RELAXATION TIME LIMITS PROBLEM FOR HYDRODYNAMIC MODELS IN SEMICONDUCTOR SCIENCE

In this article, two relaxation time limits, namely, the momentum relaxation time limit and the energy relaxation time limit are considered. By the compactness argument, it is obtained that the smooth solutions of the multidimensional nonisentropic Euler-Poisson problem converge to the solutions of an energy transport model or a drift diffusion model, respectively, with respect to different time scales.     

Global existence and asymptotic behavior for a multidimensional nonisentropic hydrodynamic semiconductor model with the heat source

In this paper, a multidimensional nonisentropic hydrodynamic model for semiconductors with the nonconstant 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. Global existence to the Cauchy problem for the multidimensional nonisentropic hydrodynamic semiconductor model with the small perturbed initial data is established, and the asymptotic behavior of these smooth solutions is investigated, namely, that the solutions converge to the general steady-state solution exponentially fast as t→+∞ is obtained. Moreover, the existence and uniqueness of the stationary solutions are investigated.     

Convergence of the nonisentropic Euler-Poisson equations to incompressible type Euler equations

In this paper, we investigate a multidimensional nonisentropic hydrodynamic (Euler-Poisson) model for semiconductors. We study the convergence of the nonisentropic Euler-Poisson equation to the incompressible nonisentropic Euler type equation via the quasi-neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and energy methods are used to rigorously justify the convergence of the limit.     

Existence analysis for a simplified transient energy‐transport model for semiconductors

A simplified transient energy‐transport system for semiconductors subject to mixed Dirichlet–Neumann boundary conditions is analyzed. The model is formally derived from the non‐isothermal hydrodynamic equations in a particular vanishing momentum relaxation limit. It consists of a drift‐diffusion‐type equation for the electron density, involving temperature gradients, a nonlinear heat equation for the electron temperature, and the Poisson equation for the electric potential. The global‐in‐time existence of bounded weak solutions is proved. The proof is based on the Stampacchia truncation method and a careful use of the temperature equation. Under some regularity assumptions on the gradients of the variables, the uniqueness of solutions is shown. Finally, numerical simulations for a ballistic diode in one space dimension illustrate the behavior of the solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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 → ∞.      

Relaxation‐time limit of the three‐dimensional hydrodynamic model with boundary effects

In this paper, we study three‐dimensional (3D) unipolar and bipolar hydrodynamic models and corresponding drift‐diffusion models from semiconductor devices on bounded domain. Based on the asymptotic behavior of the solutions to the initial boundary value problems with slip boundary condition, we investigate the relation between the 3D hydrodynamic semiconductor models and the corresponding drift‐diffusion models. That is, we discuss the relation‐time limit from the 3D hydrodynamic semiconductor models to the corresponding drift‐diffusion models by comparing the large‐time behavior of these two models. These results can be showed by energy arguments. Copyrightcopyright 2011 John Wiley & Sons, Ltd.     

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.     

The zero-electron-mass limit for a stationary nonisentropic hydrodynamic semiconductor model

In this paper, we study a general multidimensional nonisentropic hydrodynamical model for semiconductors. 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. For steady state, subsonic and potential flows, we discuss the zero-electron-mass limit of system by using the method of asymptotic expansions. We show the existence and uniqueness of profiles, and justify the asymptotic expansions up to any order.     

Asymptotic stability of solutions to the nonisentropic hydrodynamic model for semiconductors

In this paper, the asymptotic stability of smooth solutions to the multidimensional nonisentropic hydrodynamic model for semiconductors is established, under the assumption that the initial data are a small perturbation of the stationary solutions for the thermal equilibrium state, whose proofs mainly depend on the basic energy methods.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

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.     

The relaxation-time limit in the compressible Euler–Maxwell equations

The aim of this paper is to study multidimensional Euler–Maxwell equations for plasmas with short momentum relaxation time. The convergence for the smooth solutions to the compressible Euler–Maxwell equations toward the solutions to the smooth solutions to the drift–diffusion equations is proved by means of the Maxwell iteration, as the relaxation time tends to zero. Meanwhile, the formal derivation of the latter from the former is justified.     

Global existence and asymptotic behavior of solutions to the nonisentropic bipolar hydrodynamic models

In this paper, we consider a one-dimensional bipolar nonisentropic hydrodynamical model from semiconductor devices. This system takes the nonisentropic Euler-Poisson form with electric field and frictional damping added to the momentum equations. First, we prove global existence of smooth solutions to the Cauchy problem. Next, we also discuss the asymptotic behavior of the smooth solutions. We find that in large time, the densities of electron and hole tend to the same nonlinear diffusive wave, the momentums tend to the Darcy's law, and the temperatures tend to the ambient device temperature. Finally, we can obtain the algebraic decay rate of the densities to the same nonlinear diffusive wave, the momentums to the Darcy's law and the temperatures to the ambient device temperature, and the exponential decay of their difference and the electric field to zero. We can show our results by precise energy methods.     

Stability of nonconstant steady‐state solutions for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor

We consider the periodic problem for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor. By choosing a suitable symmetrizers and using an induction argument on the order of the time‐space derivatives of solutions in energy estimates, we obtain the global stability of solutions with exponential decay in time near the nonconstant steady‐states for 2‐fluid nonisentropic Euler‐Poisson equations. This improves the results obtained for models with temperature diffusion terms by using the pressure functions p^ν in place of the unknown variables densities n^ν.     

The strong relaxation limit of the multidimensional isothermal Euler equations

We construct global smooth solutions to the multidimensional isothermal Euler equations with a strong relaxation. When the relaxation time tends to zero, we show that the density converges towards the solution to the heat equation.

     

GLOBAL EXISTENCE AND EXPONENTIAL STABILITY OF SMOOTH SOLUTIONS TO A MULTIDIMENSIONAL NONISENTROPIC EULER-POISSON EQUATIONS

The global existence and large time behavior of smooth solutions to multi-dimensional nonisentropic Euler-Poisson equations are established.     

Double shocks for a hydrodynamic model of semiconductors

This paper is devoted to the study of a hydrodynamic model of drift diffusion equations. We establish the existence of a kind of discontinuous solution which consists of two multidimensional shocks. Since the system is a hyperbolic elliptic coupled one, the proof is different from that of double shocks from conservation laws in [11].      

Energy-transport and drift-diffusion limits of nonisentropic Euler–Poisson equations

Two relaxation limits in critical spaces for the scaled nonisentropic Euler–Poisson equations with the momentum relaxation time and energy relaxation time are considered. As the first step of this justification, the uniform (global) classical solutions to the Cauchy problem in Chemin–Lerner?s spaces with critical regularity are constructed. Furthermore, by the compactness argument, it is rigorously justified that the scaled classical solutions converge to the solutions of energy-transport equations and drift-diffusion equations, respectively, with respect to different time scales.