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Journal ArticleDOI

A numerical study of Coulomb interaction effects on 2D hopping transport.

15 Feb 2006-Journal of Physics: Condensed Matter (IOP Publishing)-Vol. 18, Iss: 6, pp 2013-2027
TL;DR: The supercomputer-enabled Monte Carlo simulations of hopping transport in completely disordered 2D conductors to the case of substantial electron-electron Coulomb interaction are extended, finding that the spectral density S(I)(f) of current fluctuations exhibits a 1/f-like increase which approximately follows the Hooge scaling, even at vanishing temperature.
Abstract: We have extended our supercomputer-enabled Monte Carlo simulations of hopping transport in completely disordered 2D conductors to the case of substantial electron-electron Coulomb interaction. Such interaction may not only suppress the average value of hopping current, but also affect its fluctuations rather substantially. In particular, the spectral density S(I)(f) of current fluctuations exhibits, at sufficiently low frequencies, a 1/f-like increase which approximately follows the Hooge scaling, even at vanishing temperature. At higher f, there is a crossover to a broad range of frequencies in which S(I)(f) is nearly constant, hence allowing characterization of the current noise by the effective Fano factor [Formula: see text]. For sufficiently large conductor samples and low temperatures, the Fano factor is suppressed below the Schottky value (F = 1), scaling with the length L of the conductor as F = (L(c)/L)(α). The exponent α is significantly affected by the Coulomb interaction effects, changing from α = 0.76 ± 0.08 when such effects are negligible to virtually unity when they are substantial. The scaling parameter L(c), interpreted as the average percolation cluster length along the electric field direction, scales as [Formula: see text] when Coulomb interaction effects are negligible and [Formula: see text] when such effects are substantial, in good agreement with estimates based on the theory of directed percolation.

Summary (2 min read)

1. Introduction

  • The hopping transport of quasi-localized electrons in disordered conductors and semiconductors has been studied for many years; for comprehensive reviews, see [1–3].
  • The more recent observation [4, 5] that hopping transport may implement the quasi-continuous (‘sub-electron’) charge transfer, hence providing a possible solution to the random background charge problem in single-electronics [6], has renewed interest in this phenomenon, with an emphasis on the shot noise of the hopping current [7–10].
  • Publishing Ltd Printed in the UK 2013 to the case of substantial Coulomb interaction of hopping electrons.

1.2. DC transport characteristics

  • For not very high temperatures (T T0, where kBT0 ≡ 1/ν0a2 and a is the localization radius), the ratio σ/σ0 (where σ0 is a constant characterizing the sample) depends only on two dimensionless parameters: the ratio T/T0 and parameter χ ≡ e2ν0a/κ characterizing the Coulomb interaction strength.
  • The relation between these two parameters determines two possible variable-range hopping transport regimes.
  • If the field is not extremely high (E E0 ≡ 1/eν0a3), i.e. in the variable-range hopping domain, the authors can again distinguish two different transport regimes.

1.3. Current fluctuations

  • At low temperatures1, the dynamical fluctuations of the current flowing through a mesoscopic system are more sensitive to the charge transport mechanism peculiarities than the average transport characteristics, and therefore may provide additional information about the conduction physics [17–19].
  • At very low frequencies, one can expect the 1/ f -type noise that is observed experimentally in a wide variety of conductors; see, for example, [17].
  • For the particular case of hopping conduction, two major theories of 1/ f noise have been suggested, based, respectively, on ‘carrier number’ fluctuations [21–23] and ‘mobility’ fluctuations [24, 25] as possible origins of the noise.
  • Both analytical and numerical results [7, 26] show that at 1D hopping without Coulomb interaction, α may be as low as 1/2.

