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 $\mathrm{\Delta }{F}^{\text{2D}}\propto {(1-I/I_c)}^{3/4}$. 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 $\mathrm{\Delta }{F}^{\text{2D}}$.

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• Phase slips
• Superconductivity
• Instantons
• Landau-Ginzburg theory