Bulk acoustic wave propagation characteristics in degenerately n-doped Si

The effect of degenerate level n-type doping on the bulk acoustic wave propagation properties in single crystal silicon is studied. Keyes´ theory [1] is used to calculate the electronic effect of free electrons on the elastic constants c_ij as function of temperature. Slowness curves are calculated for the bulk acoustic wave modes via solution of the Christoffel equation at several doping levels. Variation with respect to temperature is performed in order to extract the linear temperature coefficient of slowness as function of propagation direction. It is found that n-type doping has the largest effect on the slow shear mode and that temperature stable wave propagation directions exist with doping levels above 10¹⁹ cm^-3. Comparison to experiment is made via measured temperature coefficients of frequency of square plate resonators vibrating in the Lamé mode which is a pure superposition of shear bulk acoustic waves. Good agreement is found.