Stationary and transient simulations of a resonant tunneling diode in the ballistic regime are presented. The simulated model consists in a set of Schrödinger equations for the wave functions coupled to the Poisson equation for the electrostatic interaction. The Schrödinger equations are applied with open boundary conditions that model continuous injection of electrons from reservoirs. Automatic resonance detection enables reduction of the number of Schrödinger equations to be solved. A Gummel type scheme is used to treat the Schrödinger-Poisson coupling in order to accelerate the convergence. Stationary I-V characteristics are computed and the transient regime between two stationary states is simulated.

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Asymptotic quantum transport models of a two-dimensional electron gas are presented. The starting point is a singular perturbation of the threedimensional Schrödinger-Poisson system. The small parameter ε is the scaled width of the electron gas and appears as the lengthscale on which a one dimensional confining potential varies. The rigorous ε → 0 limit is performed by projecting the three dimensional wavefunction on the eigenfunctions corresponding to the confining potential. This leads to a two-dimensional Schrödinger-Poisson system with a modified Poisson equation keeping track of the third dimension. This limit model is proven to be a first-order approximation of the initial model. An intermediate model, called the "2.5D adiabatic model" is then introduced. It shares the same structure as the limit model but is shown to be a second-order approximation of the 3D model.