Completing the resonant-tunneling theory of c-axis conductivity and interlayer superconductivity in YBa2Cu3Ox versus oxygen content: chain-structure origin of the resonant centers, the extremes of x, anisotropic in-plane structure, and a predictive J_c

A. A. Abrikosov (Physica C 258, 53 (1996)) proposed that the anomalous out-of-plane (c-axis) normal-state conductivity of the cuprate superconductor YBa2Cu3Ox is dominated by resonant tunneling through localized centers associated with chain-oxygen disorder, and used the same mechanism to estimate the interlayer superconducting critical current density. The model fits measured c-axis conductivity only at moderate oxygen content x, and the author explicitly identified its open frontiers. The open problem is to turn this picture into a quantitatively validated microscopic theory over the full oxygen range: derive the resonant centers (density, level-energy distribution, level width) from the chain-oxygen structure so that the fitted quantities become predictions; extend the mechanism to the metallic extreme near x = 7 (chain band formation, chain-plane hybridization) and the insulating end near x = 6.3 (thermal excitation into the conduction band and ordinary, non-resonant tunneling with its temperature dependence); replace the isotropic in-plane Fermi-surface approximation with the anisotropic electronic structure; determine J_c from independently pinned parameters; and either justify microscopically the assumed linear-in-temperature in-plane quasiparticle scattering rate $1/\tau = \alpha T$ with $\alpha \ll 1$, or show which conclusions survive without it. No resolution of these questions is known in the literature to date.

Background

YBa2Cu3Ox (YBCO, $x$ between 6 and 7) is a cuprate high-temperature superconductor built from alternating superconducting CuO2 planes and one-dimensional CuO2-x chains running along the crystallographic b axis. The oxygen content $x$ is the doping control of the material: removing oxygen empties and fragments the chains, and transport measurements locate a metal-insulator transition near $x = 6.34$-$6.5$, below which superconductivity also disappears ($T_c \to 0$ near $x = 6.4$; $T_c$ is about 90 K near $x = 7$).

Charge transport in YBCO is strongly anisotropic. The in-plane (ab-plane) resistivity is metallic, while the out-of-plane (c-axis) conductivity is anomalously small and often decreases on warming like a semiconductor's, signaling incoherent interlayer transport. Resonant tunneling is a mechanism that can produce such behavior: when a localized electronic state sits inside a tunneling barrier at an energy matching the incident carrier's energy, the tunneling probability is enhanced by orders of magnitude over ordinary direct tunneling. The c-axis supercurrent is the Josephson current - the tunneling of Cooper pairs between adjacent superconducting CuO2 layers - and its density J_c measures the interlayer coupling that is central to interlayer-tunneling theories of high-Tc superconductivity.

Abrikosov (1996) proposed that c-axis transport in YBa2Cu3Ox is dominated by resonant tunneling via localized centers associated with defects of the chain structure, and used the same picture to estimate J_c. Modeling the in-plane quasiparticles as a Fermi liquid (Landau quasiparticles with well-defined energy and lifetime) with a phenomenological scattering rate $1/\tau = \alpha T$ ($\alpha \ll 1$, so that quasiparticles remain well defined) and an isotropic in-plane Fermi surface (the set of occupied quasiparticle momenta, approximated as direction-independent), the model reproduces the measured $\sigma_c(T, x)$ at moderate $x$ with a roughly $x$-independent resonance-level energy $E \approx 47$ meV (temperature scale $T_0 \approx 548$ K) and an amplitude fitted by $A \propto (x - 6.34)^{-1.62}$, with the divergence interpreted as the approach to the metal-insulator transition; the superconducting-state estimate gives $J_c \sim 10^5$-$10^7$ A/cm^2, compatible with the measured $1.5 \times 10^6$ A/cm^2 at $x = 6.7$ and 4.2 K (Rapp et al. 1996). The experimental basis is single-crystal c-axis transport data on oxygen-reduced YBCO from the Veal collaboration (cited in the source series partly as unpublished), of which the closest published relative is Chien et al. (1994).

