Measurements and Simulations of Athermal Phonon Transmission from Silicon Absorbers to Aluminum Sensors

Phonon reflection and transmission at the interfaces plays a fundamental role in cryogenic particle detectors, in which the optimization of the phonon signal at the sensor (in case of phonon-mediated detectors) or the minimization of the heat transmission (when the detection occurs in the sensor itself) is of primary importance to improve sensitivity. Nevertheless, the mechanisms governing the phonon physics at the interfaces are still not completely understood. The two more successful models, the acoustic mismatch model (AMM) and diffuse mismatch model (DMM) are not able to explain all the accumulated experimental data and the measurement of the transmission coefficients between the materials remains a challenge. Here, we use measurements of the athermal phonon flux in aluminum kinetic inductance detectors (KIDs) deposited on silicon substrates following a particle interaction to validate a Monte Carlo (MC) phonon simulation. We apply the Mattis-Bardeen theory to derive the phonon pulse energy and timing from the KID signal and compare the results with the MC for specular AMM and DMM reflection, finding a remarkably good agreement for specular, while diffuse reflection is clearly disfavored. For an aluminum film of 60 nm and a silicon substrate of $380\mu \mathrm{m}$, we obtain transmission coefficients $\mathrm{Si}$-$\mathrm{Al}$ in the range 0.3–0.55 and $\mathrm{Si}$-Teflon in the range 0.1–0.15.

Figure 1

Simulation of 0.1-THz phonons generated in one small spot of the $\mathrm{Si}$ wafer surface and detected in the opposite face. The panels show the flux intensity for every polarization: L (left), ST (center), and FT (right). The image spans an angle of $\pm {72}^{\circ }$. Phonon focusing structures (bright colors) are clearly formed, especially for ST and FT modes.

Figure 2

$\delta \varphi$ and $\delta a$ pulse time evolution following an energy deposition of 36 keV in the $\mathrm{Si}$ substrate. The signals are fitted to the pulse shape of Eq. (3), taking also into account $T_{\mathrm{ex}}=1\mu \mathrm{s}$ (in red for $\delta \varphi$ and blue for $\delta a$). The ${\chi }^2/\mathrm{NDF}$ for the fit in the $\delta \varphi$ component is 2.5. The resulting $\delta \varphi$ fit parameters are shown in the legend. Inset: Resonance circle that we calibrate to obtain $\delta \varphi$ and $\delta a$ components from the real and imaginary parts of the $S_{21}$ signal.

Figure 3

Left: Picture of the P1 prototype: A 60-nm-thick $\mathrm{Al}$ KID deposited on a $2\times 2$-${\mathrm{cm}}^2$, $380$-$\mu \mathrm{m}$-thick $\mathrm{Si}$ substrate. Four Teflon supports, one at each corner, fix the detector to a copper holder that is anchored to the cryostat. Right: Geometry of the single pixel: an inductor made of 30 strips with a size of $62.5\mu \mathrm{m}\times 2\mathrm{mm}$, with a gap of $5\mu \mathrm{m}$ between them, and a capacitor composed of two fingers.

Figure 4

Left: Picture of the P4 prototype. Four $\mathrm{Al}$ KIDs are deposited on a $300$-$\mu \mathrm{m}$-thick $\mathrm{Si}$ substrate with a size of $2\times 2{\mathrm{cm}}^2$. Two cylindrical Teflon supports with a contact area of around $3{\mathrm{mm}}^2$ each hold the substrate in the copper structure. Right: Geometry of the single pixel (60-nm-thick $\mathrm{Al}$ film): an inductive meander with 14 connected strips with a size of $80\mu \mathrm{m}\times 2\mathrm{mm}$ and capacitor made of 5 interdigitated fingers with a size of $1.4\mathrm{mm}\times 50\mu \mathrm{m}$. The active area of the single pixel is $1.15\times 2{\mathrm{mm}}^2$.

Figure 5

A sketch of the main components included on the MC simulation of the prototype P1 (left) and P4 (right). In both cases, the fiber spot (brown) has a diameter of 4.66 mm and fires on the opposite side of the KIDs.

Figure 6

Phonon distribution at the KIDs for P1 (upper panel) and P4 (bottom panel), corresponding to the AMM model, $T_{\mathrm{Si}\to \mathrm{Al}}=0.36$, and $T_{\mathrm{Si}\to \mathrm{Tef}}=0.4$.

Figure 7

Frequency distribution of the phonons absorbed in the different materials (Teflon, feedline, KIDs) for P1 (upper panel) and P4 (middle panel). In the bottom panel, the P4 distribution at every KID is plotted separately.

Figure 8

Comparison of the MC results with experimental data for P1 prototype and AMM model (upper panel) or DMM model (bottom panel). The red (blue) lines correspond to simulations with constant values of the T_Si→Al (T_Si→Tef) coefficient, while the points are taken from Table 3. The green error bars represent the systematic uncertainty associated to the MC parameters.

Figure 9

Same as Fig. 8 for the four KIDs of P4.