# Variable range hopping in a non-equilibrium steady state

Preeti Bhandari<EMAIL_ADDRESS>Department of Physics, Ben-Gurion
University of the Negev, Beer Sheva 84105, Israel.  Vikas Malik
<EMAIL_ADDRESS>Department of Physics and Material Science, Jaypee
Institute of Information Technology, Uttar Pradesh 201309, India.  Moshe
Schechter<EMAIL_ADDRESS>Department of Physics, Ben-Gurion University of
the Negev, Beer Sheva 84105, Israel

(August 28, 2024)

###### Abstract

We propose a Monte Carlo simulation to understand electron transport in a non-
equilibrium steady state (NESS) for the lattice Coulomb Glass model, created
by continuous excitation of single electrons to high energies followed by
relaxation of the system. Around the Fermi level, the NESS state approximately
obeys the Fermi-Dirac statistics, with an effective temperature ($T_{eff}$)
greater than the system’s bath temperature ($T$). $T_{eff}$ is a function of
$T$ and the rate of photon absorption by the system. Furthermore, we find that
the change in conductivity is only a function of relaxation times and is
almost independent of the bath temperature. Our results indicate that the
conductivity of the NESS state can still be characterized by the Efros-
Shklovskii law with an effective temperature $T_{eff}>T$. Additionally, the
dominance of phonon-less hopping over phonon-assisted hopping is used to
explain the relevance of the hot-electron model to the conductivity of the
NESS state.

††preprint: APS/123-QED

##  I Introduction

The conventional charge transport mechanism at low temperatures in insulators
is based on phonon-assisted hopping between localized states. This transport
phenomenon is termed variable-range hopping (VRH) [1, 2, 3] and is
characterized by the conductivity of the form

$\sigma\propto exp\bigg{[}-\bigg{(}\frac{T_{0}}{T}\bigg{)}^{y}\bigg{]},$ (1)

where $T$ denotes temperature and $T_{0}$ denotes a characteristic
temperature. For a two-dimensional (2D) system with a constant density of
states (DOS), one would expect to see Mott’s law of conductivity, i.e.,
$y=1/3$. Efros and Shklovskii [4] have argued that, due to the presence of
Coulomb interactions, the DOS has a so-called Coulomb gap around the Fermi
level and follows the relation

$g(\varepsilon)\propto|\varepsilon|^{d-1},$ (2)

where $d$ is the spatial dimension and $\varepsilon$, is the Hartree energy.
Efros and Shklovskii (ES) further argued that for an interacting system, the
exponent $y$ in Eq.(1) is equal to $1/2$ for both two and three dimensions.
The conductivity dominated by the VRH mechanism has been observed
experimentally [3] and numerically [5, 6].

Some interesting physics has come up when an electric field (F) is applied to
the system. Depending on the applied field’s strength, the conductivity can be
in an Ohmic or a non-Ohmic regime. When the applied field is small (low
electric field regime), we expect the conductivity to be approximately
independent of $F$. This comes under the category of “ Ohmic regime”. In this
regime, the hopping of electrons is still phonon-assisted (.i.e., depends on
the bath temperature only). However, at a high enough applied field, energy
cannot be fully dissipated to the phonons, and then the system reaches the
“non-ohmic regime”.

Within the non-ohmic regime, for a moderate electric field ($F<kT/e\xi$), the
conductivity depends on the field, the temperature, and the localization
length ($\xi$) [7, 8, 9, 10, 11]. For systems with small localization length,
the conductivity in this regime can be given by the following relation

$\sigma(T,F)=\sigma(T,0)exp\bigg{(}\frac{eFl_{0}}{kT}\bigg{)},$ (3)

where $l_{0}$ is the typical hopping length and $k$ is the Boltzmann constant.
The models satisfying Eq.(3) are called Field effect models, [12, 13, 14, 15]
where the non-linearity in the conductivity is associated with the field-
dependent tilt of the energy landscape of electron hopping.

