Calculation of the specific heat in ultra-thin free-standing silicon membranes

The specific heat of ultra-thin free-standing membranes is calculated using the elastic continuum model. We first obtain the dispersion relations of the discrete set of acoustic modes in the system. The specific heat is then calculated by summing over the discrete out-of-plane wavevector component and integrating over the continuous in-plane wavevector of these waves. In the low-temperature regime (T < 4 K), the flexural polarization is seen to have the highest contribution to the total specific heat. This leads to a linear dependence with temperature, resulting in a larger specific heat for the membrane compared to that of the bulk counterpart.