On the stationary quantum drift-diffusion model

Abstract

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)

Quantum Energy-Transport and Drift-Diffusion Models

“…The Classical Drift-Diffusion system corrected with the Bohm potential has already been used in the physics or mathematics literature [2], [3], [18], [64], [65]. Our approach provides another derivation of this model.…”

“…To our knowledge, the proof of this property is new. Other mathematical properties and numerical simulations of this model can be found in [18], [64], [65].…”

A finite-volume scheme for the multidimensional quantum drift-diffusion model for semiconductors

“…Section 3 is concerned with the proof of the existence of a solution to the numerical scheme. The proof is based on a fixed-point argument and the minimization of the discrete energy motivated by [5]. As a by-product, we show L ∞ estimates independent of the discretization parameter using a discrete Stampacchia technique.…”

“…The quantum drift-diffusion model can also be derived in the relaxation-time limit from the quantum hydrodynamic equations [22]. The existence of a weak solution to (2)-(3) was shown in [5]. Existence results for the one-dimensional transient equations can be found in [24,25].…”

An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes

“…In a diffusive context, one can find the energy transport model corrected with the Bohm potential [11] and, closer to the eQDD model studied in this paper, the Drift-Diffusion model corrected with the Bohm potential, called Density-Gradient (DG) model (also "Quantum Drift-Diffusion model" in the literature). This model was derived in [6,5] and studied in [41,28,4]. But the Bohm potential has the disadvantage of bringing higher order differential terms which are difficult to handle numerically and mathematically.…”