Generation of high-frequency coherent acoustic phonons in superlattices under hopping transport. I. Linear theory of phonon instability

In this work we consider the theory of high-frequency phonon generation in a weakly coupled doped semiconductor superlattice. Electric bias, applied to such a superlattice, destroys the electron minibands, creates electron states localized in the individual quantum wells, and forms population inversion between these states. An electric current occurs due to the phonon-induced interwell hops. We show that under such conditions the electric current produces a phonon instability: populations of phonon modes propagating almost collinearly with the superlattice axis increase exponentially in time. It is demonstrated that the population growth increment can be as high as several times ${10}^8\hspace{0.5em}{\mathrm{s}}^{-1},$ and considerably exceeds the internal phonon scattering rates. Also discussed are effects influencing the increment, such as a screening of the electron-phonon interaction and a modification of the phonon spectrum in superlattices.