Mott's spin-bipolaron scenario for the cuprate superconductors: pair binding, spin-flip-limited diffusion, crossover, disorder, and preformed pairs above $T_{\mathrm{c}}$

Mott's 1992 proposal that the normal-state carriers of the cuprate superconductors are spin bipolarons — tightly bound bosonic pairs of spin polarons, each spin polaron being a doped hole dressed by a local reorientation of the antiferromagnetic background — accounts for the linear-in-temperature resistivity through Vigren's spin-flip-limited diffusion law $D = C\,\omega_N a^{2}$ (temperature-independent $D$ combined with the Einstein relation) and for superconductivity through Bose condensation that crosses over to BCS-like behavior at higher doping. The paper and its 1990/1991 companions left the scenario's central elements open: the microscopic binding mechanism of the two spin polarons, any direct experimental test of the diffusion law, the quantitative crossover at $x > x_0$, the disorder boundary between diffusing bosons and trapped or free charge-$2e$ particles with its thermopower and Hall signatures, and whether pairs persist above $T_{\mathrm{c}}$. The open problem is the microscopic and experimental establishment or refutation of stable, mobile spin-bipolaron binding at cuprate parameters and of its spin-flip-limited diffusion.

Background

The copper-oxide (cuprate) superconductors are layered materials whose undoped parent compounds are antiferromagnetic insulators: in the quasi-two-dimensional CuO$_2$ planes, the copper spins order antiferromagnetically (neighboring spins pointing in opposite directions) below the Néel temperature $T_N$, and mobile charge carriers appear only when holes are introduced by chemical substitution, at concentration $x$ per copper site. (The parent compound is a Mott insulator: it is insulating because of electron-electron repulsion, not because all bands are filled.) Several normal-state properties of the doped compounds resist the standard theory of metals: the electrical resistivity $\rho$ is approximately proportional to the absolute temperature $T$ over a wide range, with almost no residual (temperature-independent) component; the thermopower, the voltage a material generates across a temperature gradient (also called the Seebeck coefficient), is unusually small; and the superconducting pair size (coherence length) is only a few lattice spacings, far shorter than in conventional superconductors.

In a series of papers of which the 1992 source is the transport-focused member, Nevill Mott proposed that these facts follow if the doped carriers are spin polarons and spin bipolarons. A spin polaron (or magnetic polaron) is a charge carrier that drags with it a local reorientation of the surrounding antiferromagnetic spin background, moving through the lattice by successive spin reorientations at its boundary rather than by ordinary band motion. A spin bipolaron is a tightly bound pair of two spin polarons and hence a boson (a particle of integer spin), here of charge $2e$ where $e$ is the elementary charge. For carrier concentration below a characteristic value $x_0$ at which the pair wavefunctions begin to overlap, Mott assumed that above the superconducting transition temperature $T_{\mathrm{c}}$ these bosons form a non-degenerate (classical, Maxwell–Boltzmann) gas, so that superconductivity on cooling is a Bose–Einstein condensation (the macroscopic occupation of a single quantum state by bosons) of preformed pairs, rather than the Bardeen–Cooper–Schrieffer (BCS) mechanism in which pair formation and condensation occur together at $T_{\mathrm{c}}$.

Two quantitative pillars carry the picture. First, Vigren's 1973 calculation of the mobility of a self-trapped paramagnetic spin polaron found the diffusion coefficient $D$ to be set by the dynamics of the spin background rather than by thermal activation, scaling as

$D = C\,\omega_N a^{2},$

where $C$ is a dimensionless constant of order unity, $\omega_N$ is a characteristic spin-flip angular frequency of the antiferromagnet (of order $k_{\mathrm{B}}T_N/\hbar$, with $k_{\mathrm{B}}$ Boltzmann's constant and $\hbar$ the reduced Planck constant), and $a$ is the displacement per hop, of order a lattice spacing; $D$ is then approximately temperature-independent. Mott assumed the same spin-flip-limited law for the center-of-mass diffusion of a bipolaron. Second, the Einstein relation between conductivity and diffusion for a classical gas of charged particles,

