# Interaction induced spatial correlations in a Disordered Glass

Z. Ovadyahu  Racah Institute of Physics, The Hebrew University, Jerusalem
9190401, Israel

###### Abstract

A consequence of the disorder and Coulomb interaction competition is the
electron-glass phase observed in several Anderson-insulators. The disorder in
these systems, typically degenerate semiconductors, is stronger than the
interaction, more so the higher is the carrier-concentration N of the system.
Here we report on a new feature observed in the electron-glass phase of
In${}_{\text{x}}$O with the lowest N yet studied. The feature, resolved as a
broad peak in field-effect measurements, has not been recognized in previously
studied Anderson-insulators. Several empirical facts associated with the
phenomenon are consistent with the conjecture that it reflects a correlated
charge-distribution. In particular, the feature may be turned on and off by
gate-voltage maneuvering, suggesting the relevance of charge-arrangements. It
may also be suppressed by either; temperature, non-ohmic field, or exposure to
infrared illumination. After being washed-out, the feature reappears when the
system is allowed to relax for sufficiently long time. A puzzling aspect that
arises is the apparent absence of the phenomenon when the carrier-
concentration increases above a certain value. This is reminiscent of the
glass-transition conundrum except that the role of temperature in the latter
is played by disorder. Analysis of these findings highlights several issues
that challenge our understanding of the disorder-interaction interplay in
Anderson insulators.

††preprint:

##  I Introduction

The interplay between disorder and Coulomb interactions has been the subject
of intense research, mostly devoted to degenerate Fermi systems 1 ; 2 ; 3 ; 4
. A common feature observed in disordered-interacting condensed-matter systems
is a local depression in their single-particle density-of-states
$\rho$($\varepsilon$). This feature, referred to as a zero-bias anomaly (ZBA),
is anchored to the chemical potential of the system and appears in both, the
diffusive 5 ; 6 , and insulating regimes 7 ; 8 . A zero-bias anomaly may be
observed when inserting a particle into the many-body system faster than the
time it takes existing particles to relax to its presence. This may be
accomplished in a tunneling or photoemission measurement of disordered metals
and doped semiconductors. When the system is quantum-coherent, the process is
closely related to the Anderson orthogonality catastrophe 9 .

An indirect way to monitor $\rho$($\varepsilon$), becomes possible in systems
where the disorder exceeds the critical value for Anderson-localization; the
competition between disorder and the unscreened Coulomb interaction slows down
the medium relaxation which, in turn, makes the system $\rho$($\varepsilon$)
observable in field-effect experiments 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ;
18 . In this case a ZBA appears as a modulation of the conductance versus
gate-voltage, G(V${}_{\text{g}}$) 14 . This feature, called a memory-dip (MD),
has been observed in several heavily-doped semiconductors where the disorder
necessary to render them Anderson-insulators is large enough to reduce
electronic relaxation rates many decades below the transition-times associated
with their conductivity 19 . Anderson insulators in this group exhibit glassy
dynamics and are referred to as ‘electron-glasses’. These should be
distinguished from lightly-disordered systems (sometimes called Coulomb-
glasses) that do not exhibit MD in field-effect experiments 19 . The relation
between the single-particle density-of-states and the MD has been elucidated
in 12 ; 14 ; 17 ; 18 .

Theoretical models that incorporate disorder and interactions are usually
concerned with the low-energy part of the single-particle density-of-states
The energy location of the states were expelled from the ZBA region has
received less attention. In particular, the Anderson-localized regime, where
$\rho$($\varepsilon$) has traditionally been derived via a classical Coulomb-
gap approach, the missing states are usually depicted as being evenly spread
outside the depleted region 20 ; 21 ; 22 ; 23 . Given that disorder in
Anderson insulators is in general significantly stronger than the Coulomb
interaction 24 , this picture seems plausible and is consistent with results
of tunneling 25 ; 26 , and field-effect experiments on strongly disordered
systems 27 .

Recently however, a strikingly different behavior was detected in a particular
version of amorphous indium-oxide In${}_{\text{x}}$O films; as shown in figure
1.

Figure 1: Conductance G vs. gate-voltage V${}_{\text{g}}$ for a 20nm thick
In${}_{\text{x}}$O film with carrier-concentration
N$\approx$10${}^{\text{19}}$cm${}^{\text{-3}}$. The sample is separated from a
degenerate Si:B gate by 2$\mu$m layer of SiO${}_{\text{2}}$. The figure shows
a memory-dip centered at V${}_{\text{g}}$=0 where the sample was equilibrated
for an hour after being cooled from room-temperature. The dashed line
delineates the energy dependence of $\partial$n/$\partial$$\mu$, the material
thermodynamic density-of-states..

