Third-order elastic constants and Grüneisen parameters of silicon and germanium between 3 and 300 °K

Abstract

We present a method to determine all six independent third-order elastic (TOE) constants of diamond-like solids as a function of temperature from measured values of the ultrasonic nonlinearity parameters. Ultrasonic harmonic generation experiments along the three principal directions of a cubic crystal lead to the determination of the nonlinearity parameters, which contain three linear combinations of TOE constants. These data are sufficient to determine the three anharmonic first and second neighbor force constants in the Keating model for diamond-like solids. Ultrasonic velocity data are used to determine the two harmonic force constants in this model. Since our experiments have given data on silicon and germanium between liquid helium temperature and room temperature, we are able to evaluate and plot the Keating model force constants and hence all the six third-order elastic constants between 3 and 300 °K. Most of the TOE constants of silicon and germanium are smoothly varying functions of temperature; however, C123 and C144 go through zero and assume positive values at low temperatures. To determine whether the behavior of the calculated TOE constants is reasonable, we have used them to calculate the generalized Grüneisen parameters γ in the quasiharmonic approximation and have plotted γ as a function of temperature. These curves are in better agreement with curves obtained from thermal expansion experiments than the curves plotted by earlier workers using room temperature values of third-order elastic constants. For completeness we also have calculated the pressure dependence of the SOE constants and the Anderson Grüneisen parameters over the same temperature range.