M.Baro, H.Neidhardt, J.Rehberg
Current Coupling of 
Drift-Diffusion Models and 
Dissipative Schr dinger-Poisson Systems: 
Dissipative Hybrid Models
(804K, pdf)

ABSTRACT.  A 1D coupled drift-diffusion dissipative Schr dinger model (hybrid 
model), which is capable to describe the transport of electrons and 
holes in semi-conductor devices in a non-equilibrium situation, is 
mathematically analyzed. The device domain is split into a part where 
the transport is well-described by the drift-diffusion equations (classical 
zone) and a part where a quantum description via a dissipative 
Schr dinger system (quantum zone) is used. Both system are coupled 
such that the continuity of the current densities is guaranteed. The 
electrostatic potential is self-consistently determined by Poisson's 
equation on the whole device. We show that the hybrid model is 
well-posed, prove existence of solutions and show their uniform boundedness provided 
the distribution function satisfy a so-called balance condition. 
The current densities are different from zero in the non-equilibrium case and uniformly bounded.