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Quantum hydrodynamic models for semiconductors with and without collisions

In this thesis we study quantum hydrodynamic (QHD) models, particularly the ones used in semiconductor device modeling. The QHD model consists of the conservation laws for the particle density, momentum, and energy density, including quantum corrections from the Bohm potential. We start with a review of the known results on collisionless QHD models derived from the mixed-state Schr\"odinger system or from the Wigner equation. Using the reformulation of the one-dimensional stationary QHD equations with the linear potential as a stationary Schr\"odinger equation, the semi-analytical expressions for current-voltage curves are studied. Further on, we consider viscous stabilizations of the QHD model. The numerical viscosity for the upwind finite-difference discretization of the QHD model proposed by C.~Gardner is computed. On the other side, starting from the Wigner equation with the Fokker-Planck collision operator we derive the viscous QHD model. This model contains the physical viscosity introduced by the collision operator. The existence of solutions (with strictly positive particle density) to the isothermal, stationary, one-dimensional viscous model for general data and non-homogeneous boundary conditions is shown. The estimates depend on the viscosity and do not allow to perform the inviscid limit. By numerical simulations of the resonant tunneling diode using the non-isothermal, stationary, one-dimensional viscous QHD model, we show the influence of the physical viscosity on the solution. Applying the quantum entropy minimization method, recently developed by P.~Degond and C.~Ringhofer, we derive the general QHD equations, starting from a Wigner-Boltzmann equation with the BGK-type collision operator. The derivation is based on a careful expansion of the quantum Maxwellian in powers of the scaled Planck constant. The general QHD model includes also vorticity terms and a dispersive term for the velocity. Current-voltage curve of the resonant tunneling diode for the simplified general QHD equations in one dimension is studied by numerical simulations. The results indicate that the dispersive velocity term regularizes the solution of the system.

quantum hydrodynamics; resonant tunneling diode; physical viscosity; existence of stationary solutions; entropy minimization method

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