The MD exhibited by specimen of this version reveal structure that resembles
that of a superconductor $\rho$($\varepsilon$) with its characteristic
‘coherence-peaks’ at the edges of the ZBA. The version of In${}_{\text{x}}$O
used in this work is not superconducting (at least down to $\approx$0.28K)
even when the system is in the diffusive transport regime 28 . Moreover, the
indium-rich version of In${}_{\text{x}}$O, which is superconducting when its
disorder is sufficiently small, does show a memory-dip when strongly-localized
but without these side-shoulders 27 . Different versions of In${}_{\text{x}}$O
are distinguished by their In/O composition that determines their carrier-
concentration N 28 . In terms of microstructure they are quite similar.
Electron-diffraction patterns of the superconducting version of the compound
and that of the low carrier-concentration sample used in this study are shown
in Fig.2. A scan of the radial intensity distribution of these patterns shown
in Fig.3 is required to be able to detect any difference between the two
versions. Detailed study of the In${}_{\text{x}}$O versions including x-ray
diffraction and interferometry, and Raman-spectrometry showed that in terms of
structural properties, different versions only differ by small quantitative
aspects like the position of the boson-peak 29 . However, in terms of disorder
perceived by the charge carriers, the difference may be substantial when
comparing In${}_{\text{x}}$O with similar resistivities but different carrier-
concentrations. The system with the lower carrier-concentration has a lower
disorder (given the same resistivity), and therefore it may also be more
homogeneous.

Figure 2: Electron diffraction-patterns of two versions of In${}_{\text{x}}$O
films 20nm thick. The left pattern (marked High D) has carrier-concentration
N$\approx$8x10${}^{\text{20}}$cm${}^{\text{-3 }}$ the right pattern (marked
Low D) is characteristic of the batch used in the present study with
N$\approx$10${}^{\text{19}}$cm${}^{\text{-3 }}$ Figure 3: Intensity profile
scan of the two diffraction-patterns shown in Fig.2. The first ring diameters
(in units of pixels) are shown for comparison. Note the similarity and the
small difference in the nearest-neighbor distance of the two versions.

Theoretically, piling-up of states at the edges of the ZBA is consistent with
the sum-rule for $\rho$($\varepsilon$). A non-monotonic $\rho$($\varepsilon$)
above the gap was demonstrated in numerical studies of disordered and
interacting quantum-systems, explicitly in strongly-correlated compounds 30 ;
31 ; 32 ; 33 ; 34  and in quantum-dots 35 . A ”shouldered”
$\rho$($\varepsilon$), quite similar in shape to the MD exhibited in Fig.1,
was obtained in Monte-Carlo simulations based on a random-displacement version
of the Coulomb-glass 36 ; 37 ; 38 . The shoulders in this scenario were
related to a charge-ordering process in the spirit of formation of a
”disordered Wigner crystal or, perhaps more appropriately, a ”Wigner-glass” 39
; 40 ; 41 ; 42 . The visibility of shoulders in 36  is exponentially reduced
with the disorder-strength W. Consequently, this feature has been resolved in
the simulations only for W much smaller than the Coulomb interaction amplitude
E${}_{\text{C}}$ 38 ; 43 . By contrast, the ratio W/E${}_{\text{C}}$ in
Anderson-insulators with Gaussian disorder is typically larger than unity 24 .
In the sample shown in Fig.1 for example, this ratio is estimated to be
$\approx$6, which, according to numerical simulations 36 ; 38 , seems
unfavorable for a charge-ordering mechanism to be effective. To account for a
”shouldered-MD” (SMD), it may perhaps be necessary to take additional factors
into account than included in the considerations used for the classical
Coulomb-glass.

We present in this work extensive data pertaining to this new phenomenon and
offer a heuristic interpretation for the main effects that characterize it.
Some puzzling issues that need further experimental and theoretical
elucidation are pointed out.

##  II Experimental

###  II.1 Sample preparation

The samples used in this study were amorphous indium-oxide
(In${}_{\text{x}}$O) films made by e-gun evaporation of 99.999% pure
In${}_{\text{2}}$O${}_{\text{3-x}}$ onto room-temperature Si-wafers in a
partial pressure of 1.3x10${}^{\text{-4}}$mBar of O${}_{\text{2}}$ and a rate
of 0.3$\pm$0.1Å/s. Under these conditions the carrier-concentration N of the
samples, measured by the Hall-Effect at room-temperatures, was
N$\approx$(1$\pm$0.1)x10${}^{\text{19}}$cm${}^{\text{-3}}$. Using free-
electron formula, this carrier-concentration is associated with
$\partial$n/$\partial\mu\approx$10${}^{\text{32}}$erg${}^{\text{-1}}$cm${}^{\text{-3}}$.
The Si wafers (boron-doped with bulk resistivity
$\rho\leq$2x10${}^{\text{-3}}\Omega$cm) were employed as the gate-electrode in
the field-effect experiments. A thermally-grown SiO${}_{\text{2}}$ layer,
2$\mu$m thick, served as the spacer between the sample and the conducting Si:B
substrate. The screening length of the material $\lambda$
$\approx$($\pi$e${}^{\text{2}}\partial$n/$\partial\mu$)${}^{\text{-1/2}}$ is
$\approx$2nm and therefore the voltage that actually affects the sample is
$\approx$10${}^{\text{-3}}$ of V${}_{\text{g}}$, the voltage applied between
the sample and the gate. At room-temperature the sheet resistance of the
samples used here ranged between 22.5k$\Omega$ to 25k$\Omega.$

