A quantum formalism for a terahertz acoustic laser

Abstract

The aim of this work is to improve the study of a phonon laser saser proposed by us several years ago 1 . This is a device capable to generate an intense coherent beam of acoustical phonons. Our acoustic laser consists in a double barrier heterostructure tailored such the energy di erence between the ground and the rst excited state in the well is close to the energy of the LO phonon. The electrons are directly injected into the excited level. Therefore they decay producing a high rate of LO phonons. These phonons are con ned inside the well and decay i n to a pair of phonons 2 :LO !L O + T A. The TA phonons escape the well in the 111 direction constituting an intense coherent beam. Recently were studied and sometimes realized experimentally several kinds of phonon lasers. Up to our knowledge our saser is the only that has a very short wavelength smaller than 25 A and a very long range greater than 1000 m . Because of that, such beam could have applications to acoustic nanoscopy, acoustic nanolithography and phonoelectronics. In early articles 1, 3, 4, 5, 6 we get the kinetic equations for the averaged electron and phonon populations.Quantum uctuations were not taken into account. The system Hamiltonian is H = He + Hph + He,ph+Hph,ph+He,e. T o solve this Hamiltonian we expand their eigenfunctions in the basis of the eigenstates jjn1n2n3i of the single particle part of it. We obtain a set of coupled equations for the expansion coe cients that can be solved with some approximations. The results are qualitatively similar to those obtained previously. I IntroductionIn the recent y ears, the study of mesoscopic system has steadily grown. New theoretical approaches appeared at the time when new technologies were developed to produce systems of sizes ranging from 5 nm to 500 nm. In particular, double barrier heterostructures DBH have attracted a large attention due to its potential applications as ultra-high-frequency electronic oscillators, diodes, transistors and other electronic devices 7 .After the work of Goldman, Tsui and Cunningham 8, 9 , it was realized the importance of electron-phonon interactions on the electronic properties of these structures. However, little importance has been given to the study of the phonons generated in this process, the way they propagate, their decay processes, etc. This paper is dedicated mainly to the study of phonon generation in a DBH under the e ect of an external applied bias. In previous works 3, 4, 6 this study was done through a set of phenomenological kinetic equations for the average populations. In this paper a set of quantum kinetic equations is obtained to describe the electron and phonon population using a full quantum treatment.