Mott's open questions from "Electrons in Glass" (1977): mobility edges in realistic amorphous networks, correlations in variable-range hopping, the minimum metallic conductivity, and the mechanism of Ovshinsky threshold switching

In his Nobel lecture "Electrons in Glass" (delivered 8 December 1977; Science 201, 871 (1978)), N. F. Mott enumerated five problems that remained open for the theory of electronic conduction in non-crystalline materials: (1) the position of the mobility edge separating extended from localized states had not been computed for realistic continuous random networks such as vitreous SiO2, As2Se3 or amorphous Si; (2) some materials (electrons in liquid rare gases, vitreous SiO2) show no experimental evidence for a mobility edge, and no theory predicted when the range of localized tail states should fall below the thermal energy $kT$; (3) whether a minimum metallic conductivity $\sigma_{\min}=C e^2/(\hbar a)$ exists, and the value of the constant $C$, was controversial; (4) the effect of electron correlations on variable-range hopping (the $T^{1/4}$ law) was, in Mott's words, "not yet perfectly understood"; (5) no complete model existed of the Ovshinsky threshold switch — the reversible transition of a chalcogenide-glass film to a highly conducting state. The scaling theory of localization (1979) removed the sharp minimum metallic conductivity, and Efros-Shklovskii Coulomb-gap physics (1975) showed that long-range Coulomb interactions convert the $T^{1/4}$ law into a $T^{1/2}$ law at low temperature, with extensive experimental confirmation. The surviving open core is (1)+(2) — a first-principles prediction of mobility edges (or of their absence) in realistic glassy semiconductors, with a criterion for when localized tails are thermally washed out — and (5) — a complete quantitative model of the conducting channel in threshold switches, whose mechanism is still contested in the 2025 literature.

Background

In a perfect crystal, periodicity guarantees that single-electron states are extended Bloch waves. In an amorphous ("glassy") semiconductor the atoms instead form a continuous random network (CRN): a chemically connected but disordered structure with no translational symmetry, such as the networks postulated for vitreous SiO2, As2Se3 and amorphous Si, in which each atom keeps roughly the same coordination as in the corresponding crystal. Anderson (1958) showed that sufficiently strong disorder localizes quantum eigenstates; for moderate disorder, the crucial concept is the mobility edge: a critical energy $E_c$ separating extended states (which carry band-like conduction) from localized states (which conduct only by thermally assisted hopping between sites). In many amorphous semiconductors the sharp band edges of the crystal are replaced by tails of localized states, and if the Fermi energy lies among localized states the material is a "Fermi glass" (Mott's term). Mott further argued that when the Fermi energy lies above $E_c$, the conductivity in the limit of zero temperature cannot be arbitrarily small, proposing a minimum metallic conductivity $\sigma_{\min}=C e^2/(\hbar a)$, where $a$ is an inter-atomic spacing and $C$ a constant (estimated at 0.025-0.1) fixed by the Anderson localization criterion; for $a\approx 3$ Å this gives roughly 250-1000 inverse ohm-cm. Conversely, when the Fermi energy lies below $E_c$ in a smooth density of localized states, hopping conduction follows variable-range hopping (VRH): optimizing the competition between hopping distance and activation energy gives Mott's law $\sigma=A\exp(-B/T^{1/4})$ in three dimensions ($T^{1/3}$ in two). Chalcogenide glasses (alloys of S, Se, Te with elements such as Ge and As) additionally exhibit threshold (Ovshinsky) switching: a deposited film roughly one micron thick, with an electrode on each side, switches abruptly from a high-resistance off state to a highly conducting on state as the potential across it is increased, and switches off again when the current drops below a holding value. Experiments show that in the on state the current flows in a channel in which the electron and hole densities and the current density do not depend on the total current — the channel simply widens — and that the channel is not hot enough for a thermal explanation. Mott proposed that the mechanism is double injection (holes entering at one electrode, electrons at the other) forming a degenerate electron-hole plasma (a plasma dense enough that both carrier gases are Fermi-degenerate), similar to the electron-hole droplets known in crystalline germanium, with both Fermi energies above their respective mobility edges. Conventions: conductivity units are inverse ohm-cm throughout; $kT$ is the thermal energy at absolute temperature $T$; the five questions quoted below are from the primary text of "Electrons in Glass".