Films thickness was measured in-situ by a quartz-crystal monitor calibrated
against X-ray reflectometry. Samples geometry was defined by the use of a
stainless-steel mask during deposition into rectangular strips 0.9$\pm$0.1mm
long and 1$\pm$0.1mm wide. Four different batches were made in this study and
here we report detailed results of three different samples from one of these
batches with a thickness of 20$\pm$5nm. The dimensionless parameter
k${}_{\text{F}}\ell$ for these samples is 0.27-0.29, just on the insulating
side of the critical value for this material (0.31$\pm$0.1 28 ).

###  II.2 Measurement techniques

Conductivity of the samples was measured using a two-terminal ac technique
employing a 1211 ITHACO current preamplifier and a PAR 124A lock-in amplifier
using frequencies of 64-71Hz depending on the RC of the sample-gate structure.
R is the source-drain resistance and C is the capacitance between the sample
and the gate. C in our samples was typically
$\cong$10${}^{\text{{\small-10}}}$F and R for the samples studied in this work
ranged between 1.5-2.5M$\Omega$ $at$ 4.11K. Except when otherwise noted, the
ac voltage bias in conductivity measurements was small enough to ensure near-
ohmic conditions. Except where otherwise noted, measurements were performed
with the samples immersed in liquid helium at T$\approx$4.11K held by a 100
liters storage-dewar. This allowed up to two months measurements on a given
sample while keeping it cold and in the dark. These conditions are essential
for measurements where extended times of relaxation processes are required at
a constant temperature. Fuller details of the field-effect measurements setup,
samples configuration and characterization are described in 29 . An up to date
list of the studied Anderson-insulators that exhibit a memory-dip along with a
description of their systematic dependence on the carrier-concentration of the
material are given in 19 .

##  III Results and Discussion

###  III.1 The basic facts associated with the SMD

The SMD state is sensitive to the conditions under which it is measured in the
field-effect experiment. In particular, sweeping V${}_{\text{g}}$ over a range
$|$$\Delta$V${}_{\text{g}}$$|$ that exceeds the typical width
$\Gamma_{\text{S}}$ of the shoulder results in a shoulderless MD when a
V${}_{\text{g}}$-sweep is taken again along the original interval and
polarity. The protocol used throughout this work to achieve this state
(labeled as ”reference”) involves sweeping V${}_{\text{g}}$ from 0V to +70V
and back to 0V then the sample is relaxed at this position for 20 minutes
allowing the ZBA to reform before taking the next G(V${}_{\text{g}}$) scan. An
example is illustrated in Fig.4:

Figure 4: Field-effect scans comparing the reference, shoulderless state (see
text) with the G(V${}_{\text{g}}$) of the sample 20 minutes after being
quench-cooled from room temperature to the bath-temperature. The two plots
were taken with the same scan-rate $\partial$V${}_{\text{g}}$/$\partial$t of
1.2V/s.

However, this shoulderless G(V${}_{\text{g}}$) turns out to be metastable; the
SMD state of the sample reappears if one let the sample equilibrates under the
initial V${}_{\text{g}}$ for long enough time before taking a new sweep. The
recovery of the SMD is illustrated in Fig.5 and Fig.6 for the right-hand and
left-hand side G(V${}_{\text{g}}$) scans respectively.

Figure 5: The recovery of the SMD-state with relaxation time
t${}_{\text{rel}}$ starting from a ”reference” state at t=0.08 hours. Data are
shown here for the right-hand side of the field-effect. Before taking each
plot, a new reference state was prepared by sweeping the gate voltage to +70 V
and back to 0V. The plots are displaced along the ordinate for clarity and
each was taken with the same sweep-rate of dV${}_{\text{g}}$/dt=1.2V/s. Figure
6: The recovery of the SMD-state as function of the relaxation time
t${}_{\text{rel}}$ starting from a reference state at t=0.25 hours. Data are
shown for the left-hand side of the field-effect. Before taking each plot, a
new reference state was prepared by sweeping the gate voltage to +70 V and
back to 0V. The plots are displaced along the ordinate for clarity. All plots
were taken with the same sweep-rate of dV${}_{\text{g}}$/dt=1.2V/s. The dashed
line and the red arrow delineate the way we define the magnitude of the
shoulder at a given t${}_{\text{rel}}$ (see text and Fig.7).

These time-dependent data imply that the SMD state has a lower energy than the
reference state.

