Bipolar Hydrodynamic Model For Semiconductors Research Articles

Subsonic steady-states for bipolar hydrodynamic model for semiconductors

In this paper, we study the well-posedness, ill-posedness and uniqueness of the stationary 3-D radial solution to the bipolar isothermal hydrodynamic model for semiconductors. The density of electron is imposed with sonic boundary and interiorly subsonic case and the density of hole is fully subsonic case. It is difficult to estimate the upper and lower bounds of the holes due to the coupling of electrons and holes and the degeneracy of electrons at the boundary. Thus, we use the topological degree method to prove the well-posedness of solution. We prove the ill-posedness of subsonic solution under some conditions by direct mathematical analysis and contradiction method. The ill-posedness property shows significant difference to the unipolar model. Another highlight of this paper is the application of specific energy method to obtain the uniqueness of solution in two cases. One case is the relaxation time τ=∞, namely, the pure Euler-Poisson case; the other case is jτ≪1, which means that, when the current flow is sufficiently small and the relaxation time is sufficiently large both can satisfy.

Uniformly Time-independent L∞ Estimate for a One-dimensional Hydrodynamic Model of Semiconductors

It is well known that the a-priori, uniformly time-independent L∞ estimate of solutions is the key point to ensure the large time behavior and the relaxation time limit of entropy solutions for both the unipolar and the bipolar hydrodynamic model of semiconductors. With the technical help of the artificial damping coefficient functions ai(x) ≢ 1, in the previous papers, we introduced a technique to obtain the uniform L∞ estimates of the viscosity-flux approximation solutions, in which, the maximum principle is applied to the combination of the Riemann invariants with the integrals of the density and the concentration of a fixed background charge. In this paper, we remove the auxiliary condition on the functions ai(x), and obtain the uniformly, time-independent estimate of entropy solutions for the physical case ai(x) ≡ 1.

The Time Asymptotic Expansion of the Bipolar Hydrodynamic Model for Semiconductors

The Time Asymptotic Expansion of the Bipolar Hydrodynamic Model for Semiconductors

Large-time behavior of solutions to the time-dependent damped bipolar Euler-Poisson system

This paper concerns with the Cauchy problem of the 1-D bipolar hydrodynamic model for semiconductors, a system of Euler-Poisson equations with time-dependent damping effects and for . Here, we consider a more physical case that allows the two pressure functions can be different and the doping profile can be non-zero. Different from the previous study [Li HT, Li JY, Mei. M, et al. Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping. J Math Anal Appl. 2019;437:1081-1121] which considered two identical pressure functions and zero doping profile, the asymptotic profiles of the solutions to this model are constant states rather than the nonlinear diffusion waves. When the initial perturbation around the constant states are sufficiently small in the sense of , by means of the time-weighted energy method, we prove the global existence and uniqueness of the smooth solutions to the Cauchy problem, and obtain the optimal convergence rates of the solutions toward the constant states.

Large-time behavior of entropy solutions to bipolar hydrodynamic model for semiconductors

We study the existence and large-time behavior of globally defined entropy solutions to one dimensional bipolar hydrodynamic model for semiconductors on bounded interval. By taking advantage of vanishing viscosity method and compensated compactness framework, we construct approximate solutions uniformly bounded in both space x and time t. Compared with the paper, Huang and Li (2009), a restrictive assumption on the uniform bound with respect to time t of entropy solutions is removed on bounded interval by introducing modified Riemann invariants and invariant region theory, which is a main novelty of this paper. Also, we extend the result for γ∈(1,3] in Huang et al. (2018) to the whole range of physical adiabatic exponent γ∈(1,∞) by using an elementary inequality. Furthermore, we extend the large-time behavior of weak solutions in Huang and Li (2009) from zero doping profile to non-flat one. It is shown that the entropy solution converges to the stationary solution exponentially in time under appropriate condition on doping profile. Some sharp energy estimates are derived to overcome the difficulties due to the coupling and cancellation effect between the difference of densities.

The existence and stability of smooth solutions for multidimensional isentropic bipolar hydrodynamic model of semiconductors

The present paper is concerned with the existence and stability of smooth stationary solutions to the multidimensional isentropic bipolar hydrodynamic model for semiconductors in a bounded domain Ω∈RN(N=1,2,3) with insulating boundary conditions. The existence of non-constant steady smooth solutions is firstly obtained by the calculus of variations and the maximum principle. It is further shown that the particle densities converge exponentially fast to the steady state solutions.

Large time behavior framework for the time-increasing weak solutions of bipolar hydrodynamic model of semiconductors

In this paper, we consider an isentropic Euler-Poisson equations for the bipolar hydrodynamic model of semiconductor devices, which has a non-flat doping profile and insulating boundary conditions. Using a technical energy method and an entropy dissipation estimate, we present a framework for the large time behavior of time-increasing weak entropy solutions. It is shown that the weak solutions converge to the stationary solutions in $L^2$ norm with exponential decay rate. No regularity and smallness conditions are assumed.

On the stationary solutions of multi-dimensional bipolar hydrodynamic model of semiconductors

In this note, we are interested in the existence and uniqueness of the steady solutions to N -dimensional bipolar hydrodynamic model of semiconductors. Different from the unipolar case, the time-independent model is an elliptic equation with nonlocal terms. For N = 1 , 2 , we achieve our goal by the calculus of variations, without any additional assumptions on the doping profile and the initial densities.

