Mathematical models for dispersive electronic transport in disordered semiconductor devices, derived from first principles in Mathematical Geoenergy (GEM), Chapter 18 (Geoenergy Conversion).
Chapter 18 treats two distinct but structurally parallel applications of Maximum Entropy (MaxEnt) disorder to charged-carrier transport:
Lithium-ion batteries store charge in disordered electrode particles. The diffusion of Li⁺ ions through these particles is rate-limiting for charging and discharging speed. Because the particles have a wide distribution of sizes and disordered microstructure, a single-valued diffusion coefficient is insufficient.
Applying MaxEnt to the diffusion coefficient D (only the mean D₀ is known) and integrating the Fokker-Planck Green’s function yields a dispersive diffusion kernel (Eq. 18-4) that replaces the Gaussian:
\[C(t, x) = \frac{1}{2\sqrt{D_0 t}}\cdot e^{-x/\sqrt{D_0 t}}\]Integrating this kernel over the particle geometry and over a MaxEnt distribution of particle sizes gives closed-form charge and voltage profiles (Eqs. 18-6, 18-8, 18-13) that fit experimental discharge data with only two free parameters.
Amorphous semiconductors (a-Si:H, organic polymers, quantum dots) show “anomalous” photo-response: the current pulse produced by a light impulse spreads into a long tail instead of the near-rectangular pulse seen in crystalline materials. Scher and Montroll (1975) modelled this with the Continuous Time Random Walk (CTRW), a mathematically complex framework.
The same result follows directly from MaxEnt applied to the carrier velocity distribution. For a disordered material, only the mean drift velocity ν₀ is known; MaxEnt then gives an exponential distribution (Eq. 18-28). Two equivalent derivations are presented:
Dispersive FPE (Eqs. 18-21 – 18-23): integrating the Gaussian FPE solution over the MaxEnt prior for D (Eq. 18-19) gives a dispersive carrier concentration with rate function R(t) (Eq. 18-22) that smoothly interpolates between diffusion-dominated and drift-dominated regimes.
Entropic growth-function model (Eqs. 18-27 – 18-33): using the MaxEnt exponential velocity PDF directly in a carrier-counting argument gives a charge accumulation formula C(t) = C₀·g(t)·(1 − exp(−w/g(t))) (Eq. 18-30) with the growth function g(t) = √(2Dt) + μEt (Eq. 18-31). Differentiating gives the photo-current I(t) (Eq. 18-32).
Both approaches reproduce the characteristic plateau-then-tail profile of TOF measurements in a-As₂Se₃, organic polymers (APFO, ANTH-OXA6t), a-Si:H, SiO₂, and quantum-dot composites.
Eq. 18-1 — Diffusion equation
\[\frac{\partial C(t,x)}{\partial t} - D\cdot\nabla^2 C(t,x) = 0\]Eq. 18-2 — Gaussian Green’s function (ordered medium)
\[C(t,x\mid D) = \frac{1}{\sqrt{4\pi D t}}\cdot e^{-x^2/(4Dt)}\]Eq. 18-3 — MaxEnt distribution for diffusion coefficient
\[p(D) = \frac{1}{D_0}\cdot e^{-D/D_0}\]Eq. 18-4 — Dispersive diffusion kernel (MaxEnt average over D)
\[C(t, x) = \frac{1}{2\sqrt{D_0 t}}\cdot e^{-x/\sqrt{D_0 t}}\]Obtained via the Laplace-Bessel identity (Gradshteyn & Ryzhik §3.471):
\[\int_0^\infty \frac{e^{-Ax - B/x}}{\sqrt{x}}\\,dx = \sqrt{\frac{\pi}{A}}\cdot e^{-2\sqrt{AB}}\]Eq. 18-6 — Fixed particle-size discharge
\[C(t) = C_0\cdot\frac{1 - e^{-(L-d)/\sqrt{D_0 t}}}{L - d}\]Eq. 18-8 — MaxEnt particle-size charge profile
\[C(t) \propto \frac{1}{L + \sqrt{D_0 t}}\]Eq. 18-10 — Discharge current
\[I(t) = -\frac{C\sqrt{D_0}}{2\sqrt{t}\cdot(L + \sqrt{D_0 t})^2}\]Eq. 18-13 — Constant-current voltage decline
