Abstract
We measured the temporal voltage response of NbTi superconducting filaments with varied nanoscale thicknesses to step current pulses that induce non-equilibrium superconducting states governed by a hot spot mechanism. Such detected voltage emerges after a delay time td, which is intimately connected to the gap relaxation and heat escape times. By employing time-dependent Ginzburg-Landau theory to link the delay time to the applied current, we determined that the gap relaxation time depends linearly on film thickness, aligning with the acoustic mismatch theory for phonon transmission at the superconductor-substrate interface. We thereby find a gap relaxation time of 104 ps per nm of thickness for NbTi films on polished sapphire. We further show that interfacial interaction with the substrate significantly impacts the gap relaxation time, with observed values of 9 ns on SiOx, 6.8 ns on fused silica, and 5.2 ns on sapphire for a 50 nm thick NbTi strip at T=5.75 K. These insights are valuable for optimizing superconducting sensing technologies, particularly the single-photon detectors that operate in the transient regime of nanothin superconducting bridges and filaments.