Microscopic theory of in-plane resistivity in the cuprate pseudogap state: deriving the domain size law, transport through an assembly of superconducting domains and normal boundaries, and the doping and disorder dependence of the parameters $b$, $\rho_0$, and $T^*$

In hole-doped cuprate superconductors, the in-plane resistivity $\rho(T)$ measured above the superconducting transition temperature $T_c$ is approximately linear in temperature, and its extrapolation to $T=0$ shows essentially no residual (temperature-independent) resistivity from defects. A. A. Abrikosov (Physical Review B 72, 212502 (2005)) gave a heuristic account of both features within his picture of the pseudogap state as an inhomogeneous structure on the Cu$_2$ planes: finite, locally phase-coherent superconducting domains separated by normal, resistive boundaries, with a temperature-dependent domain size $\xi(T)$ estimated from T. M. Rice's finite-size phase-fluctuation analysis plus an interpolation. The paper itself concedes that "it is very hard to construct a regular theory of such an inhomogeneous substance" and instead employs simple estimates for exactly three ingredients: an interpolation for the domain size $\xi(T)$ built on the Rice estimate; an interpolation formula for the macroscopic resistivity $\rho(T)$ in which the normal boundaries enter only through the domain-size ratio; and assumed, rather than derived, doping and disorder dependences of the parameters $b$, $\rho_0$, and $T^*$. The open task is to replace these estimates by regular derivations.

Background

High-temperature cuprate superconductors are layered copper oxides in which superconductivity resides in the copper-oxide (CuO$_2$) planes; the hole content of those planes, tuned for instance by the oxygen content in YBa$_2$Cu$_3$O$_{7-\delta}$, controls the superconducting transition temperature $T_c$. Underdoped samples (low hole content) show, above $T_c$, the pseudogap state: a partial suppression of low-energy electronic spectral weight that sets in at a characteristic temperature $T^*>T_c$ and grows as the hole content is reduced (Tallon and Loram, 2001). Two empirical facts about normal-state transport in these materials motivated the source paper. First, the in-plane resistivity $\rho(T)$ (current flowing within the CuO$_2$ planes) is approximately linear in temperature over a broad window above $T_c$. Second, when the high-temperature $\rho(T)$ line is extrapolated through the superconducting region down to $T=0$, it shows essentially no residual resistivity, i.e. no temperature-independent offset of the kind that defect scattering always produces in an ordinary metal. Abrikosov's paper 'Linear temperature dependence of the resistivity in the pseudogap state' (Physical Review B 72, 212502 (2005)) traced both facts to the spatial structure of the pseudogap state proposed in his earlier works (Abrikosov, 2001a, 2001b): below $T^*$, each CuO$_2$ plane breaks into finite superconducting domains — regions that carry a local superconducting gap and a locally phase-coherent order parameter (a fixed quantum-mechanical phase relationship across the region) but no long-range phase order — separated by normal, nonsuperconducting boundaries. Evidence that superconducting order persists locally well above $T_c$ came from vortex-like excitations detected above $T_c$ in underdoped La$_{2-x}$Sr$_x$CuO$_4$ (Xu et al., 2000), and later scanning-tunneling experiments directly imaged nanoscale pairing islands appearing below $T^*$ and phase-incoherent superconducting spectra surviving up to $T^*$ in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Gomes et al., 2007; Lee et al., 2009). In this picture, charge transport above $T_c$ is dominated by the normal boundaries between the superconducting domains: the fraction of the plane that they occupy is set by the linear domain size $\xi(T)$, which Abrikosov estimated from the classic analysis of T. M. Rice (1965) on how finite size limits superconducting phase order in one and two dimensions — the Rice estimate gives $\xi(T)\sim\hbar\nu_1/T$, with $\nu_1$ the Fermi velocity near the extended saddle-point ("flat") regions of the Fermi surface — supplemented by an interpolation $\xi(T)\sim\hbar\nu_1(1/T+1/T^*)$ [Eq. (1) of the paper], where $T^*$ is the crossover temperature from the pseudogap state to the normal state and $\xi^*=\hbar\nu_1/T^*$. Because the boundary fraction shrinks as the domains grow on cooling, the heuristic account yields a $\rho(T)$ that is linear in $T$ and extrapolates to nearly zero at $T=0$. The paper is explicit, however, that this is not a regular theory: "Since it is very hard to construct a regular theory of such an inhomogeneous substance, we will employ simple estimates." Concretely, the macroscopic resistivity follows from the interpolation formula $\rho(T)=(\xi^*/\xi)(\rho_0+bT)=T(\rho_0+bT)/(T+T^*)$ [Eq. (2) of the paper], in which the normal boundary regions contribute a resistivity $\rho_0+bT$ — $\rho_0$ being their constant residual (defect) resistivity and $bT$ their temperature-linear contribution, arising at $T>\hbar\omega_0$ mainly from low-frequency optic phonons with $\hbar\omega_0<\Delta$ — so that the boundaries enter only through the domain-size ratio $\xi^*/\xi$, without a treatment of a realistic, disordered assembly of domains and boundaries. The dependences of $b$, $\rho_0$, and $T^*$ on oxygen content and disorder are assumed qualitatively rather than derived: in the paper's words, "$T^*$ is suppressed by disorder, which increases with doping, whereas $\rho_0$ increases with disorder," so the ratio $\rho_0/bT^*$, which controls the regimes of the linear law, grows steadily with doping. Such a regular theory must additionally encompass — the roles of domain geometry, percolation (whether connected conducting paths span the sample), and phase correlations between domains. The unresolved request — to replace this heuristic account by a controlled microscopic theory — is the subject of this problem. It sits inside the broader, still-unsolved puzzle of the 'strange-metal' linear-in-$T$ resistivity of the cuprates (Keimer et al., 2015).