The dynamics associated with the SMD ‘rejuvenation’ process described by the
data in Figs. 5 and 6 differs from that exhibited by that of the electron-
glass relaxation-process monitored through the conductance G(t) which,
characteristically, is logarithmic 27 . The two types of relaxations are
compared in Fig.7 for one of the samples:

Figure 7: (a): The recovery of the shoulder with time characterized by
$\Delta$G defined by the construction shown in Fig.6 as a red arrow. The
dashed line in the figure is a fit to the stretched-exponential function with
$\beta$=0.78 and $\tau$=23,400s. (b): The relaxation of this sample
conductance monitored by recording G(t) at V${}_{\text{g}}$=45V after changing
the gate-voltage from the equilibrium state at 0V to 45V. The dashed line
delineates the logarithmic relaxation law. Note the two decades difference in
time scales between the two plots.

Note first that the SMD evolution persists and is followed for two more
decades in time than the conductance relaxation. The main part of the change
in G(t) is over while that of $\Delta$G is still building up. Secondly,
although the two processes are probably related (and begin and end essentially
at the same times), the functional time dependence of $\Delta$G(t) being
stretched exponential with exponent $\beta$=0.78, suggests a process dominated
by a relatively narrow rate-distribution 44 . This point will be clarified
below.

The disparity in the dynamics of the two types of slow relaxations is also
manifested in the dependence of the SMD magnitude on the sweep-rate. This is
shown in Fig.8 for another sample in the batch studied.

Figure 8: (a): Dependence of the SMD plots on the gate-voltage sweep-rate
dV${}_{\text{g}}$/dt. (b): The magnitude of the shoulder relative to that of
the MD, defined as shown in the inset, as function of the sweep-rate
dV${}_{\text{g}}$/dt.

Note that $\Delta$G, the depth of the ZBA in these plots is changing
substantially with the sweep-rate (Fig.8a) owing to the relative fast
relaxation dynamics of these samples. The logarithmic dependence of $\Delta$G
on the sweep-rate is a common feature in electron-glasses 45 . This dependence
carries-over to the ratio $\delta$G/$\Delta$G ( where $\delta$G is the
shoulder height, see: Fig.8b) because of the much weaker change of $\delta$G
with the sweep-rate.

###  III.2 The shoulder origin - a heuristic interpretation

Based on the data presented above, it is conjectured that the shoulders
reflect an increase in the density of states at the edges of the MD in the
same vein that the depression of G(V${}_{\text{g}}$) signifies a depletion of
charge at the chemical potential set by v${}_{\text{g}}^{\text{eq.}}$, the
equilibrium gate-voltage. The MD has been related to the soft gap at
$\rho$($\varepsilon$) resulting from the long-range Coulomb interaction 12 ;
13 ; 14 ; 17 ; 18 ; 19 ; 45 .

On its own, the interaction favors a correlated spatial distribution of the
states expelled from the low energy of $\rho$($\varepsilon$). Therefore, a
lower energy configuration should be attained when the distribution of the
nearest-neighbor distances is peaked at a value of the order
N${}^{\text{-1/3}}$ where N is the system carrier-concentration. This in turn
would show up as a peak in G(V${}_{\text{g}}$) at
E${}_{\text{C}}\approx$e${}^{\text{2}}\mathit{N}^{\text{1/3}}$/$\kappa$ where
$\kappa$ is the effective dielectric constant of the medium. The shoulder in
our samples is peaked at E${}_{\text{C}}\approx$7 meV and it tapers-off
towards $\approx$40-50 meV (Fig.1). For the batch used here,
N$\approx$10${}^{\text{19}}$cm${}^{\text{-3}}$, the observed peak of the
shoulder is consistent with
E${}_{\text{C}}\approx$e${}^{\text{2}}\mathit{N}^{\text{1/3}}$/$\kappa$ if
$\kappa\approx$45\. The bare dielectric constant of the material is
$\kappa\approx$10 but the polarization of localized states may enhance it to
be consistent with this scenario.

To further asses the plausibility of this picture, we need to consider the
role of disorder. The precondition for the MD to be observable in the field-
effect experiment is a strong enough disorder 19 . Explicitly, the disorder
has to be strong enough to Anderson-localize the system and to render its
relaxation slower 19  than the sweep-rate of V${}_{\text{g}}$. On the other
hand, too strong a disorder would defeat the electron-electron tendency to
form an ordered structure and randomize the states spatial-distribution thus
suppressing the shoulder. Intuitively, out of the systems that exhibit MD, the
best candidate to show shoulders is where the disorder to interaction ratio is
small. This favors systems with low carrier-concentration N because the
disorder necessary to Anderson-localize a system increases with its Fermi-
energy E${}_{\text{F}}\propto$N${}^{\text{{2/3}}}$ while the interaction
scales with N${}^{\text{{1/3}}}$. Note that the SMD discussed here has the
lowest N among all the electron-glasses systems yet reported 19 ; 46 . This
may be in line with our charge-ordering scenario in the sense that the present
system has more favorable parameters than previously studied electron-glasses
to show shoulders in field-effect measurements.

