We study second-harmonic generation of ultrasound in degenerate piezoelectric semiconductors using a quantum treatment which is valid at high frequencies and in strong magnetic fields. The effects of electron scattering on the electron-phonon interaction are neglected so that this quantum treatment is valid for the high-frequency region such that $\omega \tau \gg 1$, where $\omega$ is the is sound frequency and $\tau$ is the relaxation time of electrons. It is found that the magnitude of the second-harmonic generation depends strongly upon the dc magnetic field and is considerably enhanced using the nonparabolic model for the energy bands of a degenerate semiconductor. Some oscillations in the dependence of the acoustic intensity on the magnetic field can be observed for both parabolic and nonparabolic band structures at the region of high frequencies and strong magnetic fields. These oscillations can be interpreted as the so-called "giant quantum oscillations" which arise because the conduction electrons in solids can resonantly absorb or emit phonons. It can also be seen that some discontinuities occur at strong magnetic fields in the acoustic intensity of the second-harmonic generation for the nonparabolic band structure. This is due to the fact that the linear and nonlinear conductivity tensors have singularities of purely quantum origin related to the degeneracy of the electron gas.

• Received 5 December 1972

DOI:https://doi.org/10.1103/PhysRevB.8.2670