Low temperature heat capacity of phononic crystal membranes

Phononic crystal (PnC) membranes are a promising solution to improve sensitivity of bolometric sensor devices operating at low temperatures. Previous work has concentrated only on tuning thermal conductance, but significant changes to the heat capacity are also expected due to the modification of the phonon modes. Here, we calculate the area-specific heat capacity for thin (37.5 - 300 nm) silicon and silicon nitride PnC membranes with cylindrical hole patterns of varying period, in the temperature range 1 - 350 mK. We compare the results to two- and three-dimensional Debye models, as the 3D Debye model is known to give an accurate estimate for the low-temperature heat capacity of a bulk sample. We found that thin PnC membranes do not obey the 3D Debye T3 law, nor the 2D T2 law, but have a weaker, approximately linear temperature dependence in the low temperature limit. We also found that depending on the design, the PnC patterning can either enhance or reduce the heat capacity compared to an unpatterned membrane of the same thickness. At temperatures below ∼ 100 mK, reducing the membrane thickness unintuitively increases the heat capacity for all samples studied. These observations can have significance when designing calorimetric detectors, as heat capacity is a critical parameter for the speed and sensitivity of a device.