Abrikosov (1978) proposed a route to superconductivity at temperatures of up to several hundred kelvin in semiconductors or semimetals with a small or negative indirect gap and very heavy holes: the holes crystallize into a periodic 'excitonic' lattice whose vibrations mediate pairing in a coexisting electron Fermi sea, a phase he called metallic excitonium, with CuCl under pressure suggested as the candidate. Later literature settled the CuCl instantiation negatively: band-structure calculations (1979) found no indirect-gap configuration of the type required, and the reported near-100 K diamagnetism was refuted as pressure-induced disproportionation and enhanced ionic conduction (1980–1981), with the original signals still unreproduced as of 2015. What remains open is the question in its original generality: whether a controlled microscopic theory of the crystallized heavy-hole phase — its lattice dynamics, its coupling to the electron gas, and its stability against recombination, melting, and screening — yields a defensible superconducting transition temperature, and whether any real compound with a flat d- or f-derived valence band satisfies the band-structure conditions for the mechanism. Hole crystallization in flat-valence-band semiconductors is itself an established phenomenon, and exciton condensates and superconductivity in excitonic insulators have now been observed, but no result combines a crystallized excitonic phase with a coexisting metal in the way the mechanism requires, and no impossibility argument exists.

Background

A bound electron–hole pair in a semiconductor is an exciton. Halperin and Rice (1968) showed that a semiconductor with a small band gap, or a semimetal with a small overlap of the conduction and valence bands, can lower its energy by spontaneously forming a condensate of such pairs, producing an insulating ground state called an excitonic insulator. This made 'excitonic' routes to superconductivity attractive: Little (1964) proposed electron pairing mediated by electronic excitations along polarizable organic chains, and Allender, Bray and Bardeen (1973) proposed that excitons at a semiconductor–metal interface could supplement phonons as the pairing 'glue' in the Bardeen–Cooper–Schrieffer (BCS) framework, whose superconducting transition temperature TcT_c is bounded by the characteristic energy of the mediating boson.

Abrikosov (1978) proposed a distinct variant. Consider a semiconductor with a small or negative indirect gap (the valence-band maximum and conduction-band minimum at different crystal momenta) and a valence band of very heavy holes (holes being the missing electrons in an almost-filled band, with an effective mass mhm_h far larger than the conduction-electron mass mem_e, in his estimate me/mh∼0.01m_e/m_h \sim 0.01). The heavy holes crystallize into a periodic lattice — the analog of a Wigner crystal, the crystalline state electrons adopt at low density when Coulomb repulsion dominates kinetic energy — with a lattice constant set by a momentum-space matching condition between the hole lattice and the conduction electrons, and the electron–hole pair bound under a Coulomb criterion comparing the attraction to the electron kinetic energy (these two conditions are stated here only qualitatively). The vibrations of this excitonic lattice then provide the pairing interaction for the remaining electrons with characteristic energies far above acoustic-phonon scales, and Abrikosov estimated a maximum TcT_c of order (mee4/ϵ02)(me/mh)1/2(m_e e^4/\epsilon_0^2)(m_e/m_h)^{1/2}, i.e. several hundred kelvin (Gaussian electromagnetic units, as in the paper). Abrikosov called candidate materials with flat d- or f-derived valence bands and hence very heavy holes 'archconductors'. The immediate motivation was the report by Brandt et al. (1978, JETP Lett. 27) of near-perfect diamagnetism (χ=−1/4π\chi = -1/4\pi, i.e. a Meissner-like response in which a superconductor expels magnetic field) in CuCl near 100 K under pressure — though zero electrical resistance was never measured in those samples.

Three critiques appeared within months: Blount and Phillips (1978) argued the CuCl anomalies arise from pressure-induced disproportionation (2CuCl→Cu+CuCl22\mathrm{CuCl} \to \mathrm{Cu} + \mathrm{CuCl_2}, forming metallic Guinier–Preston-type copper platelets giving percolative conduction and making the Meissner reports doubtful); Wilson (1978) reviewed the material facts skeptically; and Chu et al. (1978) documented the actual anomalies — resistance drops at 40–90 kbar on pressure cycling and a transient diamagnetic signal near 180 K on warming, but no zero resistance and no stable Meissner state. The open tasks acknowledged at the time were: precise band-structure calculations for CuCl and related compounds; a microscopic theory of the crystallized excitonic phase including its lattice dynamics and electron–lattice coupling; reproducible bulk superconductivity in CuCl; and a systematic search over compounds with d- or f-character valence bands.

Problem Statement

The CuCl instantiation of Abrikosov's proposal is settled negatively, and those resolutions are recorded above. The surviving question, in the original generality the source posed it, concerns whether the metallic-excitonium mechanism is real, and comprises two open directions:

  1. Microscopic theory. A controlled microscopic theory of a crystallized heavy-hole excitonic phase coexisting with a light-electron Fermi sea in a semiconductor or semimetal with a small or negative indirect gap and a flat (d- or f-derived) valence band, replacing Abrikosov's order-of-magnitude estimate of a maximum TcT_c of order (mee4/ϵ02)(me/mh)1/2(m_e e^4/\epsilon_0^2)(m_e/m_h)^{1/2} (several hundred kelvin). Resolving this direction requires all four of the following components: (a) the ground state and stability of the hole crystal against electron–hole recombination, melting into a liquid or exciton condensate, and screening by the very electrons it is meant to pair; (b) the excitonic-lattice vibrational spectrum; (c) the induced pairing interaction and resulting superconducting TcT_c (or a bound on it); and (d) explicit validity conditions for the theory.

  2. Realization. Whether any real compound — a semiconductor or semimetal with a small or negative indirect gap and a flat d- or f-derived valence band of very heavy holes, in Abrikosov's estimate me/mh<0.01m_e/m_h < 0.01 — satisfies the mechanism's conditions, so that the heavy holes crystallize into an excitonic lattice while light conduction electrons remain mobile and available for pairing. A negative conclusion, if established, would be a scientific outcome of this open question.

Together these directions address whether superconductivity mediated by the vibrations of a crystallized excitonic phase can exist in any real material, and, if so, at what temperature. A full resolution of the open question requires both directions, not one alone; a definitive negative result in either direction — an impossibility argument for the mechanism, or a demonstration that no real compound satisfies the mechanism's conditions — resolves that direction.

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