PHYSICAL REVIEW B 85, 214202 (2012)
Logarithmic relaxation and stress aging in the electron glass
J. Bergli1 and Y. M. Galperin1,2
1
2
Department of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway
A. F. Ioffe Physico-Technical Institute RAS, 194021 St. Petersburg, Russian Federation and Centre for Advanced Study at the Norwegian
Academy of Science and Letters, 0271 Oslo, Norway
(Received 3 February 2012; revised manuscript received 2 June 2012; published 12 June 2012)
Slow relaxation and aging of the conductance are experimental features of a range of materials, which are
collectively known as electron glasses. We report dynamic Monte Carlo simulations of the standard electron
glass lattice model. In a nonequilibrium state, the electrons will often form a Fermi distribution with an effective
electron temperature higher than the phonon bath temperature. We study the effective temperature as a function
of time in three different situations: relaxation after a quench from an initial random state, during driving by an
external electric field, and during relaxation after such driving. We observe logarithmic relaxation of the effective
temperature after a quench from a random initial state as well as after driving the system for some time tw with
a strong electric field. For not too strong electric field and not too long tw we observe that data for the effective
temperature at different waiting times collapse when plotted as functions of t/tw —the so-called simple aging.
During the driving period we study how the effective temperature is established, separating the contributions
from the sites involved in jumps from those that were not involved. It is found that the heating mainly affects the
sites involved in jumps, but at strong driving, also the remaining sites are heated.
DOI: 10.1103/PhysRevB.85.214202
PACS number(s): 71.23.Cq, 72.15.Cz, 72.20.Ee
I. INTRODUCTION
At low temperatures, disordered systems with localized
electrons (e.g., located on dopants of compensated doped semiconductors or formed by Anderson localization in disordered
conductors) conduct by phonon-assisted hopping. The theory
of this process goes back to Mott1 who invented the concept
of variable range hopping (VRH). In particular, he derived the
Mott law for the temperature dependence of the conductance,
σ ∝ exp[−(T̃0 /T )1/(d+1) ],
(1)
where T is temperature, T̃0 is some characteristic temperature,
while d is the dimensionality of the conduction problem. If
Coulomb interactions are important one describes the system
as an electron or Coulomb glass due to the slow dynamics at
low temperatures.2
As is well known (see Ref. 3, and references therein), the
single-particle density of states (DOS) develops a soft gap
at the Fermi level, the so-called Coulomb gap.4–6 While this
understanding of the density of states is generally accepted, the
situation is less clear when it comes to describing dynamics in
the interacting case. Using the Coulomb gap DOS in the VRH
arguments1 in the same way as the noninteracting DOS yields
the Efros-Shklovskii (ES) law for conductance,7,8
σ ∝ exp[−(T0 /T )1/2 ].
(2)
This has been observed experimentally in many different
types of materials, such as doped semiconductors,9–13 granular
metals,14 or two-dimensional systems.15,16
In recent years, there has been increasing interest in the
nonequilibrium dynamics of hopping systems. In particular,
the glasslike behavior at low temperatures has been studied
both experimentally17–23 and theoretically.24–26 In this work,
we are interested in two features observed in the experiments. First, it was observed that the conductivity relaxes
logarithmically as a function of time after an initial quench or
perturbation.18 Secondly, if the system initially in equilibrium
1098-0121/2012/85(21)/214202(8)
is perturbed by some change in external conditions (e.g.,
temperature or electric field) for a time tw called the waiting
time, the relaxation back towards equilibrium of some quantity
such as the conductance G(t,tw ) will depend both on tw and
the time t since the end of the perturbation. It is found in
certain cases that the relaxation is in fact described by a
function G(t/tw ) of the ratio t/tw . This behavior is called
simple aging.19
While simple aging is observed in a range of different
systems, we are here particularly concerned with experiments
on disordered InO films.17 One question which has been raised
is whether the observed glassy behavior is an intrinsic feature
of the electron system, or a result of some extrinsic mechanism
such as ionic rearrangement.27 In this work, we address the
intrinsic mechanism by performing dynamical Monte Carlo
simulations of the standard lattice model of the electron glass.