Problem Statement

Construct a controlled microscopic theory of in-plane charge transport in the pseudogap state of hole-doped cuprate superconductors within the inhomogeneous superconducting-domain picture, replacing the heuristic estimates of Abrikosov (Physical Review B 72, 212502 (2005)) by regular derivations of all load-bearing ingredients. Concretely, the theory must: (1) derive, from microscopic physics of the pseudogap state, the temperature dependence $\xi(T)$ of the linear size of the locally phase-coherent superconducting domains, going beyond the Rice (1965) finite-size phase-fluctuation estimate $\xi(T)\sim\hbar\nu_1/T$ and the interpolation $\xi(T)\sim\hbar\nu_1(1/T+1/T^*)$ [Eq. (1) of the source paper], with $\nu_1$ the Fermi velocity near the extended saddle-point regions of the Fermi surface and $\xi^*=\hbar\nu_1/T^*$; (2) construct a regular (quantitative, spatially resolved) transport theory for a realistic, disordered two-dimensional assembly of superconducting domains separated by normal, resistive boundaries — a theory that must treat the domain geometry, percolative connectivity, and phase correlations between domains, which the source accounts for only through the domain-size ratio in its interpolation formula $\rho(T)=(\xi^*/\xi)(\rho_0+bT)=T(\rho_0+bT)/(T+T^*)$ [Eq. (2) of the source paper] — and derive from it the macroscopic in-plane resistivity $\rho(T)$ above $T_c$, including its observed quasilinear temperature dependence and the absence of a residual, temperature-independent term in the extrapolation of $\rho(T)$ to $T=0$; and (3) obtain microscopic, quantitative dependences of the parameters entering Eqs. (1)–(2) — $\rho_0$, the constant residual (defect) resistivity of the normal boundary regions; $b$, the coefficient of their temperature-linear resistivity contribution arising at $T>\hbar\omega_0$ from low-frequency optic phonons with $\hbar\omega_0<\Delta$; and $T^*$, the pseudogap-to-normal crossover temperature — on oxygen concentration (hole doping) and on disorder, replacing the source's assumed qualitative trends ("$T^*$ is suppressed by disorder, which increases with doping, whereas $\rho_0$ increases with disorder"), and thereby delimit the doping, disorder, and temperature regimes in which the linear law holds (governed in the source by the ratio $\rho_0/bT^*$ relative to unity). These three subgoals are components of one scientific objective: turning the source paper's model estimates into a regular theory whose ingredients are microscopically derived rather than assumed.

References

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