Phonon dynamics for light dark matter detection

The search for low-mass dark matter (DM), which has gained much interest recently, goes in parallel with the identification of new detection channels and the development of suitable detectors. The detection of the resulting small energy depositions is challenging, requiring extremely high sensitivity only achievable by cryogenic thermal detectors, which might be put to the limit. Thus, understanding the processes that can limit the performance of these detectors can be crucial for evaluating the feasibility of the proposed new detection schemes and, eventually, designing the detectors and tuning their performance. In this paper, we focus on a promising detection scheme, the excitation of single optical phonons in polar materials, to evaluate one of the possible and less understood limiting factors of cryogenic thermal detectors, i.e., the phonon dynamics in the target/absorber. We present a detailed theoretical analysis, within an entirely ab initio scheme, of the downconversion and propagation processes undergone by optical phonons, created by the interaction of a low-mass DM particle in an ${\text{Al}}_{2}{\mathrm{O}}_{3}$ target, until they reach the interface with a phonon Al collector. After a preliminary methodological survey that reveals the limitations of any relaxation time approximation (RTA)-based method, we develop a beyond-RTA phonon Monte Carlo for three-dimensional materials that allows us to introduce the spatial dimension of the device and address questions about the impact of target size and scattering position. We also analyze the effect of the phonon energy and wave vector, and show that isotopes can, perhaps counterintuitively, result in a larger heat flux by providing transport channels of higher velocities, thus favoring detection. Our results suggest that, though challenging, the direct detection of light DM via athermal phonon generation appears feasible, and phonon downconversion followed by quasiballistic propagation does not appear to be a major bottleneck in terms of reducing the signal.