It is known26 that during a quench from an initial random state
an effective electron temperature is quickly established, and
that this temperature slowly relaxes to the bath temperature. We
show that the electron temperature relaxes logarithmically over
almost three decades in time and that the system demonstrates
simple aging behavior in a stress aging protocol similar to what
is seen in the experiments.19 While this does not constitute
a proof of the intrinsic origin of the glassy behavior in the
experiments on InO films, it shows that the model can display
the observed behavior. To the best of our knowledge this was
previously only demonstrated in a mean-field approach,25 the
accuracy of which is not well understood.
II. MODEL
We use the standard tight-binding Coulomb glass
Hamiltonian,3
214202-1
H =
i
i ni +
(ni − K)(nj − K)
,
rij
i<j
(3)
©2012 American Physical Society
J. BERGLI AND Y. M. GALPERIN
PHYSICAL REVIEW B 85, 214202 (2012)
K being the compensation ratio. We take e2 /d as our unit of energy where d is the lattice constant, which we take as our unit of
distance. The number of electrons is chosen to be half the number of sites so that K = 1/2. i are random site energies chosen
uniformly in the interval [−U,U ]. In the simulations presented
here we used U = 1, which we know gives a well-developed
Coulomb gap and the ES law for the conductance.28–30 The
sites are arranged in two dimensions on a L × L lattice where
in all cases we used L = 1000, which is sufficiently large to
give a good estimate of the effective temperature in a single
state without any averaging over a set of states. We implement
cyclic boundary conditions in both directions.
To simulate the time evolution we used the dynamic Monte
Carlo method introduced in Ref. 28. The basic idea of such
simulations is to start the system in some particular configuration. The configuration can change by one electron jumping
from site i to site j (in principle, we should also take into
account transitions involving two or more electrons jumping
at the same time, but at the temperatures we consider here this
should be a minor effect; see Ref. 30, and references therein for
a discussion). The energy mismatch between the two states is
supplied by the emission or absorption of a phonon, therefore
the process is called phonon-assisted tunneling.
The rate of a phonon-assisted transition from site i to site
j is usually specified as (see, e.g., Ref. 3)
ij ∝ |γq |2 |N (Eij )|e−2rij /a .
(4)
Here Eij is the phonon energy, rij ≡ |ri − rj |, a is the
localization length of the electronic state which we choose
to be 1, while γq is a coupling constant, which in general
depends on the wave vector q of the involved phonon. Since
q ∝ |Eij | the prefactor γq leads to a power-law dependence
of the transition rate on |Eij | where the power is model
dependent. Since the power-law preexponential dependence
does not change the results in a qualitative way we assume γq
to be q independent. N (E) = (eE/T − 1)−1 is the equilibrium
phonon density. We set kB = 1 so that temperatures and
energies are measured in the same units.
In the given initial state, one should, in principle, calculate
all such rates, then select one at random weighted by the
rates. The selected transition is then performed, all Coulomb
energies recalculated, and the process repeated from the new
state. Note that the rates depend on the states through the
energy changes Eij , which include the contributions from the
Coulomb energy. All the rates therefore have to be recalculated
at each step. In practice, one restricts the possible jumps by
only considering those where the distance rij is less than some
maximal length (in our simulations we only considered jumps
of less than ten lattice units) since longer jumps become so
improbable that they are never selected anyway. The procedure
of selecting the next jump is also optimized in other ways (see
Ref. 28 and, in particular, Ref. 31 for a description of how to
calculate the proper elapsed time). The only difference from
Ref. 28 is that following Ref. 32, we use for the transition rate
ij instead of (4) the approximate formula
ij = τ0 −1 e−2rij /a min(e−Eij /T ,1),
(5)
where Eij is the energy of the phonon and rij is the distance
between the sites. τ0 contains material-dependent factors and
energy-dependent factors, which we approximate by their
average value; we consider it as constant and its value, of
the order of 10−12 s, is chosen as our unit of time. (Note that in
Ref. 28 a different approximate formula was used; we do not
believe that the difference is of great significance, although it
may change numerical values).
