---
title: Variable Range Hopping Transport
url: https://www.emergentmind.com/topics/variable-range-hopping-mechanism
type: topic
---

# Variable Range Hopping Transport

The variable range hopping (VRH) mechanism describes charge or quasiparticle transport in strongly localized, disordered systems, where electronic states near the Fermi level are not extended but spatially localized. Instead of simple thermally activated nearest-neighbor hopping, VRH involves phonon-assisted tunneling over variable spatial ranges to energetically favorable sites, optimizing the trade-off between decay of wavefunction overlap and thermally accessible energy differences. This framework underlies non-metallic conduction in a wide array of materials including amorphous oxides, granular superconductors (via vortex hopping), disordered two-dimensional electron systems, topological insulators, and functionalized graphene. The VRH phenomenology encompasses classic Mott VRH, Coulomb-gap-modified Efros–Shklovskii VRH, extensions to AC fields, spin-selective hopping, and recent approaches that model crossover between Arrhenius and Mott regimes using integral formulations.

## 1. Theoretical Foundations of Variable Range Hopping

The foundational result of VRH theory, established by N.F. Mott, asserts that in strongly disordered insulators at sufficiently low temperatures, the electrical conductivity takes the form
\[
\sigma(T) = \sigma_0\,\exp\!\left[-(T_0/T)^{1/(d+1)}\right]
\]
where:
- \( \sigma_0 \) is a weakly \( T \)-dependent prefactor,
- \( T_0 \) is the characteristic (Mott) temperature,
- \( d \) is the effective spatial dimension,
- \( T \) is temperature.

The exponent \( 1/(d+1) \) directly encodes system dimensionality: \( 1/4 \) for 3D, \( 1/3 \) for 2D, and \( 1/2 \) for 1D. For \( d \)-dimensional systems with exponential localization length \( \xi \) and density of states at the Fermi level \( N(E_F) \), the characteristic temperature is [1303.5245, 2106.08401, 1302.1411]
\[
T_0 = \frac{\beta}{k_B N(E_F) \xi^d}
\]
with \( \beta \) a numerical constant (e.g., \( 21.2 \) in 3D, \( 13.8 \) in 2D). This result arises from optimizing between the spatial decay of wavefunction overlaps \( \sim \exp(-R/\xi) \) and the thermal activation probability for an energy difference \( \varepsilon \sim \exp(-\Delta\varepsilon/k_B T) \), under the constraint that a target state is available within a combined spatial and energy shell of volume approximately one (i.e., \( N(E_F) R^d \Delta\varepsilon \sim 1 \)) [1803.05166].

In systems where Coulomb interactions open a soft gap in the density of states near the Fermi level, the Efros–Shklovskii (ES) VRH regime replaces the Mott exponent with \( 1/2 \) in all dimensions:
\[
\sigma(T) \sim \exp\!\left[-(T_{ES}/T)^{1/2}\right]
\]
with \( T_{ES} \sim e^2/(4\pi\varepsilon\varepsilon_0 k_B \xi) \), as demonstrated in FIB-fabricated Bi\(_2\)Se\(_3\) [1708.04958] and lithiated MoS\(_2\) [1801.05157].

## 2. Microscopic Mechanism: Hop Optimization and Dimensional Dependence

VRH emerges from the competition between two exponentially suppressive factors:
- **Spatial decay:** Overlap between two localized states separated by distance \( R \sim \exp(-2R/\xi) \),
- **Energy mismatch:** Phonon absorption/emission probability for intersite energy difference \( \Delta\varepsilon \), \( \exp(-\Delta\varepsilon/k_B T) \).

The optimal hopping length \( R_{\text{opt}} \) minimizes the sum of exponents \( [2R/\xi] + [\Delta\varepsilon/(k_B T)] \) subject to the percolation constraint. The optimization yields hopping distances that grow as temperature decreases, enabling transport through rare low-resistance paths [1803.05166].

A general result is:
\[
R_{\text{opt}}(T) \sim \left(\frac{\xi}{k_B T N(E_F)}\right)^{1/(d+1)}
\]
and the associated optimal energy
\[
\Delta\varepsilon_{\text{opt}}(T) \sim \frac{1}{N(E_F) R_{\text{opt}}^d}
\]
These relationships have been experimentally extracted by direct fits of resistivity or conductivity versus temperature and have been validated in granular superconducting films (vortex VRH)[1709.04128], magnetic Zintl semiconductors EuIn\(_2\)P\(_2\) [2410.05825], and oxide interfaces [1302.1411].

