Pairing mechanism of high-temperature superconductivity in the cuprates: the origin of the pairing interaction and the sufficiency of BCS-type pairing theory (from Bardeen's 1987 open questions)

In his 1987 review of the then-new 90 K superconductor YBa$_2$Cu$_3$O$_{7-\delta}$, John Bardeen posed four linked open questions: which structural unit, the CuO chains or the CuO$_2$ planes, carries the large (about $8 k_B T_c$) and small (about $2.4 k_B T_c$) superconducting energy gaps inferred from nuclear magnetic resonance; what the exact nature of the required non-phonon pairing interaction is; whether a purely phonon mechanism can account for transition temperatures near 100 K; and whether a BCS-type pairing description suffices or a fundamentally new theory such as the resonating valence bond model is necessary. Nearly four decades of subsequent literature resolved the first sub-question (single-crystal phase-sensitive and photoemission experiments establish that the large, momentum-dependent $d_{x^2-y^2}$ gap resides on the CuO$_2$ planes, while the chain band carries a distinct smaller gap), and first-principles electron-phonon calculations disfavor phonons as the sole pairing glue. The core of the question, however, remains open: the microscopic interaction that binds the Cooper pairs in these doped Mott insulators has not been identified, and whether a (generalized) BCS pairing framework suffices or a fundamentally different theory is required is undecided. The problem is to identify the pairing interaction and theoretical framework consistent with the observed high transition temperatures and the small oxygen isotope effect.

Background

Superconductivity, the lossless flow of electrical current below a critical temperature $T_c$, was explained by Bardeen, Cooper and Schrieffer (BCS) in 1957: electrons form bound pairs (Cooper pairs) via an attractive interaction that, in conventional superconductors, is mediated by lattice vibrations (phonons). In the BCS ground state an energy gap $\Delta$ opens at the Fermi surface; weak-coupling theory predicts a universal ratio $2\Delta \approx 3.5 k_B T_c$ for an isotropic gap. A classic diagnostic of phonon mediation is the isotope effect: replacing an element by a lighter or heavier isotope changes $T_c$ as $T_c \propto M^{-\alpha}$ with exponent $\alpha \approx 0.5$ when phonons do the pairing, because the phonon frequencies depend on the ionic mass $M$ while the electronic structure does not. In 1986 Bednorz and Müller discovered superconductivity near 30 K in La-Ba-Cu-O ceramics, and within months YBa$_2$Cu$_3$O$_{7-\delta}$ raised $T_c$ to about 93 K, above the boiling point of liquid nitrogen. YBa$_2$Cu$_3$O$_{7-\delta}$ has a layered structure: superconducting current flows in two-dimensional CuO$_2$ sheets (planes) and in one-dimensional Cu-O ribbons (chains), with conductivity about a hundred times larger along the layers than perpendicular to them, and with a superconducting coherence length perpendicular to the planes of only a few angstroms (about 7 angstroms in the 1987 estimates available to Bardeen), so chains and planes behave as a coupled system with a single common $T_c$. The parent compounds of these materials are Mott insulators, that is, magnetic insulators produced by strong electron-electron repulsion rather than by filled bands, and doping them with charge carriers produces the superconducting phase. Against this background, John Bardeen's 1987 MRS Proceedings review examined what could be said about the pairing origin in the new oxides. Nuclear magnetic resonance (NMR) measurements of the nuclear spin-lattice relaxation rate, which probes how fast nuclear spins relax by coupling to thermally excited electronic quasiparticles above the superconducting gap, had revealed two distinct relaxation behaviors for chain and plane copper sites, from which two pairing energies were inferred, a small gap of about $2.4 k_B T_c$ and a large gap of about $8 k_B T_c$ (roughly 19 and 63 meV for $T_c \approx 90$ K), with a controversial assignment to the two sites. The oxygen isotope effect in YBa$_2$Cu$_3$O$_{7-\delta}$ was strongly suppressed relative to the La-based cuprates, suggesting that phonons alone could not be the whole story. Bardeen proposed a two-fluid thermodynamic model coupling chains and planes to a common transition temperature, $T_c^2 = \beta T_1^2 + (1-\beta) T_2^2$ (with $T_1$ and $T_2$ the intrinsic transition temperatures of the two fluids and $\beta$ a mixing weight), tentatively assigning the large gap to the chains (intrinsic chain transition temperature near 180 K, requiring a non-phonon interaction) and the small gap to the planes (phonon-mediated, intrinsic transition temperature near 45 K), which within his model also dilutes the plane isotope exponent to the observed value near 0.05. He set this against Anderson's resonating valence bond (RVB) proposal, in which the doped antiferromagnetic Mott insulator's singlet valence bonds (spin pairs on neighboring lattice sites that resonate quantum-mechanically among many configurations), once mobile charge is introduced, provide a route to pairing that is not of the BCS type. Much is now established that was not known in 1987, and it defines the constraints on any answer. Phase-sensitive Josephson tunneling experiments established that the order parameter has predominantly $d_{x^2-y^2}$ symmetry, meaning the gap changes sign under a 90-degree rotation of the crystal and vanishes along the diagonal (nodal) directions; this symmetry resides on the CuO$_2$ planes. Angle-resolved photoemission spectroscopy (ARPES), which measures the electronic spectral function as a function of momentum, mapped the gap directly on single crystals: a strongly momentum-dependent, d-wave-like plane gap whose antinodal (maximum) value grows with underdoping, and a quasi-one-dimensional chain band that carries a distinct, smaller gap, with chain-derived states developing a pseudogap (a partial suppression of electronic states that appears above $T_c$) tied to charge order in underdoped samples. First-principles calculations of the electron-phonon interaction found the phonon-induced renormalization about an order of magnitude smaller than the observed 50-80 meV dispersion anomaly (the photoemission kink), disfavoring phonons as the exclusive pairing mechanism. What is not established is the identity of the pairing glue, the interaction that actually binds the pairs, and the correct theoretical framework: the normal state above $T_c$ is a non-Fermi-liquid strange metal (a metallic state whose properties violate the standard quasiparticle picture of ordinary metals) with a puzzling pseudogap regime, and no interaction has been identified that quantitatively produces the high transition temperatures.