III. RELAXATION AND EFFECTIVE TEMPERATURE
To get a more detailed understanding of how the effective
temperature is established, we can follow the effective temperature as a function of time after an initial quench or some
external perturbation such as a strong electric field is applied.
Since we use samples with 1000 × 1000 sites, we get good
statistics of the energies and occupation probabilities using
a single state of the system. The effective temperature is
extracted by dividing the energy axis in small bins. For each bin
we find all sites with single-particle (addition or subtraction)
energies
nj − K
Ei = i +
rij
j =i
in this bin. The fraction of these sites which are occupied
gives an estimate for the occupation probability f (E) at that
energy. If plotted as a function of energy the occupation
probability follows closely the Fermi distribution (see Figs. 3
and 4 below for examples). The corresponding temperature is
called the effective electron temperature, Teff . We extract the
effective temperature from the data by numerically integrating
the distribution
0
∞
[1 − f (E)]dE +
f (E)dE = (2 ln 2)Teff .
−∞
0
This is the same procedure as used in Ref. 26. One may well ask
about the conceptual significance of the effective temperature
and why the Fermi distribution appears in a manifestly out
of equilibrium situation. At present we do not have a full
understanding of this; for the moment we have to consider
it as an observed fact. We will discuss the issue further in
connection to the response to driving.
Let us first relax from an initial random state and measure
Teff (t). In all our simulations we used a phonon temperature
T = 0.05, which we know is well into the ES regime for VRH.
The graphs show the evolution over 108 jumps.
Shown in Fig. 1 are the energy (inset) and the effective
temperature as functions of time. As we can see, the energy
graph has almost ceased to decrease, indicating that we have
almost reached equilibrium. The same is seen by the effective
temperature, where Teff = 0.054 in the final state. We see
that the effective temperature, after an initial short time,
logarithmically decreases in time within a time domain of
about two-and-a-half orders of magnitude. The energy does
not show this behavior (as discussed in Ref. 33, it is well fitted
by a stretched exponential function). The relaxation of the
effective temperature was studied previously26 using a similar
numerical method, but having the sites at random instead of on
a lattice. A lower temperature was also used, and together this
slows down the simulations so that only much smaller samples,
up to 2000 sites, could be studied. Instead of our linear fit to
effective temperature as a function of ln t they obtain a linear
214202-2
LOGARITHMIC RELAXATION AND STRESS AGING IN . . .
PHYSICAL REVIEW B 85, 214202 (2012)
−0.3505
2
−0.348
−0.349
1.6
−0.35
−0.3515
−0.351
−0.352
1.2
T
eff
−0.353
1
−0.354 2
10
3
4
10
−0.352
Fraction of sites active
1.4
−0.351
Energy/site
Energy/site
1.8
−0.3525
10
t
0.8
−0.353
0.6
0.5
0.4
0.3
0.2
0.1
0.4
−0.3535
0
0
1000
2000
3000
t
4000
5000
6000
0.2
−0.354
0
0 0
10
1
10
2
3
10
10
100
200
300
t
4
10
400
500
600
4000
5000
6000
0.13
t
0.12
FIG. 1. Effective temperature as a function of time. The inset
shows the energy as a function of time.
0.11
0.1
Teff
0.09
0.08
0.07
0.06
0.05
0.04
0
1000
2000
3000
t
FIG. 2. Top: Energy as a function of time. Inset: The fraction
of sites involved in a jump as a function of time. Bottom: Effective
temperature as a function of time. The curves are, from top to bottom,
the temperature of the sites which were active, the temperature of all
sites, and the temperature of the sites which were not active. The
vertical lines indicate the waiting times in the stress aging protocol
(see Fig. 5).
Note that even at the latest time shown, new sites are still
being involved [see Fig. 2 (top, inset)], even if the energy and
effective temperature are more or less stable.
The fitted Fermi functions are shown in Fig. 3. The fits are
quite good, but the data for the sites which never were involved
in jumps are more noisy. Note that the number of sites which
jumped is still not much more than half the total number of
sites, so the noise is not because there are few sites, but rather
1
1
=0.115
eff
0.5
0
−0.5
0
E
1
T
f(E)
eff
0.5
=0.118
T
eff
f(E)
T
f(E)
fit to effective temperature as a function of 1/ ln t. The reason
for this difference is not clear, but it may result from our larger
samples and longer times.
