Quantitative completion of the effective-impurity model of optical absorption edges in amorphous semiconductors: donor and donor-acceptor pair transitions, realistic charge distributions, controlled approximations, and stable inversion of measured $\varepsilon_{2}(\omega)$ to the band-tail density of states

Dow and Hopfield (J. Non-Cryst. Solids 8-10, 664 (1972)) proposed an effective-impurity model for the optical absorption edges of amorphous semiconductors: spatial electrostatic potential fluctuations are represented as an ensemble of effective acceptor-like Coulomb centers of charge $-|Z|e$, and the imaginary part of the dielectric function $\varepsilon_{2}(\omega)$ for tail-to-conduction-band transitions is computed as a statistical average of generalized Elliott-type excitonic expressions, including electron-hole correlations through a Sommerfeld-form suppression factor $S(x)$. The derivation used a separable-potential approximation exact only in the infinite-hole-mass limit, an assumed distribution $\mathrm{d}N_{\mathrm{A}}/\mathrm{d}Z^{2}$ of effective charges, and the crystalline value of the momentum matrix element. The paper left open the quantitative completion of this program: extension to effective-donor and donor-acceptor pair transitions; realistic charge-strength distributions from microscopic disorder models or experiment; matrix-element corrections from short-range disorder; quantitative evaluation of the separable-potential and electrostatic-only approximations; a stable inversion of measured $\varepsilon_{2}(\omega)$ to the band-tail density of states; and quantitative comparison with Franz-Keldysh (uniform-field) predictions in specific materials. No published quantitative completion of these tasks is known: the mainstream instead adopted and experimentally supported an uncorrelated constant-matrix-element inversion for a-Si:H, leaving the quantitative role of electron-hole correlations in amorphous absorption edges, and a correlation-corrected validated inversion, without a published solution.

Background

The optical absorption edge is the spectral region in which interband absorption sets on; its strength is measured by the imaginary part $\varepsilon_{2}(\omega)$ of the dielectric function, which is proportional to the absorption coefficient $\alpha(\omega)$ divided by $\omega$. In a crystal the edge is sharp apart from excitonic structure: the Coulomb attraction between the optically excited electron and hole modifies the continuum absorption according to Elliott's classic formula, and the associated 'Sommerfeld factor' suppresses absorption when the final-state interaction is repulsive. Amorphous semiconductors — glasses and thin films lacking long-range atomic order, such as hydrogenated amorphous silicon (a-Si:H), chalcogenide glasses, and amorphous group IV or III-V films — instead show broad, usually exponential edges, $\alpha \propto \exp[(\hbar\omega-E_0)/E_U]$ (the 'Urbach' form, with $E_U$ the Urbach energy), often over many decades of $\alpha$; the relation between optical and electrical energy gaps in such materials was itself under active discussion (Fritzsche 1971).

Around 1970 two rival explanations dominated. (i) Band tails: disorder spreads each band edge into a tail of localized one-electron states whose density decays into the gap, and uncorrelated ('non-direct transition') treatments computed the edge from one-particle densities of states; Stern's band-tail model for amorphous Si (which used Halperin-Lax results for the density of states of a fluctuating potential with root-mean-square amplitude $V_{\mathrm{rms}}$ and correlation length $L$) is representative. (ii) Internal microfields: charged defects generate quasi-static electric fields, and carriers tunnel into the gap in an electric field by the Franz-Keldysh effect; Redfield had proposed defect-field broadening, Tauc applied internal fields to amorphous semiconductors, and Dow and Redfield developed the internal-field mechanism into a microfield theory of exponential edges (Urbach's rule) that stressed the necessity of including exciton (final-state electron-hole) effects.

Dow and Hopfield's 1972 paper introduced a third representation for amorphous semiconductors with predominantly electrostatic disorder. The spatial fluctuations of the local potential are modeled as an ensemble of 'effective acceptors': each spatial region carries a Coulomb potential of charge $-|Z|e$ (with $Z$ a continuous effective charge strength) that shifts the valence- and conduction-band edges in parallel. The valence-band-tail density of states is then a statistical sum over effective-acceptor 1s bound states,

$$\rho_{\mathrm{v}}(E)=\frac{1}{V}\sum_{n}\int_{0}^{\infty}\mathrm{d}Z^{2}\,\frac{\mathrm{d}N_{\mathrm{A}}}{\mathrm{d}Z^{2}}(Z^{2};n)\,\delta\!\left(E-Z^{2}R_{\mathrm{h}}\right),$$

where $E$ is measured into the gap from the valence-band edge, $R_{\mathrm{h}}=13.6\,\mathrm{eV}\times m_{\mathrm{h}}/\varepsilon_{0}^{2}$ is the hole Rydberg ($m_{\mathrm{h}}$ the effective hole mass, $\varepsilon_{0}$ the static dielectric constant), $V$ the sample volume, and $\mathrm{d}N_{\mathrm{A}}/\mathrm{d}Z^{2}$ the distribution of effective-charge strengths. The acceptor-to-conduction-band absorption is a statistical average of single-impurity contributions, each a generalized Elliott-type excitonic expression in the electron-hole relative and center-of-mass coordinates. To make this two-particle problem tractable, the full impurity potential is replaced by a separable approximation, $V_{\mathrm{I}}\approx |Z|e^{2}/\varepsilon_{0}(1-\delta)r-|Z|e^{2}/\varepsilon_{0}R$ (relative coordinate $r$, center-of-mass coordinate $R$, $\delta=m_{\mathrm{e}}/(m_{\mathrm{e}}+m_{\mathrm{h}})$), which is exact only in the limit $m_{\mathrm{h}}\to\infty$. Within this approximation the Schrödinger equation is solved in closed form, giving