But how favorable are the actual system parameters? The magnitude of the
interaction identified above with the position of the shoulder was
E${}_{\text{C}}\approx$7 meV. The relevant 47  disorder W may be estimated
from the width of the shoulder (Fig.1) yielding W/E${}_{\text{C}}\approx$6\.
The disorder to interaction ratio typical for a degenerate semiconductor
ranges between $\approx$3 to $\approx$100 24 . As anticipated, the current
system is at the ”favorable” limit in terms of parameters. Yet, this ratio is
far from the W/E${}_{\text{C}}\ll$1 limit where shoulders are observed in
classical simulations 36 ; 37 ; 38 . Hopefully, the results reported here may
assist in identifying the missing ingredients from these simulations.

The presence of disorder is also essential for understanding the time-
dependent processes depicted in Figs.5 and 6; Sweeping the gate voltage by a
$|$$\Delta$V${}_{\text{g}}$$|$$>$$\Gamma_{\text{S}}$ (where
$\Gamma_{\text{S}}$ is the shoulder-width) results in a shoulderless state
(Fig.4). However, given time, the shoulder is sluggishly recovered and
G(V${}_{\text{g}}$) eventually exhibits the SMD behavior. Presumably, the
metastable MD-state is made up of charges trapped by disorder over the
voltage-range swept by the gate. Some of the charges that were driven by the
gate to occupy higher energy-states are trapped by deep wells of the potential
while the gate travels back to v${}_{\text{g}}^{\text{eq.}}$. Charges
occupying states with shallow wells of the potential will escape these states
during the time the gate is set back in v${}_{\text{g}}^{\text{eq.}}$ to
rebuild the MD for the next G(V${}_{\text{g}}$) run. This explains both, the
shifting of charge away from the shoulder towards higher energies, and the
relative absence of the fast relaxation rates in the relaxation associated
with the shoulder recovery.

The proposed picture for creating the shoulderless state is analogous to the
process by which debris is washed ashore by sea-waves and is held on the beach
slope by friction when the wave pulls back seaward.

Despite the apparent difference in their dynamics, it is clear that the
shoulder and the memory-dip are two parts of the same phenomenon; when the
equilibrium voltage is moved, so is the position of the shoulder. An example,
illustrating the ‘two-dip-experiment’ 46  for the current batch of
In${}_{\text{x}}$O, is shown in Fig.9:

Figure 9: G(V${}_{\text{g}}$) plots taken consecutively; First,
V${}_{\text{g}}$ was swept from the original equilibrium
v${}_{\text{g}}^{\text{eq.}}$=0V to +65V (triangles). This is followed by
sweeping V${}_{\text{g}}$ to +20V and parked there for 20 minutes. Next,
V${}_{\text{g}}$ was swept to +80V (squares). V${}_{\text{g}}$ was then moved
to +20V and parked there for 20 minutes to relax. Finally, V${}_{\text{g}}$
was swept to -65V (circles). Note that the latter G(V${}_{\text{g}}$) plot was
taken less than an hour after the first sweep and the magnitude of the MD at
v${}_{\text{g}}$=0V has shrank by a factor of $\approx$60 relative to the
original value. All plots were taken with the same
dV${}_{\text{g}}$/dt=0.5V/s.

The feature that strikes the eye in Fig.9 is the barely discernible MD
produced in the third sweep at the original v${}_{\text{g}}^{\text{eq.}}$=0V.
The magnitude of this MD is smaller by almost two orders of magnitude relative
to either of the two ”fresh” MD’s. This is due to the relatively fast dynamics
of this weak-disorder batch 19 ; 46 ; 48 . A closer look at the figure reveals
a new aspect of the phenomenon; establishing a new
v${}_{\text{g}}^{\text{eq.}}$(at +20V) yields at the edge of the MD a
prominent shoulder despite being in voltage-range covered by V${}_{\text{g}}$
in a previous sweep. In a way, this is similar to that the shoulders appear
along with the MD upon the first cooldown from room-temperatures. Both the MD
and the shoulder appear in this case after a brief relaxation period
(typically 20 minutes), no days-long waiting-time is required. In fact, as
will be shown next, once a new equilibrium conditions are set, the full SMD
appears in the G(V${}_{\text{g}}$) plot.

###  III.3 Shoulder destruction and rejuvenation

As shown above, taking a G(V${}_{\text{g}}$) measurement over a
V${}_{\text{g}}$-range larger than $\Gamma_{\text{S}}$, eliminates the
shoulder from showing-up in the next scan. A prolonged relaxation of the
sample is required for its recovery. However, there are several ways to
restore the SMD without a long time-delay. Two of these were mentioned in the
paragraph above. Three other schemes effective in rejuvenating the shoulder
are described next.

The SMD’s resulting from applying the first two are illustrated in Fig.10.

Figure 10: Field-effect plots illustrating two examples of the shoulder
reappearance following a quench from a nonequilibrium, higher-energy state:
Stressing the sample with a non-ohmic field F=10${}^{\text{4}}$V/m held on for
20 minutes (circles), and exposing the sample to 1mA of the IR source for 3s
(triangles). In both cases the shown G(V${}_{\text{g}}$) plots are taken after
letting the sample relax for 20 minutes since the respective quench. These are
contrasted with the reference plot produced after a scan to +70V. All three
plots were measured in the linear-response and with the same
dV${}_{\text{g}}$/dt=1V/s.

