Variable Range Hopping Mechanism

• Variable range hopping is a transport mechanism where carriers move by phonon-assisted tunneling between spatially separated localized states.
• The mechanism features a stretched exponential resistivity law, with distinct regimes: Mott VRH for non-interacting electrons and Efros–Shklovskii VRH in the presence of Coulomb gaps.
• Experimental observations in diverse systems like graphene, amorphous semiconductors, and granular metals validate VRH predictions and highlight crossover behaviors.

Variable range hopping (VRH) is a universal transport mechanism for localized charge carriers in disordered systems, in which electrons (or other excitations) move via phonon-assisted tunneling between spatially and energetically distant localized states. Unlike nearest-neighbor hopping, VRH reflects an optimal trade-off between tunneling probability (decaying exponentially with distance) and thermal activation over energy mismatches, resulting in distinctive “stretched exponential” resistivity (or conductivity) laws with dimensionality- and interaction-dependent exponents. VRH governs low-temperature transport in amorphous semiconductors, conventional and topological insulators, disordered superconductors, granular metals, and two-dimensional electron systems in high magnetic fields, among other settings. The VRH framework bifurcates into the non-interacting Mott regime and the strongly interacting Efros–Shklovskii regime, with rich crossover physics and a range of experimentally testable predictions across diverse materials systems.

1. Theoretical Framework: Mott and Efros–Shklovskii VRH

Canonical variable range hopping is described by the resistivity law

$$\rho(T) = \rho_0 \exp{\left[\left(\frac{T_0}{T}\right)^p\right]}$$

where the exponent $p=1/(d+1)$ for non-interacting electrons in $d$ spatial dimensions (“Mott VRH”), and $p=1/2$ when long-range Coulomb interactions open a “Coulomb gap” at the Fermi level (“Efros–Shklovskii (ES) VRH”). The characteristic temperature $T_0$ encodes the density of states $g(E_F)$, localization length $\xi$, dielectric constant $\epsilon$, and geometry-dependent numerical factors:

• Mott VRH (constant DOS): $T_M = \beta_M / (g(E_F)\xi^d k_B)$, exponent $p = 1/(d+1)$ (Bennaceur et al., 2010).
  • 2D: $\rho(T) \propto \exp{[(T_M/T)^{1/3}]}$.
  • 3D: $\rho(T) \propto \exp{[(T_M/T)^{1/4}]}$.
• Efros–Shklovskii VRH (Coulomb gap): $T_{ES} = C e^2/[4\pi\epsilon\epsilon_0\xi k_B]$, exponent $p=1/2$ for all $d$. The Coulomb gap manifests as $g(E) \propto |E-E_F|$ (Bennaceur et al., 2010, Joung et al., 2012).

The optimal hopping length and energy are determined by minimizing the combination of exponential decay in spatial overlap and thermal activation probability, under the constraint that an available final site exists within a certain distance and energy window. In the ES regime, the Coulomb gap suppresses low-energy density of states, favoring longer hops in real space but over smaller energy gaps (Joung et al., 2012, Bennaceur et al., 2010).

2. Experimental Realizations and Regime Crossovers

VRH conduction is widely observed in a range of disordered and correlated systems. In monolayer graphene in the quantum Hall (QH) regime, the temperature dependence of the longitudinal conductivity $\rho_{xx}(T)$ exhibits a clear crossover: at fixed filling on QH plateaus, ES–VRH (exponent $1/2$) describes conduction up to 80 K; upon tuning the localization length to exceed twice the gate–sample distance (screening length), the system crosses to Mott VRH with the exponent $1/3$ (Bennaceur et al., 2010). Similar crossovers are observed in oxygen-deficient polycrystalline ZnO films, FeS$_2$ thin films, and reduced graphene oxide, with transitions from Mott to ES hopping at low temperatures, frequently captured by universal scaling laws (Huang et al., 2017, Joung et al., 2012, Shukla et al., 2021).

Vortex VRH, a conceptual mapping of vortex tunneling in granular or amorphous superconducting films onto electronic VRH, is directly evidenced in InO$_x$ films, with resistance traces perfectly described by 2D Mott VRH (exponent $1/3$) for the vortex channel (Percher et al., 2017).

Table 1: Key VRH exponents and crossover physics

Regime	Exponent $p$	DOS condition	Notable example systems
Mott VRH	$1/(d+1)$	Constant $g(E_F)$	Bulk Si, MoSe$_2$, Na$_2$IrO$_3$ (Suri et al., 2017, Jenderka et al., 2013)
ES VRH	$1/2$	$g(E)\sim|E-E_F|$	Graphene QH, RGO, topological insulators (Bennaceur et al., 2010, Bhattacharyya et al., 2017)

The criterion for the ES–Mott crossover is that the localization length $\xi$ exceeds the screening length (typically set by sample–gate separation): for $\xi>2d$, the system transitions from Coulomb-interaction controlled (ES) to constant-DOS (Mott) hopping (Bennaceur et al., 2010).