2. Model

  • The authors have studied broad 2D rectangular conductors (W Lc) with ‘open’ boundary conditions on the interfaces with well-conducting electrodes [9, 10].
  • The conductors are assumed to be ‘fully frustrated’,with a large number of localized sites randomly distributed over the conductor area.
  • Such exponential dependence on the length of a hop is standard for virtually all theoretical studies of hopping2.
  • Notice that in contrast with some prior works, the authors do not include the factor 2 into the definition of the exponent.
  • The spectral density of current fluctuations is calculated using the advanced algorithm described in detail in [11].

3. Results

  • In order to classify the physical regimes of hopping behaviour, it is useful to note that their model has four relevant energy scales: (i) 1/ν0a2 describes the energy spectrum discreteness, (ii) eEa is the scale of the electric field strength, (iii) e2/κa = χ/ν0a2 characterizes the Coulomb interaction strength, and (iv) kBT is the scale of thermal fluctuations.
  • (21) We are not interested in the case of extremely high temperatures, so that the authors will always assume that T T0, i.e. ET E0.the authors.

3.2. DC transport characteristics

  • The results for χ = 0 coincide with those discussed in their previous work [10].
  • (Unfortunately, the above values had no uncertainty reported.).
  • For very high temperatures (ET E0), the exponential temperature dependence of variable-range hopping theory cannot give a good description of the results, because in this case transport is dominated by very short hops with lengths of the order of the localization radius.
  • The corresponding results for χ = 0.1 and χ = 0.5 show that increasing Coulomb interaction strength suppresses the nonlinear DC conductivity, just as in the low-field case.

3.3. Current fluctuations

  • Figure 4 shows typical results of their calculations of current noise at zero temperature, finite Coulomb interaction strength and fixed electric field, for several values of conductor length.
  • The frequency fk of the 1/ f noise ‘knee’ (the crossover from this noise to a quasi-flat spectral density) is relatively constant (or at most grows slowly with decreasing conductor length).
  • Figure 6(b) shows that both results can be collapsed onto a universal scaling curve by the introduction of certain length scales: Lc for F and Lh for F∞.
  • For not too high fields (ET E E0), the results in figure 7 for negligible Coulomb interaction are in good agreement with the variable-range hopping scaling described by equation (7), while for substantial Coulomb interaction they follow scaling similar to equation (9).

4. Discussion

  • To summarize, the authors have carried out numerical simulations of 2D hopping within a broad range of temperature, electric field and Coulomb interaction strength.
  • For average (DC) transport characteristics, their results are in general agreement with the variable-range hopping theories, except for the (model-dependent) cases of ‘ultra-high’ electric field and/or temperature, where the hopping length becomes of the order of the localization radius.
  • For the spectrum of current fluctuations, their results are more significant.
  • In hindsight, this result does not seem too surprising.
  • The authors second important result is that in the presence of significant Coulomb interaction, the quasi-white noise above the 1/ f noise knee is suppressed according to the scaling law (13) with α = 1 (within the accuracy of their numerical experiment).

