Completing and testing Abrikosov's resonant interlayer tunneling scenario for the cuprate pseudogap: microscopic derivation, granular-phase dynamics, parameter determination, and experimental verification

In 2001 A. A. Abrikosov proposed an explanation of the pseudogap regime of high-temperature cuprate superconductors built on three ingredients: resonant electron tunneling between CuO2 planes through localized oxygen-centered states in the intermediate layers, which sets the temperature of three-dimensional interlayer phase coherence; disorder-induced pair breaking of d-wave superconductivity, which suppresses the single-layer pairing temperature; and quasi-one-dimensional order-parameter fluctuations near the extended saddle points ('hot spots') of the Fermi surface, which let locally superconducting grains survive above the macroscopic transition, giving a spatially inhomogeneous, percolative pseudogap phase with vortex-like excitations. The construction was qualitative: $T_c$ entered through a phenomenological interpolation formula, the granular phase was described only as a physical picture, parameters were illustrative, and the central approximations were asserted rather than computed. The problem is to complete and test this scenario: derive the interlayer-coherence relation for $T_c$ and the assumed proportionality between resonant-center concentration and quasiparticle scattering from microscopic electronic structure and dopant chemistry; develop the time-dependent statistical description of the granular pseudogap phase (grain statistics, phase dynamics, vortex kinetics); determine the model parameters and fit the cuprate phase diagram in the setting the paper treats; compute the Ginzburg-Landau cross-term parameter $\eta$ underlying the one-dimensional fluctuation approximation; and verify the predicted nanoscale granular structure and test the oxygen-resonant-state attribution of the 'c gap' experimentally.

Background

The high-temperature cuprate superconductors (for example Bi2Sr2CaCu2O8+δ, YBa2Cu3O7−δ, La2-xSrxCuO4) are layered oxides in which superconductivity appears in the CuO2 planes upon hole doping. Their phase diagram is nonmonotonic: the superconducting transition temperature $T_c$ rises from low doping to a maximum near optimal doping and falls again on the overdoped side. Above $T_c$ on the underdoped side lies the 'pseudogap' regime: angle-resolved photoemission spectroscopy (ARPES, a technique measuring the electrons' energy and momentum distribution) observes a partial, strongly momentum-dependent suppression of spectral weight resembling the superconducting $d$-wave gap (a gap that changes sign under a 90-degree rotation within the CuO2 plane; Ding et al. 1996; Loeser et al. 1996); nuclear magnetic resonance sees a reduced spin susceptibility; and the Nernst effect (an electric field induced by a transverse temperature gradient in a magnetic field) signals mobile vortex-like excitations well above $T_c$ (Xu et al. 2000). The pseudogap opens at a crossover temperature $T^*$ that decreases with doping and merges with $T_c$ near optimal doping. Whether the pseudogap is a superconducting precursor (preformed electron pairs whose phases fluctuate, as in short-coherence-length superconductors; Randeria et al. 1992) or a competing order remains among the central open questions of the field (Keimer et al. 2015; Vishik 2018).

A. A. Abrikosov developed, through the late 1990s, a scenario combining his extended saddle-point model with resonant tunneling, summarized in his Nobel lecture (Abrikosov 2004). The cuprate band structure was taken to contain anomalously flat regions - extended van Hove (saddle-point) singularities near the $(\pi,0)$ points of the Brillouin zone - which the 2001 paper calls 'hot spots' (a different usage from the modern one, where hot spots denote intersections of the Fermi surface with the antiferromagnetic zone boundary); near them quasiparticle motion is effectively one-dimensional. Electrons cannot tunnel coherently between neighboring CuO2 planes except through localized resonant states at oxygen atoms of the intermediate layers (BiO planes or CuO chains): such a resonant center, whose wave functions overlap both planes, mediates coherent interlayer tunneling. Three-dimensional superconductivity then requires interlayer phase locking, controlled by the surface concentration $c$ of resonant centers, with coherent tunneling lost below a critical concentration $c^{(0)}$. At the same time the dopant oxygens scatter quasiparticles with probability $w=1/\tau$ (proportional to the concentration of dopant defects), breaking pairs of a $d$-wave order parameter: disorder suppresses the single-layer pairing temperature $T^*$ through the standard pair-breaking relation (with the digamma function $\psi$; taken by the paper as formula (9) of Abrikosov 1996)

$$\ln\frac{1}{y}-\psi\!\left(\frac{1}{2}+\frac{x}{\pi y}\right)+\psi\!\left(\frac{1}{2}\right)=0,\qquad y=\frac{T^*}{T_{c0}},\;x=\frac{w}{T_{c0}},$$

where $T_{c0}$ is the clean-layer value. The competition between improving interlayer coherence and degrading intralayer pairing as doping increases produces the nonmonotonic $T_c$(doping) dome.