Both schemes produce shoulders that exhibit higher visibility than would be
expected by allowing the system to relax at v${}_{\text{g}}^{\text{eq.}}$ for
the time required for completing the scheme (compare with Fig. 5).

The third scheme is the easiest to implement and it turns out to be the most
effective in pulling back the shoulder from the reference state. In this case,
V${}_{\text{g}}$ is first wept to
V${}_{\text{g}}$=-$|$v${}_{\text{g}}^{\text{bias}}$$|$${}_{\text{1}}$ staying
there for $\approx$2 seconds and then set back at V${}_{\text{g}}$=0V for 20
minutes relaxation before recording a G(V${}_{\text{g}}$) plot from 0V to
+60V. As noted before, this trip erases the shoulder. This
”v${}_{\text{g}}^{\text{bias}}$” protocol is repeated for different (negative)
bias-values thus generating plots as function of
-$|$v${}_{\text{g}}^{\text{bias}}$$|$. In each such plot the shoulder is
”rejuvenated” by pre-biasing the sample with a finite
-$|$v${}_{\text{g}}^{\text{bias}}$. These bias values were taken at random
order to verify that ‘history’ does not play a role. A series of such plots
with $|$v${}_{\text{g}}^{\text{bias}}$$|$ values ranging from 5V to 80V is
shown in Fig.11a.

Figure 11: The reappearance of the shoulder in the G(V${}_{\text{g}}$) plots
recorded by the ”v${}_{\text{g}}^{\text{bias}}$ protocol” described in the
text. These are labeled by their pre-scan (negative) bias (a). The plots were
taken consecutively using the same sweep rate dV${}_{\text{g}}$/dt=1V/s. (b)
The magnitude of the shoulder vs. the pre-bias voltage. This magnitude is
taken as the distance from the peak of the shoulder at 6.8V to the value of
the reference-curve at this voltage.

Note (Fig.11b) that the relative magnitude of the shoulder $\delta$G increases
with $|$v${}_{\text{g}}^{\text{bias}}$$|$ and tends to saturate for
v${}_{\text{g}}^{\text{bias}}\simeq$-50V. The range over which the gate-
voltage bias affects $\delta$G is similar to the typical width of the shoulder
$\Gamma_{\text{S}}$ that, in turn, is related to the disorder associated with
the phenomenon W. It is emphasized that these data are generated using random
values for $|$v${}_{\text{g}}^{\text{bias}}$$|$ in the consecutive
measurements. Therefore the correlation between
$|$v${}_{\text{g}}^{\text{bias}}$$|$ and $\delta$G is meaningful and
suggestive. Actually, these results lead us to recognize the common element in
the four different schemes that produces a SMD from a ”reference” state: They
all involve a quench from a state where the charge-carriers are endowed with
excess energy E${}_{\text{excess}}$ relative to the equilibrium state. These
schemes differ by how E${}_{\text{excess}}$ is structured (distributed in
energy) relative to W, which presumably determines its efficiency in pulling-
up the shoulder. When the excess-energy of charge-carriers exceeds W, they are
no longer trapped by potential wells of the disorder. Rather, their spatial
distribution is controlled by the Coulomb repulsion forcing them apart. In
this stage the system is in a fluid, spatially-correlated state that,
following a quench and brief relaxation, is well primed to form SMD.

The efficiency of a scheme to produce SMD is not necessarily reflected in the
conductance enhancement accompanying the process. For example, stressing the
sample with a large longitudinal field enhances the sample conductance by a
factor of $\approx$2 while the shoulder that results from this scheme is
visibly weaker (Fig.10) than that obtained by the brief IR illumination that
had a much smaller impact on the conductance (see Fig.12a,b).

Figure 12: The effects produced on the sample by a brief IR illumination and a
large applied electric-field. (a) The conductance increase and subsequent
relaxation in response to a 3s exposure to the 1mA IR source. This is compared
to the similar effect produced by a sudden change of the gate-voltage from 0V
to 45V (a 5 seconds trip-time). (b) The sample resistance as function of the
applied electric field.

####  III.3.1 Temperature dependence

Temperature has a marked effect on the electron-glass features. Both the width
of the memory-dip and its relative magnitude are systematically affected by
raising the temperature 49 . Here it is shown that the shoulder follows the
same trend; its peak is slightly shifted towards a higher V${}_{\text{g}}$
upon increasing the bath temperature, and its magnitude is exponentially
suppressed (Fig.13b). This temperature dependence was obtained before in the
electron-glass phase of In${}_{\text{2}}$O${}_{\text{3-x}}$ films 49 .