However, systems with large localization lengths are not well characterized by
Eq.(3), but rather by what is denoted by the “hot electron model (HEM) [16].
Within the HEM, it is assumed that the conductivity of the system at some
temperature (T) and electric field (F) is equal to the linear conductivity at
an effective temperature ($T_{eff}$),

$\sigma(T,F)=\sigma(T_{eff},0)=\sigma_{0}\,exp\bigg{[}-\bigg{(}\frac{T_{0}}{T_{eff}}\bigg{)}^{1/2}\bigg{]}\,.$
(4)

Eq.(4) represents the so-called Hot-electron Model, where $\sigma_{0}$ is the
proportionality constant and $T_{0}$ is the characteristic temperature of the
system when the ES law is obeyed in the Ohmic regime at temperature T. The
assumption here is that the conductivity of the system for a given electric
field and temperature depends only on the effective temperature of the
electrons ($T_{eff}$) and not on the bath temperature ($T$). Several
experiments [17, 18, 19, 20] and numerical simulations [6] on systems that in
equilibrium obey VRH have been interpreted in terms of the HEM.

One can see an activationless hopping when the electric field increases even
further, $F>kT/e\xi$ (strong electric field regime) [21, 9, 22, 23, 24, 25,
26, 27]. In this regime, the field plays a role similar to that played by the
bath temperature in the “Ohmic regime”, and the conductivity is given by

$\sigma\propto exp\bigg{[}-\bigg{(}\frac{F_{0}}{F}\bigg{)}^{1/2}\bigg{]}\ ,$
(5)

where $F_{0}=bkT_{0}/(e\xi)$. Here $T_{0}$ is the characteristic temperature
of the ES law, and $b$ is a constant of order unity.

The variation in the applied electric field has attracted much attention as it
is the formal way of formulating the non-linearity in the conductivity. This
method has provided much helpful and reliable information, as discussed
before. The present work presents an alternative approach, obtained by
creating a non-equilibrium steady state (NESS) by irradiating the sample
continuously with high-frequency photons. The conductivity of the NESS state
is calculated in the Ohmic regime. The conceptual advantage of this approach
is that it decouples the formation of a non-equilibrium state from
conductivity calculations.

In this paper, we study the Coulomb glass model (see details in Sec. II), with
localization length twice the lattice spacing, and the equilibrium conductance
is in agreement with the ES law. Following the HEM, for each NESS state,
dictated by the temperature and the excitation rate, we calculate the
effective temperature of the system ($T_{eff}$) related to the occupation of
the single electron states near the Fermi level [28]. We then analyze the
conductivity ($\sigma$) and density of states at the Fermi level ($g(0)$)
through their dependence on $T_{eff}$. Our results are in agreement with HEM
results for the field-driven NESS [6]. For large enough $T_{eff}$, we find
deviations from the form of Eq.(4). We explain these deviations within the
picture of the HEM, their appearance signaling the initiation of the regime
where phonon-less conductivity becomes dominant.

As a second approach, and in an attempt to relate to experimental protocol
[29], we also analyze the various observables by fitting the equilibrium
equations for each observable directly, assigning the temperature as a free
parameter. We find that different effective temperatures can be assigned to
each observable within this approach. Specifically, we find that the effective
temperature for the conductivity is smaller than the effective temperature for
the density of states at the Fermi energy. This result is in qualitative
agreement with a recent experiment on Indium oxide films where the NESS state
under the influence of IR radiation was studied. However, the experimental
results show larger deviations between the effective temperatures of the
conductivity and of the memory dip; see the discussion below.

Our paper is organized as follows. In Sec. II, the Coulomb Glass lattice model
is introduced. We present the numerical techniques used here in Sec. III. In
Sec. IV, we present our numerical results, and finally, in Sec. V, we conclude
the paper with a summary of our results.

