Current dependence of the red boundary of superconducting single-photon detectors in the modified hot-spot model

We find the relation between the energy of the absorbed photon and the threshold current at which the resistive state appears in the current-carrying superconducting film with probability about unity. In our calculations we use the modified hot-spot model, which assumes a different strength of suppression of the superconducting order parameter in the finite area of the film around the place where the photon is absorbed. To find the threshold current we solve the Ginzburg-Landau equation for the superconducting order parameter, which automatically includes the current continuity equation and it allows us to consider the back effect of current redistribution near the hot spot on the stability of the superconducting state. We find quantitative agreement with recent experiments, where we use the single fitting parameter which describes what part of the energy of the photon goes for the local destruction of the superconductivity in the film.

Figure 1

(a) Distribution of $\left|\Delta \right|$ and (b) superconducting current density along the white lines marked in the insets (where we plot the contour plot of $\left|\Delta \right|$ in the film). The region where we put $\alpha =0$ is bordered by the white circle in the insets in (a) and at all radiuses the transport current is close to the threshold value.

Figure 2

Threshold current as a function of the radius of the spot $\left(\alpha =0\right)$ for films with different widths. Solid curves correspond to Eq. (12) from Ref. [8] with $\gamma =0$. Dotted line corresponds to the dependence $I_{\text{thr}}\left(R\right)$ for the film with width $w=30\xi$ in the model with spatially uniform current distribution in the sidewalks near the hot spot.

Figure 3

Dependence of the threshold current on the energy difference between the states of the superconducting film with and without the hot spot (we did not take into account the difference in the energy of quasiparticles).

Figure 4

Dependence of the threshold current, when $\text{IDE}\to 1$, on the energy of the absorbed photon. The experimental results were found for TaN based SSPD in Refs. [6, 7].

Figure 5

Symbols: Experimental results from Ref. [14] found when $\text{IDE}\sim 0.25$. Solid curves: Theoretical results calculated in GL and London models when $\text{IDE}\simeq 1$. Dashed line: Corresponds to the model with uniform current distribution in the sidewalks near the normal spot.