$$\varepsilon_{2}(\omega)=\frac{16}{\pi}\left(\frac{m_{\mathrm{e}}^{3}m_{\mathrm{h}}^{6}}{M^{9}}\right)^{\!1/2}C\int_{1-\delta^{2}}^{\infty}\mathrm{d}Z^{2}\,\rho_{\mathrm{v}}(Z^{2}R_{\mathrm{h}})\,|Z|^{-3}\,\tilde{E}^{1/2}\,S(x)\,\theta(\tilde{E}),$$

with $\tilde{E}=Z^{2}(M/m_{\mathrm{h}})+(\hbar\omega-E_{\mathrm{gap}})/R_{\mathrm{h}}$, $x=\tilde{E}m_{\mathrm{h}}^{2}/[M m_{\mathrm{e}}(|Z|-1+\delta)^{2}]$, $M=m_{\mathrm{e}}+m_{\mathrm{h}}$, $\theta$ the step function, and

$$S(x)=\frac{2\pi}{\sqrt{x}}\left[\exp\!\left(2\pi/\sqrt{x}\right)-1\right]^{-1},$$

a Sommerfeld-form factor that here acts as an energy-dependent suppression of deep-tail transitions; $C$ is the momentum matrix element factor, taken at its crystalline value. A numerical example (an exponential band tail of width $W=0.1$ eV, $m_{\mathrm{e}}=0.3$, $m_{\mathrm{h}}=1$, $\varepsilon_{0}=10$) showed that including the correlation factor $S(x)$ substantially suppresses the deep-tail absorption amplitude but does not turn a broad density-of-states tail into a sharp optical edge.

The paper was explicit about what this construction left undone, and those acknowledged gaps define the problem addressed here: the derivation covers only effective-acceptor-to-conduction-band transitions (the donor-side and donor-acceptor pair cases are asserted to be analogous but never derived); the distribution $\mathrm{d}N_{\mathrm{A}}/\mathrm{d}Z^{2}$ is a free input rather than a consequence of a microscopic disorder model or experiment; the matrix element is taken at its crystalline value; the separable-potential and electrostatic-only approximations carry no error estimate; the integral expression is offered as a forward model but no stable inversion from measured $\varepsilon_{2}(\omega)$ to the band-tail density of states is constructed; and no quantitative comparison with the Franz-Keldysh (uniform-field) representation is carried out for any specific material. The paper also restricts its own premise: for carefully prepared amorphous group IV and III-V materials, non-electrostatic ('elastic', antisymmetric) fluctuations that shift the two band edges differently may dominate.

Problem Statement

Complete, with quantitative control, the program left open by Dow and Hopfield's effective-impurity model of optical absorption edges in amorphous semiconductors. All components serve one common objective: to determine whether the fluctuating-Coulomb-field picture with electron-hole correlations quantitatively describes the optical absorption edges of real amorphous semiconductors, and whether it supports reliable inversion from measured spectra to the underlying band-tail density of states. The components are linked — the correlation-corrected kernel produced by the first and third components is exactly what the inversion of the fourth must use, and the comparisons of the fifth test the picture against alternative representations — and each corresponds to a gap the source paper itself acknowledged (see Background).

1. Determine whether the correlated effective-impurity calculation extends to the remaining transition classes — valence-band to effective-donor transitions and donor-acceptor pair clusters — by a derivation and numerical evaluation analogous to the acceptor-to-conduction-band case.
2. Replace the assumed effective-charge-strength distribution $\mathrm{d}N_{\mathrm{A}}/\mathrm{d}Z^{2}$ with realistic forms derived from microscopic disorder models or from independent experiments, and quantify the corrections to the momentum matrix element factor $C$ caused by short-range disorder.
3. Quantify the model's uncontrolled approximations: bound the error introduced by the separable-potential approximation (exact only for $m_{\mathrm{h}}\to\infty$) at finite mass ratios, and determine the effect of the neglected non-electrostatic (elastic, antisymmetric) potential components on the factor $S(x)$ and on the predicted suppression of the absorption tail.
4. Determine whether a stable inversion procedure that recovers the band-tail density of states from measured $\varepsilon_{2}(\omega)$ (or $\alpha(\omega)$) can be constructed using the derived integral expression with the correlation factor included, or whether a different forward model is required; whenever such a procedure can be constructed, test it against experimental data using an independently determined density of states as the ground-truth validation target.
5. Carry out the quantitative comparison with Franz-Keldysh (uniform-field) predictions and with at least one related internal-field (microfield) representation discussed in the Background (Redfield, Tauc, or Dow and Redfield) for at least one specific material, matching the parameters common to the compared models (energy gap, effective masses, static dielectric constant, and field strength).

Each component is to be settled with quantitative control: by an explicit derivation, evaluation, or bound, or by a demonstrated obstruction showing that the required object does not exist, rather than merely asserted or discussed qualitatively.

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