Heat transport in ultrathin dielectric membranes and bridges

Phonon modes and their dispersion relations in ultrathin homogeneous dielectric membranes are calculated using elasticity theory. The approach differs from the previous ones by a rigorous account of the effect of the film surfaces on the modes with different polarizations. We compute the heat capacity of membranes and the heat conductivity of narrow bridges cut out of such membranes, in a temperature range where the dimensions have a strong influence on the results. In the high-temperature regime we recover the three-dimensional bulk results. However, in the low-temperature limit the heat capacity CV is proportional to T (temperature), while the heat conductivity κ of narrow bridges is proportional to T3∕2, leading to a thermal cutoff frequency fc=κ/CV∝T1∕2.