Behavior of the Hall coefficient and thermopower across the Anderson metal-insulator transition (the unresolved questions of Mott, Pepper, Pollitt, Wallis and Adkins, 1975)

Mott, Pepper, Pollitt, Wallis and Adkins' 1975 review 'The Anderson transition' (Proc. R. Soc. Lond. A 345, 169) closed with a list of unsolved problems. The source's two-dimensional critical-parameter questions (the critical disorder ratio $(V_0/B)_{\mathrm{crit}}$ and a universal 2D minimum metallic conductance) were resolved by the 1979 one-parameter scaling theory of localization, which shows that all states of a non-interacting two-dimensional system are localized for arbitrarily weak disorder. The open core concerns the behavior of the transport coefficients across an Anderson-type metal-insulator transition: whether the Hall coefficient $R_H$ diverges critically at the mobility edge or stays classical ($R_H = 1/ne$; conflicting theoretical predictions and split experimental evidence across materials); whether the Hall response in the variable-range-hopping regime admits a controlled quantitative description; and whether the diffusion thermopower diverges or stays finite at the transition, together with the origin of the anomalous kinetic constant $B \approx -2$ in semiconducting expanded fluid mercury. A 2022 experimental study still describes the Hall-coefficient behavior across the metal-insulator transition as a longstanding unresolved problem, with conflicting theoretical predictions and split experimental evidence.

Background

In a crystalline solid, band theory distinguishes metals from insulators by the filling of allowed energy bands. When the periodic lattice potential is replaced by a static random potential, a second mechanism appears: Anderson localization, the confinement of an electron wave function by destructive interference among multiply scattered waves. As the disorder strength grows, states near the band edges become localized while states near the band center remain extended; the boundary energy $E_c$ is called a mobility edge. If the Fermi energy $E_F$ (the energy of the highest occupied electron state at zero temperature) is moved through a mobility edge by changing composition, carrier density, or pressure, the system undergoes a metal-insulator transition of the 'Anderson type': on the metallic side, conduction is by extended states; on the insulating side, only by thermally assisted hopping between localized states (variable-range hopping, giving a conductivity $\sigma \propto \exp(-A/T^{1/4})$ in three dimensions).

The 1970s literature attached several quantitative expectations to this picture. There was a minimum metallic conductivity $\sigma_{\min}$, the smallest conductance a metal was thought to support just above the mobility edge; the dimensionless parameter $g = N(E_F)/N_{\mathrm{free}}(E_F)$, the ratio of the density of states at the Fermi energy to its free-electron value, which measures the disorder-induced reduction of the conduction band; a thermopower $S$ (the open-circuit voltage generated per unit applied temperature difference) in the activated regime of the form $S = (k/e)[(E_c - E_F)/kT + B]$, where $k$ is Boltzmann's constant, $e$ the electron charge, and $B$ a kinetic constant reflecting the energy dependence of the transport; and, for the Hall effect (the transverse voltage induced by a magnetic field), Friedman's random-phase theory, which predicted that the Hall mobility $\mu_H$ of a disordered metal should scale with $g$, while in the localized regime the Hall response should arise from Holstein's three-site interference mechanism (quantum interference between hopping amplitudes around closed triangles of three localized sites) and be extremely small. In a classical metal the Hall coefficient equals $R_H = 1/ne$, with $n$ the carrier density.

In 1975 Mott, Pepper, Pollitt, Wallis and Adkins reviewed this theory and the experimental evidence across the known material classes - doped semiconductors such as Si:P and Ge:Sb (the notation marks host and dopant: phosphorus in silicon, antimony in germanium), expanded fluid mercury, liquid Te-Se alloys, oxide and sulfide compounds, and the Si/SiO$_2$ inversion layer (a two-dimensional electron gas at a field-effect interface whose density is tuned by a gate voltage) - and closed with a list of unsolved problems. Two of their questions concerned the critical parameters of the two-dimensional transition: the critical disorder ratio $(V_0/B)_{\mathrm{crit}}$, where $V_0$ is the amplitude of the random potential and $B$ the tight-binding band width (not the thermopower constant of the same symbol), measured as about 0.8 for the (100) Si surface against a predicted 4/3 for a square lattice; and a universal two-dimensional minimum metallic conductance. These questions were answered shortly afterwards by the one-parameter scaling theory of localization: in two dimensions all states are localized for arbitrarily weak disorder (absent spin-orbit coupling or interactions), so no finite two-dimensional critical disorder ratio and no universal minimum conductance exist in the thermodynamic limit; the 1970s 'transitions' are crossovers whose apparent critical parameters track sample quality. The behavior of the transport coefficients - above all the Hall coefficient and the thermopower - at and across the Anderson transition remains open. The source noted that doped semiconductors obey the classical formula $R_H = 1/ne$ in their metallic region, contrary to Friedman's prediction $\mu_H \propto g$; that the Hall effect is essentially unmeasurable in the hopping regime; that the thermopower kinetic constant takes anomalous values (about $-2$ in semiconducting expanded fluid mercury); and it raised related material-specific puzzles (thermopower anomalies in La$_{1-x}$Sr$_x$VO$_3$, Hall mobility in liquid Te-Se, hopping laws and low-temperature thermopower in the vacancy compounds Ce$_{3-x}\square_x$S$_4$ and CeS, where $\square$ denotes a cation vacancy).

Problem Statement

Determine, quantitatively and predictively, the behavior of the charge-transport coefficients of disordered electronic systems across an Anderson-type metal-insulator transition, as posed by the unresolved-issues list of Mott, Pepper, Pollitt, Wallis and Adkins (1975). The open core consists of three independently posed open questions on the behavior of the transport coefficients at and across the mobility edge, each resolvable on its own:

(1) Hall coefficient in the metallic and critical region. Establish whether the Hall coefficient $R_H$ diverges critically as the Fermi energy approaches the mobility edge (and with what critical exponent), or remains non-critical and close to the classical value $R_H = 1/ne$. Theoretical predictions conflict between smooth and divergent behavior, and the experimental evidence is split across materials.

(2) Hall effect in the hopping regime. Determine whether the Hall response in the localized, variable-range-hopping regime admits a controlled quantitative description, and if so what it is.

(3) Thermopower across the transition. Determine whether the diffusion thermopower $S$ (the contribution to $S$ from carrier diffusion, as distinct from the phonon-drag contribution, in which heat-carrying lattice vibrations drag carriers along) remains finite or diverges at the transition, and the behavior of the kinetic constant $B$ in $S = (k/e)[(E_c - E_F)/kT + B]$, including the origin of its negative value $\approx -2$ in the semiconducting regime of expanded fluid mercury.

The two-dimensional critical-parameter questions raised in the same source (the critical disorder ratio $(V_0/B)_{\mathrm{crit}}$ and a universal two-dimensional minimum metallic conductance) were resolved by the 1979 scaling theory and are background, not open problem. The source's remaining material-specific puzzles (La$_{1-x}$Sr$_x$VO$_3$ thermopower, liquid Te-Se Hall mobility, Ce$_{3-x}\square_x$S$_4$ hopping laws, CeS low-temperature thermopower, the field-induced transition of Gd$_{3-x}\square_x$S$_4$, and the disorder-correlation condition in amorphous Ge-Fe films) are background rather than part of the open core; $\square$ denotes a cation vacancy.

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