Figure 1:  Energy of the non-equilibrium steady state as a function of the
number of excitation steps at different relaxation steps. Different graphs
correspond to a different number of relaxation steps ($x$) following each
excitation step. The initial state here is the annealed state at $\beta=80$.

.

##  II Model

We consider the standard two-dimensional (2d) Coulomb Glass (CG) lattice model
with the Hamiltonian

$H=\sum_{i}\phi_{i}S_{i}+\frac{1}{2}\sum_{i\neq j}\frac{S_{i}S_{j}}{r_{ij}},$
(6)

where $S_{i}=n_{i}-1/2$ is the pseudo spin variable at site $i$,
$n_{i}\in\\{0,1\\}$ is the electron occupation number, $\phi_{i}$ are the
random on-site energies chosen from a box distribution with the interval
$\bigg{[}-W/2,W/2\bigg{]}$ and $r_{ij}$ is the distance between sites $i$ and
$j$ under periodic boundary conditions. All energies and temperatures are
calculated in units of $e^{2}/a$, where $a$ is the lattice constant. We have
used a square lattice of size $64\times 64$ and disorder $W=2$.

##  III Numerical details

Experimentally, the NESS state is created by irradiating a well-equilibrated
sample with high-frequency radiation ($\nu$), having energy greater than the
Coulomb gap width ($\delta$), $h\nu$ $>>\delta$. Numerically, to create a NESS
state, we first use the simulated annealing technique [30] to thermalize the
Hamiltonian (6). All the observables were averaged over time (i.e.,
logarithmic binning of Monte Carlo steps) to test the equilibration. The
system is in thermal equilibrium once the last three bins agree within errors.
A two-step approach is required to create a NESS condition at a specific
temperature. The first step involves inducing excitation in the system (i.e.,
$\Delta E=\varepsilon_{i}-\varepsilon_{j}-1/r_{ij}>0$), followed by a
relaxation procedure. These two steps are then repeated continuously.

Excitation process: We start with an annealed state (we have considered the
annealed states at $\beta=1/T=20,40$ and $80$) and choose a site ‘$i$’ at
random. Then we pick another site ‘$j$’, now with a probability,
$e^{-2r_{ij}/\xi}$, where $\xi=2$ is the localization length. In addition, we
make certain that $S_{i}\neq S_{j}$. Finally, we swap the two spins, creating
an excitation in the system. This corresponds to an electron making a $\Delta
E>0$ transition between site $i$ and $j$ by absorbing a photon.

Relaxation process: Experimentally, the time between two photons striking the
system (relaxation time, $x$) varies as a function of the power.

$Power\propto\frac{1}{relaxation\,time}\quad.$ (7)

Between two-photon absorptions, the system may relax toward the equilibrium
state. To simulate the relaxation process, we allow the system to relax via
the kinetic monte carlo algorithm [5] between two subsequent excitations. In
our simulation, we vary the number of relaxation steps
($x=1,3,7,10,30,70,100$) after each excitation process ($ex$) and trace its
effect on the energy, DOS, occupation probabilities, and conductivity of the
NESS state. A single relaxation step corresponds to $L^{2}$ Monte Carlo steps.

A NESS state is reached once the rate of energy gained by the excitations
becomes equal to the energy lost in the relaxation process, and the system
achieves steady-state energy. The energy of the NESS state increases as the
relaxation time decreases, as seen in Fig.(1).

To simulate the conductivity of the NESS state, we apply a small electric
field (F) in the x-direction by adding a term $\sum_{i}Fx_{i}$ (where
$F=T/10$) to the Hamiltonian (6), and perform Kinetic Monte Carlo simulations
as described in Ref.[5] The NESS state is maintained during conductivity
calculations, and results were obtained after averaging over 500 disorder
realizations.

Figure 2:  (a) The behavior of the density of states at the Fermi level (for
the NESS state) as a function of relaxation steps ($x$) at different bath
temperatures ($T=1/\beta$). (b) The conductivity of the NESS state as a
function of $x$ at different bath temperatures.

