---
title: Physics-Informed Impact-Ionization Model
url: https://www.emergentmind.com/topics/physics-informed-impact-ionization-model
type: topic
---

# Physics-Informed Impact-Ionization Model

Searching arXiv for the cited impact-ionization and NEGF papers to ground the article in current literature.
arXiv search query: 2503.19167 OR 2605.01244 OR 1704.02200 OR 2603.02391 OR 1912.11549 OR 1705.09203
A physics-informed impact-ionization model is a description of carrier multiplication in which the ionization process is tied to microscopic band structure, carrier transport, and conservation laws rather than represented only by fitted local coefficients such as \(\alpha(E)\) and \(\beta(E)\). In reverse-biased avalanche structures, impact ionization converts the kinetic energy of a primary carrier into an additional electron–hole pair; in the formulation developed for avalanche photodiodes, this means that dead space, multiplication region, and gain emerge from non-equilibrium carrier distributions in space and energy rather than from ad hoc rate laws. Across the literature, the term also covers compact transport-calibrated laws, stochastic detector-response models, and first-principles scattering calculations, but the shared idea is the same: the ionization law is constrained by the underlying Hamiltonian, available phase space, and competing relaxation channels [2503.19167][2603.02391][1912.11549].

## 1. Conceptual scope

In the matrix quantum-kinetic formulation for avalanche photodiodes, a physics-informed impact-ionization model means starting from a microscopic Hamiltonian \(H\) in real space, using the non-equilibrium Green’s function formalism with open boundaries and non-equilibrium bias, including electron–electron scattering through higher-order self-energies involving products of multiple Green’s functions, enforcing energy conservation explicitly by delta functions and momentum conservation through the interaction correlator \(D\), and resolving non-equilibrium carrier distributions in space and energy so that dead space and multiplication are calculated rather than imposed. In the cryogenic Ge internal-charge-amplification literature, the same label is used for a compact coefficient \(\alpha(E,T)\simeq A(T)\exp[-B(T)/E]\) whose parameters are explicitly tied to energy-dependent scattering, Kane nonparabolicity, intervalley transfer, and the lucky-drift tail rather than treated as purely empirical Chynoweth parameters. In the SuperCDMS HVeV context, the model is much more reduced: impact ionization is represented by a per-pair stochastic probability \(A_+\), trapping by \(A_-\), and the observable signal by analytic convolutions of single-pair response functions, source Poisson statistics, and detector resolution [2503.19167][2603.02391][1912.11549].

This suggests that the phrase does not identify a single mathematical formalism. It instead identifies a modeling strategy: empirical closure is minimized, while the dominant physics of threshold formation, phase-space restriction, and energy redistribution is made explicit at the level appropriate to the problem.

## 2. Matrix quantum-kinetic formulation

For nanoscale avalanche devices, the most explicit realization is the matrix NEGF treatment. The stationary retarded Green’s function is written as
\[
G(E)=\Bigl[E I - H - U - \Sigma_1(E)-\Sigma_2(E)-\Sigma_S(E)\Bigr]^{-1},
\]
with \(H\) the real-space Hamiltonian, \(U\) the electrostatic potential matrix, \(\Sigma_{1,2}\) the contact self-energies, and \(\Sigma_S\) the scattering self-energy. The lesser correlation function is
\[
G^n(E)\equiv G^{<}(E)=G^R(E)\,\Sigma^{in}(E)\,G^A(E),
\]
and the diagonal elements of \(G^n(E)\) and \(G^p(E)\) give the position- and energy-resolved carrier distributions \(n(x,E)\) and \(p(x,E)\). The current is then evaluated with the Meir–Wingreen formula, which separates inflow into empty states from outflow from filled states. In the 2025 APD treatment, the basis is a real-space dimer chain with a block-tridiagonal Hamiltonian; in the 2026 atomistic extension, the same structure is implemented in a localized orbital basis fitted to silicon bands and used to resolve LDOS and occupied DOS in a reverse-biased depletion region [2503.19167][2605.01244].

The decisive departure from standard self-consistent Born scattering is that impact ionization is not represented by a self-energy proportional to a single \(G^{<}\). Because the process involves two initial and two final states, the self-energies depend on products of multiple Green’s functions. In compact band-resolved notation,
\[
\Sigma_{S,b}^{in}(E)=D_0\otimes \bigl[G_b^{min} * G_b^{maj} G_{b'}^{maj}\bigr],
\]
\[
\Sigma_{S,b}^{out}(E)=D_0\otimes \bigl[G_{b'}^{maj} * G_b^{min} G_b^{min}\bigr],
\]
where \([A*BC](E)\) is a triple energy convolution with a delta function enforcing
\[
E+E'=E''+E'''.
\]
The four-level model used to motivate this structure shows explicitly that these kernels conserve current: the virtual scattering terminal has zero net current. In the device calculation, the same structure allows multiplication to emerge from the available and occupied states of the biased device spectrum rather than from a prescribed local coefficient [2503.19167][2605.01244].