## 3. Extensions, Crossover Regimes, and Integral VRH Models

Standard VRH with discrete Arrhenius and Mott regimes neglects the smooth crossover observed in real materials. The Integral Variable Range Hopping (IVRH) model [2601.10226] replaces ad hoc regime boundaries by a single physically motivated integral over all possible hopping distances:
\[
\sigma(T) = A \int_{R_0}^\infty \exp\!\left[ -2\alpha R - \frac{\beta}{D_0 V(R) k_B T} \right] dR
\]
where \( V(R) \) encodes the system geometry (e.g., \( \pi R^2 \) for 2D, \( \frac{4\pi}{3} R^3 \) for 3D) and the energy penalty is dynamically determined by the effective density of accessible states in each volume. The IVRH integral reduces to Arrhenius or Mott limits at high or low temperatures, respectively, with smooth interpolation in the crossover regime. Monte Carlo simulation validates this form and reveals reduced parameter variance and robust extraction of physically meaningful quantities such as the localization length [2601.10226].

## 4. Experimental Signatures and Parameter Extraction

The operational test of VRH is the observation of linear dependence in plots of \( \ln \rho \) (or \( \ln \sigma \)) versus \( T^{-1/(d+1)} \) over extended temperature ranges. Extracted slopes yield \( T_0 \), which, with an independent estimate of \( N(E_F) \), provides the localization length \( \xi \) [1303.5245, 2106.08401, 1904.11962]. Representative values are:
- \( T_0 \sim 10^5-10^8 \) K in transition-metal oxides and disordered thin films,
- \( \xi \sim 0.1-3 \) nm, in agreement with interatomic or nanoscale distances.

The dimensionality is generally established by the value of the characteristic exponent, with 3D (\( 1/4 \)) found in thick granular films and bulk nanocrystalline samples [1303.5245, 2106.08401, 1904.11962], 2D (\( 1/3 \)) at oxide interfaces and nanowires [1302.1411, 1708.04958], and 1D (\( 1/2 \)) in EFros–Shklovskii and strictly one-dimensional systems [1803.05166, 1605.03445, 2404.19000]. Deviations from the Mott exponent signal, for example, power-law localized impurity states rather than exponential (as established for graphene with resonance-state impurities) [1208.5026].

Other transport measurements provide corroborating evidence:
- **Magnetotransport:** Negative magnetoresistance scaling as \( T^{-1} \) or quadratic in field, characteristic of interference effects in VRH [1302.1411, 2410.05825].
- **Impedance spectroscopy:** Crossover between Arrhenius and VRH in Nyquist plots, dielectric relaxation, and modulus formalisms [1904.11962].
- **Spectroscopic signatures:** Raman phonon lifetimes and FWHM anomalies correlating with crossover into the VRH regime [2410.05825].
- **AC response and scaling:** Universal scaling of \( \sigma(\omega) / \sigma(0) = F(\omega / [T \sigma(0)]) \) as illustrated by dynamical Monte Carlo [1409.0671].

## 5. Modifications: Coulomb Gap, Power-law Localization, and Memory Effects

Efros–Shklovskii VRH modifies the Mott scenario by considering the suppression of the density of states near the Fermi level, leading to the characteristic exponent \( 1/2 \) and the appearance of the interaction energy scale in \( T_{ES} \). This has been experimentally confirmed in topological insulator nanowires [1708.04958] and strongly disordered MoS\(_2\) [1801.05157].

In systems where localized states decay as a power law rather than exponentially—such as adatom-doped graphene—the temperature dependence of conductivity becomes a pure power law, \( \sigma(T) \sim T^{\eta} \), where \( \eta \) is directly related to the decay exponent [1208.5026].

Memory effects, specifically dynamical correlations in occupation numbers of localized sites, introduce subleading exponential corrections to the Mott law through a two-color percolation problem; this effect subtly alters the prefactor and can manifest in experimental deviations from VRH fits [1404.3981].

## 6. Generalizations: Vortex Hopping, Spin Selectivity, and Field-Driven Transport

The VRH paradigm extends to non-electronic quasiparticle transport. In disordered superconducting films, vortex dynamics analogously undergo 2D Mott VRH among random pinning sites, leading to a resistance minimum below which hyperbolic cooling reveals VRH-type exponential temperature dependence [1709.04128].

Spin-dependent VRH arises in chiral systems such as DNA, where spin–phonon coupling mediated by chiral phonon vorticity yields universal \( T^{-3/2} \) laws for spin polarization in 1D VRH chains [2404.19000].