##  IV Results

The perturbation leading to the NESS state drives the system out of
equilibrium. Specifically, the relaxation of the electrons near the Fermi
energy is limited by the scarcity of available states to relax to. This leads
to a change of the occupation probability of electronic states near the Fermi
energy [28], which is well approximated by a Fermi function with $T_{eff}\neq
T$,

$f_{i}=\frac{1}{[exp(\varepsilon_{i}/T_{eff})+1]}\quad.$ (8)

The supplemental material contains more information on calculating $T_{eff}$
using Eq.(8). This effective temperature is used to express the various
quantities below.

Figure 2 shows the behavior of the DOS at the Fermi-level ($g(0)_{NESS}$) and
the conductivity ($\sigma_{NESS}$) of the NESS state as a function of $x$ at
different temperatures.

Let us first consider the NESS state’s conductivity in more detail. In
Fig.(3), we plot $\sigma_{NESS}$ as a function of $x$. At long relaxation
times, we find

$\sigma_{NESS}\approx\sigma(T)+\frac{c}{x}\ ,$ (9)

.i.e. $\sigma_{NESS}-\sigma(T)$ is nearly independent of $T$, as shown in the
inset. Here $c$ is a constant, and $\sigma(T)$ is the conductivity of the
thermal state.

Figure 3:  The conductivity of the NESS state at various bath temperatures as
a function of intensity $(1/x)$. The behavior of change in conductivity as a
function of intensity for the NESS state at different bath temperatures is
presented in the inset.

We now analyze our results for the conductivity with regard to the Hot
Electron Model. Our analysis of the HEM is presented in Fig.(4), where we have
used $T_{eff}$ computed using Eq.(8) and plotted $\sigma_{NESS}$ vs
$T_{eff}^{-1/2}$ for different temperatures and relaxation times. The solid
line in Fig.(4) corresponds to the conductivity as given by Eq.(4).

For $\beta=20$ and small enough intensity (large relaxation times $x>7$),
Eq.(4) is approximately satisfied. For $\beta=40$ and $80$, the deviation from
the solid line in Fig.(4) leads us to a conclusion that for low temperatures
(and for cases where $\Delta\sigma>>\sigma$), the conductivity of the NESS
state can be explained using the following relation

$\sigma_{NESS}=\sigma_{0}^{\prime}\,exp\bigg{[}-\bigg{(}\frac{T_{0}^{\prime}}{T_{eff}}\bigg{)}^{1/2}\bigg{]}\
,$ (10)

where $T_{0}^{\prime}\approx T_{0}$ and $\sigma_{0}^{\prime}\geq\sigma_{0}/2$
approaching $\sigma_{0}/2$ for $T_{eff}\textgreater\textgreater T$ [3]. This
variation of $\sigma_{0}$ from the constant value given in Eq.(4) provides a
generalization of the HEM. Let us now discuss its physical origin. In the case
of a well-equilibrated thermal state, up transitions ($\Delta E>0$) and down
transitions ($\Delta E<0$) contribute equally to the conductivity of the
system, and the transition from a site $i$ to $j$ is given by

$\Gamma_{ij}\sim\gamma_{0}\,exp\bigg{[}-\frac{2r_{ij}}{\xi}-\frac{\Delta
E_{ij}}{kT}\bigg{]}\quad{\color[rgb]{1,0,0}.}$ (11)

Minimizing the exponent in Eq.(11) leads to the ES law of conductivity
(Eq.(1)). Note that $\Gamma_{ij}=\Gamma_{ji}$ for a system at a steady state
under the influence of a small electric field. On the limited range of
temperatures available in our simulation, the temperature dependence of the
conductivity in equilibrium is in agreement with the ES law (as shown by the
solid line in Fig.(4).

