Field-Effect-Aware Fluctuation Tunnelling (FEAFIT)

• Field-Effect-Aware Fluctuation-Induced Tunnelling is a framework describing electron transport in disordered semiconductors and granular materials through quantum tunnelling across insulating gaps.
• It integrates field-effect modulation with stochastic thermal and electromagnetic fluctuations to extract material-specific transport properties directly from device measurements.
• The model explains the transition from low-temperature tunnelling plateaus to high-temperature Arrhenius conduction, aiding optimization in systems like IGZO and granular metals.

Field-Effect-Aware Fluctuation-Induced Tunnelling (FEAFIT) is a unified theoretical framework for understanding electronic transport in highly disordered semiconductors and granular materials. It rigorously accounts for tunnelling across nanoscale gaps, incorporating both stochastic thermal and electromagnetic fluctuations and field-induced modulation of the tunnelling barriers. FEAFIT enables extraction of material-specific transport properties directly from device measurements, providing quantitative links between experimental observations, first-principles calculations, and microscopic disorder. Its predictive capacity has been validated in amorphous oxide semiconductors such as IGZO and in granular metal systems subjected to high-frequency fields (Zhao et al., 26 Dec 2025, Hirlimann, 2013).

1. Physical Basis: Partially Coherent Domains and Tunnelling Junctions

FEAFIT is fundamentally built on three physical assumptions, with their realization most clearly elucidated in amorphous IGZO thin films (Zhao et al., 26 Dec 2025):

1. Partially coherent electronic states: Electrons retain spatial coherence only within discrete nanometer-scale domains or "puddles." Within a domain, electronic states are well described by plane waves; coherence is lost at domain boundaries, precluding globally delocalized transport.
2. Non-degenerate conduction: The Fermi level $E_F$ is positioned within a dense manifold of localized tail states below the threshold for fully delocalized states ($E_{dloc}$). Global free-electron conduction does not occur; instead, transport is dominated by transfer across energetic barriers.
3. Insulating gaps as tunnelling junctions: Electrical conduction proceeds by quantum tunnelling across narrow, energetically defined insulating gaps that separate coherent domains. These junctions have a width $d$ and barrier height $\Phi$, both modulated by external electric fields such as gate voltages in transistors.

This paradigm is supported by both Hall effect measurements, which reveal the coexistence of coherent and incoherent carriers, and first-principles simulations showing separated spatial pockets of conduction-band states whose separation matches the extracted tunnelling gap width (Zhao et al., 26 Dec 2025).

2. Tunnelling Action, Fluctuations, and Field Modulation

WKB Transmission and Fluctuation-Induced Enhancement

The tunnelling probability for an individual junction is governed by the WKB transmission action. At zero temperature, the action is given by

$$S_0 = 2\int_0^d \sqrt{2m^*[\,\Phi - E\,]} \, dx \approx 2d \sqrt{2m^*\Phi}/\hbar$$

where $m^*$ is the effective mass and $E$ the incident electron energy.

Thermal and electromagnetic field fluctuations transiently reduce the barrier height, thus boosting tunnelling rates. The junction can be mapped to a nanoscale parallel-plate capacitor (capacitance $C = \epsilon A/d$), so the variance of the voltage drop across the barrier, $\langle(\delta V)^2\rangle = k_B T / C$, characterizes the amplitude of thermal fluctuations. Applying Sheng’s theory, averaging over these fluctuations gives a smooth interpolation between elastic (low-$T$) tunnelling and high-$T$ Arrhenius activation (Zhao et al., 26 Dec 2025).

Field-Effect Modulation

For field-effect devices, barrier parameters are modulated by the external field (e.g., gate voltage $V_G$):

• Effective barrier height: $\Phi_{eff}(V_G) = \Phi_0 - \alpha (V_G - V_{FB})$
• Barrier width: $d(V_G) = d_0 - \beta (V_G - V_{FB})$

Here, $\alpha$ and $\beta$ parameterize the sensitivity of the respective quantities to field, and $V_{FB}$ is the flat-band voltage.

3. Analytical Model: FEAFIT Conductivity and Resistance Expressions

The FEAFIT formalism provides closed-form analytical expressions for observables. For channel conductivity in a thin-film transistor:

$$\sigma(V_G,T) = \sigma_0 \exp\left[ -\frac{T_1(V_G)}{T} \, S_1(T) - \frac{T_1(V_G)}{T_0(V_G)} \, S_0(T) \right]$$

where:

• $\sigma_0$ is the domain-scale conductance prefactor,
• $T_1(V_G) = \Phi_{eff}(V_G) / k_B$ is the gate-dependent effective activation energy,
• $T_0(V_G)$ encodes the characteristic energy scale set by barrier width,
• $S_1(T)$, $S_0(T)$ are crossover functions defined via a reduced fluctuation variable $n^*(T) = T / [\,T + T_0(V_G)\,]$, with $S_1(T) = [n^*(T)]^2$ and $S_0(T) = n^*(T)\,[1 - n^*(T)]$.

This form naturally recovers both the high-$T$ Arrhenius-like behavior ($\sigma\sim \exp[-T_1/T]$ for $T\gg T_0$) and the low-$T$ tunnelling plateau ($\sigma \sim \exp[-T_1/T_0]$ for $T\ll T_0$) without artificial regime splicing (Zhao et al., 26 Dec 2025).

