Interaction correction to the conductivity tensor of a quasi-two-dimensional metal across the 2D-3D dimensional crossover

Abrikosov's 2000 analysis of quantum corrections in quasi-two-dimensional metals — motivated by logarithmic low-temperature resistivity upturns in layered cuprates and by an almost isotropic $\sqrt{H}$ magnetoresistance in layered manganites — computed the weak-localization (quantum-interference) correction to both components of the conductivity tensor across the crossover between two-dimensional (2D) and three-dimensional (3D) behavior, but for the electron-electron interaction (Altshuler-Aronov) correction it derived only the density-of-states anomaly, with crossover parameter $a=\Omega/(2\alpha^{2}\tau)$, and left the interaction contribution to the conductivity tensor itself uncalculated, estimating only its order of magnitude. The open problem is the completion of that program: a full calculation of the interaction correction to the in-plane and c-axis conductivities for arbitrary $a$ with controlled prefactors, and quantitative validation against experiments, deciding whether interference or interaction corrections dominate the observed low-temperature transport anomalies of layered metals. The core of this problem remains open: later work extended the interference part (spin-dependent scattering, numerical fitting methods) and confirmed the density-of-states part experimentally (the $\sqrt{E}$ tunneling anomaly in layered manganites), and the analogous quasi-one-dimensional problem has been solved, while current experimental analyses of quasi-2D materials still combine strict-2D and strict-3D interaction formulas in an ad hoc way.

Background

Layered metals — the high-$T_c$ cuprates such as La$_{2-x}$Sr$_x$CuO$_4$ and YBa$_2$Cu$_3$O$_{6+x}$, the bilayer manganites La$_{2-2x}$Sr$_{1+2x}$Mn$_2$O$_7$, and many organic and van der Waals conductors — conduct well within their layers and only weakly between them. A minimal band model for such a quasi-two-dimensional (quasi-2D) metal, used by Abrikosov in his 1994 and 2000 papers, is $\varepsilon = p_t^2/2m_t + \alpha\cos(p_l d)$: an in-plane parabolic band with in-plane momentum $p_t$, interlayer hopping amplitude $\alpha$, momentum $p_l$ along the layer normal, and layer spacing $d$, giving a weakly corrugated cylindrical Fermi surface. Impurity scattering is characterized by the elastic scattering time $\tau$, and quantum phase coherence survives over the phase-coherence time $\tau_\varphi \gg \tau$, which shortens as the temperature rises.

On top of the classical (Boltzmann) conductivity, disordered metals carry two kinds of quantum corrections. The first is the quantum-interference, or weak-localization, correction: constructive interference between time-reversed diffusive electron paths enhances backscattering, so the resistivity rises as the temperature is lowered; in a strictly 2D metal the correction grows logarithmically, $\propto\ln(\tau_\varphi/\tau)$, and in a 3D metal it grows as $\sqrt{\tau_\varphi^{-1}}$ (Altshuler, Khmel'nitskii, Larkin, and Lee, 1980). A magnetic field suppresses this interference and produces a negative magnetoresistance. The second is the electron-electron interaction, or Altshuler-Aronov, correction (Altshuler and Aronov, 1979): disorder-enhanced Coulomb interactions deplete the single-particle density of states (DOS, the number of electronic states per unit energy and volume) near the Fermi level and add a singular correction to the conductivity — logarithmic in 2D and proportional to $\sqrt{T}$ in 3D. In strictly 2D or 3D isotropic metals both corrections are textbook material; the open question here concerns their fate in an anisotropic layered metal whose effective dimensionality itself crosses over as parameters change.

The experimental puzzles that motivated Abrikosov (2000) were: (i) in cuprates with superconductivity suppressed by a magnetic field, the resistivity upturns grow logarithmically with decreasing temperature and are much more pronounced along the c axis (perpendicular to the layers) than in the plane (Ando et al., 1995; Gantmakher et al., 1997); and (ii) in layered manganites the relative magnetoresistance correction is nearly isotropic and proportional to $\sqrt{H}$ regardless of field direction — the signature of a 3D metal despite extreme anisotropy. Abrikosov computed the interference correction to both tensor components of the conductivity, at zero field and in magnetic fields perpendicular and parallel to the layers, starting from an integral expression (his Eq. (2), derived in his 1994 paper under the assumptions $v_t q_t \ll \tau^{-1}$ and $\alpha|\sin(q_l d/2)| \ll \tau^{-1}$, with $v_t$ the in-plane Fermi velocity and $\mathbf{q}$ the diffusive momentum transfer). He found a crossover governed by the parameter $p = (2\alpha^2\tau_\varphi\tau)^{-1/2}$: at higher temperatures ($p \gg 1$, quasi-2D) the relative correction is extremely anisotropic — logarithmic in the plane and a power law along the c axis — while at lower temperatures ($p \ll 1$, effectively 3D) the relative correction is isotropic and proportional to $\sqrt{\tau_\varphi^{-1}}$ and $\sqrt{H}$, with the coefficient depending only on the field orientation, in agreement with the manganite data.

