Hydrodynamic Model For Semiconductors Research Articles

Global solutions and relaxation limit to the Cauchy problem of a hydrodynamic model for semiconductors

In this paper, we study the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation. First, the existence of global entropy solutions is proved by using the vanishing artificial viscosity method, where, a special flux approximate is introduced to ensure the uniform boundedness of the electric field E and the a-priori L∞ estimate, 0<2δ≤ρε,δ≤M(t),|uε,δ|≤M(t), where M(t) could tend to infinity as the time t tends to infinity, on the viscosity-flux approximate solutions (ρε,δ,uε,δ); Second, the compensated compactness theory is applied to prove the pointwise convergence of (ρε,δ,uε,δ) as ε,δ go to zero, and that the limit (ρ(x,t),u(x,t)) is a global entropy solution; Third, a technique, to apply the maximum principle to the combination of the Riemann invariants and ∫−∞xρε,δ(x,t)−2δdx, deduces the uniform L∞ estimate, 0<2δ≤ρε,δ≤M,|uε,δ|≤M, independent of the time t and ε,δ; Finally, as a by-product, the known compactness framework [14,29] is applied to show the relaxation limit, as the relation time τ and ε,δ go to zero, for general pressure P(ρ).

The subsonic or sonic–subsonic solution of the electric potential driven problem to the HD model for semiconductors

This paper concerns the stationary subsonic or sonic–subsonic solution of the electric potential driven problem to the isentropic hydrodynamic model for semiconductors. We give a necessary and sufficient condition to ensure the existence of this kind of solution. Specifically, there exists a subsonic or sonic–subsonic solution to this problem if and only if 0≤ϕr≤ϕ̄, with the number ϕ̄=O1τ. Here, ϕr is the right side electric potential and τ is the relaxation time. Moreover, there is a number ϕ̃∈(0,ϕ̄] such that when ϕr=ϕ̃, there exists a sonic–subsonic solution. In this way, we show the influence of the relationship between the boundary value of the electric potential and the semiconductor effect on the existence of this kind of solution. Finally, we prove the uniqueness of subsonic solution by the energy method under the assumption that ϕr and ϕrτ are both small enough.

Structural stability of subsonic steady states to the hydrodynamic model for semiconductors with sonic boundary

The hydrodynamic model for semiconductors with sonic boundary, represented by Euler–Poisson equations, possesses the various physical steady states including interior-subsonic/interior-supersonic/shock-transonic/C 1-smooth-transonic steady states. Since these physical steady states result in some serious singularities at the sonic boundary (their gradients are infinity), this makes that the structural stability for these physical solutions is more difficult and challenging, and has remained open as we know. In this paper, we investigate the structural stability of interior subsonic steady states. Namely, when the doping profiles are as small perturbations, the differences between the corresponding subsonic solutions are also small. To overcome the singularities at the sonic boundary, we propose a novel approach, which combines the weighted multiplier technique, local singularity analysis, monotonicity argument and squeezing skill. Both the result itself and the technique developed here will give us some truly enlightening insights into our follow-up study on the structural stability of the remaining types of solutions. A number of numerical approximations are also carried out, which intuitively confirm our theoretical results.

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.

Quasi-Neutral Limit to Steady-State Hydrodynamic Model of Semiconductors with Degenerate Boundary

Quasi-Neutral Limit to Steady-State Hydrodynamic Model of Semiconductors with Degenerate Boundary

Structural stability of subsonic solutions to a steady hydrodynamic model for semiconductors: From the perspective of boundary data

In this paper, we are concerned with a one-dimensional isothermal steady hydrodynamic model for semiconductors driven by boundary data. In the purely subsonic setting, we obtain the existence, uniqueness and structural stability of purely subsonic solutions. Moreover, when the boundary data range from the subsonic region to the sonic line, we further study the degenerate problem from the perspective of boundary data, and prove that there exists a unique interior subsonic solution to the degenerate problem. As a byproduct, we also establish the structural stability between purely subsonic solution and interior subsonic solution in a relatively weak sense. These results provide us with a completely new perspective to understand the singularity caused by the boundary degeneracy. A number of numerical simulations are also carried out, which confirm our theoretical results.

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.

Global Solutions to a Hydrodynamic Model for Semiconductors with Velocity Relaxation

Global Solutions to a Hydrodynamic Model for Semiconductors with Velocity Relaxation

Asymptotic behavior of solutions to the unipolar hydrodynamic model of semiconductors with time-dependent damping in bounded domain

Asymptotic behavior of solutions to the unipolar hydrodynamic model of semiconductors with time-dependent damping in bounded domain

The Time Asymptotic Expansion of the Bipolar Hydrodynamic Model for Semiconductors

The Time Asymptotic Expansion of the Bipolar Hydrodynamic Model for Semiconductors

Nonlinear structural stability and linear dynamic instability of transonic steady-states to hydrodynamic model for semiconductors