Now for a NESS state, where the change in conductivity is mainly due to the
downward energy transitions, $\Gamma_{ij}\neq\Gamma_{ji}$. Here, the downward
energy transitions between two sites, $i$ and $j$, given by

$\Gamma_{ij}=\gamma_{0}\,exp\bigg{[}-\frac{2r_{ij}}{\xi}-\frac{\Delta
E_{ij}}{k\,T_{eff}}\bigg{]}.$ (12)

The reverse upward energy transition from $j$ to $i$ is given by Eq.(11). When
$T_{eff}>>T$, the energy-lowering transitions dominate over energy-gaining
ones, and the system’s conductivity is defined by Eq.(10). When the downward
transitions completely dominate the system (we see this at very low
temperature $\beta=80$), then

$\sigma_{NESS}(T_{eff},x)\approx\frac{1}{2}\sigma(T_{eff})\ .$ (13)

In Fig.(4), the deviation of the NESS conductivity (dashed line) from the
equilibrium conductivity (solid line) is a feature of the HEM and not its
failure. The change in Fig.(4) from $\sigma_{0}$ being similar to its thermal
value to being about half its thermal value notes the relaxation time (value
of $x$ in our case) where the phonon-less conductivity becomes dominant. We
note that for each temperature, there is a one-to-one correspondence between
the effective temperature and the relaxation time ($x$).

Figure 4:  Conductivity ($\sigma^{*}$) as a function of temperature ($T^{*}$)
is log-real plot. Here $\sigma^{*}=\sigma$ and $T^{*}=T$ for the equilibrium
data and $\sigma^{*}=\sigma_{NESS}$ and $T^{*}=T_{eff}$ for the NESS data. The
red dots correspond to the conductivity of the thermal state. This data has
been fitted using the relation, $\sigma=\sigma_{0}\ exp\\{-(T_{0}/T)^{1/2}\\}$
(solid line). Blue cross ($\beta=20$), green square ($\beta=40$), and orange
triangle ($\beta=80$) correspond to the conductivity of the NESS state as a
function of $T_{eff}^{-1/2}$. The orange triangles are fitted (dashed line)
using the relation $\sigma_{ness}=\sigma^{\prime}_{0}\
exp\\{-(T_{0}/T_{eff})^{1/2}\\}$, where $T_{eff}$ is calculated from the Fermi
Dirac distribution. Figure 5:  Density of state at the Fermi-level
($g^{*}(0,T^{*})$) as a function of temperature ($T^{*}$). Here
$g^{*}(0,T^{*})=g(0,T)$ and $T^{*}=T$ for the equilibrium data and
$g^{*}(0,T^{*})=g^{NESS}(0,T_{eff})$ and $T^{*}=T_{eff}$ for the NESS data.
The red dots correspond to the single-particle density of states (DOS) at the
Fermi-level of the thermal states $g(0,T)$ at disorder $W=2$. This data has
been fitted using the relation, $g(0,T)=c\,T^{\alpha}$ (solid line). Blue
cross ($\beta=20$), green square ($\beta=40$), and orange triangle
($\beta=80$) correspond to DOS of the NESS state as a function of $T_{eff}$.
The orange triangles are fitted (dashed line) using the relation
$g^{NESS}(0,T_{eff})=c^{\prime}\,T_{eff}^{\alpha}$, where $T_{eff}$ is
calculated from the Fermi Dirac distribution.

We can now also explain the results shown in Fig.(3). The system gains excess
energy (excitations) by photon absorption and attempts to attain equilibrium
by energy-lowering transitions. The number of excitations created by photon
absorption is proportional to the radiation power. Thus, the excess
conductivity, i.e., $\Delta{\sigma}$, dominated by energy-lowering transitions
facilitated by excited electrons, depends only on the radiation power, not on
the bath temperature (see the supplementary material for more details).

