Time analysis of the voltage response created by a critical current pulse of superconducting NbTi nanowires

We have investigated in time domain the voltage response of superconducting NbTi filaments with different thickness to a supercritical (I>Ic) step-pulse of electrical current. The resistive state is created in the filament for exceeding the critical current, where a hot spot is found far below to the critical temperature and a permanent phase-slip center close to Tc. In both cases, the resistive response appears after a certain delay time t_d depending upon the temperature and the ratio I/I_c(T) which can been analyzed through a Time-Dependent Ginzburg-Landau equation. We found that the experimental data can be fitted by using a gap relaxation time, independently of the sample widths. However, it depends strongly on the filament thickness. Assuming proportionality to sample thickness, this indicates a phonon escape time of about 80 ps/nm for a NbTi film sputtered on polished crystalline Al₂O₃.