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 18 (2006) 2013–2027 doi:10.1088/0953-8984/18/6/016
A numerical study of Coulomb interaction effects on
2D hopping transport
Yusuf A Kinkhabwala, Viktor A Sverdlov and Konstantin K Likharev
Department of Physics and Astronomy, Stony Brook University, Stony Brook,
NY 11794-3800, USA
Received 25 July 2005, in final form 12 October 2005
Published 27 January 2006
Online at stacks.iop.org/JPhysCM/18/2013
Abstract
We have extended our supercomputer-enabled Monte Carlo simulations of
hopping transport in completely disordered 2D conductors to the case of
substantial electron–electron Coulomb interaction. Such interaction may
not only suppress the average value of hopping current, but also affect its
fluctuations rather substantially. In particular, the spectral density S
I
( f ) of
current fluctuationsexhibits, at sufficiently low frequencies, a 1/ f -like increase
whichapproximatelyfollowsthe Hooge scaling, even at vanishingtemperature.
At higher f ,thereisacrossover to a broad range of frequencies in which
S
I
( f ) is nearly constant, hence allowing characterization of the current noise
by the effective Fano factor F S
I
( f )/2e
I
.For sufficiently large conductor
samples and low temperatures, the Fano factor is suppressedbelowthe Schottky
value (F = 1), scaling with the length L of the conductor as F = (L
c
/L)
α
.
The exponent α is significantly affected by the Coulomb interaction effects,
changing from α = 0.76 ± 0.08 when such effects are negligible to virtually
unity when they are substantial. The scaling parameter L
c
,interpreted as
the average percolation cluster length along the electric field direction, scales
as L
c
E
(0.98±0.08)
when Coulomb interaction effects are negligible and
L
c
E
(1.26±0.15)
when such effects are substantial, in good agreement with
estimates based on the theory of directed percolation.
1. Introduction
The hopping transport of quasi-localized electrons in disordered conductors and
semiconductors has been studied for many years; for comprehensive reviews, see [1–3]. The
more recent observation [4, 5]that hopping transport may implement the quasi-continuous
(‘sub-electron’)charge transfer, hence providing a possible solution to the random background
charge problem in single-electronics [6], has renewed interest in this phenomenon, with an
emphasis on the shot noise of the hopping current [7–10]. The objective of this paper is to
present the results of an extension of our previous numerical studies of 2D hopping [9, 10]
0953-8984/06/062013+15$30.00 © 2006 IOP Publishing Ltd Printed in the UK 2013

2014 YAKinkhabwala et al
to the case of substantial Coulomb interaction of hopping electrons. Just as in the case of
negligible interaction [10], the use of advanced algorithms of spectral density calculation [11]
and modern supercomputer facilities has allowed us not only to obtain more complete and exact
results for average characteristics of hopping transport (including the dependence of the DC
current on temperature and electric field), but also to calculate the spectral density of current
fluctuations at low temperatures.
In order to explain our new findings, we have to start with a brief summary of the basic
prior results.
1.1. Coulomb gap
Most theoretical discussions of the Coulomb interaction effects on hopping are based on
the notion of the Coulomb gap in the electron energy spectrum. Generally speaking,
substantial Coulomb interaction makes the single-particle energy meaningless. However, the
introduction [2]oftheeffective single-particle energy ε,whichincludes the contribution from
the Coulomb interaction with other electrons, immediately leads to a ‘soft’ gap in the single-
particle density of states ν
(
ε
)
at ε µ,whereµ is the Fermi level. In the case of 2D
conductors with the 3D Coulomb interaction law, which is the focus of our current work,
simple arguments [2, 3]yield
ν
(
ε
)
= c
κ
2
e
4
|
ε µ
|
, (1)
where e is electron charge, κ is the dielectric constant of the insulating environment and c is
adimensionless constant. Equation (1)isvalid only when the 2D density of states ν
(
ε
)
is
much smaller than the ‘seed’ density of states ν
0
;forlargerε there is a continuous crossover
to ν
0
.Theeffectivewidth of the Coulomb gap can be estimated from the natural condition
ν
(
)
= ν
0
,resulting in
=
e
4
ν
0
cκ
2
. (2)
Aself-consistent approach [3]allows a more rigorous evaluation of the Coulomb gap width,
giving c = 2.
1.2. DC transport characteristics
At low applied electric fields E,theaverage current
I
is a linear function of E,i.e.the2D
(‘sheet’) DC conductivity σ
(
T, E
)
I
/EW (where W is thewidth of the conductor)
is independent of E.Fornot very high temperatures (T T
0
,wherek
B
T
0
1
0
a
2
and a
is the localization radius), the ratio σ/σ
0
(where σ
0
is a constant characterizing the sample)
depends only on two dimensionless parameters: the ratio T/T
0
and parameter χ e
2
ν
0
a
characterizing the Coulomb interaction strength. The relation between these two parameters
determines two possible variable-range hopping transport regimes.
If the Coulomb interaction is weak (χ
3
T/T
0
), the average length r
(
T, E
)
of the
so-called ‘critical hops’, which connect percolation clusters and hence determine the current,
may be found from the Mott theory [1–3]:
r
(
T, 0, 0
)
T
0
T
1/3
a. (3)
In this case the conductivity is [1–3]
σ
σ
0
A
(
T, 0, 0
)
exp
B
(
T, 0, 0
)
T
0
T
1/3
, (4)

