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

## 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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