Another type of experiment is to relax the system at low
temperature and then apply a strong electric field to pump
energy into the system. To equilibrate the system, we simulated
the system with a localization length of 100, which facilitates
long jumps and faster equilibration. When the energy did not
decrease anymore, we switched to the normal localization
length of 1 and observed that the energy remained constant
with fluctuations and Teff = 0.05 ± 0.001 after each 106 jumps
for a total of 7 × 106 jumps. We then applied an electric field
E = 0.1 (in units of e/d 2 ). We know that Ohmic conduction
takes place when E T /10, so this should be well into the
non-Ohmic regime and we expect the electron temperature
to increase above the phonon temperature. Shown in the top
panel of Fig. 2 is the energy per site as a function of time.
The effective temperature as a function of time is shown in
the lower panel (middle curve). Noting the difference in the
time scales we conclude that the energy stabilizes at a new
value much faster than the effective temperature. The exact
relation between the relaxation of energy and temperature
depends on several factors. The energy is determined by both
the occupation probability and the density of states, both of
which change during driving. In addition, there may be a
number of sites which do not follow the Fermi distribution.
If the number of these sites is small they may not affect the
effective temperature noticeably, but still influence the total
energy. How the ratio of these time scales depends on the
parameters of the model deserves further investigation.
We have also plotted the effective temperature taking into
account only those sites which were involved in a jump [Fig. 2
(bottom, upper curve)], and those which did not jump [Fig. 2
(bottom, lower curve)]. As we can see, the sites which are not
involved in jumps are still at a temperature close to the phonon
temperature. This is not a trivial statement, since the energies
of the sites which are not involved in jumps also change due to
the modified Coulomb interactions with the sites that jumped.
0.5
0
−0.5
0
E
0.5
=0.0526
0.5
0
−0.5
0
E
0.5
FIG. 3. Fermi functions. Left: All sites. Center: Sites which took
part in a transition. Right: Sites which did not take part in a transition.
The points are the data and the curve is the fitted function.
214202-3
J. BERGLI AND Y. M. GALPERIN
PHYSICAL REVIEW B 85, 214202 (2012)
adjust to the new energy level. In our case, the occupation
numbers of the sites not involved in transitions are fixed,
while their energies are pushed around by the interactions
with the jumping sites. A natural guess would be that there is
no correlation between the energy of a site before the driving
starts (and hence the occupation probability of that site) and
the shift it receives during driving. One would then guess that
the initial Fermi distribution at the real phonon temperature
would be scrambled during the driving, and that for the sites
not involved in jumps we would not find a well-defined Fermi
function. The fact that this does not happen indicates that there
is certain structure to the nature of the shifts of the energies,
and would be an interesting point for further study.
1.4
1.2
1
T
eff
0.8
0.6
0.4
0.2
IV. STRESS AGING
0
0
10
20
30
40
50
60
70
t
1
1
eff
0.5
0
−5
0
E
1
T
f(E)
f(E)
eff
=0.722
5
=0.786
T
eff
f(E)
T
0.5
0
−5
0
E
5
=0.295
0.5
0
−5
0
E
5
FIG. 4. When driving with electric field E = 1. Top: Teff as a
function of time. The curves are, from top to bottom, the temperature
of the sites which were active, the temperature of all sites, and the
temperature of the sites which were not active. Bottom: Fitted Fermi
functions at time t = 19. Left: All sites. Center: Sites which took part
in a transition. Right: Sites which did not take part in a transition. The
points are the data and the curve is the fitted function.
(as could be expected) that the sites which did not jump are
those which are far from the Fermi level. These are either
filled or empty and do not contribute much to the effective
temperature. The noise indicates that the energy shifts are not
correlated with the original energy of the site, so that there is
no systematic change in the occupation probability giving a
change in the effective temperature.
