Finite-temperature longitudinal conductivity of a strongly localized quasi-one-dimensional metal in the quantum-phonon regime and the interchain-hopping crossover

A quasi-one-dimensional (Q1D) metal — a crystal of parallel conducting chains — has all electron states localized by any random potential (the Mott-Twose result), so its longitudinal dc conductivity vanishes at zero temperature; at finite temperature, inelastic electron-phonon scattering delocalizes the electrons and generates a dc conductivity. Abrikosov and Ryzhkin (Solid State Communications 24, 319, 1977) computed this conductivity in the classical-phonon regime $T\gg\omega_D$ for the experimentally common case $\omega_Dl^{*}/v\ll1$, where all diagrams with intersecting phonon and impurity lines contribute and only a scaling ansatz yields the dc limit, finding $\sigma\propto T$ for $l_i\ll l_{ph}$ and $\sigma\propto T^{-2}$ for $l_i\gg l_{ph}$. They explicitly left open the quantum regime $T\lesssim\omega_D$ (where the classical-field phonon treatment fails and graphs containing virtual electron-hole pairs must be kept), the crossover at the lowest temperatures to transport controlled by interchain transitions, and a microscopic justification of the scaling ansatz itself.

Background

Quasi-one-dimensional (Q1D) metals — crystals of parallel conducting chains whose electron spectra are nearly flat transverse to the chains and approximately linear along them, $\epsilon(\mathbf{p})\approx v(|p_z|-p_0)$ with $v$ the Fermi velocity along the chain — were studied intensively in the 1970s in connection with organic charge-transfer conductors such as TTF-TCNQ (tetrathiafulvalene-tetracyanoquinodimethane, the prototype organic charge-transfer salt). A foundational result of localization theory (Mott and Twose, 1961) is that in a strictly one-dimensional (1D) conductor any random potential localizes all electron states, so the dc conductivity of an infinite isolated disordered chain vanishes; the impurity-only 1D chain has a nonzero finite-frequency response only (Mott's law $\sigma(\omega)\sim\omega^2$ up to logarithmic factors in $\omega$), and exact treatments of that problem are summarized in Abrikosov and Ryzhkin (1978). Abrikosov and Ryzhkin (1977) asked how such a conductor acquires a finite dc conductivity at finite temperature when electrons are scattered by static impurities and by phonons. Writing $l_i$ for the impurity mean free path, $l_{ph}=v/\lambda T$ for the phonon mean free path ($\lambda$ a dimensionless electron-phonon coupling constant, $T$ the temperature), $l^{*}=(l_i^{-1}+l_{ph}^{-1})^{-1}$ for the effective mean free path, $S$ for the area per chain, and $\omega_D$ for the limiting (Debye) phonon frequency (units with $\hbar=1$), they worked in the high-temperature regime $T\gg\omega_D$, where thermal phonon modes are heavily occupied and the phonon field can be treated as a classical random field. Earlier work had covered the regime $\bar{\omega}l^{*}/v\gg1$ ($\bar{\omega}$ a typical phonon frequency) keeping only nonintersecting phonon lines; Abrikosov and Ryzhkin addressed the opposite, experimentally common regime $\omega_Dl^{*}/v\ll1$, in which diagrams with intersecting phonon and impurity lines all contribute comparably, no perturbative diagram selection is possible, and the conductivity exists only because inelastic electron-phonon scattering delocalizes the electrons, making the dc limit nonanalytic in the delocalization strength. Since a direct summation of the intersecting diagrams was unavailable, they introduced a scaling ansatz $\sigma=\sigma_0 f[\gamma(v/(i\omega l^{*}))^{\nu}]$, with $\sigma_0$ the impurity-only 1D result, $\gamma$ a delocalization parameter, and $\nu$ an exponent fixed by heuristic power counting, and obtained the dc limits $\sigma(0)\sim(e^2l^{*}/\pi S)\gamma^{1/\nu}$, hence $\sigma\propto T$ for $l_i\ll l_{ph}$ and $\sigma\propto T^{-2}$ for $l_i\gg l_{ph}$ at $T\gg\omega_D$. The paper itself states what this construction does not cover. The regime $\bar{\omega}l^{*}/v\gg1$ is set aside as exotic (it had been analyzed in the earlier literature, which the paper cites). For $T\ll\omega_D$ the phonon population is small, quantum effects become essential, and graphs containing virtual electron-hole pairs — diagrams in which a quantized phonon is exchanged between an electron line and a hole line, the Kohn-anomaly-type quantum corrections that a classical-field treatment discards — must be retained; the finite-temperature conductivity in that quantum regime is not computed. Transitions between different chains are neglected, although they become essential at the lowest temperatures, so the crossover from strictly one-dimensional transport to interchain-hopping-controlled transport is outside the scope. Finally, the conductivity in the delocalization regime rests on the scaling assumption rather than on an explicit nonperturbative summation of the relevant intersecting diagrams.

Problem Statement

Derive, from a microscopic model of electrons on parallel disordered chains coupled to quantized phonons — electron spectrum $\epsilon(\mathbf{p})\approx v(|p_z|-p_0)$ along the chain ($v$ the Fermi velocity, $p_0$ the Fermi momentum, $p_z$ the momentum component along the chain), elastic impurity scattering with forward- and back-scattering mean free paths $l_1$ and $l_2$ entering the impurity mean free path $l_i$, electron-phonon coupling of dimensionless strength $\lambda$ with phonon mean free path $l_{ph}=v/\lambda T$ at temperature $T$, limiting (Debye) phonon frequency $\omega_D$; units with $\hbar=1$; interchain transitions neglected in the quantum-phonon computation of sub-questions (1)-(2) below, with finite interchain hopping entering the model only as the crossover parameter at the lowest temperatures — the longitudinal dc conductivity $\sigma(T)$ of an infinite quasi-one-dimensional metal in the quantum-phonon regime $T\lesssim\omega_D$, where the classical-field treatment of the phonons fails and graphs containing virtual electron-hole pairs (diagrams in which a quantized phonon is exchanged between an electron line and a hole line) must be retained, together with the crossover from this regime to the classical-phonon regime $T\gg\omega_D$ treated by Abrikosov and Ryzhkin (1977). Specifically, the problem is: (1) for the strictly one-dimensional chain with interchain transitions neglected (the setting of the Abrikosov and Ryzhkin (1977) computation), keep the quantum phonon dynamics and the virtual electron-hole-pair diagrams in the transport calculation for $T\lesssim\omega_D$, and establish (or disprove) the existence and value of a finite dc conductivity in the $\omega\to0$ limit of the infinite sample, where $\omega$ is the external frequency of the conductivity $\sigma(\omega)$ and the limit must be taken in the correct order with respect to the sample length $L\to\infty$; (2) in the intersecting-diagram regime $\omega_Dl^{*}/v\ll1$ with $l^{*}=(l_i^{-1}+l_{ph}^{-1})^{-1}$ the effective mean free path, either derive the scaling form $\sigma=\sigma_0 f[\gamma(v/(i\omega l^{*}))^{\nu}]$ microscopically — with $\sigma_0$ the impurity-only one-dimensional result, $\gamma$ the delocalization parameter, $\nu$ the exponent and $f$ the scaling function, all to be fixed by the derivation — or replace it by a controlled nonperturbative result for the same regime; and (3) determine the temperature dependence of $\sigma(T)$ as $T$ is lowered through and below $\omega_D$, including the crossover, at the lowest temperatures, to transport controlled by transitions between chains — where the finite interchain hopping enters as the crossover parameter — i.e. the interplay of inelastic delocalization and interchain escape.

References

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