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 of 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 cutoff for the phonon 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.

Figure 1

(a) Full, dielectric membrane. (b) Cut membrane for better thermal insulation.

Figure 2

The ratio ${\tau }_{h,0,k_{\Vert }\left(\omega \right)}∕{\tau }_{a,0,k_{\Vert }\left(\omega \right)}$ as function of the ratio $c_t^2∕c_l^2$ for different values of $C_l∕C_t$. The value $C_l∕C_t=4∕3$ is the minimum possible value and, thus, all other curves lie above the solid curve in the plot.