Quantum kinetic equations for electrons in random impurities

Abstract

The problem of the linear response of a collection of dynamically independent electrons interacting with a random set of scattering centers to an arbitrary electromagnetic field is considered on the basis of quantum kinetic equations. First a projection technique is used to derive a general kinetic equation for that part of the density matrix that is averaged over the uncorrelated impurity distribution. This procedure is free of the criticisms of the earlier methods of treating the impurity averaging and is at the same time much simpler. For the case of weak scattering, the importance of the interference term of the kinetic equation is demonstrated by studying the indirect power absorption. The case of uniform electric field is studied in detail for various simple systems, for which more specialized kinetic equations are derived simply as matrix elements of the general equation. For an arbitrary electromagnetic field the gauge invariance of the results is demonstrated by deriving a kinetic equation in terms of the physical field strengths rather than their potentials. A “local” density matrix is introduced for the evaluation of the current density and its kinetic equation derived. From this a transport equation for a quantum-mechanical distribution function is obtained for the case of free electrons. The conditions under which it reduces to the Boltzmann equation for the classical distribution function are investigated.