Pinning effects on hot-electron vortex flow instability in superconducting films

The hot-electron vortex flow instability in superconducting films in magnetic field $B$ at substrate temperature $T_0 \ll T_c$ is theoretically considered in the presence of pinning. The magnetic field dependence of the instability critical parameters (electric field $E^\ast$, current density $j^\ast$, resistivity $\rho^\ast$, power density $P^\ast$ and vortex velocity $v^\ast$) is derived for a cosine and a saw-tooth washboard pinning potential and compared with the results obtained earlier by M.\,Kunchur [Phys. Rev. Lett. \textbf{89} (2002) 137005] in the absence of pinning. It is shown that the $B$-behavior of $E^\ast$, $j^\ast$ and $\rho^\ast$ is monotonic in $B$, whereas the $B$-dependence of $v^\ast$ is quite different, namely $dv^\ast/dB$ may change its sign twice with decreasing $B$ as it was observed in some experiments. The simplest heat balance equation for electrons in low-$T_c$ superconducting films in the two-fluid approach is considered. It allows one to show that the instability critical temperature $T^\ast \approx 5T_c/6$ at $T_0 < T^\ast/2$ with $T^\ast$ independent of $B$.