Another experimentally relevant quantity is the DOS at the Fermi-level $g(0)$,
which, theoretically for 2D systems, is proportional to the phonon-bath
temperature (T) [31]. Numerical simulations claims that $g(0,T)\propto
T^{\alpha}$ where $\alpha\neq 1$ [32, 33]. As shown in Fig.(5) (red circles),
our data show that the density of states of the thermal states at the Fermi-
level $g(0)$ follows the following relation

$g(0)=cT^{\alpha}\ ,$ (14)

where $\alpha=1.29$ and $c$ is the proportionality constant. For the NESS
state, we find (see Fig.(5)) that the density of states at the Fermi-level,
$g_{0}^{NESS}(x,T)$, as a function of relaxation time $(x)$ and bath
temperature $T$ follows the relation

$g_{0}^{NESS}(x,T)=c^{\prime}(T_{eff})^{\alpha}\ ,$ (15)

where $c\approx c^{\prime}$ at $\beta=20$ and $c\neq c^{\prime}$ as the
temperature and the relaxation time decreases. Also, here, we find a decrease
in the proportionality constant in the regime where phonon-less hopping
dominates, albeit this effect here is smaller than for the conductivity.

Figure 6:  Effective temperature ($T_{eff}^{*}$) of the NESS state as a
function of relaxation time ($x$) calculated using: conductivity ($\sigma$),
Fermi-Dirac distribution around the Fermi-level (FD), and DOS at the Fermi-
level ($g(0)$)) at the bath temperature $T=0.025$. $T_{eff}^{*}$ here
corresponds to $T_{eff}^{\sigma}$ for the blue line, $T_{eff}^{g}$ for the
orange line and $T_{eff}^{FD}$ for the green line.  Figure 7:  Effective
temperature for a NESS state at beta = 20, 40, and 80 as a function of
relaxation times ($x$). $T^{g}_{eff}$ and $T^{\sigma}_{eff}$ are different
when the system is out of equilibrium, which gradually merges to a single
temperature ($T$) as the relaxation time increases (system approaching
equilibrium). The lower the temperature, the longer the relaxation times where
the two effective temperatures merge.

Finally, we now make an attempt to connect our numerical simulation to recent
experimental results on the NESS state of amorphous indium oxide [29]. Here,
following the analysis in the experiment, we adopt a different method to
determine the effective temperature for conductivity ($T^{\sigma}_{eff}$) and
DOS at the Fermi level ($T^{g}_{eff}$). (The detailed explanation of the
procedure used to calculate the effective temperature is provided in the
supplementary material). We calculate $T^{\sigma}_{eff}$ and $T^{g}_{eff}$ of
the NESS state by using Eq.(4) and Eq.(14) respectively at different
temperatures and relaxation times. The values of the parameters $c$,
$\sigma_{0}$, $T_{0}$ are kept equal to their equilibrium values in this
calculation. For clarity, we denote the $T_{eff}$ calculated using FD
distribution in Eq.(8) as $T^{FD}_{eff}$ for the rest of the discussion. We
observe that at short relaxation times, the effective temperatures describing
each physical quantity are different. In Fig.(6), we plot the effective
temperatures for the different observables as a function of relaxation time at
$T=0.025$ (data for $T=0.05$ and $T=0.0125$ are shown in supplementary
material). Note that there is a threshold degree of non-equilibrium, where
$\Delta\sigma>\sigma$ (which depends on the bath temperature and relaxation
time of the system) only beyond which effective temperatures of different
observables appears. Unlike the situation in equilibrium, when the system is
pushed far enough out of equilibrium, it acquires multiple time scales
affecting differently various measurable quantities and consequently dictating
observable specific effective temperatures.

