Irradiation Tests of Superconducting Detectors and Comparison with Simulations

Abstract

For the future satellite mission at the second sun–earth Lagrangian point (L2), we need to mitigate phonon propagation created by cosmic rays to superconducting detectors. We simulate phonon propagation in silicon substrate and show that putting a metal layer on the substrate or making hole in the substrate reduces the propagation. We also show a function which shows the response of a TES bolometer on a substrate. To validate these theoretical expectations, we make irradiation tests using two types of superconducting detectors: transition edge sensor bolometers and kinetic inductance detectors. From the tests, we show that putting metal can reduce correlations between detectors and number of hit events from charged particles.

Appendices

Appendix 1: Modelling of a TES Bolometer

From the model of Fig. 2, we created simultaneous differential equations of temperatures of a TES bolometer and a silicon substrate:

$$\begin{aligned} \frac{\text{d}}{\text{d}t} \begin{pmatrix} \Delta T_1 (t) \\ \Delta T_2 (t) \end{pmatrix} = \begin{pmatrix} -\frac{G_1 +G_2}{C_1} &{} \quad \frac{G_2}{C_1} \\ \frac{G_2}{C_2} &{} \quad -\frac{G_2}{C_2} \end{pmatrix} \begin{pmatrix} \Delta T_1 (t) \\ \Delta T_2 (t) \end{pmatrix} \end{aligned}$$

(2)

$$\begin{aligned} \begin{pmatrix} \Delta T_1 (t) \\ \Delta T_2 (t) \end{pmatrix} = \alpha _1 \begin{pmatrix} \frac{G_2}{C_1} \\ \varLambda _1 \end{pmatrix}e^{\lambda _1 t} +\alpha _2 \begin{pmatrix} \frac{G_2}{C_1} \\ \varLambda _2 \end{pmatrix}e^{\lambda _2 t}, \end{aligned}$$

(3)

where $\Delta T_1$ and $\Delta T_2$ are change in temperature of silicon substrate and TES bolometer, $\varLambda _{1,2} = \lambda _{1,2}+\frac{G_1+G_2}{C_1}$,

$$\begin{aligned} \begin{aligned} \lambda _1&= -\frac{1}{2}\left[ \frac{G_1+G_2}{C_1} + \frac{G_2}{C_2} +\sqrt{D} \right] \\ \lambda _2&= -\frac{1}{2}\left[ \frac{G_1+G_2}{C_1} + \frac{G_2}{C_2} -\sqrt{D} \right] , \end{aligned} \end{aligned}$$

(4)

and

$$\begin{aligned} D = \left( \frac{G_1 + G_2}{C_1}+ \frac{G_2}{C_2}\right) ^2 -4 \frac{G_1 G_2}{C_1C_2}. \end{aligned}$$

(5)

If we consider an initial state that the energy ($E_{\rm in}$) is injected into a silicon substrate at $t_0$,

$$\begin{aligned} \Delta T_2(t) = \frac{1}{\lambda _2-\lambda _1}\varLambda _1\varLambda _2\frac{E_{\rm in}}{G_2}\left( e^{\lambda _1 (t- t_0)}- e^{\lambda _2 (t-t_0)}\right) \end{aligned}$$

(6)

If we consider an initial state that the energy ($E_{\rm in}$) is injected into a TES bolometer at $t_0$,

$$\begin{aligned} \Delta T_2(t) = \frac{E_{\rm in}}{C_2 \left( \lambda _2-\lambda _1\right) }\left( -\varLambda _1 e^{\lambda _1 (t-t_0)} + \varLambda _2 e^{\lambda _2 (t-t_0)}\right) \end{aligned}$$

(7)

Appendix 2: Finding Dropped KID

Because we only have nine resonances out of ten designed KIDs, we develop a way to identify one dropped KID. For this analysis, we apply the same veto as described in Sec3.2.

After the veto, we calculate the correlation between ith and jth KIDs with triggered time index from ith KID, $\mathrm {Corr}(d_i (t_i), d_j (t_i))$. This means that $\mathrm {Corr}(d_i (t_i), d_j (t_i))$ and $\mathrm {Corr}(d_j (t_j), d_i (t_j))$ are different. Then, we calculate all the correlations between all bolometers. The correlations without f1 are plotted in Fig. 10. We checked all the cases with correlation plots and finally find f1 KID is dropped. As shown in Fig. 10, f2, f4, f7, and f9 have strong correlations as we expect from the design 4.

Fig. 10

Digraphs of correlations between KIDs. The vector from ${fi}$ to ${fj}$ shows the correlation of ${\mathrm {Corr}}(d_i (t_i), d_j (t_i))$. The red vector shows the case that $i\le j$, and the blue vector shows the other case. The width of vectors is related to $\propto 1.3\times 5.0^{\mathrm {Corr}/0.5}$ (Color figure online)