3. Transport Signatures and Scaling Laws

VRH resistivity follows stretched exponential or, under particular conditions, power-law temperature dependences. In two-dimensional systems with power-law localized impurity states, as in defect-functionalized graphene, the VRH conductivity $\sigma(T)\propto T^\eta$ with a power related to the spatial decay exponent of the wavefunction—a fundamental distinction from standard exponential VRH (Liang et al., 2012).

Peak widths of the QH conductance traces scale as

$\Delta\nu\propto T^\kappa,\quad \kappa=1/(2\gamma)$

with $\gamma$ the localization-length critical exponent ($\gamma=2.3$ in high-field graphene QH experiments), and under current bias $\Delta\nu\propto I^\mu$ with $\mu=1/(3\gamma)$ (Bennaceur et al., 2010). The experimentally measured exponents agree precisely with the Polyakov–Shklovskii “VRH-only” scenario, ruling out the need for a separate metallic phase at QH plateau transitions.

AC transport in VRH regimes for interacting electrons (e.g., 2D Coulomb gap) shows universal scaling collapse: the complex conductivity at frequency $\omega$ obeys

$$\sigma(\omega)/\sigma(0) = F(\omega/\omega_c),\quad \omega_c = T\sigma(0)$$

where $F$ is a universal scaling function independent of material details until the microscopic hopping rate cutoff (Bergli et al., 2014).

4. Microscopic Mechanisms and Role of Disorder

VRH requires both strong localization and disorder, which can arise from structural randomness, compositional inhomogeneity, or magnetic fluctuations. In FeS$_2$ thin films, DFT calculations correlate Mott VRH to sulfur-vacancy–induced midgap states whose spatial extent and density set the effective localization length and density of states, governing $T_M$ (Shukla et al., 2021). In layered or anisotropic materials like MoSe$_2$, VRH dimensionality can switch from 2D (intralayer) to 3D (interlayer) with temperature, and the Mott $T_0$ encodes the strong anisotropy in hopping probability (Suri et al., 2017).

In magnetic materials (EuIn$_2$P$_2$), short-range magnetic order provides a dynamically fluctuating random potential, localizing carriers and producing 3D Mott VRH. Negative magnetoresistance scales with field squared and normalized magnetization, as expected from polaronic models within VRH (Tolinski et al., 2024).

5. Advanced Developments: Memory Effects, Interactions, and Nonlinearities

Memory effects in occupation of localized states introduce correlations beyond the Markovian approximation, yielding a subleading exponential correction to the Mott VRH law, analytically described via a two-color percolation mapping. The modified conductivity has the form

$$\sigma(T) \sim \exp\left[-(T_0/T)^{1/(d+1)} + \alpha_d (T_0/T)^{\mu_d}\right]$$

where the exponent $\mu_d$ is determined by percolation cluster exponents and $\alpha_d$ is $O(1)$, providing a quantitative description of the subleading temperature dependence (Agam et al., 2014).

In electron glasses near metallic gates, dynamical polaronic effects due to coupling to the gate electrons can suppress phonon-assisted hopping, leading to large reductions in $\sigma(T)$ without changing the functional form of the VRH exponent (Asban et al., 2020). In ultranarrow geometries or marginally localized phonon baths, as in 1D wires, VRH rates are further suppressed and require many-phonon processes, resulting in highly singular prefactors multiplying the conventional Mott law (Banerjee et al., 2015).

6. Thermoelectric and Nonlinear Response in VRH Regimes

In the VRH regime, thermoelectric coefficients exhibit universal scaling. For Mott VRH with energy-independent localization lengths, the Seebeck coefficient follows $S(T)\propto T^{(d-1)/(d+1)}$; but when the Anderson-localization length diverges near the mobility edge, critical scaling modifies this to $S(T)\propto T^{d/(d+1)}$, as demonstrated in thiospinel CuCrTiS$_4$ and MoSe$_2$ (Yamamoto et al., 2022, Suri et al., 2017). Nonlinear transport (current- or field-driven) in ES VRH is characterized by an effective temperature or field scale, leading to $\sigma \propto \exp{[-(E_0/E)^{1/2}]}$ (Bennaceur et al., 2010, Papadopoulos et al., 2018).

7. Universality and Open Problems

VRH is a unifying paradigm for electronic conduction in disordered and correlated electronic materials, bridging classical and quantum regimes, and integrating the effects of interactions, dimensionality, and screening. The universal character of VRH exponents across geometries and materials is robustly confirmed in experiment and simulation, but notable open problems include the role of collective interactions in vortex VRH, breakdown of standard VRH in non-exponential (power-law) localized scenarios, anomalous prefactor effects due to occupation memory, and the accurate microscopic description of VRH–metal crossovers (Percher et al., 2017, Liang et al., 2012, Agam et al., 2014).

These questions point to continued efforts to refine the theoretical foundation of VRH and to probe regimes where VRH phenomenology may fail or require new theoretical approaches in the presence of strong correlations, topological order, or unconventional disorder distributions.