Problem Statement

Identify the microscopic pairing mechanism of high-temperature superconductivity in the cuprates, the question posed (but left open) in Bardeen's 1987 review of YBa$_2$Cu$_3$O$_{7-\delta}$. The question has three linked parts: determine the exact nature of the interaction that binds the Cooper pairs, that is, whether it is phonon-mediated, non-phonon (electronic or magnetic), or a combination, and what that interaction is; establish whether a purely phonon mechanism can account for transition temperatures near 100 K (the observed transitions being at about 90 K); and decide whether a BCS-type pairing description, generalized to an anisotropic $d_{x^2-y^2}$ order parameter, is sufficient, or whether a fundamentally different theory such as the resonating valence bond theory of the doped Mott insulator is necessary. A generalized (anisotropic) BCS-type pairing description suffices when it is quantitatively consistent with the pairing observables (the pairing energy scale at the transition temperature near 90 K at optimal doping, the momentum-dependent $d_{x^2-y^2}$ plane gap and the small oxygen isotope exponent) and at least qualitatively consistent with the non-Fermi-liquid normal state these materials show above $T_c$, and a fundamentally different theory is necessary only if no such description can. The pure-phonon part is the special case in which the identified interaction is phonon-mediated; when the identified interaction is not phonon-mediated, whether a purely phonon mechanism could reach transition temperatures near 100 K remains a separate part of the question. Consistency with the observed high transition temperatures, about 90 K at optimal doping in YBa$_2$Cu$_3$O$_{7-\delta}$, and with the small oxygen isotope effect is a constraint the source itself places on any proposed mechanism. Later-established experimental facts constrain candidate mechanisms as established context: the doping dependence of the transition temperature; the large, momentum-dependent, d-wave-like superconducting gap on the CuO$_2$ planes together with the distinct smaller gap on the CuO chains (the resolved part of the source's gap-assignment question, established by phase-sensitive and single-crystal photoemission experiments); and the momentum and doping dependence of the gap. The gap-assignment sub-question raised alongside this in the source (which of chains or planes carries the large gap) has been settled by later single-crystal work; the open target is the identification of the pairing interaction and the theoretical framework. No such identification or decision exists in the literature to date, so the question remains open. The question is to identify the proposed pairing interaction and its mediator or order-parameter symmetry, to reach a verdict on whether a purely phonon mechanism can account for transition temperatures near 100 K, and to reach the framework verdict of the third part.

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