If we drive with a stronger field E = 1 (Fig. 4), we observe
two new features: First, the temperature of the sites which
took part in a jump changes nonmonotonously, first increasing
rapidly and then decreasing towards a stationary value. Note,
however, that at very short times, before the peak is reached,
the distribution is not close to a Fermi distribution and the
temperature is not a meaningful concept. On the decreasing
part of the curve the Fermi function was already established.
Second, we now observe significant heating also of the sites,
which did not take part in a jump. Because of the Coulomb
interaction it is not surprising that these sites are also affected
by the transitions on other sites. What is surprising is that the
occupation numbers still follow the Fermi distribution quite
closely. Notice that the data are not noisy like in the case of
weaker driving (Fig. 3), so the shifts in the energy levels are
systematically adjusting to the Fermi distribution. Normally,
we think of energy levels as fixed and occupation numbers
changing dynamically in such a way as to maintain the Fermi
distribution in the time-averaged occupation probabilities. If
the energy levels are shifted by some process, the transition
rates also have to change, and the occupation numbers will
Motivated by the logarithmic relaxation of effective temperature in Fig. 1 and using the heating curve of Fig. 2 we will try
to reproduce the experimental results19 using the stress aging
protocol. This means applying a non-Ohmic field for a certain
time tw and then turning this field off (keeping only an Ohmic
measuring field, which supposedly does not appreciably
perturb the system). The heating process of Fig. 2 is exactly
such a non-Ohmic driving, and we need only start simulations
with zero fields at different points along this curve. (In the
experiments they kept an Ohmic measuring field, but since we
are monitoring the effective temperature and not the current,
this is not needed. We also ran simulations in Ohmic fields
and confirmed that this did not appreciably affect the effective
temperature as was expected.) We have chosen six tw , which
correspond to the points marked in Fig. 2 with vertical lines.
Figure 5 (top) shows the effective temperature as a function
of time after the end of the driving period. Note that the
time dependence of the energy is very similar in all cases,
as shown in Fig. 5 (bottom, inset). The effective temperature
as a function of t/tw is shown in Fig. 5 (bottom).
From Fig. 5 we conclude that there is a logarithmic
relaxation of the effective temperature after a driving by a
non-Ohmic field just as in the case of relaxation from a
random initial state (Fig. 1). Furthermore, we see that the
curves for different tw collapse when time is scaled with
tw when tw is smaller than some critical value tw(c) ≈ 2500.
The curve for tw = 2865 seems to lie a little to the left of
the collapse curve, and for tw = 5696 this tendency is clear.
The collapse of the curves for short tw is similar to what is
observed in the experiments both on indium oxide films19 and
porous silicon.23 In the case of porous silicon, a departure from
the simple aging at longer waiting times was also observed,
while sufficiently long times were never reached in the case
of indium oxide. If we compare to Fig. 2 (bottom) we see
that the critical value tw(c) corresponds to the time where the
effective temperature stabilizes. Comparing to Fig. 2 (top) we
see that this is a time much longer than that which is needed
for the energy to stabilize. To check that this behavior is not
particular to one specific sample we repeated the procedure
(relaxing to equilibrium using large localization length, then
the stress aging protocol) on a different sample which gave
similar results.
For the first sample we also repeated the stress aging
protocol at different driving fields, E = 0.05, 0.2, 0.5, and 1.
214202-4
LOGARITHMIC RELAXATION AND STRESS AGING IN . . .
0.12
PHYSICAL REVIEW B 85, 214202 (2012)
0.08
tw = 373
t = 466
w
t = 729
w
0.11
tw = 915
t = 1441
w
t = 2153
w
0.1
tw = 1813
0.07
t = 2865
w
tw = 2711
Teff
tw = 5696
T
eff
0.09
0.06
0.08
0.07
0.05
0.06
1
2
10
0.05
3
10
4
10
t
10
0.08
10
t
3
4
5
10
10
0.08
0.12
0.07
−0.351
−0.3515
0.1
0.09
−0.3525
0.06
0.04
eff
−0.352
T
Energy/site
0.11
eff
2
10
T
0.04 1
10
0
0.06
1000 2000 3000
t
−0.353
Teff
−0.3535
−0.354 0
10
0.08
0.05
1
2
10
3
10
4
10
10
5
10
t
−1
10
0.07
0
10
t/tw
1
10
2
10
0.06
FIG. 6. (Color online) After driving with E = 0.05. Top: Teff as a
function of time. Bottom: Teff as a function of t/tw . Inset: The heating
curve.
