|The quantum hydrodynamical model (QHD) coupled with the Poisson equation is able to describe the quantum effect appearing at nanoscale in many modern semiconductor devices. It can be obtained adding the Bh\"om potential to the classical hydrodynamical equations (HD) or directly from the Schroedinger equation.
Here we derive a physically reasonable set of boundary conditions (BCs) for the QHD-Poisson system. We just consider the unipolar case, thus the holes concentration is neglected.
These new BCs have two interesting explanations from the physical viewpoint. Firstly, if we consider the Bohm term as a correction for the pressure functional, it implies the conservation of the generalized enthalpy at the interface metal-semiconductor. Alternatively, assuming the Bohm term works together to the electrical potential, the BCs imply the equilibrium between diffusive and quantum forces.
The existence and the uniqueness of a regular solution for the QHD-Poisson system is then discussed using these new BCs.
The model is tested numerically on a toy device and the linear stability of the solution is discussed in a special case.
The same consideration can be used to derive interface conditions between QHD and HD in the contest of the hybrid models.