A numerical study of Coulomb interaction effects on 2D hopping transport 2015
where A
(
T, E
)
is a dimensionless, model-dependent, slow function of its arguments, while
B
(
T, E
)
is usually treated as a constant, but in general may be also a weakly dependent
function of its arguments.
On the other hand, if the Coulomb interaction is strong (χ
3
T/ T
0
), the critical hops
are longer [2, 3]:
r
(
T, 0
)
χ T
0
T
1/2
a, (5)
and the DC conductivity is suppressed [2, 3]:
σ
σ
0
A
(
T, 0
)
exp
B
(
T, 0
)
χ T
0
T
1/2
. (6)
In the case of relatively high electric fields (E E
T
,whereE
T
k
B
T/ea), the DC
current is a highly nonlinear (exponential) function of the applied electric field E.Iftheeld
is not extremely high (E E
0
1/eν
0
a
3
), i.e. in the variable-range hopping domain, we can
again distinguish two different transport regimes.
If the Coulomb interaction is weak (χ
3
E/E
0
), one can neglect the effects of Coulomb
interaction to evaluate the critical hop length
r
(
0, E, 0
)
E
0
E
1/3
a. (7)
In this case, the DC conductivity is [12–16]
σ
σ
0
A
(
0, E, 0
)
exp
B
(
0, E, 0
)
E
0
E
1/3
. (8)
In the opposite limit (χ
3
E/E
0
),
r
(
0, E
)
χ E
0
E
1/2
a, (9)
and the DC conductivity is lower [15]:
σ
σ
0
A
(
0, E
)
exp
B
(
0, E
)
χ E
0
E
1/2
. (10)
1.3. Current fluctuations
At low temperatures
1
,the dynamical fluctuations of the current flowing through a mesoscopic
system are more sensitive to the charge transport mechanism peculiarities than the average
transport characteristics, and therefore may provide additional information about the
conduction physics [17–19]. If we refrain from the discussion of the quantum fluctuations
at extremely high frequencies, two basic frequency ranges have to be distinguished. At very
low frequencies, one can expect the 1/f -type noise that is observed experimentally in a wide
variety of conductors; see, for example, [17]. In most cases the noise scales approximately
in accordance with the phenomenological Hooge formula [17, 20]. For a 2D conductor, this
formula can be presented as
S
I
( f )
I
2
=
a
2
LW
C
(
f
)
f
, (11)
1
In the opposite limit of thermal noise, the broadband current fluctuations are described by the fluctuation–dissipation
theorem and hence do not provide any information not already available from average transport characteristics.

