Adiabatic approximation of the Schrödinger–Poisson system with a partial confinement: The stationary case

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Asymptotic quantum transport models of a two-dimensional gas are presented. The models are the stationary versions of those introduced in a previous paper by Ben Abdallah, Méhats, Pinaud. The starting point is a singular perturbation of the three-dimensional stationary Schrödinger–Poisson system posed on bounded domain. The electron injection in the device is modeled thanks to open boundary conditions. Under a small density assumption, the asymptotics lead to a full two-dimensional first-order approximation of the initial model. An intermediate model, called the “2.5D adiabatic model” in Ben Abdallah, Méhats, Pinaud is then introduced. It shares the same structure as the limit but is shown to be a second-order approximation of the three-dimensional model.