For the interaction correction Abrikosov followed the schematic of Altshuler and Aronov (1979) but completed the calculation only for the DOS: he obtained $\delta\nu/\nu_0$ as a function of temperature with the crossover parameter $a = \Omega/(2\alpha^2\tau)$, where $\Omega$ is the external frequency and, at $\Omega \sim T$, this parameter sets the 2D-3D crossover of the interaction correction. The 3D limit $a \ll 1$ gives $\delta\nu \propto \sqrt{T}$, while the 2D limit gives a $\ln^2 T$ anomaly — the square of the naive logarithm. He also observed that the interaction crossover criterion differs strongly from the interference one ($p$), so that at experimentally accessible temperatures the interaction correction should sit in its 3D regime even when the interference correction is quasi-2D. For the interaction correction to the conductivity tensor — the quantity actually measured in transport — he performed no calculation, writing only that one can "speculate" that the in-plane relative correction is of the same order as the DOS correction and that the c-axis correction is proportional to $\alpha^2$; on that basis he concluded that the in-plane upturn should be interference-dominated while the c-axis anomaly could be of either origin. Completing this missing calculation and validating the result quantitatively against experiment is the open problem.

Problem Statement

Complete the theory of quantum corrections to the conductivity tensor of the quasi-two-dimensional disordered metal with band spectrum $\varepsilon = p_t^2/2m_t + \alpha\cos(p_l d)$, by calculating the electron-electron interaction (Altshuler-Aronov) contribution to transport — the piece Abrikosov (2000) left as an order-of-magnitude speculation — and validating it quantitatively. Specifically:

(1) Full interaction correction to the conductivity tensor. Compute the interaction contribution to the relative corrections $\Delta\sigma_t/\sigma_{0t}$ (in-plane) and $\Delta\sigma_l/\sigma_{0l}$ (along the layer normal), where $\sigma_{0t}$ and $\sigma_{0l}$ are the corresponding classical (Boltzmann) conductivities, as functions of temperature — that is, for arbitrary crossover parameter $a = \Omega/(2\alpha^2\tau)$ (with $\Omega \sim T$), spanning the entire crossover from 2D to 3D behavior — with explicit prefactors, in the same microscopic model and diffusive approximation scheme as the existing interference calculation. The result must reduce to the isotropic Altshuler-Aronov $\sqrt{T}$ law in the 3D limit $a \ll 1$, exhibit the correct logarithmic behavior in the 2D limit $a \gg 1$, and quantify the anisotropy between the in-plane and c-axis components across the crossover, replacing the speculation that the in-plane correction tracks the DOS correction while the c-axis correction scales as $\alpha^2$ with a derivation.

(2) Quantitative confrontation with experiment. Using the completed theory, and confronting it with the measured temperature and field dependences of both conductivity components in layered metals, decide quantitatively whether the observed low-temperature in-plane and c-axis anomalies of quasi-2D metals are interference- or interaction-dominated — in particular the c-axis anomaly, for which Abrikosov's order-of-magnitude estimate left either origin possible. The confrontation is against the published measurements of the cited materials — the in-plane and c-axis resistivities of the cuprates (Ando et al., 1995; Gantmakher et al., 1997) and the magnetoresistance of the layered manganite La$_{2-2x}$Sr$_{1+2x}$Mn$_2$O$_7$ (Li, Gray, and Mitchell, 2000) — with the microscopic parameters of the completed theory fitted to the data to decide quantitatively whether the interference or the interaction correction dominates each observed anomaly.

These deliverables jointly serve one objective: a complete, quantitatively validated account of the anisotropic low-temperature transport of quasi-2D metals. The objective covers both tensor components across the full 2D-3D crossover.

References

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