values an apparently linear
Coulomb gap forms (glassy/microcrystalline structure) which vanishes for strong disorder
(disordered electron distribution). The crystalline structure is stable for U < Uc , which
manifests itself in a linear decrease of Δh with a breakdown at Uc where Δh drops to zero.
The value found numerically for Uc is in agreement with a simple consideration of the
stability of the addition energy in infinite systems.
An interesting observation is the deviation of the Coulomb gap from the classical
Efros–Shklovskii behavior at intermediate disorder strength. However, further analytical
investigation of this result is needed.