$\sigma = \frac{N q^{2} D}{k_{\mathrm{B}} T},$

with $N$ the pair density and $q = 2e$, then gives $\sigma \propto 1/T$, hence $\rho \propto T$ with no residual resistivity, since a particle that moves by spin flips is not scattered by impurities in the usual quasiparticle sense. The 1992 paper adds a phenomenology of disorder: a single trap parameter $W$, characterizing the depth of potential wells that can immobilize a pair, produces an additive residual-like contribution to the resistivity, used to interpret substituted (for example Zn- or Co-doped) and thin-film samples; in the strong-disorder limit the pairs are predicted to behave instead as free non-degenerate particles with thermopower $S \propto (k_{\mathrm{B}}/2e)\ln T$ and an approximately temperature-independent Hall coefficient (the proportionality between transverse voltage, current, and magnetic field), whereas localized pairs that occupy lattice sites obey the Heikes form $S = (k_{\mathrm{B}}/2e)\ln[(1-z)/z]$ with $z$ the fraction of occupied sites. The observed dome-shaped curve of $T_{\mathrm{c}}$ versus $x$ is read as a crossover: beyond $x_0$, overlapping pairs should recover BCS-like behavior.

The rival descriptions contemporary with Mott kept the carriers fermionic: in the spin-bag picture of Schrieffer, Wen, and Zhang, two holes attract because each locally disturbs the antiferromagnetic order, but the pairs remain momentum-space Cooper pairs rather than real-space bosons; and the later mainstream reading of the cuprates — a large Fermi surface and a predominantly $d$-wave order parameter (a superconducting gap that changes sign under a $90^{\circ}$ rotation of the crystal axes) — sits uneasily with a simple real-space-pair condensate. What the source paper and its 1990/1991 companions defer, and what therefore constitutes the open problem, is everything on which the picture rests: the binding of the two spin polarons is assumed rather than derived, with the relative roles of phonons and spin exchange undetermined; the transfer of Vigren's law from single polarons to bipolarons is untested; and the crossover, disorder, and above-$T_{\mathrm{c}}$ pair physics are qualitative.

Problem Statement

Determine whether Mott's spin-bipolaron scenario for the cuprate superconductors is correct — establish it or refute it at the microscopic and experimental level. The 1992 source paper and its 1990/1991 companions leave open five linked questions about the same scenario:

(1) Pair binding. Derive from a microscopic Hamiltonian of the doped antiferromagnetic CuO$_2$ plane — or exclude by controlled calculation or experiment — the existence of a stable, mobile, tightly bound pair of two spin polarons, of spatial extent comparable to or smaller than the superconducting coherence length; quantify the separate contributions of the electron-phonon interaction and the antiferromagnetic spin exchange to the binding, and the resulting pair effective mass.

(2) Diffusion law. Determine the diffusion coefficient of such pairs and test whether it is spin-flip-limited and approximately temperature-independent, $D = C\,\omega_N a^{2}$ with $C$ of order unity.

(3) Crossover. Provide the quantitative microscopic description, missing in the source, of the regime $x > x_0$ in which the pair wavefunctions overlap and a BCS-like description should emerge, including the prediction of $x_0$ and of the dome-shaped dependence of $T_{\mathrm{c}}$ on $x$.

(4) Disorder and transport discriminants. Delineate quantitatively the conditions under which strong disorder (substitutional defects, chain disruption, film boundaries) traps the pairs or makes them behave as free charge-$2e$ particles rather than diffusing bosons, and test the predicted thermopower and Hall signatures of the two regimes across cuprate materials and sample qualities.

(5) Pairs above $T_{\mathrm{c}}$. Determine whether the pairs persist as distinct entities above $T_{\mathrm{c}}$ in the cuprates, as the bosonic-condensation picture requires, and reconcile the picture with spectroscopic probes of pair formation in the pseudogap regime (the partially gapped normal state above $T_{\mathrm{c}}$) and with the established $d$-wave symmetry of the superconducting order parameter.

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