0.05
0.04 −3
10
−2
10
−1
10
0
10
t/tw
1
10
2
10
3
10
FIG. 5. (Color online) After driving with E = 0.1. Top: Teff as
a function of time. Bottom: Teff as a function of t/tw . Inset: Time
dependencies of the energy for different tw .
Note that to be in the Ohmic regime we should have E T /10,
so all the fields are well outside of this. For E = 0.05 the
heating curve is shown in Fig. 6 (inset) and the effective
temperature as a function of time after the end of the driving
period is shown in Fig. 6. As we can see, the general behavior
is the same as when driving with E = 0.1. For E = 0.2 the
heating curve is shown in Fig. 7 (inset) and the effective
temperature as a function of time after the end of the driving
period is shown in Fig. 7. We see that the curves do not collapse
satisfactorily, even for tw shorter than the time at which Teff
stabilizes. This behavior is also observed in the experiments,19
and is characteristic for stronger fields.
Overall, what we observe is qualitatively very close to
what is seen in experiments, but it should be noted that in
experiments it is always the conductance that is measured,
whereas we have studied the effective temperature. If we
believe that the nonlinearity of the conductance is mainly due
to heating of the electrons, then the nonlinear conductance
can be found as the linear conductance, σ0 , at the effective
temperature. It is known that this is approximately true.34
Based on this assumption and expecting that σ0 (Teff ) is an
analytical function, one would expect that σ ≡ σ − σ0 (T ) ∝
Teff − T at Teff − T T . This would be sufficient to show that
the behavior of the conductivity would be the same as what
we see for the effective temperature.
However, in our case Teff is significantly greater than T , and
the above consideration seemingly does not work. Therefore
we made an attempt to study conductivity directly. Instead
of turning off the field after tw we switched to E = 0.005,
which should be more or less the highest field which is still
in the Ohmic range. The effective temperature shows behavior
which is similar to that seen at E = 0, which is what we expect
since this field is so weak as to hardly affect the effective
temperature.
To simulate the relaxation of conductivity is not so easy.
First, the system has to be followed over some time and the
transported charge measured to find the conductance. Usually
it is noisy and one needs to average over a considerable time.
With a large system like the one we have used, it is better, but
still difficult to follow changes in the current. Secondly, and
more importantly, when the strong, non-Ohmic field is applied,
the system is polarized by charges shifting locally. These are
charges which are not on the conducting path, and therefore
they do not contribute to the dc current. But when the field is
switched to a small, non-Ohmic value, these charges will flow
backward. Since we find the current by counting transferred
charge in the direction of the field for each jump, this will
lead to a negative current for some time until this polarization
214202-5
J. BERGLI AND Y. M. GALPERIN
PHYSICAL REVIEW B 85, 214202 (2012)
0.07
0.15
tw = 45
tw = 144
0.11
0.05
tw = 275
eff
−3
0.04
0.09
eff
T −T
T
0.06
tw = 78
0.13
5
x 10
4
0.03
(T −T)
2
0.07
3
eff
0.02
0.05
2
0.01
1
2
10
3
10
1
4
10
10
t
0.15
eff
T
−0.01
0
0.15
eff
0.05
0
0.09
100
200
300
t
0.07
0.05
−1
10
10
0
2
E (λ t) −E (λ
1
10
t/tw
1
2
min
3
4
E1(λmint) −E1(λmin(t+tw))
1
4
(t+t ))
6
5
6
min
w
FIG. 8. (Color online) δT as a function of E1 (λmin t) −
E1 [λmin (t + tw )]. Inset: δT 2 as a function of E1 (λmin t) − E1 [λmin (t +
tw )] after driving with E = 0.1. The data are the same as in Fig. 5
and the color labels are the same.
