Diffusion relaxation limit of a bipolar hydrodynamic model for semiconductors

In the paper, we discuss the relaxation limit of a bipolar isentropic hydrodynamical models for semiconductors with small momentum relaxation time. With the help of the Maxwell iteration, we prove that, as the relaxation time tends to zero, periodic initial-value problems of a scaled bipolar isentropic hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the bipolar hydrodynamic model.     

Relaxation limit and initial layer to hydrodynamic models for semiconductors

We study a relaxation limit of a solution to the initial-boundary value problem for a hydrodynamic model to a drift-diffusion model over a one-dimensional bounded domain. It is shown that the solution for the hydrodynamic model converges to that for the drift-diffusion model globally in time as a physical parameter, called a relaxation time, tends to zero. It is also shown that the solutions to the both models converge to the corresponding stationary solutions as time tends to infinity, respectively. Here, the initial data of electron density for the hydrodynamic model can be taken arbitrarily large in the suitable Sobolev space provided that the relaxation time is sufficiently small because the drift-diffusion model is a coupled system of a uniformly parabolic equation and the Poisson equation. Since the initial data for the hydrodynamic model is not necessarily in “momentum equilibrium”, an initial layer should occur. However, it is shown that the layer decays exponentially fast as a time variable tends to infinity and/or the relaxation time tends to zero. These results are proven by the decay estimates of solutions, which are derived through energy methods.     

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.     

Relaxation-time limit of the multidimensional bipolar hydrodynamic model in Besov space

In this paper, we study a multidimensional bipolar hydrodynamic model for semiconductors or plasmas. This system takes the form of the bipolar Euler-Poisson model with electric field and frictional damping added to the momentum equations. In the framework of the Besov space theory, we establish the global existence of smooth solutions for Cauchy problems when the initial data are sufficiently close to the constant equilibrium. Next, based on the special structure of the nonlinear system, we also show the uniform estimate of solutions with respect to the relaxation time by the high- and low-frequency decomposition methods. Finally we discuss the relaxation-time limit by compact arguments. That is, it is shown that the scaled classical solution strongly converges towards that of the corresponding bipolar drift-diffusion model, as the relaxation time tends to zero.     

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.     

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 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.     

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.     

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.     

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.     

Long-time self-similarity of classical solutions to the bipolar quantum hydrodynamic models

In this paper, a bipolar transient quantum hydrodynamic model (BQHD) for charge density, current density and electric field is considered on the one-dimensional real line. This model takes the form of the classical Euler-Poisson system with additional dispersion caused by the quantum (Bohn) potential. We investigate the long-time behavior of the BQHD model and show the asymptotical self-similarity property of the global smooth solution. Namely, both of the charge densities tend to a nonlinear diffusion wave in large time, which is not a solution to the BQHD equation, but to the combined quasi-neutral, relaxation and semiclassical limiting model. Next, as a by-product, we can compare the large-time behavior of the bipolar quantum hydrodynamic models and of the corresponding classical bipolar hydrodynamic models. As far as we know, the nonlinear diffusion phenomena about the 1D BQHD is new.     

A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension

In this paper, we consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.     

Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors

The global in-time semiclassical and relaxation limits of the bipolar quantum hydrodynamic model for semiconductors are investigated in R³. We prove that the unique strong solution exists and converges globally in time to the strong solution of classical bipolar hydrodynamical equation in the process of semiclassical limit and that of the classical drift-diffusion system under the combined relaxation and semiclassical limits.     

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 diffusive relaxation limit of non-isentropic Euler-Maxwell equations for plasmas

This paper concerns the non-isentropic Euler-Maxwell equations for plasmas with short momentum relaxation time. With the help of the Maxwell-type iteration, it is obtained that, as the relaxation time tends to zero, periodic initial-value problem of certain scaled non-isentropic Euler-Maxwell equations has unique smooth solutions existing in the time interval where the corresponding classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the non-isentropic Euler-Maxwell equations.     

Localized minimizers of flat rotating gravitational systems

We study a two-dimensional system in solid rotation at constant angular velocity driven by a self-consistent three-dimensional gravitational field. We prove the existence of stationary solutions of such a flat system in the rotating frame as long as the angular velocity does not exceed some critical value which depends on the mass. The solutions can be seen as stationary solutions of a kinetic equation with a relaxation-time collision kernel forcing the convergence to the polytropic gas solutions, or as stationary solutions of an extremely simplified drift-diffusion model, which is derived from the kinetic equation by formally taking a diffusion limit. In both cases, the solutions are critical points of a free energy functional, and can be seen as localized minimizers in an appropriate sense.     

Drift-diffusion in electrochemistry: thresholds for boundary flux and discontinuous optical generation

We consider an extension of the classical drift-diffusion model, which incorporates thermodynamic switching rules for generation and boundary flux. The motivation is the important case of the splitting of water molecules upon photonic irradiation of a semiconductor electrode located in an electrochemical cell. The solid state electrode forms the spatial domain of the model. The rules are motivated by the fact that the valence band of the semiconductor, which supplies positive charge to solution, has to be located at a lower energy level than the electrochemical potential of O₂ evolution in solution, and the conduction band, which supplies electrons to solution, has to be positioned at a higher energy level than the electrochemical potential of H₂ evolution. This defines thresholds in terms of electrochemical potentials before boundary flux is activated. The optical generation rate is affected, due to the increased carrier relaxation time, when these thresholds are crossed, and may be discontinuous. We thus consider a self-consistent model, in which ‘switching’ occurs only in principal variables. The steady-state model is considered, and trapping regions are derived for the solutions.     

Global smooth solutions for a one-dimensional nonisentropic hydrodynamic model with non-constant lattice temperature

In this paper, a one-dimensional nonisentropic hydrodynamic model for semiconductors with non-constant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. The existence and uniqueness of the corresponding stationary solutions are investigated carefully under proper conditions. Then, global existence of the smooth solutions for the Cauchy problem with initial data, which are perturbations of stationary solutions, is established. It is shown that these smooth solutions tend to the stationary solutions exponentially fast as t → ∞.      

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.     

Stability of transonic strong shock waves for the one-dimensional hydrodynamic model for semiconductors

In this paper we study the stability of transonic strong shock solutions of the steady-state one-dimensional unipolar hydrodynamic model for semiconductors. The approach is based on the construction of a pseudo-local symmetrizer and on the paradifferential calculus with parameters, which combines the work of Bony-Meyer and the introduction of a large parameter.