The semiclassical limit in the quantum drift-diffusion model

Semiclassical limit to the solution of isentropic quantum drift-diffusion model in semicon- ductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical drift-diffusion model. In addition, we also proved the global existence of weak solutions.     

SEMICLASSICAL LIMIT FOR BIPOLAR QUANTUM DRIFT-DIFFUSION MODEL

Semiclassical limit to the solution of transient bipolar quantum drift-diffusion model in semiconductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical bipolar drift-diffusion model. In addition, the authors also prove the existence of weak solution.     

Weak solutions to the stationary quantum drift-diffusion model

The existence of weak solutions to the stationary quantum drift-diffusion equations for semiconductor devices is investigated. The proof is based on minimization procedure of non-linear functional and Schauder fixed-point theorem. Furthermore, the semiclassical limit ε→0 from the quantum drift-diffusion model to the classical drift-diffusion model is discussed.     

Drift-dominated current flow in forward-biased semiconductor devices

The drift-diffusion equations of semiconductor physics, allowingfor field-dependent drift velocities, are analysed by the methodof matched asymptotic expansions for one-dimensional PN andPNPN forward-biased structures. The analysis is relevant todescribing the structure of the solutions to the drift-diffusionequations for large electric fields when drift velocity saturationeffects become significant. In this high-field limit, the boundarylayer structure for the solutions to the drift-diffusion equationsis seen to differ substantially from that near equilibrium.In particular, boundary layers for the carrier concentrationscan occur near the contacts. The asymptotic solutions and thecurrent-voltage relations, constructed in the high-field limit,are found to agree well with direct numerical solutions to thedrift-diffusion equations.     

Quasi-neutral limit to the drift-diffusion models for semiconductors with physical contact-insulating boundary conditions

In this paper the limit of vanishing Debye length in a bipolar drift-diffusion model for semiconductors with physical contact-insulating boundary conditions is studied in one-dimensional case. The quasi-neutral limit (zero-Debye-length limit) is proved by using the asymptotic expansion methods of singular perturbation theory and the classical energy methods. Our results imply that one kind of the new and interesting phenomena in semiconductor physics occurs.     

QUASI-NEUTRAL LIMIT AND THE INITIAL LAYER PROBLEM OF THE DRIFT-DIFFUSION MODEL

In this article we study quasi-neutral limit and the initial layer problem of the drift-diffusion model. Different from others studies, we consider the physical case that the mobilities of the charges are different. The quasi-neutral limit with an initial layer structure is rigorously proved by using the weighted energy method coupled with multi-scaling asymptotic expansions.     

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R³ is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R³ close to the steady-state is obtained.     

Asymptotic Analysis of a Semiconductor Model Based on Fermi–Dirac Statistics

The quasi-hydrodynamic carrier transport equations for semiconductors extended to Fermi–Dirac statistics are considered. It is shown that in the high injection case, these equations reduce to a drift-diffusion model with non-linear diffusion terms. The limiting procedure is proved rigorously and error estimates are shown. We compute numerically static voltage–current characteristics of a forward biased pn-junction diode and compare the curves with the corresponding characteristics obtained from the standard drift-diffusion model based on Boltzmann statistics. It turns out that there exists a so-called threshold voltage at which the behaviour of the characteristic changes. Under high injection conditions, the dependence of the current on the bias appears to be approximately polynomial. The characteristics are studied analytically for a unipolar device.     

Diffusive semiconductor moment equations using Fermi�CDirac statistics

Diffusive moment equations with an arbitrary number of moments are formally derived from the semiconductor Boltzmann equation employing a moment method and a Chapman?CEnskog expansion. The moment equations are closed by employing a generalized Fermi?CDirac distribution function obtained from entropy maximization. The current densities allow for a drift-diffusion-type formulation or a ??symmetrized?? formulation, using dual-entropy variables from nonequilibrium thermodynamics. Furthermore, drift-diffusion and new energy-transport equations based on Fermi?CDirac statistics are obtained and their degeneracy limit is studied.     