Let us note that for the regime where the effective temperatures are
different, $T^{\sigma}_{eff}$ is always less than $T^{FD}_{eff}$. This can be
explained as follows: Comparing the expressions for the conductivity within
the two approaches discussed above, we find

$\sigma_{0}\,exp\bigg{[}-\bigg{(}\frac{T_{0}}{T_{eff}^{\sigma}}\bigg{)}^{1/2}\bigg{]}=\sigma_{0}^{\prime}\,exp\bigg{[}-\bigg{(}\frac{T_{0}}{T^{FD}_{eff}}\bigg{)}^{1/2}\bigg{]}\
.$ (16)

At low temperatures, one finds that $\sigma_{0}^{\prime}\approx\sigma_{0}/2$
(this is true in our case at $\beta=80$ where the phononless hopping
completely dominates). Using this relation in Eq.(16), we get

$\bigg{(}\frac{1}{T_{eff}^{\sigma}}\bigg{)}^{1/2}\approx\bigg{(}\frac{1}{T_{eff}^{FD}}\bigg{)}^{1/2}+\bigg{(}\frac{ln(2)}{\sqrt{T_{0}}}\bigg{)}\
,$ (17)

i.e. $T_{eff}^{\sigma}<T_{eff}^{FD}$.

We now compare our findings with the experiment [29], where it was found that
far from equilibrium, the effective temperature of the conductivity is much
smaller than the effective temperature of the memory dip. Following this
experimental finding, we concentrate on the effective temperatures of the NESS
state that we determined using $\sigma$ and $g(0)$; the latter is believed to
be proportional to the memory dip. We compare the two effective temperatures
at T=0.05, 0.025, and 0.0125 for different relaxation times, as shown in
Fig.(7). The dashed line represents the bath temperature and indicates how far
the NESS state is from the equilibrium state. When the system is further from
equilibrium, we do observe that the effective temperature computed using
conductivity ($T^{\sigma}_{eff}$) is lower than the one computed using the
density of states at the Fermi level ($T^{g}_{eff}$). However, this difference
is much smaller than the difference between the experimentally obtained
effective temperatures for the conductivity and the DOS. Fig.(7) also shows
that as the system moves away from equilibrium, the two effective temperatures
converge at a larger relaxation time value.

##  V Discussion

We demonstrate the electron transport in a non-equilibrium steady state in the
context of understanding the general theory of the Hot-electron model. The
non-equilibrium steady state is created by irradiating the system with high-
frequency photons. Experiments of this kind have been recently conducted on
indium oxide films [29].

We have calculated the conductivity ($\sigma$) and the single-particle density
of states at the Fermi-level ($g(0)$) of the NESS state. At relatively high
temperatures, our results are remarkably similar to the non-equilibrium
simulation results by Caravaca et al [6] done in high electric fields at
$\xi=2$. At lower temperatures, at first glance (in our work) and as reported
by Caravaca et al [6] for smaller localization length ($\xi=1$ case), the
conductivity of the NESS state deviates from the HEM. The deviation happens
because of slow relaxation due to small $\xi$ in Ref. ([6]) and small $T$ in
our case. We explain this deviation as a feature of the HEM in terms of the
dominance in this regime of phonon-less hopping over phonon-assisted hopping.
We show that both $g(0)$ and $\sigma$ of the non-equilibrium steady state obey
the HEM model.

Our results, therefore, provide a robust way to understand the crossover from
phonon-less hopping to phonon-assisted hopping, which can be tested
experimentally by varying the intensity of radiation on the target sample or
decreasing the temperature of the system.

Our results are also qualitatively in agreement with the experimental finding
in [29] that $T^{\sigma}_{eff}$ $<$ $T^{g}_{eff}$ when the NESS state is far
from equilibrium. One of the possible reasons for the quantitative difference
could be that the measurement in the experiment (memory dip) is not directly a
measurement of the DOS (in simulations) but is believed to be proportional to
it. Another possibility can be quantum effects which are beyond the current
study in this paper.

In the future, it will be interesting to explore how our results alter if we
move from the ES to the Mott regime. It would also be interesting to see how
positional disorder affects the system.

## Acknowledgement

P.B. acknowledges the Kreitman School of Advanced Graduate Studies for
financial support. M.S. acknowledges support from the Israel Science
Foundation (Grant No. 2300/19). Illuminating discussions with Z. Ovadyahu is
gratefully acknowledged.

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