Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - October 2006 - Volume 49, Issue 03  

Research Article

DIFFUSIVE TRANSPORT OF PARTIALLY QUANTIZED PARTICLES: EXISTENCE, UNIQUENESS AND LONG-TIME BEHAVIOUR

N. Ben Abdallaha1, F. Méhatsa1 and N. Vaucheleta1

a1 Mathématiques pour l’Industrie et la Physique (UMR 5640), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 4, France (naoufel@mip.ups-tlse.fr; mehats@mip.ups-tlse.fr; vauchel@mip.ups-tlse.fr)

Abstract

A self-consistent model for charged particles, accounting for quantum confinement, diffusive transport and electrostatic interaction is considered. The electrostatic potential is a solution of a three-dimensional Poisson equation with the particle density as the source term. This density is the product of a two-dimensional surface density and that of a one-dimensional mixed quantum state. The surface density is the solution of a drift–diffusion equation with an effective surface potential deduced from the fully three-dimensional one and which involves the diagonalization of a one-dimensional Schrödinger operator. The overall problem is viewed as a two-dimensional drift–diffusion equation coupled to a Schrödinger–Poisson system. The latter is proven to be well posed by a convex minimization technique. A relative entropy and an a priori $L^2$ estimate provide sufficient bounds to prove existence and uniqueness of a global-in-time solution. In the case of thermodynamic equilibrium boundary data, a unique stationary solution is proven to exist. The relative entropy allows us to prove the convergence of the transient solution towards it as time grows to infinity. Finally, the low-order approximation of the relative entropy is used to prove that this convergence is exponential in time.

(Online publication January 25 2007)

(Received September 20 2004)

Keywords

  • Primary 35Q40;
  • 76R99;
  • 74H40;
  • 49S05;
  • 49K20;
  • Schrödinger equation;
  • drift–diffusion system;
  • relative entropy;
  • long-time behaviour;
  • subband method;
  • convex minimization