Figure 13: (a): The dependence of the G(V${}_{\text{g}}$) plot on temperature
for a typical SMD sample. Each plot was taken after first sweeping the gate to
-70V and back to 0V to allow shoulder rejuvenation (see text) and relaxation.
Then the bath temperature was changed and the same protocol repeated with
V${}_{\text{g}}$/dt=0.5V/s. (b): The relative magnitude of the ZBA and the
shoulder extracted from the data in (a). $\Delta$G/G for the ZBA is the value
of G(V${}_{\text{g}}$) at V${}_{\text{g}}$=7V. $\Delta$G/G values for the
shoulder are taken as in Fig. 6 relative to the dashed lines in (a). Dotted-
lines fit exp[-T/1.4] (see text). Note that both axes in plot (b) use
logarithmic scale.

Apart from reaffirming that the shoulder is an integral part of the memory-
dip, it illustrates how sensitive is its visibility to the addition of high-
energy components.

####  III.3.2 Nonequilibrium steady-state under IR illumination

Even more striking is the effect of a continuous IR illumination which
suppresses the shoulder despite the extremely weak power-load on the sample
(Fig.14):

Figure 14:  Sample conductance versus time while being exposed it to a IR
radiation with a constant power of 2·10${}^{\text{-7}}$Watt. Note that G(t)
tends to reach a saturated value for t$\gtrsim$1 hour (b): The sample
G(V${}_{\text{g}}$) plot in the dark (squares) compared with the
G(V${}_{\text{g}}$) plot taken under the constant IR illumination after the
conductance reached a ”saturated” conditions (explicitly, the change with time
for the duration of the G(V${}_{\text{g}}$) plot is negligible). Both curves
were taken after first sweeping V${}_{\text{g}}$ to -70V, then relaxing the
sample for 20 minutes, and using the same sweep-rate
dV${}_{\text{g}}$/dt=0.5V/s.

Under a continuous IR illumination the sample SMD magnitude was reduced to
half its dark-value while the conductance increased by only $\simeq$2.8% (see,
Fig.14a). Based on the conductance versus temperature of this sample shown in
Fig.15 this is the equivalent of raising the temperature by $\approx$20mK. To
achieve a similar reduction of the SMD magnitude of the same sample, the bath
temperature must be raised to T$\simeq$5.5K (see Fig.13) which increases the
conductance by $\simeq$300%.

The difference between these two protocols is in how the energy added to the
system is distributed among the available degrees of freedom. Under the IR
illumination, the system is populated with an excess of high-energy phonons.
These are generated by a cascade process: Electrons are excited to high-
energy, then relax by optical-phonons emission 50 . This nonequilibrium
steady-state has much fewer low-energy phonons than when the sample is heated-
up to $\approx$5.5K. The density of high-energy phonons is also increased by
raising the bath temperature, but most of the absorbed energy is spent on
enhancing the conductance. The indication is that the SMD-state is sensitive
to the presence of high-energy phonons. This may hint on an underlying
mechanism that counteracts the disorder in the competition with interactions.
High-energy phonons is a potent source of dephasing which raises the
possibility that quantum effects play a role. Effects that hinge on quantum-
coherence and may be relevant to the problem at hand are those that promote
spatially-extended wavefunctions like resonances 51 ; 52 ; 53  and other
wavefunction forms resulting from hybridization. The existence of extended
wavefunctions naturally mimics a weaker disorder. There is already evidence
for quantum coherence in this system 50 ; 55  so there is reason to further
explore this direction.

Figure 15: The temperature dependence (on a double-log scale) of the
conductance for the sample used in figures 8-14 inclusive

###  III.4 Discussion

There are several issues related to the appearance of the shoulders that need
to be further clarified. First is the absence of this feature in other
versions of In${}_{\text{x}}$O. In contrast with the samples reported here,
all studied Anderson-insulating In${}_{\text{x}}$O samples with N ranging from
$\approx$4x10${}^{\text{19}}$cm${}^{\text{-3}}$ to
$\approx$8x10${}^{\text{21}}$cm${}^{\text{-3}}$ exhibit electron-glass
characteristics with MD magnitude and width varying systematically with N  but
without shoulders 19 ; 27 ; 48 . It seems therefore that, for
In${}_{\text{x}}$O, as N increases from
$\approx$1x10${}^{\text{19}}$cm${}^{\text{-3}}$ to
$\approx$4x10${}^{\text{19}}$cm${}^{\text{-3}}$, the structure manifested by
the shoulders melts-away. In terms of In/O ratio, which is the way N is
determined in the compound, this range is merely $\approx$2% wide 28 . It
would be interesting to fine-tune the system N and track the evolution of the
phenomenon across this region. This is a much harder undertaking than it may
seem and will require establishing a new level of material deposition and
post-treatment control. Nonetheless, on the basis of what is already
established, this SMD$\rightarrow$MD transition is surprisingly sharp.

