Thermal phase slips in superconducting films

TL;DR

This paper analyzes thermal phase slips in superconducting films, deriving exact saddle-point configurations and activation energies near the critical current, with implications for understanding dissipation and dark counts in photon detectors.

Contribution

It provides an exact solution for the saddle-point configuration in 2D superconducting films near the critical current using the Boussinesq equation and Hirota's method.

Findings

The saddle-point configuration is described by the integrable Boussinesq equation.

The instanton size scales as _x \u221d (1 - I/I_c)^{-1/4} and _y  (1 - I/I_c)^{-1/2}.

Activation energy scales as  F^{2D}  (1 - I/I_c)^{3/4}.

Abstract

A dissipationless supercurrent state in superconductors can be destroyed by thermal fluctuations. Thermally activated phase slips provide a finite resistance of the sample and are responsible for dark counts in superconducting single photon detectors. The activation barrier for a phase slip is determined by a space-dependent saddle-point (instanton) configuration of the order parameter. In the one-dimensional wire geometry, such a saddle point has been analytically obtained by Langer and Ambegaokar in the vicinity of the critical temperature, $T_c$, and for arbitrary bias currents below the critical current $I_c$. In the two-dimensional geometry of a superconducting strip, which is relevant for photon detection, the situation is much more complicated. Depending on the ratio $I/I_c$, several types of saddle-point configurations have been proposed, with their energies being obtained numerically. We demonstrate that the saddle-point configuration for an infinite superconducting film at $I\to I_c$ is described by the exactly integrable Boussinesq equation solved by Hirota's method. The instanton size is $L_x\sim\xi(1-I/I_c)^{-1/4}$ along the current and $L_y\sim\xi(1-I/I_c)^{-1/2}$ perpendicular to the current, where $\xi$ is the Ginzburg-Landau coherence length. The activation energy for thermal phase slips scales as $\Delta F^\text{2D}\propto (1-I/I_c)^{3/4}$. For sufficiently wide strips of width $w\gg L_y$, a half-instanton is formed near the boundary, with the activation energy being 1/2 of $\Delta F^\text{2D}$.