Microscopic origin of the pseudogap and the giant BCS ratio $2\Delta(0)/k_BT_c$ in strongly underdoped $\mathrm{Bi}_2\mathrm{Sr}_2\mathrm{CaCu}_2\mathrm{O}_{8+\delta}$: quantitative validation of Abrikosov's resonant-tunneling-center and phase-fluctuation mechanism

The open question left by A. A. Abrikosov's 1997 explanation of the giant BCS ratio $2\Delta(0)/k_BT_c>20$ in strongly underdoped Bi2Sr2CaCu2O8+δ: what is the microscopic origin of the pseudogap and its remarkable thermal stability, and is the proposed mechanism — $T_c$ suppressed toward the Berezinskii–Kosterlitz–Thouless scale by loss of resonant-tunneling-center-mediated interlayer coupling while a local gap $\Delta(0)$ persists without long-range order — correct at the quantitative level? The source paper leaves the resonant centers unidentified (suspected misplaced oxygen atoms), their concentration $c_j$ and coupling $\delta_0$ undetermined, the phenomenological constants ($b$, $\alpha$) and the validity of the Ginzburg–Landau/Efetov–Larkin treatment far below the mean-field scale unfixed, and no direct test performed. The qualitative phase-fluctuation paradigm has become mainstream and components of it have been tested piecemeal, but no work to date identifies the centers, determines the parameters, quantitatively tests the $T_c$ versus interlayer-coupling relation, or settles the pseudogap origin: the problem remains open.

Background

High-temperature superconductivity in the copper oxides ('cuprates') arises in weakly doped CuO2 planes separated by insulating blocks; the number of doped holes ('doping') tunes the critical temperature $T_c$ along a dome, and samples on the low-doping side of the dome are called underdoped. In weak-coupling BCS (Bardeen–Cooper–Schrieffer) theory the zero-temperature energy gap $\Delta(0)$ — the energy needed to break a Cooper pair — and $T_c$ are locked together, giving the universal 'BCS ratio' $2\Delta(0)/k_BT_c=3.52$. Cuprates violate this: the ratio is about 5 at optimal doping, and angle-resolved photoemission spectroscopy (ARPES — an experiment resolving electron energy as a function of momentum) on strongly underdoped $\mathrm{Bi}_2\mathrm{Sr}_2\mathrm{CaCu}_2\mathrm{O}_{8+\delta}$ (Bi2212) found values exceeding 20, together with a pseudogap — a partial gap in the one-particle spectrum that persists to temperatures far above $T_c$ and is remarkably stable against warming (Ding et al. 1996; Loeser et al. 1996).

Conventional explanations tie a large BCS ratio either to strong electron–boson coupling (Eliashberg theory, the strong-coupling generalization of BCS, in which the ratio can exceed 3.52) or to 'preformed pairs' — locally bound pairs forming at a high temperature and condensing only at a much lower Bose-condensation temperature. Both routes require substantial changes in the interaction strength or in the Fermi energy; but the same ARPES experiments found essentially no shrinkage of the Fermi surface (the boundary between occupied and unoccupied electron states in momentum space) upon underdoping, making the Fermi energy — and with it both explanations — problematic.

Abrikosov (1997) proposed instead a phase-fluctuation mechanism. Its ingredients: in a superconductor with small superfluid density, as emphasized by Emery and Kivelson (1995), the phase of the superconducting order parameter fluctuates strongly, so that pairing (mean-field scale $T_c^{(0)}$) and global phase coherence (the true $T_c$) need not coincide; in an isolated two-dimensional superfluid the transition occurs by the Berezinskii–Kosterlitz–Thouless (BKT) mechanism — the unbinding of vortex–antivortex pairs at $T_{\mathrm{BKT}}=\pi n_s/2m$, with $n_s$ the two-dimensional superfluid density; and the Ginzburg–Landau (GL) free energy — the standard expansion of the superconducting free energy in powers of the order parameter — supplies the phase-only description. Abrikosov's specific addition concerns the interlayer coupling. The $c$-axis (out-of-plane) conductance of Bi2212 had been argued to be dominated by resonant tunneling — tunneling dramatically enhanced when the electron energy coincides with a localized defect state situated between the CuO2 bilayers — with the centers suspected to be wrongly positioned ('misplaced') oxygen atoms that are destroyed first under oxygen depletion. Drawing on his earlier work he took the effective interlayer coupling as $\gamma=\delta_0 c_j^2$ (with $\gamma$ the dimensionless interlayer coupling, distinct from the oxygen excess $\delta$ of the formula $\mathrm{Bi}_2\mathrm{Sr}_2\mathrm{CaCu}_2\mathrm{O}_{8+\delta}$), with $c_j$ the atomic concentration of resonant centers and $\delta_0\ll 1$ a microscopic constant, and estimated, using the GL functional combined with the Efetov–Larkin mean-field treatment of phase fluctuations (a self-consistency method, introduced in 1974 for quasi-one-dimensional superconductors, that computes how fluctuations depress the transition temperature), that decoupling the bilayers drives the transition toward

