Electrodynamic instabilities in the deformation-potential interaction

Abstract

This paper is concerned with several electrodynamic effects associated with the interaction of hypersonic waves with conduction electrons in crystals of n-type germanium. One aspect of this interaction involves the creation of coherent microwave current by a sound wave under the influence of a dc electric field. We study this aspect in relation to current oscillation, amplification, and traveling-wave effects with a view toward engineering applications. Electromagnetic radiation and internal transmission of energy are similarly discussed. We have constructed a model for the interaction of electron current with hypersonic waves ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w \geq 10_{9}</tex> ) through shear-deformation potential coupling in moderately doped n-type germanium subject to a dc electric field. From a Boltzmann equation treatment of the charge transport, and a continuous-system Lagrange equation for the sound field, we obtained the dispersion relation. By means of the Bers-Briggs criteria, we found a convective instability (spatial growth) for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">10_{10} &lt; w &lt; 10_{12}</tex> for supersonic drift. Since no absolute instability (spontaneous oscillation) was detected, this interaction can provide stable amplification of ac electron current and of hypersonic waves. Or, if one is an input to the crystal, the other will be generated. The gain is of order hundreds of decibels per centimeter. Solving the eigenvalue problem yielded an infinite discrete set of complex frequencies whose imaginary parts indicated growth rates of order 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sup> dB/s, which amounts to a usable gain of order 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> dB. Finally, Maxwell's equations were used to estimate the electro-magnetic effects of the various electrodynamic instabilities. Sample calculations indicated current densities of order 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> A/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> .