Transport models for partially quantized systems

Transport of electrons in extremely confined semiconductor devices like Double Gate Mosfets (DGMOS) involves a variety of scales that need to be included in the mathematical modelling. Since the lengthscale in the confined direction is comparable to de Broglie wavelength of the electrons, quantum effects do appear in this direction, while in the remaining transport direction, the lengthscale is usually much larger and is dealt with semiclassically. We shall start this course by analyzing the confinement direction through the study of a Schrödinger-Poisson system. We shall begin by the one di-mensional case using the properties of eigenfunctions and eigenvalues of the Schrödinger operator [5] and using a reformulation of the problem as a min-imization problem of a strictly convex function [3]. The multidimensional case is then treated by means of functional calculus and by the so-called Sjöstrand-Hellfer formula [4]. The coupling with the classical direction will be done first in the kinetic context by combining the above studies with the energy and entropy estimates for kinetic equations [1] and for diffusion type models by using entropy and relative entropy techniques [2]