Under strong external fields, 1D Mott VRH becomes a mathematically tractable nonequilibrium random walk with rigorously established criteria for ballisticity and sub-ballisticity; sharp transitions in the drift velocity as a function of field bias and environmental disorder are demonstrated [1605.03445, 1803.05166].

## 7. Thermoelectric, Nonlinear, and Multi-parameter Extensions

Thermoelectric response in VRH systems is determined by the interplay of the spectral conductivity and the statistical distribution of hopping parameters. Recent works show that allowing for an energy-dependent localization length \( \xi(\varepsilon) \) consistent with Anderson localization scaling theory modifies the Seebeck coefficient's low-temperature scaling to \( S(T) \sim T^{d/(d+1)} \), contrasting with the noninteracting prediction \( S(T) \sim T^{(d-1)/(d+1)} \) [2204.02764].

Nonlinear transport in VRH, arising from high applied fields or current densities, modifies the hopping statistics and requires integral or advanced percolation approaches to capture transitions from regime-to-regime and parameter extraction [2601.10226].

## Table: VRH Parameters in Model Systems

| System/Material                   | Dim. | \( T_0 \) (K)     | \( \xi \) (nm) | Characteristic Regime                 | Reference         |
|-----------------------------------|------|-------------------|---------------|---------------------------------------|------------------|
| Na\(_2\)IrO\(_3\) thin films      | 3D   | \(10^5\)-\(10^8\) | 0.74–4.7      | Mott VRH (exp 1/4)                    | [1303.5245]      |
| Iron pyrite (FeS\(_2\)) films     | 3D   | \(10^5\)-\(10^6\) | 0.9–2.7       | Mott VRH, NNH crossover               | [2106.08401]     |
| Bi\(_2\)Se\(_3\) nanowires        | 2D   | \(10^1\)-\(10^3\) | 0.5–20        | Efros–Shklovskii (ES) VRH (exp 1/2)   | [1708.04958]     |
| Pr\(_2\)ZnMnO\(_6\) electrodes    | 3D   | \(5.4\times10^7\) | 0.1           | NNH (high T)/Mott VRH (low T)         | [1904.11962]     |
| NdAlO\(_3\)/SrTiO\(_3\) interface | 2D   | not stated        | not stated    | Mott VRH (exp 1/3), negative MR       | [1302.1411]      |
| DNA (spin VRH)                    | 1D   | not stated        | 1–5           | Chiral phonon–induced, exp 1/2        | [2404.19000]     |

## References

- "Mott Variable Range Hopping and Weak Antilocalization Effect in Heteroepitaxial Na₂IrO₃ Thin Films" [1303.5245]
- "Universal scaling form of AC response in variable range hopping" [1409.0671]
- "Evidence of robust 2D transport and Efros-Shklovskii variable range hopping in disordered topological insulator (Bi2Se3) nanowires" [1708.04958]
- "Charge Carrier Transport in Iron Pyrite Thin Films: Disorder Induced Variable Range Hopping" [2106.08401]
- "Vortex Variable Range Hopping in a Conventional Superconducting Film" [1709.04128]
- "Evidence of variable range hopping in the Zintl phase EuIn2P2" [2410.05825]
- "Existence of nearest-neighbor and variable range hopping in Pr₂ZnMnO₆ oxygen-intercalated pseudocapacitor electrode" [1904.11962]
- "Efros-Shklovskii variable range hopping and nonlinear transport in 1T/1T$^{\prime}$-MoS$_{2}$" [1801.05157]
- "Memory effects, two color percolation, and the temperature dependence of Mott's variable range hopping" [1404.3981]
- "Evolution of variable range hopping in strongly localized two dimensional electron gas at NdAlO3/SrTiO3 (100) heterointerfaces" [1302.1411]
- "Thermoelectric Effect in Mott Variable-Range Hopping" [2204.02764]
- "Integral Variable Range Hopping for Modeling Electrical Transport in Disordered Systems" [2601.10226]
- "The velocity of 1D Mott variable range hopping with external field" [1605.03445]
- "Chirality-induced spin selectivity by variable-range hopping along DNA double helix" [2404.19000]
- "1D Mott variable-range hopping with external field" [1803.05166]
- "Impurity State and Variable Range Hopping Conduction in Graphene" [1208.5026]
- "Poole-Frenkel effect and Variable-Range Hopping conduction in metal / YBCO resistive switching devices" [1505.05813]

Source: https://www.emergentmind.com/topics/variable-range-hopping-mechanism