Effect of surface roughness on the universal thermal conductance

We explain the reduction of the thermal conductance below the predicted universal value observed by Schwab et al. [Nature (London) 404, 974 (2000)] in terms of the scattering of thermal phonons off surface roughness using a scalar model for the elastic waves. Our analysis shows that the thermal conductance depends on two roughness parameters: the roughness amplitude δ and the correlation length a. At sufficiently low temperatures the ratio of the conductance to the universal value decreases quadratically with temperature at a rate proportional to δ²a. Values of δ equal to $22%$ and a equal to about $75%$ of the width of the conduction pathway give a good fit to the data.