Integral Variable Range Hopping for Modeling Electrical Transport in Disordered Systems

TL;DR

This work addresses the limitations of traditional VRH by introducing integral variable range hopping (IVRH), which replaces the empirical temperature exponent with a physics-based integral over hopping distances controlled by an effective hopping volume $V(R)$. The method yields a conductivity $\sigma(T)$ that naturally transitions between Arrhenius and Mott behaviors and extends to multi-layer and confined geometries via $V(R)$, with a consistent dimensional crossover. Monte Carlo simulations validate the model and reveal dimension-specific $\beta$ values, while IVRH demonstrates improved fitting robustness over Mott fits. Application to experimental data on MoS$_2$ and WS$_2$ shows a unified description across regimes and gate-tunable delocalization, highlighting the method's physical interpretability and practical relevance for disordered thin films and layered materials.

Abstract

The variable range hopping (VRH) model has been widely applied to describe electrical transport in disordered systems, providing theoretical formulas to fit temperature-dependent electric conductivity. These models rely on oversimplified assumptions that restrict their applicability and result in problematic fitting behaviors, yet their overusing situation is becoming increasingly serious. In this work we formulate an integral variable range hopping (IVRH) model, which replaces the empirical temperature power-law dependence in standard VRH theories with a physics-inspired integral formulation. The model builds upon the standard hopping probability $ω(R)$ w.r.t. hopping distance $R$ and incorporates the density of accessible electronic states through an effective volume function $V(R)$, which reflects the influence of system geometry. The IVRH formulation inherently reproduces both the Mott behavior at low temperatures and the Arrhenius behavior at high temperatures, respectively, and enables a smooth transition between the two regimes. We apply the IVRH model to two-dimensional, three-dimensional, and multi-layered systems. Monte Carlo simulations validate the model's predictions and yield consistent values for the fitting parameters, with substantially reduced variances compared to fitting using the standard VRH model. Furthermore, the improved robustness of IVRH also extends to the transport measurements in monolayer MoS$_2$ system and monolayer WS$_2$ system, enabling more physically meaningful interpretation.IVRH model offers a more stable and physically sound framework for interpreting hopping transport in low-dimensional amorphous materials, providing deeper insights into the universal geometric scaling factors that govern charge transport in disordered systems.

Figures (4)

• Figure 1: IVRH in 2D and 3D systems and comparison with Mott's VRH model. (a) Dependence of the hopping probability $\omega$ on the hopping radius $R$ at different temperatures in 2D systems. From top to bottom, the temperatures of different curves decrease. (b) The temperature dependence of the IVRH conductivity in 2D and 3D systems. (c,d) First derivatives of the $\sigma(T)$ curves in 2D systems (c) and 3D systems (d) w.r.t. temperature $T$, respectively. The solid line corresponds to the IVRH model, and the red dashed line corresponds to Mott’s law, which is independent of $T$. Parameters used: $\alpha=\beta=D_0=k_B=1$ as defined in Eq. \ref{['e2']}.
• Figure 2: Monte Carlo simulation and conductivity extraction on 2D and 3D systems. (a) The relationship between total displacement and field intensity. The slope is proportional to conductivity. From bottom to top, the temperature $T$= 0.01, 0.0145, 0.019, 0.0235, 0.028, 0.0325, 0.037, 0.0415, 0.046, 0.0505, respectively. (b) Monte Carlo results (dots) and corresponding IVRH model fits (curves) for 2D (blue) and 3D (green) systems. Parameters used in the simulation: $\alpha = 0.5$, $D_0 = 1$ as defined in Eq. \ref{['e2']}.
• Figure 3: The application of IVRH in multi-layer systems and test with Monte Carlo results (a) Schematic of hopping geometry in multilayer systems. Here, the center of the sphere is located at the $h_i$th layer. The lower part of the sphere traverses the system boundary, thus $a_i=\min(h_i, R)=h_i$. The upper part is inside the system, so $b_i=\min(H-h_i, R)=R$. The shaded part corresponds to the effective hopping volume. (b) Temperature dependence of conductivity for various numbers of layers given by IVRH model. Parameters: $\alpha = \beta = D_0 = 1$. (c)(d)Based on Monte Carlo simulation, the effective dimension of multi-layer systems predicted by the (c) IVRH model and (d) Mott's model, respectively.
• Figure 4: Application of IVRH to experimental measurements. (a) The temperature dependence of conductivity in a single-layer MoS$_2$ device. The data (dots) are fitted by the IVRH model (lines). In the high-temperature (red) regime, the conductivity exhibits Arrhenius behavior. As the temperature is reduced, a crossover occurs into a low-temperature (blue) regime, where transport is governed by VRH and the conductivity follows Mott’s law. From top to bottom, the carrier density equals to 25, 20, 15, 10, 1$\times10^{11}$cm$^{-2}$, respectively. Experimental data are extracted from ref.qiu2013hopping. (b) The value and fitted error bar of localization length under different gate voltages in a monolayer WS$_2$ device. Experimental data are extracted from ref.ovchinnikov2014electrical.