Thermal phase slips in superconducting films

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 $L_x\simξ(1-I/I_c)^{-1/4}$ along the current and $L_y\simξ(1-I/I_c)^{-1/2}$ perpendicular to the current, where $ξ$ is the Ginzburg-Landau coherence length. The activation energy for thermal phase slips scales as $ΔF^\text{2D}\propto (1-I/I_c)^{3/4}$. 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 $ΔF^\text{2D}$.

Figures (2)

• Figure 1: Saddle-point configuration of the current $\mathbf{j}(x,y)$ responsible for the thermal phase slip at $I/I_c=0.98$. The supercurrent follows the solid lines, and its magnitude (normalized by the current density $j_0$ at infinity) is shown by the color scale. Contours of constant phase appear dashed. In the regions encircled by thick solid lines, $j$ exceeds the nominal critical current density $j_c$ for a uniform flow.
• Figure 2: Numerical results for the variational Boussinesq-like ansatz: (a) energy barrier (dots) and its asymptotic behavior \ref{['DF-res']} (dashed line); (b) order parameter at the center of the instanton (dots) and $\Delta_\text{B}(0)$ (dashed line) normalized by its value at infinity.