2016 YAKinkhabwala et al
where S
I
( f ) is the current spectral density, L is thelength of the conductor (along the current
flow) and C
(
f
)
is either a dimensionless constant or a weak function of the observation
frequency f .Inparticular, many studies [17]havefound that C
(
f
)
/ f 1/ f
p
,wherep is
typically between 1 and 2. For the particular case of hoppingconduction, two major theories of
1/ f noise have been suggested, based, respectively, on ‘carrier number’ fluctuations [21–23]
and ‘mobility’ fluctuations [24, 25]aspossible origins of the noise. Unfortunately, both
theories have been developed for the case of substantially nonvanishing temperatures, for
which an accurate numerical study of noise is difficult even with currently available advanced
simulation algorithms and supercomputer resources.
At relatively high frequencies, the noise spectral density is a very slow (practically
constant) function of f ,andmaybeconsidered as a mixture of thermal fluctuations and shot
noise. In the most interesting case of sufficiently low temperatures, the thermal fluctuations are
negligible, and the broadband fluctuations are completely due to electric charge discreteness
(shot noise).
An emphasis of most recent studies has been on the suppression of the shot noise with
respect to its Schottky value 2e
I
.Inparticular, such suppression is a necessary condition for
quasi-continuous charge transfer at relatively high frequencies [4, 5]. If the current spectral
density S
I
( f ) is flat at f 0, it may be characterized by the Fano factor
F
S
I
(0)
2e
I
, (12)
so that the term ‘shot noise suppression’means that F < 1. Previous theoretical studies of shot
noise at hopping in artificial (space-ordered) 1D [7, 26]and (both space-ordered and random)
2D [9, 10]systems have shown that the shot noise may be, indeed, suppressed, obeying
F =
L
c
L
α
, L L
c
, (13)
where L
c
is a scaling constant interpreted as the average percolation cluster length (i.e. the
average distance separating critical hops [2, 27]) and α is a positiveexponent. In fact, at T 0
in the limit of negligible Coulomb interactions, our prior results [10]showthat L
c
obeys the
law
L
c
= J
E
0
E
µ
a, (14)
where J is adimensionless constant of the order of 1, and the value of the numerical exponent
is µ = 0.98 ± 0.08, consistent with the estimate µ 0.91 based on directed percolation
theory [10, 27–29].
Considering a very long conductor, one might suspect that the electron motion in distant
parts should not be correlated. This assumption immediately leads to α = 1[19]. However,
both analytical and numerical results [7, 26]show that at 1D hopping without Coulomb
interaction, α may be as low as 1/2. This nontrivial result may be interpreted as a consequence
of an essentially infinite correlation length in 1D conductors, due to the on-site interaction
of hopping electrons. Even more surprisingly, the exponent α may be substantially below 1
even in 2D conductors. For systems on a regular lattice, and without the Coulomb interaction,
numerical modelling [9]yields α = 0.85 ±0.02. In our recent work [10], this finding has been
confirmed for 2D hopping in conductorswith completely random distribution of localized sites
both in space and in energy. Our most accurate result was α = 0.76 ±0.08, i.e. significantly
below 1.
It has not been immediately clear how the inclusion of Coulomb interaction effects might
affect this result. For 1D hopping with increasing strength of the Coulomb interaction,

A numerical study of Coulomb interaction effects on 2D hopping transport 2017
numerical results [7]showα crossing over from nearly 1/2upto1. Thesimilar behaviour
might be expectedfor 2D hopping, because the long-rangecorrelations, apparently responsible
for the difference between α and 1, should be suppressed by Coulomb interaction effects,
provided that the conductor length L is larger than a certain crossover length determined by
the interaction constant χ .Unfortunately, recent experiments [8, 30, 31]could not help in
answering this question; while giving a reliable confirmation of the shot noise suppression
in longer conductors, their accuracy is not sufficienttoresolve a possible (relatively minor)
deviation of α from 1.
The resolution of the problem of shot noise suppression in long conductors has been the
main motivation for the numerical experiment described in this paper. However, since the
calculation of DC transport characteristics is computationally much less demanding than that
of current noise, we have used this opportunity to obtain accurate values for the slow functions
A and B for the same model of 2D hopping.
2. Model
We have studied broad 2D rectangular conductors (W L
c
)with ‘open’ boundary conditions
on the interfaces with well-conducting electrodes [9, 10]. The conductors are assumed to be
‘fully frustrated’,with a large numberoflocalizedsites randomly distributed overthe conductor
area. At the sites, the corresponding electron ‘seed’ eigenenergies ε
(
0)
are also random, being
uniformly distributed over a sufficiently broad energy band 2B,sothatthe 2D density of states
ν
0
is constant at all energies relevant for conduction.
The carriers are permitted to hop from any site j to any other site k with the rate
γ
jk
=
jk
exp
r
jk
a
, (15)
where r
jk
is the site separation distance and
jk
contains the energy dependence (see below).
Such exponential dependence on the length of a hop is standard for virtually all theoretical
studies of hopping
2
.Following our prior work [9, 10], we take equation (15) literally even at
small distances r
jk
a.Theenergy dependence of
jk
is given by the usual formula [10]
¯h
jk
U
jk
= g
U
jk
1 exp
U
jk
/k
B
T
, (16)
where g is a dimensionlessparameter whichdetermines the 2D conductivity scale σ
0
ge
2
/¯h,
3
while U
jk
is the difference of the total system energy before and after the hop from site j to
site k:
U
jk
U
j
U
k
+ eEr
jk
. (17)
Here U is the total internal energy of the system, including the effects of Coulomb interaction:
U
l
n
l
ε
(0)
l
+
e
2
2κ
n
l
1
2
l
=l
n
l
1
2
G
(
r
l
, r
l
)
, (18)
where n
l
= 0or1isthe occupation number of the lth localized site. (Similar to earlier
studies [2, 3]ofthe Coulomb effect on hopping, we keep the system electroneutral by adding
2
Notice that in contrast with some prior works, we do not include the factor 2 into the definition of the exponent.
This difference should be kept in mind at the level of result comparison.
3
Parameter g must be small (g 1) in order to keep coherent quantum effects (leading to weak localization and
metal-to-insulator transition) negligible.

