Thermal Properties of $\mathrm{NbN}$ Single-Photon Detectors

We investigate thermal properties of a $\mathrm{NbN}$ single-photon detector capable of unit internal detection efficiency. Using an independent calibration of the coupling losses, we determine the absolute optical power absorbed by the $\mathrm{NbN}$ film and, via resistive superconductor thermometry, the temperature dependence of the thermal resistance $Z(T)$ of the $\mathrm{NbN}$ film. In principle, this approach permits simultaneous measurement of the electron-phonon and phonon-escape contributions to the energy relaxation, which in our case is ambiguous because of the similar temperature dependencies. We analyze $Z(T)$ with a two-temperature model and impose an upper bound on the ratio of electron and phonon heat capacities in $\mathrm{NbN}$, which is surprisingly close to a recent theoretical lower bound for the same quantity in similar devices.

Figure 1

(a) Two-temperature model of a metal-superconducting film on an insulating substrate. Only a fraction of the incident optical power $P_{\mathrm{in}}$ is absorbed by the film as determined by the coupling efficiency of the apparatus $P_{\mathrm{abs}}={\eta }_{\mathrm{couple}}{P}_{\mathrm{in}}$. The electron subsystem under illumination can be described by an equilibrium Fermi-Dirac distribution with an effective electron temperature $T_e$ exceeding the phonon temperature in $\mathrm{NbN}$ $T_{\mathrm{ph}}$ and the bath temperature $T_b$. Further, it is assumed that $T_e$ is the only parameter that controls the resistivity of the superconducting film. $Z_{e\mathrm{-ph}}$ and $Z_{\mathrm{esc}}$ are the thermal resistances, which determine the heat exchange of the electrons with the acoustic phonons in $\mathrm{NbN}$ and the heat exchange of the phonons in $\mathrm{NbN}$ with the substrate, respectively. (b) Measured SSPD detection efficiency $D_E$ at a wavelength of $1.55\mu \mathrm{m}$ at $T_b=1.7$ K versus the bias current in the unit of the critical current. The saturation of the bias-current dependence signals the regime of unit internal efficiency, which enables us to calibrate the coupling losses as $D_E={\eta }_{\mathrm{couple}}$. The inset shows a sketch of the experimental setup. ATT, attenuator; LD, laser diode; PC, polarization controller; PM, power meter; S, sample.

Figure 2

The temperature dependencies of the sample resistance in equilibrium (a) and of the measured voltage response under optical radiation (b) for a set of magnetic fields (see the legend). For (b) a laser wavelength of 1.55 $\mu \mathrm{m}$, an input laser power of $P_{\mathrm{in}}=320$ nW, and a dc bias current of $I=100$ nA are used. The inset shows the temperature dependence of the upper critical field $B_{c2}(T)$ used to evaluate the density of states in $\mathrm{NbN}$. The temperature dependencies in (a),(b) are obtained by our measuring the signals during warm-up at discrete temperatures (15–20 points per curve). At each point the temperature was stabilized for at least 10 min with an accuracy of about $\pm 1$ mK.

Figure 3

Temperature dependence of the thermal resistance $Z$ for a $\mathrm{NbN}$ SSPD at the resistive transition. The experimental datasets are obtained from the measured power response of our SSPD in different magnetic fields, as explained in Sec. 2 2d. The dashed guide line represents the power-law dependence $Z\propto T^{-3}$ observed. The $B=0$ critical temperature of the resistive transition is marked by an arrow. The inset shows the ratio of the electron (normal state) and phonon thermal capacitances as a function of temperature. The solid line is the upper bound imposed by our experiment, as discussed in the text. The star is the lower bound, which follows from the theory in Ref. [13] for parameter values similar to those of our SSPD.