A WKB approach to the quantum multiband electron dynamics in the kinetic formalism

We derive a WKB-like asymptotic expansion of the multiband Wigner function. The model, derived in the envelope function theory, is designed to describe the dynamics in semiconductor devices when the interband conduction-valence transition cannot be neglected.
We derive a hierarchy of equations that to lowest order consist of two Hamilton-Jacobi equations corresponding to the classical dynamics of point particles with positive and negative kinetic energy. Our methodology is based on the Van Vleck approach and a WKB-like asymptotic expansion procedure is used to reduce the numerical complexity of the Wigner multiband evolution system. An approximate closed-form solution is obtained by an iterative procedure that exploits the different time scales on which the intraband and interband dynamical variables evolve. The interband tunneling mechanism appearing to the first order of the expansion is expressed in a very simple mathematical form. By exploiting the highly oscillating behaviour of the multiband Wigner functions we derive a asymptotic expression of the interband transition probability.
The resulting formulation reveals particularly close to the classical description of the particles motion and this formal analogy is useful to gain new physical insight and to profit of the numerical method developed for classical systems. The approximates evolution equations are used to simulate the evolution of the Wigner quasi-distribution function in a IRTD diode