Heat Exchange in Liquid Helium by Phonon Tunneling Through Very Thin Plates

Thin plates with a thickness d smaller than the phonon mean-free path can be used to increase the heat exchange in He-3-He-4 refrigerators at very low temperatures. The resistance R for the heat current density of these plates is theoretically investigated. For plates with d still larger than the dominant thermal-phonon wavelength lambda(T) in the plate, R is given by twice the Kapitza resistance. For d approximate to lambda(T) the resistance R decreases due to phonon tunneling. For d << lambda(T) there exists a region where R varies as T-1, in contrast to the T-3 dependence for thicker plates. Numerical calculations are carried out for copper plates immersed in pure He-4. The results show that one obtains a large increase of the heat exchange due to phonon tunneling for plates with a thickness of a few microns at temperatures in the millidegree range.