Ultrasonic-harmonic generation in bismuth

Abstract

Second-harmonic generation of ultrasound in a semimetal like bismuth is investigated using a quantum-mechanical treatment which is valid at high frequencies and in strong magnetic fields. It is shown that the amplitude of the second harmonic can be expressed in terms of the fundamental using the linear and nonlinear conductivity tensors for ultrasonic waves propagating parallel to a dc magnetic field. At first we assume that the energy-band structure of bismuth is the Cohen nonellipsoidal nonparabolic (NENP) model. It is found that the magnitude of the second-harmonic generation oscillates with the dc magnetic field. Some suboscillations with the dc magnetic field are also observed. These suboscillations occur owing to the spin-splitting effect of the Landau levels in bismuth. When the sound frequency increases, these oscillations will be diminished due to the breakdown of the screening. If the energy-band structure is assumed to be the ellipsoidal nonparabolic (ENP) model, the oscillations and suboscillations can also be observed. However, the suboscillations will be diminished very rapidly with the sound frequency increasing. In fact, these suboscillations are washed out at the region of high frequencies. When the energy-band structure is assumed to be the ellipsoidal parabolic (EP) model, no suboscillations can be observed. It is also shown that the second-harmonic generation increases with increasing the sound frequency for the NENP model. The second-harmonic generation will peak in the neighborhood of $\ensuremath{\omega}=2\ifmmode\times\else\texttimes\fi{}{10}^{11}$ rad/sec owing to the effect of highly anisotropic Fermi surfaces in bismuth. However, no peaks occur at the high-frequency region for the ENP or EP models.