Abrikosov himself identified the model's open frontiers, which the task set here must address: the fit degrades at both extremes - near $x = 7$, where the chains are metallic and their levels broaden into a band so that chain-plane hybridization (mixing of chain and plane electronic states, seen in photoemission) must matter, and near the insulating end, where direct thermal excitation of carriers into the conduction band and ordinary (non-resonant) tunneling compete with the resonant channel, whose temperature dependence in the insulating phase was left unexplored; the isotropic in-plane approximation ignores the real anisotropy between the $a$ and $b$ directions, which entails direction-dependent densities of states and lifetimes; the input $1/\tau = \alpha T$ has no microscopic explanation; the $J_c$ estimate rests on poorly known quantities (the resonance level width, the level energy, the interlayer barrier height, and the frequency dependence of the superconducting gap); and the identification of the resonant centers with specific chain-oxygen configurations, behind the fitted $(x - 6.34)^{-1.62}$ divergence, remains to be derived rather than fitted. Note also that the later literature uses both the resonant-center picture and competing mechanisms (boson-assisted tunneling, hopping and polaronic transport) for the c-axis anomaly.

Problem Statement

Construct and quantitatively validate a microscopic theory of the c-axis normal-state charge transport and interlayer superconducting coupling of YBa2Cu3Ox as a function of oxygen content x, completing the resonant-tunneling program beyond its self-identified limits. The single common objective - a mechanism-based, parameter-accounted microscopic description of the c-axis channel over the full oxygen range, in which the phenomenological constants of the 1996 model become derived quantities - requires the following results:

(i) Derive the resonant centers from the chain-oxygen structure: their areal density n(x), level-energy distribution, and level width, so that the empirically fitted quantities of the original analysis - the amplitude $A(x)$ including its divergence at the metal-insulator boundary $x = 6.34$ (fitted as $A \propto (x - 6.34)^{-1.62}$, an exponent the source attributes partly to a decreasing Fermi velocity rather than to n(x) alone) and the essentially x-independent resonance level energy ($E \approx 47$ meV, $T_0 \approx 548$ K) - become derived quantities of the microscopic model rather than fitted numbers, quantitatively establishing the identification of the resonant centers with the specific chain-oxygen configurations that the source argues only qualitatively.

(ii) Extend the transport mechanism to both extremes of x: near $x = 7$, where the chain levels broaden into a metallic band and chain-plane hybridization must be included; and near the insulating end ($x \lesssim 6.5$), where direct thermal excitation of carriers into the conduction band and ordinary non-resonant tunneling contribute - including the temperature dependence of non-resonant tunneling in the insulating phase, which the original model left unexplored.

(iii) Replace the isotropic in-plane Fermi-surface approximation with the anisotropic in-plane electronic structure of the planes and chains, and quantify how the resulting direction-dependent densities of states and quasiparticle lifetimes modify $\sigma_c$ and $J_c$.

(iv) Determine the magnitude and oxygen dependence of the interlayer superconducting critical current density $J_c(x)$ from microscopic parameters pinned independently of the transport data being explained (the resonance level width, level energy, the interlayer barrier height, and superconducting gap function), and confront the result with measured values, e.g. $J_c = 1.5 \times 10^6$ A/cm^2 at $x = 6.7$ and 4.2 K.

(v) Either provide a microscopic justification of the assumed in-plane quasiparticle scattering rate $1/\tau = \alpha T$ with $\alpha \ll 1$ (required for the Fermi-liquid treatment), or demonstrate which of the model's conclusions survive without it.

The theory must explain the measured single-crystal $\sigma_c(T, x)$ and $J_c(x)$ data (e.g. the oxygen-deficiency series of Chien et al. and the critical-current data of Rapp et al.) over the ranges of x and T for which those datasets are measured, with the parameters fixed independently of the data being explained, so that it covers the full oxygen range rather than only a restricted window of x or T in which fitted parameters are reintroduced. Quantitatively, the theory must reproduce the fitted $A(x)$ and level energy $E$; must reproduce the cited $\sigma_c(T, x)$ data over the full oxygen range $6 \le x \le 7$ and over that dataset's measured temperature range; must yield a $J_c(x)$ consistent with the measured values; and must establish its chosen branch of (v) rather than merely assert it.

References

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