Resistivity of Inhomogeneous Superconducting Wires

TL;DR

This paper investigates how quantum phase fluctuations and inhomogeneities in superconducting wires influence their low-temperature resistivity, revealing a crossover from power-law to granular behavior and matching experimental observations.

Contribution

It introduces a perturbative approach to model resistivity in inhomogeneous superconducting wires considering spatial thickness fluctuations and quantum phase slips.

Findings

Resistivity exhibits a crossover from power-law to granular behavior with increasing temperature.

Inhomogeneity enhances the resistivity prefactor exponentially.

The model's predictions align with experimental data on long, narrow superconducting wires.

Abstract

We study the contribution of quantum phase fluctuations in the superconducting order parameter to the low--temperature resistivity $\rho(T)$ of a dirty and inhomogeneous superconducting wire. In particular, we account for random spatial fluctuations of arbitrary size in the wire thickness. For a typical wire thickness above the critical value for superconductor--insulator transition, phase--slips processes can be treated perturbatively. We use a memory formalism approach, which underlines the role played by weak violation of conservation laws in the mechanism for generating finite resistivity. Our calculations yield an expression for $\rho(T)$ which exhibits a smooth crossover from a homogeneous to a ``granular'' limit upon increase of $T$, controlled by a ``granularity parameter'' $D$ characterizing the size of thickness fluctuations. For extremely small $D$, we recover the power--law dependence $\rho(T)\sim T^\alpha$ obtained by unbinding of quantum phase--slips. However in the strongly inhomogeneous limit, the exponent $\alpha$ is modified and the prefactor is {\em exponentially enhanced}. We examine the dependence of the exponent $\alpha$ on an external magnetic field applied parallel to the wire. Finally, we show that the power--law dependence at low $T$ is consistent with a series of experimental data obtained in a variety of long and narrow samples. The values of $\alpha$ extracted from the data, and the corresponding field dependence, are consistent with known parameters of the corresponding samples.