Resistivity of inhomogeneous superconducting wires

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 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 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)similar to T-alpha obtained by unbinding quantum phase slips. However in the strongly inhomogeneous limit, the exponent alpha is modified and the prefactor is 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, which earlier studies have attempted to fit by an exponential trial function. The values of alpha extracted from the data, and the corresponding field dependence, are consistent with known parameters of the corresponding samples.