This paper shows that various models of electron transport in semiconductors that have been previously proposed in the literature can be connected one with each other by the diffusion approximation methodology. We first investigate the diffusion limit of the semiconductor Boltzmann equation towards the so-called ‘‘spherical harmonic expansion model,’’ under the assumption of dominant elastic scattering. Then, this model is again connected, either to the energy-transport model or to a ‘‘periodic spherical harmonic expansion model’’ through a diffusion approximation, respectively making electron–electron or phonon scattering large. We provide the mathematical background which makes the Hilbert expansions associated with these various diffusion limits rigorous.

The nonlinear Schrödinger equation with general nonlinearity of polynomial growth and harmonic confining potential is considered. More precisely, the confining potential is strongly anisotropic; i.e., the trap frequencies in different directions are of different orders of magnitude. The limit as the ratio of trap frequencies tends to zero is carried out. A concentration of mass on the ground state of the dominating harmonic oscillator is shown to be propagated, and the lower-dimensional modulation wave function again satisfies a nonlinear Schrödinger equation. The main tools of the analysis are energy and Strichartz estimates, as well as two anisotropic Sobolev inequalities. As an application, the dimension reduction of the three-dimensional Gross-Pitaevskii equation is discussed, which models the dynamics of Bose-Einstein condensates.

Abstract. An efficient and accurate numerical method is presented for the solution of highly oscillatory differential equations. While standard methods would require a very fine grid to resolve the oscillations, the presented approach uses first an analytic (second order) WKB-type transformation, which filters out the dominant oscillations. The resulting ODE is much smoother and can hence be discretized on a much coarser grid, with significantly reduced numerical costs.In many practically relevant examples, the method is even asymptotically correct w.r.t. the small parameter ε that identifies the oscillation wave length. Indeed, in these cases, the error then vanishes for ε → 0, even on a fixed spatial mesh. Applications to the stationary Schrödinger equation are presented.

A bipolar Quantum Drift Diffusion Model including generation-recombination terms is considered. Existence of solutions is proven for a general setting including the case of vanishing particle densities at some parts of the boundary. The proof is based on a Schauder fixed point iteration combined with a minimization procedure. It is proven that, contrary to the classical drift-diffusion model, vacuum can only appear at the boundary. In the case of nonvanishing boundary data, the semiclassical limit is carried out rigorously. The variational structure of the model allows to prove strong H 1 convergence of particle densities, Fermi levels and electrostatic potential. Mathematics Subject Classification (1991). 35J50, 35J55, 35J70, 35Q40, 35Q55, 49J45, 49K20, 81Q20. Keywords. Bipolar quantum hydrodynamic model, stationary states, existence of solutions, elliptic boundary value problems of degenerate type, voltage current characteristics, variational formulation, minimization of energy functionals, semi-classical limit, drift-diffusion model. 252 N. Ben Abdallah and A. Unterreiter ZAMP els on bounded domains, an increasing effort has been made during the very last years. First, equilibrium states of the Schrödinger-Poisson systems were analyzed by minimization techniques [21, 22, 23]. Current carrying models and absorbing boundary conditions were recently investigated [24, 2, 1, 16, 5].The incorporation of collisions in quantum models is one of the important issues of mesoscopic semiconductor modelling. It would imply a better understanding of quantum macroscopic models (hydrodynamic, drift-diffusion). However, macroscopic "Quantum Hydrodynamic Models (QHD)", based on moment expansions, are derived from the Wigner equation or the many particle Schrödinger equation [9,11]. Being numerically rather tractable [10,12] the capability of QHD to simulate ultra small semiconductor devices is a field of intensive mathematical research. Also the consistency problem is of importance: It has to be expected that the solutions of QHD converge to solutions of semiclassical models as the Planck constant tends to zero. Ancona and Iafrate proposed in [4, 3] a stationary "Quantum Drift Diffusion Model (QDD)" dedicated to describe the behaviour of electrons in the vicinity of strong inversion layers. We shall extend this model to bipolar devices including generation-recombination processes. We prove the existence of solutions for fixed Planck constant in the case of vanishing particle densities at the strong inversion layer. The proof is based on a minimization argument coupled with Schauder's fixed point theorem. The essential estimates concern lower bounds away from zero in the interior of the domain. In the case of nonvanishing particle density on the boundary the semiclassical limit is performed leading to the classical drift-diffusion model.The scaled QDD stated on a bounded domain Ω ⊂ R d , d = 1, 2 or d = 3 readsJ n = µ n n∇F , J p = −µ p p∇G ∇ · J n = R 0 (n, p)R 1 (F, G), ∇ · J p = −R 0 (n, p)R 1 (F, G)