Quantization of the elastic modes in an isotropic plate

We quantize the elastic modes in a plate. For this, we find a complete, orthogonal set of eigenfunctions of the elastic equations and we normalize them. These are the phonon modes in the plate and their specific forms and dispersion relations are manifested in low temperature experiments in ultra-thin membranes.Comment: 14 pages, 2 figure

Interaction of Lamb modes with two-level systems in amorphous nanoscopic membranes

Using a generalized model of interaction between a two-level system (TLS) and an arbitrary deformation of the material, we calculate the interaction of Lamb modes with TLSs in amorphous nanoscopic membranes. We compare the mean free paths of the Lamb modes with different symmetries and calculate the heat conductivity $\kappa$. In the limit of an infinitely wide membrane, the heat conductivity is divergent. Nevertheless, the finite size of the membrane imposes a lower cut-off for the phonons frequencies, which leads to the temperature dependence $\kappa\propto T(a+b\ln T)$. This temperature dependence is a hallmark of the TLS-limited heat conductance at low temperature.Comment: 9 pages, 2 figure

Heat transport in ultra-thin dielectric membranes and bridges

Phonon modes and their dispersion relations in ultrathin homogenous 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, $C_V$, is proportional with $T$ (temperature), while the heat conductivity, $\kappa$, of narrow bridges is proportional to $T^{3/2}$, leading to a thermal cut-off frequency $f_c=\kappa/C_V\propto T^{1/2}$.Comment: 6 pages and 6 figure