In Abrikosov (2001) these elements were combined into a proposal for the pseudogap phase itself. Between $T_c$ and $T^*$ the pairing amplitude $|\Psi|$ stays finite inside local 'grains' while the interlayer phase correlation $\langle e^{i\varphi}\rangle$ is lost. Because fluctuations at the hot spots are one-dimensional, Rice's classic result on one-dimensional superconductivity (Rice 1965) gives an in-plane correlation length

$$X_0\sim\frac{\hbar\Delta^2 v}{T^{*2}T}$$

($\Delta$ the local gap amplitude, $v$ the hot-spot quasiparticle velocity, $\hbar$ the reduced Planck constant), which at temperatures well below $T^*$ exceeds the coherence length $\xi\sim\hbar v/T^*$ and produces grains large and long-lived enough to support vortex-like excitations - explaining the Nernst signal - while thermodynamic quantities vary smoothly through the resistive $T_c$, the macroscopic transition being a percolation threshold reached when the superconducting area fraction is about one half. For $T_c$ the paper postulates the interpolation formula

$$T_c=\left[T^{*-1/2}+\left(c^{(0)}/c\right)\mu_1^{-1/2}\right]^{-2},$$

where $\mu_1$ is the Fermi energy at the hot spots, together with the proportionality $c=aw$ between resonant-center concentration and scattering probability; in dimensionless variables this yields a dome $z=(1/\sqrt{y}+A/x)^{-2}$ with a single parameter $A=(c^{(0)}/a)(T_{c0}\mu_1)^{-1/2}$, plotted for the illustrative value $A=0.1$. In the Ginzburg-Landau functional (the standard phenomenological free-energy description of superconductivity) for the two saddle-point components $\Psi_a$, $\Psi_b$ (constructed in Abrikosov 1997), the cross term $\alpha\eta|\Psi_a+\Psi_b|^2$ is neglected relative to the gradient terms on the estimate $\eta\ll(T/T^*)^2$. Finally, the gap seen in $c$-axis tunneling (tunneling along the crystallographic $c$ axis, perpendicular to the CuO2 planes; the 'c gap'), nearly independent of magnetic field and of temperature across $T_c$, is attributed to the energy offset between the Fermi level and the localized oxygen resonant states, making it a single-particle feature distinct from the in-plane pseudogap.

The paper presents this construction as qualitative: $T^*$ enters via pair-breaking formulas, the $T_c$ relation as an interpolation, $c^{(0)}$ only as a single-compound estimate ($4/15$ for Bi2Sr2CaCu2O8+δ, from about four resonant centers per 120 square angstroms of interlayer area), and the granular phase as a physical picture without a statistical theory. The open problem is the completion and experimental test of this scenario. For the granular-structure test the paper names scanning tunneling microscopy rather than decoration, since the structure may be too fine for the latter, feasible only if the structure does not vary too fast in time. Subsequent work by the same author has not closed the program: his 2006 percolation theory derived a related interlayer-coherence relation and a static percolation description but left the microscopic derivation, the time-dependent dynamics, parameter calibration, and mechanism-specific experimental tests open; scanning-tunneling experiments have found nanoscale pairing regions above $T_c$ (evidence for a granular pseudogap in general, not for the resonant-center mechanism); and no later work is known to have computed $\eta$, fitted phase diagrams within the framework, or spectroscopically identified the resonant oxygen states, so the scenario has been neither completed nor refuted.

