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.     

The isentropic quantum drift-diffusion model in two or three space dimensions

We investigate the isentropic quantum drift-diffusion model, a fourth order parabolic system, in space dimensions d = 2, 3. First, we establish the global weak solutions with large initial value and periodic boundary conditions. Then we show the semiclassical limit by delicate interpolation estimates and compactness argument. Supported by the Natural Science Foundation of China (No.10401019 and No.10626030).     

Existence and semiclassical limit for the quantum drift-diffusion model

In this paper, we construct a weak solution to the unipolar quantum drift-diffusion model in a bounded domain with Neumann boundary conditions, thereby extending an existence result in Gianazza et al. (Arch Rational Mech Anal 194:133–220, 2009) to the case where the boundary of the domain is only Lipschitz. We also obtain the limit of the solution as the scaled Planck constant \(\varepsilon \) in the problem goes to \(0\) .     

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.     

Quasi-neutral limit of the drift-diffusion model for semiconductors with general sign-changing doping profile

The quasi-neutral limit of time-dependent drift diffusion model with general sign-changing doping profile is justified rigorously in super-norm (i.e., uniformly in space). This improves the spatial square norm limit by Wang, Xin and Markowich.     

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.     

Existence,semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion model

The existence, semiclassical limit and long-time behavior of weak solutions to the transient quantum drift-diffusion model are studied. Using semi-discretization in time and entropy estimate, we get the global existence and semiclassical limit of nonnegative weak solutions to one-dimensional isentropic model with nonnegative initial and homogeneous Neumann (or periodic) boundary conditions. Furthermore, by a logarithmic Sobolev inequality, we obtain an inequality of the periodic weak solution to this model (or its isothermal case) which shows that the solution exponentially approaches its mean value as time increases to infinity.     

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.     

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.     

A natural map from a quantized space onto its semiclassical limit and a multi-parameter Poisson Weyl algebra

A natural map from a quantized space onto its semiclassical limit is obtained. As an application, we see that an induced map by the natural map is a homeomorphism from the spectrum of the multiparameter quantized Weyl algebra onto the Poisson spectrum of its semiclassical limit.     

Quasineutral limit of bipolar quantum hydrodynamic model for semiconductors

This paper is concerned with the quasineutral limit of the bipolar quantum hydrodynamic model for semiconductors. It is rigorously proved that the strong solutions of the bipolar quantum hydrodynamic model converge to the strong solution of the so-called quantum hydrodynamic equations as the Debye length goes to zero. Moreover, we obtain the convergence of the strong solutions of bipolar quantum hydrodynamic model to the strong solution of the compressible Euler equations with damping if both the Debye length and the Planck constant go to zero simultaneously.     

Quasineutral limit of the viscous quantum hydrodynamic model for semiconductors

The quasineutral limit (zero-Debye-length limit) of viscous quantum hydrodynamic model for semiconductors is studied in this paper. By introducing new modulated energy functional and using refined energy analysis, it is shown that, for well-prepared initial data, the smooth solution of viscous quantum hydrodynamic model converges to the strong solution of incompressible Navier-Stokes equations as the Debye length goes to zero.     

Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model

Francis Filbet Université de Lyon, Université Lyon 1, CNRS, UMR 5208, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France Received on 28 June 2006. Revised on 6 December 2006. In this paper, we propose a finite-volume discretization formultidimensional nonlinear drift-diffusion system. Such a systemarises in semiconductors modelling and is composed of two parabolicequations and an elliptic one. We prove that the numerical solutionconverges to a steady state when time goes to infinity. Severalnumerical tests show the efficiency of the method.     

The Dirichlet problem of the quantum drift-diffusion model

We study the quantum drift-diffusion model, a fourth-order parabolic system, with Dirichlet boundary conditions. Using a semi-discretization approximate method with a compact argument and applying a new entropy estimate, we prove the existence of global regular weak solutions.     

The Semiclassical Limit in the Quantum Drift-Diffusion Equations with Isentropic Pressure

The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supplemented with homogeneous Neumann boundary conditions. It is shown that the semiclassical limit of this solution solves the classical drift-diffusion model. In the meanwhile, the global existence of weak solutions is proved.     

Some limit analysis in a one-dimensional stationary quantum hydrodynamic model for semiconductors

In this paper, we study the steady-state hydrodynamic equations for isothermal states including the quantum Bohn potential. The one-dimensional equations for the electron current density and the particle density are coupled self-consistently to the Poisson equation for the electric potential. The quantum correction can be interpreted as a dispersive regularization of the classical hydrodynamic equations. In a bounded interval supplemented by the proper boundary conditions, we investigate the zero-electron-mass limit, the zero-relaxation-time limit, the Debye-length (quasi-neutral) limit, and some combined limits, respectively. For each limit, we show the strong convergence of the sequence of solutions and give the associated convergence rate.