For granular metallic contacts under electromagnetic stimulation, the resistance is modeled as

$$R(E_0, \omega, T) = R_{00} \exp\left[\,\frac{2d}{\hbar}\sqrt{2mU_0} \Bigl(1 - \gamma E_0^2 f(\omega)\Bigr) + \frac{U_0}{k_B T}\,\right]$$

with $E_0$ the field amplitude, $\omega$ the frequency, $U_0$ the nominal barrier, and $f(\omega)=1/(1+\omega^2\tau_{ref}^2)$ a frequency-domain cutoff controlled by the electron reflection time $\tau_{ref}$ (Hirlimann, 2013).

4. Extraction and Correlation of FEAFIT Parameters

Comprehensive device data enable direct extraction of the key FEAFIT parameters:

• Barrier height and width ($\Phi_0$, $d_0$): Obtained by global nonlinear least-squares fitting of conductivity data as a function of gate voltage and temperature. The dependence of $T_1(V_G)$ and $T_1(V_G)/T_0(V_G)$ on $V_G$ quantifies $\Phi_0$, $d_0$, and their field couplings $\alpha$, $\beta$.
• Coherence length $L_\varphi$: Inferred from the extracted effective area $A_{eff} = \epsilon d_0/T_0(0)$, using $A_{eff} \sim L_\varphi^2$.
• Energetic disorder: Tail-state density-of-states analyses yield disorder parameters (widths $E_{0,1}$, $E_{0,2}$), which are found to tightly correlate with the empirical barrier energies ($\Phi_0 \approx k_B E_{0,1}$).

These parameters track the material’s composition. For example, first-principles supercell calculations for IGZO with varying Ga content demonstrate that as Ga fraction increases, the mean separation between coherent domains ($\sim$4 to 8 nm) grows in parallel with fitted $d_0$, and the energetic disorder extracted from density of states matches the FEAFIT-determined barriers (Zhao et al., 26 Dec 2025).

5. Universality and Applicability Across Materials Systems

The FEAFIT model generalizes beyond amorphous oxide semiconductors to describe granular metals and classical Branly-effect systems. For each metal–insulator–metal (MIM) junction, the application of field-dependent induced tunnelling and stochastic fluctuation terms directly recovers measured scaling laws:

• Low field/temperature: pure tunnelling regime with resistance $R \sim R_{00}\exp(\beta/T)$.
• Intermediate field: field-induced enhancement, $R \sim R_0\exp(-\alpha E_0^2 f(\omega))$, matching observed quadratic-in-field resistance decreases up to the threshold for barrier suppression.
• High field/low frequency: saturation or irreversible drops due to contact welding (Hirlimann, 2013).

The framework thus covers continuous variation from pure quantum tunnelling, fluctuation-assisted activation, to strong field-driven nonlinearity. In field-effect transistors, it explains monotonic enhancement of conductivity with gate voltage, the crossover from Arrhenius to plateau forms, and composition-dependent suppression or enhancement of low-temperature conduction plateaus (Zhao et al., 26 Dec 2025).

6. Limitations, Boundary Conditions, and Materials Engineering Implications

FEAFIT, while predictive, is subject to several key boundary conditions:

• Assumes 1D square barriers; real systems present distributions in area and thickness, requiring statistical averaging or network representation for full accuracy (Hirlimann, 2013).
• Neglects heating and feedback below the threshold for contact welding; sustained field application may introduce additional nonlinearities.
• The induced-tunnelling enhancement is contingent on sub–$\tau_{ref}$ (fs-scale) coherent field spikes; long-pulse fields produce only WKB modulation, setting a frequency cutoff at $\omega_c \sim 1/\tau_{ref}$.
• In low-temperature regimes, quantum (zero-$T$) fluctuations rather than thermal noise set the variance, requiring appropriate modification of $\sigma_U$.
• The coherence domain framework is inapplicable if global (delocalized) band conduction is restored, i.e., $E_F$ rises above $E_{dloc}$.

From a materials engineering perspective, FEAFIT quantitatively demonstrates that raising carrier mobility and on-current in AOS or granular systems hinges on maximizing the size of coherent domains (e.g., by increasing In fraction or optimizing O content) and minimizing tail-state energetic disorder through compositional control (e.g., reducing Ga-induced fluctuation). All extracted parameters are directly interpretable in terms of microscopic disorder and coherence, providing an actionable roadmap for device optimization (Zhao et al., 26 Dec 2025).

7. Tables of Key FEAFIT Quantities and Relationships

The following table summarizes the key variable dependencies for FEAFIT transport in a-IGZO and granular systems:

Quantity	Physical Meaning	Functional Dependence
$\Phi_{eff}(V_G)$	Effective barrier height	$\Phi_0 - \alpha(V_G-V_{FB})$
$d(V_G)$	Effective barrier width	$d_0 - \beta(V_G-V_{FB})$
$T_1(V_G)$	Activation scale	$\Phi_{eff}(V_G)/k_B$
$T_1(V_G)/T_0(V_G)$	Tunnelling scale	$\propto d(V_G)\sqrt{\Phi_{eff}(V_G)}$
$S_1(T), S_0(T)$	Crossover functions	$S_1=[n^*]^2$, $S_0=n^*(1-n^*)$ with $n^*=T/[T+T_0]$
$R(E_0,\omega,T)$	Contact resistance (Branly type)	See above; fields, frequency, and temperature

Empirical correspondence with first-principles calculations and observed device behavior underpins the universality and robustness of the FEAFIT approach (Zhao et al., 26 Dec 2025, Hirlimann, 2013).