## 3. Thresholds, conservation laws, and dead space

A physics-informed impact-ionization model is distinguished from a purely fitted rate law by how it treats thresholds. In the APD NEGF formulation, energy conservation is imposed directly by delta functions in the self-energy convolutions, and momentum conservation is enforced by the spatial structure of the interaction correlator \(D\). For parabolic bands, the electron- or hole-initiated threshold is
\[
E_{TH}^{e,h}=\frac{2\mu+1}{\mu+1}E_G,
\qquad
\mu=\frac{m^*_{c,v}}{m^*_{v,c}},
\]
and the corresponding dead-space estimate is
\[
d_{e,h}\simeq \frac{E_{TH}^{e,h}}{q\mathcal{E}}.
\]
In the simulations, the spatial onset of multiplication aligns with this dead-space prediction, and changing effective masses shifts the onset through the threshold rather than through a fitted ionization coefficient [2503.19167].

For direct-gap semiconductors at \(\Gamma\), the microscopic near-threshold rate has the form
\[
\mathcal{W}(E_0,\mathbf{u})=
A(\mathbf{u})(E_0-E_{th})^2+B(E_0-E_{th})^3.
\]
The quadratic term is strongly anisotropic and vanishes along \([100]\) and \([111]\); the cubic term is isotropic. The same analysis concludes that at room temperature the cubic contribution dominates the integrated rate for most direct-gap materials with \(E_g\) up to \(1.5\) eV. This directly challenges the common use of a single effective exponent \(n\) in \(\mathcal{W}(E)=C(E-E_{th})^n\) [1704.02200].

A distinct but related threshold mechanism appears in narrow-gap CdHgTe quantum wells with “resonant” subband structure. There the condition \(E_{c2}-E_{c1}\approx E_g\) permits \(c2\to c1\) impact ionization with small momentum transfer, while analogous small-\(q\) processes are impossible for \(c1\)-initiated ionization. The calculated result is an approximately two-orders-of-magnitude enhancement for electrons in the second subband relative to the first. Here the threshold is controlled not only by the gap but by a subband-alignment condition that reopens low-\(q\) phase space [2601.11198].

## 4. Compact and reduced models

Not all physics-informed models are fully microscopic. In cryogenic high-purity Ge, the compact impact-ionization coefficient is written as
\[
\alpha(E,T)\simeq A(T)\exp\!\left[-\frac{B(T)}{E}\right],
\]
with
\[
B(T)=\frac{1}{q}\int_0^{\varepsilon_i}\frac{d\varepsilon}{v(\varepsilon)\tau_{\rm inel}(\varepsilon,T)},
\qquad
A(T)\sim \frac{1}{\lambda_{\rm eff}}.
\]
The resulting critical field for a multiplication region of width \(d\) is
\[
E_{\rm crit}^{(\mathrm{PI})}(T,d)=\frac{B(T)}{\ln[A(T)d]},
\]
while the single-free-flight upper bound is
\[
E_{\rm crit}^{(\mathrm{SFF})}=\frac{\sqrt{2\varepsilon_i/m^\ast}}{\mu}.
\]
Because \(B(T)\) is an energy-relaxation integral containing nonparabolic dispersion and energy-dependent inelastic scattering, this compact law remains explicitly tied to microscopic transport. In that sense it is a reduced but still physics-informed model [2603.02391].

The SuperCDMS HVeV model is more phenomenological, but still constrains the signal by a physical drift picture. A single generated pair undergoes exactly one of three outcomes: trapping with probability \(A_-\), impact ionization with probability \(A_+\), or unhindered drift with probability \(A_1=1-A_--A_+\). The single-pair response is
\[
{}^{(1)}h(x)=
A_-\,\Theta(x)\Theta(1-x)+
A_+\,\Theta(x-1)\Theta(2-x)+
A_1\,\delta(x-1),
\]
with trapping modeled as a uniform distribution on \([0,1]\) and impact ionization as a uniform distribution on \([1,2]\). Multi-pair events are obtained by repeated convolution, and the detector response is then convolved with Gaussian resolution and a leakage-background model. The model is therefore “physics-inspired” at the single-pair level, although it does not attempt a microscopic scattering calculation [1912.11549].

