Resistivity of inhomogeneous superconducting wires

We study the contribution of quantum phase fluctuations in the superconducting order parameter to the low-temperature resistivity ρ (T) of a dirty and inhomogeneous superconducting wire. In particular, we account for random spatial fluctuations of arbitrary size in wire thickness. For a typical wire thickness above the critical value for a superconductor-insulator transition, phase-slip processes can be treated perturbatively. We use a memory formalism approach, which underlines the role played by a weak violation of conservation laws in the mechanism for generating finite resistivity. Our calculations yield an expression for ρ (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 ρ (T) ∼ Tα obtained by unbinding quantum phase slips. However in the strongly inhomogeneous limit, the exponent α is modified and the prefactor is exponentially enhanced. We examine the dependence of the exponent α 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, which earlier studies have attempted to fit by an exponential trial function. The values of α extracted from the data, and the corresponding field dependence, are consistent with known parameters of the corresponding samples.