Ultrasonic attenuation by impurities in semiconductors

We have calculated the ultrasonic attenuation by donors and acceptors in semiconductors by solving the equation of motion for one-particle density matrix and the equation of sound. Explicit expressions of the attenuation coefficient are given for $n-\mathrm{Ge}$ and $p-\mathrm{Si}$. The crucial point in our theory is to take account of the relaxation of the system into an instantaneous local thermal equilibrium. In contrast to the theories for $n-\mathrm{Ge}$ by Kwok and for $p-\mathrm{Si}$ by Suzuki and Mikoshiba, who calculated the attenuation from the self-energy function in the Green's-function method, our formula is written as the sum of three terms: the classical Zener relaxation term and the usual resonance and antiresonance terms with the Lorentzian line shape. It is pointed out that the diagram technique used by Kowk is not justified for the impurity system. Our theory seems to be valid when the angular frequency of ultrasonic waves is smaller than the inverse of the relaxation time of the system.