Quantum Theory of Lattice-Wave Amplification in Semiconductors

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Abstract
The amplification of lattice waves (acoustical or optical) in semiconductors is considered from the viewpoint of the absorption and stimulated emission of phonons by conduction electrons. An adaptation of the Weisskopf-Wigner theory is used to calculate the effect of scattering in permitting transitions for which wave vector is conserved between initial and final states, but energy is not. No restriction on the frequency or wave vector of the phonon is made. A finite probability of emission and absorption of a given phonon is found for every initial electronic state; emission exceeds absorption when the group velocity of the state exceeds the phase velocity of the phonon. An expression is given for the net rate of emission into a particular mode by a distribution of electrons, uniform in real space but arbitrary in wave-vector space, and rates of lattice-wave amplification are calculated for some simple cases. These rates are compared with previously known macroscopic calculations based on transport theory, and the reason for the differences found are discussed. It is concluded that the present results describe the contribution to the amplification by the spatially uniform part of an electron distribution, whereas the macroscopic ones are associated with bunching phenomena.