Application of a Slowly Varying Envelope Function Onto the Analysis of the Wigner Transport Equation

The Wigner function formalism leads to several difficulties in order to investigate quantum effects in heterostructure devices. The corresponding spatial variation of the effective mass distribution is neglected in most methods. Furthermore, conventional methods for the numerical analysis of the Wigner transport equation disregard the quantum effects at the boundaries caused by the nonlocal kinetic operator. A method is proposed, which considers the spatially varying effective mass distribution for the stationary equilibrium case in the ballistic regime and considers an adequate inclusion of the nonlocal kinetic operator. Due to the interaction between the nonlocal kinetic operator and the quantum statistical distribution function at the boundaries, the boundary conditions inherently include the nonlocal character of quantum mechanics. For this purpose a slowly varying envelope function is introduced, which considers the modification of the Wigner function for the spatially varying effective mass distribution on the basis of the Wigner function for the case of a spatially constant effective mass distribution. The envelope function approach is presented and validated.