Universal Model for the Turn-on Dynamics of Superconducting Nanowire Single-Photon Detectors

We describe an electrothermal model for the turn-on dynamics of superconducting nanowire single-photon detectors (SNSPDs). By extracting a scaling law from a well-known electrothermal model of SNSPDs, we show that the rise-time of the readout signal encodes the photon number as well as the length of the nanowire with scaling $t_\text{rise}\propto \sqrt{\ell/n}$. We show that these results hold regardless of the exact form of the thermal effects. This explains how SNSPDs have inherent photon-number resolving capability. We experimentally verify the photon number dependence by collecting waveforms for different photon number, rescaling them according to our predicted relation, and performing statistical analysis that shows that there is no statistical significance between the rescaled curves. Additionally, we use our predicted dependence of rise time on detector length to provide further insight to previous theoretical work by other authors. By assuming a specific thermal model, we predict that rise time will scale with bias current, $t_\text{rise}\propto \sqrt{1/I_b}$. We fit this model to experimental data and find that $t_\text{rise}\propto 1/(n^{0.52 \pm 0.03} ~I_b^{0.63 \pm 0.02})$, which suggests further work is needed to better understand the bias current dependence. This work gives new insights into the non-equilibrium dynamics of thin superconducting films exposed to electromagnetic radiation.