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Generalized Hopping Mechanism

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The generalized hopping mechanism refers to an expanded phenomenological model in solid-state physics and physical chemistry, developed to describe temperature-dependent transport phenomena — such as electrical conductivity and ionic diffusion — in materials where the classical Arrhenius and conventional hopping models prove insufficient.

While traditional descriptions predict that conductivity exhibits a strict exponential relationship with the inverse of temperature, modern experimental results in diverse materials (non-linear conductors, semiconductors, disordered systems) reveal decay rates frequently governed by power laws. To reconcile these experimental observations, researchers implemented d-exponential functions as a statistical tool to generalize thermal charge transport mechanisms.

Introduction and Historical Context

Historically, the conduction mechanism via hopping (quantum or thermal jumps of charge carriers between localized states) introduced by Nevill Francis Mott — famous for the Variable Range Hopping (VRH) model — assumes that the jump probability of an electron (and the subsequent electrical conductivity) decays exponentially as the inverse of temperature.

In the conventional form (Arrhenius equation and Mott's laws), the activation energy or diffusion barrier is considered constant with respect to temperature. However, robust experiments covering wide temperature ranges and new materials such as carbon nanotube networks, quasicrystals, and doped polymers, demonstrate that the actual dependence between ${\textstyle log(\sigma )}$ and the inverse of temperature (${\textstyle 1/T}$) presents curvatures (concavities and convexities). This indicates that the activation energy is not strictly constant; instead, the system's behavior is better described by a decay in the form of a power law.

The d-Hopping Model

To mathematically accommodate deviations from linearity in Arrhenius and traditional Hopping plots, physicists use a deformation of the Euler exponential function, often linked to non-extensive thermodynamics (such as the Tsallis generalization (1988)), and for chemical kinetics (as Aquilanti-Mundim (2010)). The model formulated this way is known as d-hopping (deformed hopping) he generalized electrical conductivity takes the following analytical form:

           ${\textstyle \sigma _{d}(T)=\sigma _{0}\left[1-d\left({\frac {\tau _{0}}{T}}\right)^{\gamma }\right]^{\frac {1}{d}}}$

Where:

• ${\textstyle \sigma _{0}}$ is the pre-exponential factor or the conductivity limit at high temperatures.
• ${\textstyle \tau _{0}=E_{0}/k}$ is a characteristic temperature associated with the basic thermal activation energy of the system (${\textstyle k}$ is the Boltzmann constant).
• ${\textstyle \gamma }$ is the fractional exponent of the model (for example, ${\textstyle \gamma =1}$ for Arrhenius, ${\textstyle \gamma =1/4}$ for 3D Mott-VRH).
• ${\textstyle d}$ is a dimensionless real number (deformation parameter) that directly affects the thermodynamic curvature of the distribution.

The d-hopping model flowchart

The crucial aspect of this formalism is that the conventional Euler exponential is recovered when taking the limit $d\to 0$. Under these ideal conditions:

$\lim _{d\to 0}\sigma _{d}(T)=\sigma _{0}\exp \left[-\left({\frac {\tau _{0}}{T}}\right)^{\gamma }\right]$

The thermodynamic activation energy in this generalized context ${\textstyle (E_{a}(T))}$ is not constant, but inherently depends on the temperature itself and on the parameter ${\textstyle d}$.

Normalized d-hopping conductivity (σ₀ = 1), log σ vs. 1000/T, for −0.03 ≤ d ≤ 0.03. Dashed: usual hopping (d → 0). Blue curves (d < 0, sub-hopping) lie above it; red curves (d > 0, super-hopping) lie below.

Conduction Regimes The algebraic sign of the parameter dictates the functional behavior and the regime of the transport process in the material, indicating how structural disorder affects charge migration.

Sub-hopping (${\textstyle d<0}$)

Occurs in systems that present an exceptionally high level of structural disorder. Instead of the traditional freezing of conductivity at low temperatures, materials exhibit conductivity rates higher than exponentially expected. The Arrhenius plot exhibits a characteristic concavity (concave down).

Physically, high disorder provides multiple low-energy paths in the energy landscape, allowing charge carriers to migrate predominantly "downhill" (downhill hopping). Experimental examples: Thin semiconductor films of Titanium Dioxide (${\textstyle TiO_{2}}$), PAni/PSS polymer blends, and amorphous carbon nanoparticle films.

Conventional (${\textstyle d=0}$) The conventional regime corresponds to systems with moderate disorder. This is the particular limit in which d-hopping falls back into the standard models of Mott Hopping, VRH, Efros-Shklovskii or thermally activated (linear Arrhenius). The carrier executes standard stochastic random walks. Traditional models perfectly serve this regime in certain delimited temperature ranges.

Super-hopping (${\textstyle d>0}$) Associated with low disorder systems, approaching quasi-crystalline lattices. Here, carriers encounter sparse states separated by well-defined energy barriers, relying heavily on thermal excitation to make transitions ("jumps" upward in energy space or uphill hopping).

The resulting curve is convex and the conductivity is markedly lower at low temperatures than predicted by classical VRH. Experimental examples: Granular superconductors, complex single crystals, and compounds like ${\textstyle Yb_{1-x}Gd_{x}Ba_{2}Cu_{3}O_{7-y}}$

Density of States (DOS) and Disorder

To provide microscopic and physical foundation to the phenomenology of the d-Hopping model, the Density of States (DOS) is frequently modeled as a Gaussian distribution. This approximation elegantly captures the variation and variability of local (or site) energies caused by molecular and structural imperfections in disordered networks.

The state distribution function can be described as: ${\textstyle g(E)={\frac {1}{\sigma _{D}{\sqrt {2\pi }}}}\exp \left[-{\frac {(E-E_{0})^{2}}{2\sigma _{D}^{2}}}\right]}$

In this case, ${\textstyle E_{0}}$ represents the center of the distribution (usually coinciding with the Fermi Level) and ${\textstyle \sigma _{D}}$ describes the structural disorder parameter (in electron-volts, eV), representing the width (or spread) of the available energy bands caused by amorphous local fluctuations.

The generalized theory demonstrates that the phenomenological parameter ${\textstyle d}$ is closely and inversely linked to the statistical level of disorder ${\textstyle (\sigma _{D})}$, where roughly ${\textstyle d\propto -\sigma _{D}}$.

Highly disordered materials (wide ${\textstyle \sigma _{D}}$) induce pronounced Sub-hopping (${\textstyle d<0}$) behaviors, favoring the optimization of cryogenic or low-temperature electronics. Ordered materials (narrow ${\textstyle \sigma _{D}}$ ) boost Super-hopping (${\textstyle d>0}$), being desired in thermoelectric applications and electronic components operating under high-temperature stress.

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