Resonant Amplification of Sound by Conduction Electrons

Abstract

A Boltzmann equation technique is used to calculate the angular distribution of sound amplified by conduction electrons under various conditions. It was found that if ${\mathrm{v}}_d$ is the drift velocity of the electrons, and $\hat{q}$ a unit vector in the direction of propagation of sound, amplification occurs for all $\hat{q}$ such that $\hat{q}·{\mathrm{v}}_d>v_s$, where $v_s$ is the velocity of sound. Resonance peaks in the amplification occur at certain directions of propagation. For a semimetal in crossed electric and magnetic fields resonant amplification occurs when $\hat{q}·{\mathrm{v}}_d-v_F\left(\hat{q}·\hat{H}\right)\approx v_s$ (where $\hat{H}$ is a unit vector in the direction of the magnetic field, and $v_F$ the Fermi velocity) if $\mathrm{ql}\gg 1$, where $l$ is the mean free path of the electrons and $q$ the wave number of the sound wave; if $\mathrm{ql}\ll 1$ resonance peaks occur for $\hat{q}·{\mathrm{v}}_d\approx v_s$. The peak amplification is rather insensitive to the value of $\frac{v_d}{v_s}$, and varies with frequency as $\mathrm{ql}$. For a metal or semiconductor in an applied electric field only, resonant amplification occurs for those directions of propagation such that $\hat{q}·{\mathrm{v}}_d\approx v_s$, if $\mathrm{ql}\ll 1$, and the peak amplification is independent of $\frac{v_d}{v_s}$. There are no resonances in this case of $\mathrm{ql}\gg 1$.