The Asymptotic Behavior of Global Smooth Solutions to the Macroscopic Models for Semiconductors

The authors study the asymptotic behavior of the smooth solutions to the Cauchy problems for two macroscopic models (hydrodynamic and drift-diffusion models) for semiconductors and the related relaxation limit problem. First, it is proved that the solutions to these two systems converge to the unique stationary solution time asymptotically without the smallness assumption on doping profile. Then, very sharp estimates on the smooth solutions, independent of the relaxation time, are obtained and used to establish the zero relaxation limit.     

Existence of weak solution and semiclassical limit for quantum drift-diffusion model

The existence and semiclassical limit of the solution to one-dimensional transient quantum drift-diffusion model in semiconductor simulation are discussed. Besides the proof of existence of the weak solution, it is also obtained that the semiclassical limit of this solution solves the classical drift-diffusion model. The key estimates rest on the entropy inequalities derived from separation of quantum quasi-Fermi level.     

Existence of weak solution and semiclassical limit for quantum drift-diffusion model

The existence and semiclassical limit of the solution to one-dimensional transient quantum drift-diffusion model in semiconductor simulation are discussed. Besides the proof of existence of the weak solution, it is also obtained that the semiclassical limit of this solution solves the classical drift-diffusion model. The key estimates rest on the entropy inequalities derived from separation of quantum quasi-Fermi level.     

Perturbation analysis of the MNA equations coupled with the drift-diffusion equations

We perform a perturbation analysis on a coupled system modeling an integrated circuit. The modified nodal analysis equations are coupled with the drift-diffusion equations. An index 1 estimate is proven under the assumptions that the network graph contains neither loops of voltage sources and capacitors nor cutsets of current sources and inductors. Additionally, it is assumed that the Dirichlet boundaries of the drift-diffusion modeled region are connected by capacitive paths. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasi-neutral Limit of a Nonlinear Drift Diffusion Model for Semiconductors

The limit of the vanishing Debye length (the charge neutral limit) in a nonlinear bipolar drift-diffusion model for semiconductors without a pn-junction (i.e., with a unipolar background charge) is studied. The quasi-neutral limit (zero-Debye-length limit) is determined rigorously by using the so-called entropy functional which yields appropriate uniform estimates.     

半导体中非线性漂流扩散模型的拟中性极限:快扩散情形

本文研究无Pn-联结的非线性双极半导体漂流扩散模型的消失Debye长度极限(即粒子中性极限)问题．使用熵方法和弱紧性方法从数学上严格证明了快扩散情形的拟中性极限．     

Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.     

Vacuum solution and quasineutral limit of semiconductor drift-diffusion equation

The first half of this paper is concerning with the nonlinear drift-diffusion semiconductor model in d (d?3) dimensional space. The global estimate is achieved on the evolution of support of solution and the finite speed of propagation. The proof is based on the estimate of the weighted norm with special designed weight functions. In the second half, we prove the quasineutral limit locally for 1-dimensional standard drift-diffusion model with discontinuous, sign-changing doping profile.     

The bipolar quantum drift-diffusion model

A fourth order parabolic system, the bipolar quantum drift-diffusion model in semiconductor simulation, with physically motivated Dirichlet-Neumann boundary condition is studied in this paper. By semidiscretization in time and compactness argument, the global existence and semiclassical limit are obtained, in which semiclassieal limit describes the relation between quantum and classical drift-diffusion models, Furthermore, in the case of constant doping, we prove the weak solution exponentially approaches its constant steady state as time increases to infinity.     

Quasineutral limit of a standard drift diffusion model for semiconductors

The limit of vanishing Debye length (charge neutral limit) in a nonlinear bipolar drift-diffusion model for semiconductors without pn-junction (i.e. without a bipolar background charge) is studied. The quasineutral limit (zero-Debye-length limit) is performed rigorously by using the weak compactness argument and the so-called entropy functional which yields appropriate uniform estimates.     

Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.Present address: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal