Abstract
We consider the problem of finite resistance $R$ in superconducting films with geometry of a strip of width $W$ near zero temperature. The resistance is generated by vortex configurations of the phase field. In the first type of process, quantum phase slip, the vortex worldline in 2+1 dimensional space-time is space-like (i.e., the superconducting phase winds in time and space). In the second type, vortex tunneling, the worldline is time-like (i.e., the phase winds in the two spatial directions) and connects opposite edges of the film. For moderately disordered samples, processes of second type favor a train of vortices, each of which tunnels only across a fraction of the sample. Optimization with respect to the number of vortices yields a tunneling distance of the order of the coherence length $\xi$, and the train of vortices becomes equivalent to a quantum phase slip. Based on this theory, we find the resistance $\ln R \sim -g W/\xi$, where $g$ is the dimensionless normal-state conductance. Incorporation of quantum fluctuations indicates a quantum phase transition to an insulating state for $g \lesssim 1$.
Highlights
Instead, we concentrate on the different experimental situation of homogeneous films for which the impurity-induced reduction and ultimate annulment of Tc is well described by fermionic theories [8,9] and the study of resistivity below Tc constitutes a separate, subsequent question: At finite temperature, but infinite system size, resistance is established by vortex proliferation above a renormalized BerezinskiiKosterlitz-Thouless temperature [10,11]
The obtained resistance R is dominated by cotunneling of N ∼ W/ξ > 1 vortices, a process that is equivalent to a quasi-1D quantum phase slip, see Eqs. (3) and (7)
The exponential dependence of resistance on W is reminiscent of exponential suppression of conductance with the system size on the insulating side of superconductor-insulator transition (SIT)
Summary
Experimental evidence [41] for quantum tunneling of vortices in 2D superconductors at finite current bias, in particular in the context of dark photon counts [42,43], augmented interest in this research field Modern experimental tools, such as SQUID-on-tip microscopy [44], allow accessing both vortex motion and energy dissipation (local heating). We find that the combined tunneling of several vortices dominates over single-vortex events and demonstrate that, for the optimal number of involved vortices, the tunneling process has the same contribution as quantum phase slips This allows us to determine the linear-response resistance of superconducting strips, which is exponentially small but finite. Beyond the strictly 2D limit, which is the focus of all calculations, our results apply to the experimentally relevant situation of quasi-2D films of type-II superconductors, which are thinner than the correlation length
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