# Thermal elastic-wave attenuation in low-dimensional SiN$_{x}$ bars at low temperatures

## TL;DR

This study measures the elastic-wave attenuation length in low-dimensional SiN$_{x}$ bars at millikelvin temperatures, revealing the transition from ballistic to diffusive heat transport and informing thermal noise control in quantum devices.

## Contribution

It provides the first measurement of the ballistic to diffusive transition scale in low-dimensional amorphous dielectric bars at very low temperatures.

## Key findings

- Attenuation length of 20 μm for 6-7 elastic modes.
- Heat transport transition scale identified between 1 μm and 400 μm.
- Insights into scattering processes affecting thermal fluctuation noise.

## Abstract

At low temperatures, < 200 mK, the thermal flux through low-dimensional amorphous dielectric bars, < 2 $\mu$m wide and 200 nm thick, is transported by a small number of low-order elastic modes. For long bars, L > 400 $\mu$m, it is known that the conductance scales as 1/L, where L is the length, but for short bars, 1 $\mu$m < L < 400 $\mu$m, the length dependence is poorly known. Although it is assumed that the transport must exhibit a diffusive to ballistic transition, the functional form of the transition and the scale size over which the transition occurs have not, to our knowledge, been measured. In this paper, we use ultra-low-noise superconducting Transition Edge Sensors (TESs) to measure the heat flux through a set of SiN$_{x}$ bars to establish the characteristic scale size of the ballistic to diffusive transition. For bars supporting 6 to 7 modes, we measure a thermal elastic-wave attenuation length of 20 $\mu$m. The measurement is important because it sheds light on the scattering processes, which in turn are closely related to the generation of thermal fluctuation noise. Our own interest lies in creating patterned phononic filters for controlling heat flow and thermal noise in ultra-low-noise devices, but the work will be of interest to others trying to isolate devices from their environments, and studying loss mechanisms in micro-mechanical resonators.