Two-dimensional phononic thermal conductance in thin membranes in the Casimir limit

We discuss computational analysis of phononic thermal conduction in the suspended membrane geometry, in the case where heat can flow out radially in two dimensions from a central source. As we are mostly interested in the low-temperature behavior where bulk scattering of phonons becomes irrelevant, we study the limit where all phonon scattering takes place at the membrane surfaces. Moreover, we limit the discussion here to the case where this surface scattering is fully diffusive, the so called Casimir limit. Our analysis shows that in the two-dimensional case, no analytic results are available, in contrast to the well known 1D Casimir limit. Numerical solutions are presented for the temperature profiles in the membrane radial direction, for several different membrane thicknesses and heater diameters. Our results can be applied, for example, in the design of membrane-supported bolometric radiation detectors.