It appears that one has to decide between two possibilities: Either the
transition is a real phase-change, or it is a crossover to a regime with ultra
slow dynamics. Explicitly, as N increases, the shoulders-dynamics becomes too
slow to be resolved on a realistic time scale. This dilemma is similar to the
quandary that has haunted the glass community 54 , and perhaps it may take as
long to reach a conclusion. In the present case the possibility of a diverging
relaxation time has empirical support; The relaxation time of
In${}_{\text{x}}$O electron-glasses with
N$\approx$4x10${}^{\text{19}}$cm${}^{\text{-3}}$ to
N$\approx$10${}^{\text{20}}$cm${}^{\text{-3}}$ was shown to increase
dramatically with N 45 ; 46 , and as seen here. the dynamics associated with
the shoulders may actually be slower.

Yet, the possibility that the SMD to MD changeover is a true phase-transition
cannot be ruled-out either. If the MD$\rightarrow$SMD transition is indeed a
change between two distinct phases what are their nature? The SMD state,
characterized by a shoulder in the G(V${}_{\text{g}}$) plot has been
interpreted here as a preponderance of charge at energy of the order
E${}_{\text{C}}$ relative to the chemical-potential. It is illuminating to
contrast this picture with the spatial-order of the amorphous structure
manifested in the electron-diffraction patterns (Fig.2, and Fig.3 above). The
shoulders and the diffraction-pattern represent two amorphous structures; one
composed of ions, and the other of electronic-charges. The first ring in the
former is related to the nearest-neighbor distance between ions which is the
analog of the shoulder position in energy. Both the diffraction-ring and the
shoulder have a width that is a measure of the disorder associated with the
respective phenomenon. The diffraction-ring is relatively much sharper than
the shoulder-width. This is due to the difference in the two-particle
potential that constrain spatial-freedom of the constituents; Lennard-Jones
versus Coulomb-interaction for the ions and charge-carriers respectively. For
energies outside the Coulomb gap, this description depicts the SMD-state as a
”frozen-liquid” while the MD-state, lacking ‘medium-range-order’, is a
”frozen-gas”. The experiments that were performed up to and including this
work indicate that there is a transition between these states controlled by
the carrier-concentration N. This was ascribed to the competition between
disorder and interaction that is tilted in favor of the former when N
increases. To further test this picture one needs to change disorder, or the
interaction, independently which is a challenging task.

A related issue is the use of lower carrier-concentrations than
N$\approx$10${}^{\text{19}}$cm${}^{\text{-3}}.$ As mentioned above, in the
Anderson-localized regime (a precondition to show electron-glass features 19
), both disorder W and the Coulomb-interaction E${}_{\text{C}}$ increase with
N but W$\propto$N${}^{\text{2/3}}$ while
E${}_{\text{C}}\propto$N${}^{\text{1/3}}$ and therefore a lower disorder-
interaction ratio is achievable by reducing N. This however has a limited
scope; the ratio is a weak function of N while the dynamics, mainly controlled
by W 19 ; 46 , depends on it exponentially. When W decreases the relaxation
will quickly become too fast for the experimental
$\partial$V${}_{\text{g}}$/$\partial$t to expose a memory-dip in the field-
effect experiments. To our knowledge, there is no Anderson-insulator with
N$\lesssim$10${}^{\text{18}}$cm${}^{\text{-3 }}$that exhibits either MD or
SMD.

There are other factors that may be detrimental for charge-ordering besides
too strong disorder. Obviously, sample inhomogeneity beyond that inherent in
the sample disorder (estimated ‘globally’, usually on the basis of measuring
the resistance) will smear-out the shoulders even when all other conditions
are favorable. In particular, an excess of randomly distributed deep
potential-wells, should be avoided. Generally, the detailed structure of the
background potential, not just its average amplitude, is relevant for the
charge-ordering scenario.

A subtle question that is yet to be answered is the effect of a superimposed
periodic potential. The effect of underlying crystallinity on the disorder-
interaction balance is a pertinent question that may be tested by numerical
simulations. However, numerical studies usually use a lattice-model with half-
filling, which probably introduces a spatial constraint that likely is
incommensurate with the optimal charge arrangement. It is not clear that it
may suffice to employ just a very small filling-factor in the simulation to
capture the physics of charge-ordering.

Attempts to answer some of these questions may benefit from further
experimental work. Systems that exhibit electron-glass features, such as GeTe,
Tl${}_{\text{x}}$O, In${}_{\text{2}}$O${}_{\text{3-x}}$,
GeSb${}_{\text{2}}$Te${}_{\text{5}}$, and Be 19  do not show shoulders in
their G(V${}_{\text{g}}$) plots. It would be useful to know whether this is
due to unfavorable W/E${}_{\text{C}}$ or because of their crystallinity.
Filling factor in these compounds is naturally small but their relatively high
carrier-concentration may be unfavorable. It is worth trying to tweak this by
varying their composition. These systems should also be tested for quantum-
coherence as this may play a role as alluded to above. A methodical study of
these aspects may establish a new platform for the interplay between disorder
and interaction in Anderson insulators.

###### Acknowledgements.

Discussions with Oded Agam, and Ady Vaknin are gratefully acknowledged.

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