Problem Statement

The problem is to convert Abrikosov's qualitative resonant-center/granular scenario for the cuprate pseudogap into a controlled, quantitative, and experimentally tested theory of the pseudogap regime and of the nonmonotonic $T_c$(doping) dome - or to refute the scenario at a specified step. The components below concern one and the same microscopic scenario and trace the open directions the 2001 paper itself states and the assumptions it explicitly labels as crude:

(i) Controlled microscopic derivation. Derive from the microscopic electronic structure of the layered cuprates and the chemistry and placement of the dopant oxygens the interlayer-coherence-controlled relation between $T_c$, the single-layer pairing temperature $T^*$, and the resonant-center concentration $c$ - the relation the 2001 paper states only as the interpolation formula $T_c=[T^{*-1/2}+(c^{(0)}/c)\mu_1^{-1/2}]^{-2}$ and its 2006 sequel derived only at the level of percolation geometry - together with the assumed proportionality $c=aw$ between resonant-center concentration and quasiparticle scattering probability, and the pair-breaking suppression of $T^*$ entering that relation.

(ii) Dynamics and statistics of the granular pseudogap phase. Develop the mathematical description of the spatially inhomogeneous, time-dependent pseudogap state - the description the paper itself identifies as its open theoretical problem, for a phase that is slowly varying in time and contains grains with a finite order parameter and mutually incoherent phases: the statistics of grain sizes and lifetimes, the phase dynamics within and between grains, the percolation geometry of the superconducting regions, and the nucleation and kinetics of vortex-like excitations in the granular medium that would generate the observed Nernst signal.

(iii) Parameter determination and phase-diagram fit. Establish quantitative values of the model parameters ($a$, $c^{(0)}$, $A$, $\mu_1$, and the microscopic relation between dopant placement and resonant-center density; the paper's fitting parameters are $T_{c0}$ and $A$) and compare the resulting dimensionless dome $z=(1/\sqrt{y}+A/x)^{-2}$ and the pair-breaking suppression of $T^*$ with the measured $T^*(p)$ and $T_c(p)$ curves (with $p$ the hole doping) for Bi2Sr2CaCu2O8+δ - the compound for which the paper estimates $c^{(0)}=4/15$ from the 120-square-angstrom characteristic area and the four-center assumption - and for the analogous layered cuprates in which the dopant-oxygen interlayer picture applies - those whose intermediate layers are themselves the resonant oxygen centers, the BiO planes of the bismuth cuprates or the CuO chains of YBa2Cu3O7−δ; extension to cuprate families whose intermediate layers are not such resonant oxygen centers, for example the single-layer La2-xSrxCuO4, is optional and not part of the target.

(iv) Justification of the hot-spot fluctuation approximation. Justify the assumption that quasi-one-dimensional fluctuations at the extended saddle points ('hot spots') dominate the pseudogap regime - in particular compute the parameter $\eta$ of the Ginzburg-Landau cross term $\alpha\eta|\Psi_a+\Psi_b|^2$ and establish the regime in which the cross terms can be neglected (the paper assumes $\eta\ll(T/T^*)^2$ except very near $T_c$).

(v) Experimental verification. Specify, on the length and time scales the scenario predicts, the measurement that would establish the nanostructured (granular) composition of the pseudogap phase and the signature it should register, and state the test of the mechanism's attribution of the $c$-axis tunneling ('c') gap to the offset between the Fermi level and the localized oxygen resonant states of the intermediate layers, whose predicted near-independence of magnetic field and of temperature across $T_c$ distinguishes it from the in-plane pseudogap; compare the cited spatially resolved and interlayer-tunneling data (Gomes et al. 2007; Suzuki et al. 2001) against these predictions. Performing new experiments lies outside the scope of the analysis.

The scenario may be ruled out at a specified step: the independently testable steps are the proportionality $c=aw$ between resonant-center concentration and quasiparticle scattering probability (item i), the interlayer-coherence relation for $T_c$ (item i), the pair-breaking suppression of $T^*$ (item i), the neglect of the Ginzburg-Landau cross term under $\eta\ll(T/T^*)^2$ in the one-dimensional-fluctuation approximation (item iv), and the attribution of the $c$-axis tunneling gap to the localized oxygen resonant states (item v). A documented failure of any one of these steps refutes the scenario as stated at that step; item (v) identifies the decisive experimental signature that would confirm or refute the mechanism the 2001 paper itself proposes.

References

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