$$T_c \sim \frac{\varepsilon_0}{[\ln(1/\gamma)]^2},$$

where $\varepsilon_0=\pi n_s^{(0)}/2m$ is set by the electron density associated with the extended saddle points (nearly flat bands of the Fermi surface near $(\pi,0)$, whose dominance was a key assumption of his theory). In the strongly decoupled limit $T_c\approx T_{\mathrm{BKT}}\ll T_c^{(0)}$, while the one-particle gap $\Delta(0)$ — a local spectral feature that does not require long-range order, just as long-range order does not guarantee a one-particle gap — survives essentially unchanged: this is the pseudogap. The same estimate predicts that $T_c$ stays pinned near $T_c^{(0)}$ unless $\gamma$ is very small, explaining why $T_c$ is reproducible across samples even when the normal-state $c$-axis conductivity varies strongly.

The paper closes by acknowledging what it does not explain: the microscopic origin and temperature stability of the pseudogap ('We have no explanation at present of the pseudogap and its stability with temperature'); the identity, concentration, and coupling strength of the resonant centers; the numerical constants entering the estimates — the constant $b$ in the BKT correlation length $\xi=\xi_0\exp\{b[T_{\mathrm{BKT}}/(T-T_{\mathrm{BKT}})]^{1/2}\}$ and the parameter $\alpha\sim T_c^{(0)2}/\mu_1$, with $\mu_1$ the chemical potential measured from the saddle point — together with the validity of the GL/Efetov–Larkin treatment in the regime $T_c\ll T_c^{(0)}$ far from the GL region of applicability; and the absence of any direct experimental test of the mechanism.

Problem Statement

Determine, at the microscopic and quantitative level, why the superconducting transition temperature and the one-particle energy gap decouple so strongly in strongly underdoped $\mathrm{Bi}_2\mathrm{Sr}_2\mathrm{CaCu}_2\mathrm{O}_{8+\delta}$ — where ARPES measures $2\Delta(0)/k_BT_c>20$ with little change of the Fermi surface — and, in particular, establish or refute Abrikosov's proposed explanation: suppression of $T_c$ by phase fluctuations (down toward the Berezinskii–Kosterlitz–Thouless scale $T_{\mathrm{BKT}}$) caused by oxygen-depletion-driven loss of the localized resonant tunneling centers that mediate interlayer coupling, with the local gap $\Delta(0)$ (the pseudogap) persisting without long-range order. The objective is a quantitatively tested microscopic account of the decoupling of the pairing gap from $T_c$ and of the pseudogap in strongly underdoped cuprates, comprising the following linked components:

(1) a microscopic identification of the origin of the pseudogap and of its stability with temperature — local pairing without phase coherence, a competing or intertwined order, or another mechanism;

(2) a microscopic identification of the localized resonant tunneling centers (the misplaced-oxygen hypothesis or an alternative), with the resonant-center concentration $c_j$ and the microscopic coupling constant $\delta_0$ entering the assumed relation $\gamma=\delta_0 c_j^2$ for the effective interlayer coupling — or a derived replacement of that relation — determined from first principles or from independent experiment;

(3) fixed values or derivations of the phenomenological constants entering the $T_c$ estimate — the constant $b$ in the Berezinskii–Kosterlitz–Thouless correlation-length formula $\xi=\xi_0\exp\{b[T_{\mathrm{BKT}}/(T-T_{\mathrm{BKT}})]^{1/2}\}$ and the parameter $\alpha\sim T_c^{(0)2}/\mu_1$, where $T_c^{(0)}$ is the mean-field pairing scale and $\mu_1$ the chemical potential measured from the extended saddle points of the Fermi surface — together with an assessment of the range of validity of the Ginzburg–Landau/Efetov–Larkin mean-field treatment in the regime $T_c\ll T_c^{(0)}$ where the mechanism operates, so that accurate numerical predictions for $T_c$, $\Delta(0)$, and $c$-axis transport as functions of oxygen content and doping become possible for real materials.

Settling this objective requires the three components together; establishing that only one component holds (for example, that phase fluctuations suppress $T_c$ in some regime) does not by itself settle whether the resonant-center mechanism explains the giant BCS ratio and the pseudogap of strongly underdoped Bi2212.

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