Ballistic phonon transport in dielectric membranes

We have calculated the ballistic phononic heat transport in dielectric membranes as a function of radiator temperature and membrane thickness. The phonon modes of such membranes are known as Lamb-modes from elasticity theory. The striking result is that, for a fixed temperature, the radiated power first decreases with decreasing membrane thickness, but then develops a minimum when the transition to two dimensionality is reached. Further decrease of the membrane thickness in the 2D limit leads to increasing radiated power.