Large time behavior of solutions to multi-dimensional bipolar hydrodynamic model of semiconductors with vacuum

In this paper, a multi-dimensional hydrodynamic model for the bipolar semiconductor device with insulating boundary conditions and a non-flat doping profile is investigated. The model is assumed to be spherically symmetrical. A large time behavior framework is builded for any L∞ bounded radial symmetry weak solutions, that is, the weak entropy solutions are shown to converge to the corresponding stationary solutions in L2 norm and an exponential decay rate is also derived. No smallness and regularity conditions are assumed and the doping profile are permitted to be of large variation.

Weak solutions of the isothermal bipolar hydrodynamic model for semiconductors with large data

This paper is devoted to weak solutions of Cauchy problem to the isothermal bipolar hydrodynamic model with large data. The model takes the bipolar Euler-Poisson form, with electric field and relaxation terms added to the momentum equations. Using Glimm scheme to the hyperbolic part and the standard theory to the ordinary differential equations, we first construct the approximation solutions, then from the facts that the total charge is quasi-conservation, we can obtain a uniform estimate of the total variation of the electric field, which allows to prove the L∞ estimate of densities and velocities, and the convergence of the scheme. Then we can prove the global existence of weal solution to Cauchy problem with large data.

Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors

We consider the initial boundary value problem of the one dimensional full bipolar hydrodynamic model for semiconductors. The existence and uniqueness of the stationary solution are established by the theory of strongly elliptic systems and the Banach fixed point theorem. The exponentially asymptotic stability of the stationary solution is given by means of the energy estimate method.

Asymptotic behavior of solutions to Euler–Poisson equations for bipolar hydrodynamic model of semiconductors

In this paper we study the Cauchy problem for 1-D Euler–Poisson system, which represents a physically relevant hydrodynamic model but also a challenging case for a bipolar semiconductor device by considering two different pressure functions and a non-flat doping profile. Different from the previous studies (Gasser et al., 2003 [7], Huang et al., 2011 [12], Huang et al., 2012 [13]) for the case with two identical pressure functions and zero doping profile, we realize that the asymptotic profiles of this more physical model are their corresponding stationary waves (steady-state solutions) rather than the diffusion waves. Furthermore, we prove that, when the flow is fully subsonic, by means of a technical energy method with some new development, the smooth solutions of the system are unique, exist globally and time-algebraically converge to the corresponding stationary solutions. The optimal algebraic convergence rates are obtained.

Long-time Behavior of Solutions to the Bipolar Hydrodynamic Model of Semiconductors with Boundary Effect

For a bipolar hydrodynamic model of semiconductors in the form of Euler–Poisson equations with Dirichlet or Neumann boundary conditions, in this paper we first heuristically analyze the most probable asymptotic profile (the so-called diffusion waves) and then prove this long-time behavior rigorously. For this, we construct correction functions to show the convergence of the original solution to the diffusion wave with optimal convergence rates by the energy method. Moreover, in the case with Dirichlet boundary condition, when the initial perturbation is in some weighted $L^1$ space, a faster and optimal convergence rate is also given.

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.

Large Time Behavior of Solutions ton-Dimensional Bipolar Hydrodynamic Models for Semiconductors

In this paper, we study the n-dimensional ($n\geq1$) bipolar hydrodynamic model for semiconductors in the form of Euler–Poisson equations. In the 1-D case, when the difference between the initial electron mass and the initial hole mass is nonzero (switch-on case), the stability of nonlinear diffusion waves has been open for a long time. In order to overcome this difficulty, we ingeniously construct some new correction functions to delete the gaps between the original solutions and the diffusion waves in $L^2$-space, so that we can deal with the 1-D case for general perturbations, and prove the $L^\infty$-stability of diffusion waves in the 1-D case. The optimal convergence rates are also obtained. Furthermore, based on the 1-D results, we establish some crucial energy estimates and apply a new but key inequality to prove the stability of planar diffusion waves in n-D case. This is the first result for the multidimensional bipolar hydrodynamic model of semiconductors.

Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic model of semiconductors

We consider a one-dimensional bipolar hydrodynamic model of semiconductors. Although some results exist for the bipolar case, almost their conditions (the boundary condition, the doping profile, etc.) are far from practical application. In the present paper, under a condition appropriate for engineering, we shall prove the existence and the uniqueness of classical solutions for the stationary problem. The most difficult point is to obtain the bounded estimate and the energy estimate.

Relaxation limit and initial layer analysis of a bipolar hydrodynamic model for semiconductors

This paper is concerned with a bipolar hydrodynamical model for semiconductors or plasma with short momentum relaxation time in several space variables. We first construct formal approximations of the initial layer solution to the nonlinear problem by the matched expansion method. Then, assuming some regularity of the solution to the reduced problem and using the energy method, we prove the global existence of classical solutions, and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval and give the convergence rate up to any order.

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.

The relaxation limits to the bipolar hydrodynamic model for semiconductors

Zero relaxation limit results of weak solutions of the initial-boundary value problem to the bipolar hydrodynamic model with the general pressure-density relation for semiconductors are established by the method of high energy estimates.

Global Existence and Asymptotic Limits of Weak Solutions of the Bipolar Hydrodynamic Model for Semiconductors

For the case of the adiabatic exponents being larger than \(\), we establish the global existence of entropy weak solutions of the Cauchy problem to the bipolar hydrodynamic model for semiconductors. Using the theory of compensated compactness, we hence give finally a complete answer on the related existence problems with the “γ-law” pressure relation. A new kind of singular limit of the modified entropy weak solution is discussed. To some extent, the limit of this sort can provide some information about the uniform boundedness of the scaled solution sequences. The quasineutral-relaxation limit of the entropy weak solutions is also investigated.