\[V(t) = V_0 - k_v\cdot I_\text{const}\cdot(L + \sqrt{D_0 t})\cdot 2\sqrt{t}\]Eq. 18-14 — Ideal TOF current (crystalline reference)
\[I(t) = K\cdot[u(t) - u(t - t_T)]\]Eq. 18-15 — MaxEnt TOF charge (inspection result)
\[q(t) = q\cdot N\cdot\exp\left(-\frac{w}{\sqrt{(\mu E t)^2 + 2Dt}}\right)\]Eq. 18-17 — Gaussian FPE solution (ordered semiconductor)
\[n_0(x,t) = \frac{A}{\sqrt{4\pi D t}}\cdot\exp\left(-\frac{(x + \mu E t)^2}{4Dt}\right)\]Eq. 18-18 — Einstein relation
\[D = V_t\cdot\mu, \qquad V_t = kT/q\]Eq. 18-21 — Dispersive FPE carrier concentration
\[n(x,t) = \frac{e^{-x\cdot R(t)}}{\sqrt{t\cdot(4D_0 + (\mu E)^2 t)}\cdot R(t)}\]Eq. 18-22 — Rate function R(t)
\[R(t) = \sqrt{\frac{1}{D_0 t} + \frac{E^2}{4V_t^2}} - \frac{E}{2V_t}\]Eq. 18-28 — MaxEnt carrier velocity PDF
\[p(\nu) = \frac{1}{\nu_0}\cdot e^{-\nu/\nu_0}\]Eq. 18-29 — Fraction of carriers past position x
\[C(t\mid x) = \int_{x/t}^\infty p(\nu)\\,d\nu = e^{-x/(\nu_0 t)}\]Eq. 18-30 — Entropic charge accumulation
\[C(t) = C_0\cdot g(t)\cdot\Bigl(1 - e^{-w/g(t)}\Bigr)\]Eq. 18-31 — Growth function
\[g(t) = \sqrt{2Dt} + \mu E t\]Eq. 18-32 — Entropic dispersive current
\[I(t) = C_0\cdot\frac{dg}{dt}\cdot\left(1 - e^{-w/g(t)}\Bigl(1 + \frac{w}{g(t)}\Bigr)\right)\]Eq. 18-36 — Generalised Einstein relation
\[D = \frac{\mu}{q}\cdot\frac{N(E_c)}{dN(E_c)/dE}\]For a Maxwell-Boltzmann density of states this reduces to D = μkT/q. For a MaxEnt (exponential) distribution of energy states with wide variance, D = μ(kT + E_c)/q, explaining anomalously high diffusivities in disordered materials.
| File | Purpose |
|---|---|
electron_symbolic.py |
Symbolic derivation of all Chapter 18 equations using SymPy |
electron_numerical.py |
Numerical implementation, validation, and composite figure |
electron_model_output.png |
Output figure (6 panels) generated by electron_numerical.py |
Install dependencies (from models/requirements.txt):
pip install -r ../requirements.txt
Run symbolic derivation (all assertions print ✓):
python electron_symbolic.py
Run numerical model and generate figure:
MPLBACKEND=Agg python electron_numerical.py
Same MaxEnt disorder, different physics: both the Li-ion battery (disorder in D and particle size) and the PV semiconductor (disorder in carrier velocity and D) lead to the same dispersive exponential kernel (Eq. 18-4). The principle unifies battery, solar-cell, and porous-media transport (Chapter 20).
Fat-tail transport is not anomalous: the long tails of TOF measurements in amorphous semiconductors arise naturally from the MaxEnt exponential velocity distribution — no fractal arguments, Lévy flights, or CTRW formalism are required.
Two-regime current profile: the dispersive FPE (Eq. 18-21) and the entropic model (Eq. 18-32) both predict a diffusion-dominated initial spike (∝ 1/√t) followed by a drift-dominated power-law tail (∝ 1/t²). The knee separating the two regimes is the characteristic Scher-Montroll feature.
Universal dispersive kernel: the formula C(t, x) = exp(−x/√(D₀t))/(2√(D₀t)) appears in Li-ion battery discharge (Part A), semiconductor TOF (Part B), and porous-media contaminant transport (Ch. 20). It is the MaxEnt analogue of the Gaussian Green’s function.
Generalised Einstein relation: in a maximally disordered semiconductor the effective diffusivity D = μ(kT + E_c)/q exceeds the thermal value μkT/q. This explains why D fitted from experimental data often appears larger than the Einstein prediction.