Problem Statement

As stated in the source (N. F. Mott, "Electrons in Glass", Nobel lecture 1977; Science 201, 871 (1978)), five problems on electronic conduction in non-crystalline materials share one scientific objective — a predictive, material-specific theory of transport and switching in glassy semiconductors. Subsequent work has substantially resolved parts (3) and (4), leaving (1) and (2) together with (5) as the open core of the task. (1) compute the position of the mobility edge $E_c$ (the critical energy separating extended from localized states) for realistic continuous random networks such as SiO2, As2Se3 and amorphous Si — in the source's words, this "has not yet proved possible" and "is going to be quite a challenge for the theoreticians"; (2) explain why some materials (electrons in liquid rare gases, vitreous SiO2) show no experimental evidence for a mobility edge at all — in the source, the range of localized states $\Delta E$ in such materials must be smaller than $kT$ at room temperature — and predict when this should be so ("we await theoretical predictions of when this should be so"); (3) determine whether a minimum metallic conductivity $\sigma_{\min}=C e^2/(\hbar a)$ (with $a$ an inter-atomic spacing and $C$ a dimensionless constant) exists and fix the constant $C$; (4) determine the effect of electron correlations on variable-range hopping ("perhaps the effect of correlation is not yet perfectly understood"); (5) construct a complete model of the Ovshinsky threshold switch, including the formation, structure and dynamics of its conducting channel ("we are still far from a full understanding of the behaviour of this fascinating device"). Parts (3) and (4) have since been substantially resolved — the scaling theory of localization shows there is no sharp minimum metallic conductivity (all non-interacting states are localized in two dimensions, and the conductivity vanishes continuously at the mobility edge in three), and Efros-Shklovskii Coulomb-gap physics shows that unscreened long-range electron-electron interactions convert the $T^{1/4}$ law into a $T^{1/2}$ law at low temperature, both with extensive experimental support. The surviving open core is therefore stated precisely as follows. (i) From first principles, predict the position — or establish the absence — of mobility edges in realistic continuous random networks of actual glassy semiconductors (amorphous Si, SiO2 and As2Se3), and derive a general criterion that tells for which materials the localized-tail energy range is smaller than $kT$ at room temperature, consistent with the documented classes where no mobility edge is observed. The material-specific prediction is required for each of the three named networks — amorphous Si, SiO2 and As2Se3 — and the general criterion is also required. The scientific target is, for each of the three named networks, the mobility-edge energy — or a demonstrated absence — obtained by a stated first-principles method applied to an explicitly specified, relaxed continuous random network, together with a general criterion that reproduces the documented classes where no mobility edge is observed (electrons in liquid rare gases and vitreous SiO2) and yields a checkable prediction for at least one material outside them. (ii) Construct a complete quantitative model of Ovshinsky threshold switching — channel formation, channel structure, and switching dynamics — consistent with the experimentally established electronic (non-thermal) nature of the on state and with the established fact that, in the on state, the electron and hole densities and the current density in the channel do not depend on the total current. A complete quantitative model must quantitatively reproduce these two established features of the on state and account for channel formation, channel structure and switching dynamics; any microscopic mechanism consistent with them is admissible, including double injection (holes entering at one electrode, electrons at the other) forming a degenerate electron-hole plasma — Mott's own proposal in the source ("This is still my opinion") — as well as the competing mechanisms of the cited threshold-switching literature (trap-limited transport with impact ionization, field-induced nucleation, conductive filaments, and percolation).

References

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