Elastic constants and lattice anharmonicity of GaSb and GaP from ultrasonic-velocity measurements between 4.2 and 300 K

Measurements have been made of the transit times of pulses of 30-MHz longitudinal and transverse ultrasonic waves in sulfur-doped $n$-type single crystals of GaSb and GaP down to 4.2 K. Length-versus-temperature measurements have been made for GaP between 80 and 300 K using a silica dilatometer. Values are presented for the elastic constants $C_{\mathrm{ij}}$ at various temperatures. The $C_{\mathrm{ij}}$ for GaP below room temperature are the first ever reported. From the low-temperature elastic constants are deduced elastic Debye temperatures of 269.4 K for GaSb and 443.8 K for GaP. Martin's relation between the elastic constants is found to be satisfied no better when the "harmonic" elastic constants rather than room-temperature data are used in it. The temperature dependence of each $C_{\mathrm{ij}}$ of GaSb and GaP can be fitted in most of our temperature range by a function having the form suggested by Leibfried and Ludwig to account for lattice anharmonicities. The function contains the average energy of a harmonic lattice oscillator multiplied by a factor $K_{\mathrm{ij}}$ whose value is chosen to yield agreement with data at 240 and 4.2 K. For GaSb the value of $K_{12}$ is accounted for mainly by a term containing the thermal expansion, the pressure derivative of $C_{12}$, and the bulk modulus; whereas $K_{11}$ and $K_{44}$ each has a much larger value than can be attributed to the thermal-expansion term. Data on other semiconductors are reviewed and it is found that $K_{12}>K_{11}>K_{44}$ for all III-V semiconductors for which sufficient data are available.