Modelling electron injection in kinetic theory can be done through the prescription of inflowing distribution function at the considered device boundary. In the quantum context, this can be done thanks to transparent boundary conditions for the Schrodinger equation.

In this talk, we present existence results for the stationary coupled Schrodinger Poisson problem in one an multi-dimensional setting and show in the one-dimensional case that the semi-classical limit leads to the standard inflow boundary conditions for the Vlasov equation.

Then, we present and briefly analyze two time-dependent models which apply in the one-dimensional case to situations where the injection profile are time-dependent and the electrostatic potential is stationary, or the potential is time dependent while the injection profiles are stationary. We conclude by presenting numerical simulations of quantum waveguides in the stationary regime.

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