At the device-simulation level, the same philosophy appears in 3D finite-element drift–diffusion. Method A in the finite-element treatment rewrites the DD current in quasi-Fermi-potential form and uses a harmonic-average conductivity to obtain a genuine 3D extension of Scharfetter–Gummel. The same framework is then used in an off-state n-MOS simulation including impact-ionization generation, and the resulting model computes the \(I\)-\(V\) characteristic accurately until drain-to-bulk junction breakdown [1412.3691].

## 5. Extensions across materials and many-body regimes

The idea extends well beyond APDs. In warm dense matter, a physics-informed collision-ionization module couples Lotz impact-ionization cross sections, three-body recombination derived from detailed balance and Saha–Boltzmann equilibrium, and Stewart–Pyatt ionization-potential depression inside a PIC code. The key point is not only that impact ionization is included, but that its inverse process and plasma-modified thresholds are treated self-consistently, so the kinetic evolution relaxes toward the intended equilibrium rather than toward an over-ionized state [1607.02724].

In wide-bandgap \(\beta\)-Ga\(_2\)O\(_3\), the impact-ionization rate is built from screened Coulomb matrix elements evaluated with maximally localized Wannier functions and then propagated through full-band Monte Carlo together with electron–phonon scattering. The resulting electron ionization coefficients are computed up to \(8\) MV/cm, show strong orientation dependence, and are then fitted in Chynoweth form for use in device simulators. Here “physics-informed” means that the coefficient is extracted from ab initio band structure, dielectric screening, and full-band transport rather than inferred directly from breakdown data [1705.09203].

In strongly correlated systems, the same term acquires a different meaning. In a quasi-2D small-gap model with coherent-phonon gap quenching, impact ionization is generated by Coulomb scattering in a time-dependent band structure, and carrier multiplication is amplified by feedback between nonequilibrium carriers and the phonon-driven reduction of \(E_g(t)\). In the ionic Hubbard model, a distinct pathway appears: excess ionic potential energy, not only excess kinetic energy, can be converted into additional double occupancy after photoexcitation. In small Hubbard clusters, generalized Loschmidt amplitudes resolve which many-body eigenstates contribute to the delayed increase in double occupation and show that spin-energy loss is of little importance for the effect [1808.07815][2412.05798][2112.00685].

These examples suggest that the unifying content of a physics-informed impact-ionization model is not the use of any single transport equation, but the retention of the actual microscopic channel that supplies the pair-creation energy.

## 6. Relation to empirical models and unresolved issues

The main contrast throughout the literature is with empirical field-dependent coefficients and simplified Monte Carlo formulas. Standard approaches use \(\alpha(E)\) and \(\beta(E)\) fitted to experiment, Keldysh-like rates on simplified parabolic bands, or drift–diffusion source terms such as \(G_{\text{II}}\) inserted directly into continuity equations. Those descriptions become problematic when bands are strongly non-parabolic, anisotropic, or multi-valley; when tunneling, quantum confinement, or interference matter; or when impact ionization is itself the dominant inelastic process rather than a perturbative correction [2503.19167][2605.01244].

A common misconception is that any Chynoweth law is necessarily empirical. The cryogenic-Ge framework shows that a compact law can still be physics-informed if \(A(T)\) and \(B(T)\) are explicitly linked to scattering spectra, mobility, and nonparabolic transport. A second misconception is that a lowest-order Born self-energy proportional to a single \(G^{<}\) is a sufficient microscopic treatment. The APD NEGF analysis rejects that for avalanche multiplication, because the relevant effective rates scale like products such as \(n^2p\) or \(p^2n\) and therefore require multi-Green’s-function kernels [2603.02391][2503.19167].

Several unresolved directions recur across the papers. The APD and SPAD literature repeatedly points toward fully self-consistent Poisson–NEGF with realistic atomistic \(H\) and screened interaction matrices \(D\); the cryogenic-Ge work points toward calibration of \(A(T)\) and \(B(T)\) from measured \(M(V)\) with propagated uncertainty; the WDM and detector literatures point toward better treatment of inverse processes, backgrounds, and noise; and first-principles rate calculations still require practical downfolding from full band structure to device-level compact forms. A plausible implication is that future physics-informed impact-ionization models will increasingly be multiscale: microscopic enough to preserve conservation laws and phase-space selection, but reduced enough to be used inside transport, circuit, and detector simulations [2503.19167][2603.02391][2605.01244].

Source: https://www.emergentmind.com/topics/physics-informed-impact-ionization-model