Modeling electrothermal feedback of superconducting nanowire single photon detectors in SPICE

TL;DR

The paper addresses the challenge of modeling electrothermal feedback in SNSPDs within SPICE by introducing a compact 0-D thermal model that maps the nanowire temperature dynamics to a parallel RC network. Key elements include temperature-dependent heat capacities for normal and superconducting states, a photon-energy deposition input, and a switching/retrapping framework with I_sw(T) and I_rt(T) relations, such as I_sw(T) = I_sw(0) (1 - (T/T_c)^2)^{3/2} and I_rt(T) = sqrt(2/psi) I_sw(T). The model reproduces hotspot growth, after-pulsing, and latching, and demonstrates how electronic reset time and thermal time shape device behavior, with analytic boundaries for regime transitions. Compared to finite-element approaches, the SPICE model achieves substantial speedups with reasonable parameter fitting, enabling rapid, scalable circuit design and integration with modular SNSPD architectures like SNAPs and thermally coupled detectors.

Abstract

Superconducting nanowire single-photon detectors (SNSPDs) exhibit complex switching behaviors due to electrothermal feedback during the detection process. Modeling and understanding these behaviors is integral for designing superconducting devices; however, many models often prioritize accuracy over computational speed and intuitive integration for circuit designers. Here, we build upon a growing architecture of SPICE tools for superconducting nanowire devices by capturing complex residual heating effects in a compact thermal model of an SNSPD. We demonstrate that our model is comparable to more complicated thermal models of superconducting nanowire devices, including finite-element simulations, and is applicable for the fast development of SNSPD circuits.

Figures (5)

• Figure 1: (a) SPICE SNSPD model from Ref. Berggren2018, showing hotspot growth $f(i_\mathrm{D})$ and added electrothermal feedback (blue dashed RC network mapping heat capacitance and thermal resistance to $C(T)$ and $R_{\rm th}$), with a pulsed source $hv$ and Joule heating source $i_\mathrm{D}^2R_{\rm hs}$. (b) Simulated $I$–$V$ curve marking the zero-temperature retrapping current $I_{\rm rt}(0)$, switching current $I_{\rm sw}(0)$, and the superconducting (SC) vs. normal (N) regions.
• Figure 2: The electronic heat capacity fitting function used in the model compared to the numerically solved BCS heat capacity. The plot shows the BCS characteristic jump in electronic heat capacity at $T = T_\mathrm{c}$ such that $\Delta c_\mathrm{el}~=~c_\mathrm{el,sc}(T_\mathrm{c})~-~c_\mathrm{el,n}(T_\mathrm{c})~=~1.43\gamma T_\mathrm{c}$.
• Figure 3: Response to a photon pulse at time $t = 2 \, \mathrm{ns}$ for various electronic reset times $\tau_\mathrm{e} =2.0\,\mathrm{ns}$ (single pulse), $1.5 \,\mathrm{ns}$ (after-pulse), and $2.0\,\mathrm{ns}$ (latching). The thermal reset time is kept constant, so increasing the electronic reset time determines the device effects. (a) shows the output voltage, (b) shows the device temperature, and (c) shows the hotspot resistance.
• Figure 4: The current-temperature relationship for a single pulse, after-pulsing, and latching is shown. For $\tau_\mathrm{e} = 1.0 \,$ns, the nanowire cools down and the full bias current returns at the bath temperature (i). For $\tau_\mathrm{e} = 1.5 \,$ns, the nanowire cools down, but switches once (ii) before retrapping at the bath temperature. For $\tau_\mathrm{e} = 2.0 \,$ns, the nanowire does not retrap and remains at the retrapping current (iii).
• Figure 5: After-pulsing time by varying $\tau_\mathrm{e}$ and $I_\mathrm{B}$ generated by the electrothermal SPICE model. The contour lines distinguishes single pulses, after-pulsing, and latching.