N. F. Mott's 1934 wave-mechanical theory of the resistance of liquid metals gave a formula for the roughly twofold resistivity increase on melting of normal metals that was, in his words, 'in quantitative agreement with experiment for normal metals', while for the 'abnormal' metals — bismuth, gallium and antimony, poor solid conductors whose conductivity increases on melting — 'no quantitative theory can as yet be given'. The later literature (Ziman/Faber-Ziman diffraction theory, Hall-effect and transport measurements, pseudopotential calculations for liquid Bi and Ga, and modern first-principles molecular-dynamics plus Kubo-Greenwood transport) has treated the liquid-side resistivity semi-quantitatively with empirical pseudopotentials, but no work provides a quantitative theory in agreement with experiment for the sign and magnitude of the melting conductivity change of the abnormal metals. That core remains open.

Background

When a normal metal (Na, Cu, Pb, ...) melts, its electrical resistivity typically jumps by roughly a factor of two, because the loss of long-range crystalline order makes the ion arrangement a strong scatterer for the conduction electrons, whose mean free path collapses to a few interatomic distances. N. F. Mott, in 'The resistance of liquid metals' (Proc. R. Soc. Lond. A 146, 465 (1934)), gave a wave-mechanical (quantum-mechanical electron-wave) treatment of this change of resistance for normal metals, obtaining a formula in quantitative agreement with experiment. The paper's abstract describes the near-melting liquid as atoms vibrating about mean positions that, though not fixed, move slowly compared with the thermal vibration velocity — i.e. short-lived local order — and singles out the 'abnormal' metals, bismuth, gallium and antimony, which are rather poor conductors in the solid state yet increase their conductivity on melting; for these 'no quantitative theory can as yet be given', only a qualitative explanation. The physical reason these solids are poor conductors is semimetallic or covalent character: crystalline Bi and Sb are semimetals, in which the valence and conduction bands slightly overlap so that the number of conduction electrons is tiny compared with a normal metal, and solid Ga has a complex structure with covalent features. On melting, the semimetal band structure is destroyed and the melt behaves more nearly like a simple nearly-free-electron metal (electrons approximated as plane waves weakly scattered by the ions), which is the standard qualitative account of the anomaly — but not a quantitative one. The later machinery that made liquid-metal resistivity quantitative is the diffraction theory of J. M. Ziman (1961) and its extensions: the resistivity is an integral over the squared electron-ion pseudopotential (an effective weak scattering potential for the electron-ion interaction) times the static structure factor S(q)S(q), the x-ray/neutron diffraction measure of spatial correlations between atoms in the liquid. This Faber-Ziman formula works quantitatively for simple (normal) liquid metals; its assumptions are strained for semimetallic melts, where scattering is strong and covalent local order survives. Experimentally, the effective number of conduction electrons in a liquid is probed by the Hall coefficient (the transverse voltage induced by a current in a magnetic field), which in many liquid metals matches the free-electron value for the metal's valence. The modern first-principles route computes the conductivity from the Kubo-Greenwood formula (a linear-response expression evaluated from the electronic eigenstates) on atomic configurations generated by density-functional-theory molecular dynamics (DFT-MD). Mott's 1934 question for the abnormal metals — a quantitative account of why and by how much these metals conduct better as liquids than as solids — survives, in narrowed form, at the intersection of these methods: no work combines them into a quantitative theory, in agreement with experiment, of the melting anomaly of Bi, Ga and Sb.

Problem Statement

Provide a quantitative microscopic theory of the electrical transport of the abnormal metals — bismuth, gallium and antimony — across the melting transition, succeeding where Mott's 1934 treatment succeeded only for normal metals. The objective is a quantitative description that agrees with experiment, as Mott's 1934 formula did for normal metals, of why and by how much the resistance of these semimetallic poor solid conductors changes at the melting point: the predicted sign and magnitude of the resistivity change on melting for each of Bi, Ga and Sb, i.e. the solid and liquid resistivities near the melting point and their ratio, delivered by the same microscopic theory rather than by separate fits. Agreement with experiment here means consistency, for each of Bi, Ga and Sb, with the measured resistivities on both sides of the melting point reported in the cited experimental references: the orientation-averaged solid resistivity where the solid is anisotropic, together with the liquid resistivity just above the melting point, within the scatter of those measurements. The effective number of conduction electrons in the melt and the role of local (crystal-like) atomic order in the liquid are physically relevant context for such a theory, but the open question posed by the source is the quantitative account of the melting resistance change itself.

References

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