0.1
0.11
0
0
1
0.2
0.13
T
0
2
10
FIG. 7. (Color online) After driving with E = 0.2. Top: Teff as a
function of time. Bottom: Teff as a function of t/tw . Inset: The heating
curve.
has relaxed. One can ask whether this would also be seen in
real experiments which only measure current in an external
circuit. It seems that even if the localized charges are not on
the conducting path, they would be capacitively coupled to the
external circuit, and thereby detectable in real experiments. No
such effect has been reported.
To see the relaxation of the conductivity, we took the
accumulated transferred charge and subtracted the same in
the absence of measuring field, which should contain only
the relaxation of the polarization. The result was smoothed
and the derivative calculated to find the current. The resulting
curve showed some relaxation of the current, but the noise
was too large and further work is needed in order to draw any
clear conclusions.
V. DISCUSSION
The phenomenology observed in our simulations agrees
with what is seen in the experiments19,23 in virtually all
essential aspects: We observe logarithmic relaxation of the
effective temperature after a quench from a random initial
state. We also observe logarithmic relaxation after driving the
system for some time tw with a strong electric field. When the
driving field is not too strong and tw not too long we observe
simple aging in the sense that the curves for different waiting
times collapse on a common curve when plotted as functions
of t/tw . When tw exceeds some critical value tw(c) , the scaled
curves do not follow the common curve. When the driving field
is large the curves do not collapse even for short times. The
only difference is that while in the experiments conductance
was measured, we have studied the effective temperature. We
have to rely on the assumption that the conductance is given
by the linear conductance at the effective temperature to relate
our results to the experiments.
The aging protocol discussed here is the subject of a
recent mean-field analysis.23,25 It predicts the relaxation of
the average occupation number after the field is turned off. It
is then argued that the excess current should follow the same
law,
λmax
dλ
(1 − e−λtw )e−λt ,
δσ ∼
λ
λmin
where λmin and λmax are the slowest and fastest modes. If we
assume that λmax t 1 we can set λmax = ∞ and we get
δσ ∼ E1 (λmin t) − E1 [λmin (t + tw )],
(6)
∞ −1 −t
where E1 (x) = x t e dt is the exponential integral function. Its series expansion is E1 (x) = − ln x − γ + x + · · ·
so that for t,tw 1/λmin we get δσ ∼ ln(1 + tw /t). When
t tw it reduces to what we have used before. The idea is
that the failure to collapse the curves for long tw is because
λmin tw 1 and we have to use the full exponential integral
instead of only the leading logarithmic term. Assuming the
excess effective temperature follows the same law as the
conductance we should plot δT = Teff − T as a function of
E1 (λmin t) − E1 [λmin (t + tw )] and obtain a collapse of the
curves to a straight line (Fig. 8). This was indeed observed
in the experiments on porous silicon.23 λmin is now a fitting
parameter, which we fit by hand to find the best collapse
when λmin = 2.5 × 10−4 . This agrees at least approximately
with the idea that 1/λmin = 4000 should be not far from the
tw(c) = 2500 that we found above. The collapse is not to a
straight line, but rather close to a square root. We can see
this clearer if we plot (Fig. 8, inset) δT 2 as a function of
E1 (λmin t) − E1 [λmin (t + tw )]. All the curves fall close to the
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PHYSICAL REVIEW B 85, 214202 (2012)
straight line indicating that the scaling relation predicted by the
mean-field theory is respected by our data. However, we can
ask why δT 2 rather than δT falls on a straight line. If we believe
that our numerical model is applicable to the experiments, it
seems to indicate that δσ ∝ δT 2 instead of δσ ∝ δT as we
obtained above from the assumption that the conductance
is the linear conductance at the effective temperature and
δT /T 1.
A debated issue in the context of glassy dynamics is the
question whether coherent multielectron transitions (transitions where several electrons simultaneously change their
positions with the emission or absorption of a single phonon)
play an important role in the dynamics or not (see Ref. 35,
and references therein). In our simulations we have not
included such processes. As demonstrated in Ref. 30 these
processes should not be important at the temperatures we
consider, and we interpret our results as indicating that
multielectron transitions are not necessary in order to observe
the phenomenology discussed here.
