Steady states and interface transmission conditions for heterogeneous quantum-classical 1-D hydrodynamic model of semiconductor devices

We study a hybrid model linking the quantum hydrodynamics equation with classical hydrodynamics deriving the transmission conditions between the two PDE systems modeling the quantum and the classical dynamics. These conditions are derived under the assumption of constant scaled temperature and assuming a current jump across the interface between the classical and the quantum region. In Section 2.2 we shall give a heuristic and physically plausible explanation of this assumption.Because of this fact we produce not a weak solution on the whole device domain, but a piecewise smooth solution.