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"A numerical study of Coulomb intera..." refers background or methods in this paper

  • ...[10], we may use the theory of directed percolation [ 28 , 29, 30] to predict the following scaling:...

    [...]

  • ...the average distance separating critical hops [2, 28 ]) and � is a positive exponent....

    [...]

  • ...where J is a dimensionless constant of the order of 1, and the value of the numerical exponent is µ = 0.98 ± 0.08, consistent with the estimate µ ≈ 0.91 based on directed percolati on theory [10, 28, 29, 30]....

    [...]

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Book
01 Jan 1991
TL;DR: In this article, the authors present a model for the distribution of mesoscopic fluctuations and relaxation processes in disordered conductors, which is based on random matrix theory and maximum entropy models.
Abstract: Preface. 1. Aharonov-Bohm effects in loops of gold (S. Washburn). 2. Mesoscopic fluctuations of current density in disordered conductors (B.Z. Spivak and A.Yu. Zyuzin). 3. Interference, fluctuations and correlations in the diffusive scattering of light from a disordered medium (M.J. Stephen). 4. Conductance fluctuations and 1/f noise magnitudes in small disordered structures: theory (S. Feng). 5. Conductance fluctuations and low-frequency noise in small disordered systems: experiment (N. Giordano). 6. Single electronics: A correlated transfer of single electrons and Cooper pairs in systems of small tunnel junctions (D.V. Averin and K.K. Likharev). 7. Ballistic transport in one dimension (G. Timp). 8. Transmittancy fluctuations in randomly non-uniform barriers and incoherent mesoscopics (M.E. Raikh and I.M. Ruzin). 9. Random matrix theory and maximum entropy models for disordered conductors (A.D. Stone et al). 10. Distribution of mesoscopic fluctuations and relaxation processes in disordered conductors (B.L. Altshuler, V.E. Kravtsov and I.V. Lerner). Author index. Subject index. Cumulative index.

1,339 citations


"A numerical study of Coulomb intera..." refers result in this paper

  • ...This result is similar to that for negligible Coulomb interaction [10], and may be interpreted as a result of ‘capacitive division’ of the discrete increments of externally measured charge jumps resulting from single-electron hops through the system [43]....

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Frequently Asked Questions (1)
Q1. What have the authors contributed in "A numerical study of coulomb interaction effects on 2d hopping transport" ?

The authors have extended their supercomputer-enabled Monte Carlo simulations of hopping transport in completely disordered 2D conductors to the case of substantial electron–electron Coulomb interaction. In particular, the spectral density SI ( f ) of current fluctuations exhibits, at sufficiently low frequencies, a 1/ f -like increase which approximately follows the Hooge scaling, even at vanishing temperature.