It is instructive to compare the times of our simulations with
the Maxwell τM time of the system. In two dimensions this
is dependent on some length scale L and is given by 1/τM =
L/2π σ . In our units (where time is measured in units of τ0 and
energy in units of e2 /d) we have σ = 1/(T ) exp[−(T0 /T )1/2 ]
where it is known28 that T0 ≈ 5.8. The Maxwell time is the
time at which an initial excess charge fluctuation spreads over
a distance L. In a disordered medium, the conductivity is not
well defined on length scales shorter than the correlation length
ξ of the percolation cluster. Let us for the moment use this as
the length scale L. We know36 that at this temperature the
conductivity of samples show clear mesoscopic fluctuations if
the sample has a size L 100, while for samples with L 100
the conductivity stabilizes and becomes independent of size.
We can therefore assume that a sample with L = 100 already
contains a number of correlations cells, let us say ten, which
gives ξ ≈ 30. In this way we obtain at T = 0.05, τM ≈ 104 .
Comparing to Figs. 1 and 5 we see that this corresponds
to the time scales at which we observe the relaxation. We
believe, however, that the Maxwell time as estimated above
constitutes an upper estimate of the relevant time scale in
our situation for two reasons. First, most of the relaxation
of the initial charge fluctuations are on a length scale much
less than ξ and the conductivity of small regions should
be larger since the ξ is the typical distance between the
critical resistors, which are the least conductive bonds on the
percolation network. Second, the conductivity is a function
of the effective temperature, and during most of the time
it is larger than the equilibrium conductivity. For example,
at the effective temperature Teff = 0.1 we get τM ≈ 103 . We
conclude therefore that our simulations cover times in excess
of the Maxwell time, but not by orders of magnitude.
In the experiments19,23 the time scales are much longer
than the Maxwell time, and one can ask about the relevance of
our simulations to these experiments. The relative change in
effective temperature which we observe would also correspond
1
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to a change in the conductance much larger than what is
observed in the experiments. The latter fact is probably the
least problematic. In order to get equilibration in a reasonable
computer time we cannot work at too low temperatures.
However, we are well into the regime where the energy
self-correlation function departs from a simple exponential,
indicating that the dynamics is constrained by the structure of
the configuration space.33 Also, we need to perturb the system
sufficiently strongly as to see a clear signal. The question
of time scale is more difficult to address. If we reduce the
temperature, both the Maxwell time and our relaxation times
would increase. One can imagine three scenarios: First, the
logarithmic relaxation and aging behavior that we observe
is continuously transformed into the one observed in the
experiments. The typical relaxation times would then have
to exceed the Maxwell time by larger and larger amounts as
temperature decreased. Second, there may be two independent
logarithmic and aging regimes, one short time which we
observe in simulations and one long time which is observed
in the experiments. The long-time behavior may also well
be seen at the temperatures we are using, but the limitations
in simulation time and the noise in the data prevents us
from observing the small and slow relaxation in this regime.
Third, as temperature is decreased, the short-time dynamics
which we have studied will be modified and the logarithmic
relaxation and aging properties will be lost, and the long-time
relaxation will develop the same phenomenology but with
different physical origins. Which of these are true can only
be determined by further work.
VI. CONCLUSIONS
We have demonstrated logarithmic relaxation of the effective temperature after a quench from a random initial state. The
same is observed after driving by some non-Ohmic electric
field. In the latter case we also observe simple aging when the
driving field is not too strong and the waiting time not too long.
At longer waiting times or after driving with stronger electric
fields we observe departure from the simple aging qualitatively
similar to what is seen in experiments.
The Monte Carlo approach allows us to access several
properties, which are not available either in the experiments
or the mean-field theory. When applying a non-Ohmic field
both the average energy and the effective temperature increase
and saturate at a level above the equilibrium one. We find that
the saturation of energy is much faster than the saturation of
effective temperatures.
We also see that the heating mainly affects those sites
which were involved in jumps. At moderate driving fields
the energies of the remaining sites are shifted by the changing
Coulomb interactions, but there are no systematic shifts, and
the best-fitting Fermi distribution is still at or close to the bath
temperature. At stronger fields there is also some heating of
the sites which were not involved in jumps, and the distribution
still follows a Fermi function.
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