For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, and the boundary data are in subsonic region and supersonic region separately, the system possesses the shock transonic steady-states and the smooth transonic steady-states. In this paper we study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. For any relaxation time: 0<τ≤+∞, by means of elaborate singularity analysis, we first investigate the structural stability of the C1-smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. We note that, when the C1-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the proof of structural stability. Moreover, when the relaxation time is large enough τ≫1, under the condition that the electric field is positive at the shock location, we prove that the transonic shock steady-states are structurally stable with respect to small perturbations of the supersonic doping profile. Furthermore, we show the linearly dynamic instability for these transonic shock steady-states provided that the electric field is suitable negative. The proofs for the structural stability results are based on singularity analysis, a monotonicity argument on the shock position and the downstream density, and the stability analysis of supersonic and subsonic solutions. The linear dynamic instability of the steady transonic shock for Euler-Poisson equations can be transformed to the ill-posedness of a free boundary problem for the Klein-Gordon equation. By using a nontrivial transformation and the shooting method, we prove that the linearized problem has a transonic shock solution with exponential growths. These results enrich and develop the existing studies.

Existence and uniqueness of steady states to semiconductor bipolar full quantum hydrodynamic model

In this paper, we are concerned with the existence and uniqueness of subsonic steady states to the one-dimensional bipolar full quantum hydrodynamic model for semiconductors in a bounded interval. The main result is proved by perturbation arguments combined with Banach's Fixed Point Theorem. Moreover, we also obtain the uniform estimates of steady states with respect to the physical parameters appear in the model, which comes in useful for discussing the relaxation time limit and semi-classical limit in future study.

On a One-Dimensional Hydrodynamic Model for Semiconductors with Field-Dependent Mobility

We consider a one-dimensional, isentropic, hydrodynamical model for a unipolar semiconductor, with the mobility depending on the electric field. The mobility is related to the momentum relaxation time, and field-dependent mobility models are commonly used to describe the occurrence of saturation velocity, that is, a limit value for the electron mean velocity as the electric field increases. For the steady state system, we prove the existence of smooth solutions in the subsonic case, with a suitable assumption on the mobility function. Furthermore, we prove uniqueness of subsonic solutions for sufficiently small currents.

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.

Radial solutions of hydrodynamic model of semiconductors with sonic boundary

The purpose of this paper is to study the multi-dimensional steady hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary. We prove that, the steady Euler-Poisson equations with sonic boundary possess a unique subsonic solution and at least one supersonic solution in the radial form. The adopted approach for proof is the energy method combining the compactness analysis. For the n-D radial supersonic solutions, since it is more challenging to get the crucial energy estimates due to the effect by the multiple dimensions and the restriction by the sonic boundary, we propose a brand new two-steps iteration scheme to build up the key energy estimates. This is the first attempt to study the n-D steady-states with the sonic boundary, and the results obtained essentially improve and develop the previous studies in the one-dimensional case.

Smooth Transonic Steady States of Hydrodynamic Model for Semiconductors

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 1 February 2021Accepted: 01 June 2021Published online: 30 August 2021KeywordsEuler--Poisson equations, transonic solutions, hydrodynamic model of semiconductors, doping profileAMS Subject Headings35R35, 35Q35, 76N10, 35J70Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

Existence and Large Time Behavior of Entropy Solutions to One-Dimensional Unipolar Hydrodynamic Model for Semiconductor Devices with Variable Coefficient Damping

In this paper, we investigate the global existence and large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Possion equations with time and spacedependent damping in a bounded interval. Firstly, we prove the existence of entropy solutions through vanishing viscosity method and compensated compactness framework. Based on the uniform estimates of density, we then prove the entropy solutions converge to the corresponding unique stationary solution exponentially with time. We generalize the existing results to the variable coefficient damping case.

Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode at 300 K

The smooth quantum hydrodynamic model is an extension of the classical hydrodynamic model for semiconductor devices that can handle in a mathematically rigorous way the discontinuities in the potential energy that occur at heterojunction barriers in quantum semiconductor devices. Smooth quantum hydrodynamic model simulations of a GaAs resonant tunneling diode at 300 K with a 5-nm quantum well and 2.5-nm 280 meV double quantum barriers are presented that successfully produce realistic negative differential resistance and hysteresis in the current–voltage curve. This is the first quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode at 300 K, demonstrating that fluid dynamical concepts can model this fundamental quantum mechanical effect even at room temperature. A dispersive quantum term $$\hbar ^2 n u_{xxx}/(8 m)$$ in the energy transport equation, where n is the electron density, u is the electron velocity, and m is the effective electron mass, is essential here in obtaining a realistic hysteresis loop in the current–voltage curve.

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.

Steady hydrodynamic model of semiconductors with sonic boundary and transonic doping profile

As shown in [17,18], for the hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary and subsonic/supersonic doping profile, the structure of stationary solutions are very complicated. It may possess various solutions like subsonic/supersonic/transonic flows. In this paper, we consider a more challenging case where the doping profile is transonic, which is categorized into two types: subsonic-dominated and supersonic-dominated. In the subsonic-dominated case, we show that the system has a unique interior subsonic solution, at least one interior supersonic solution and infinitely many transonic solutions under the suitable assumptions. However, the difference with the case of subsonic doping profile is that the interior subsonic solution and interior supersonic solution may not exist in special cases when the relaxation time is small. In the supersonic-dominated case, the non-existence and existence of all types of solutions are also obtained. The approach adopted is the technical compactness analysis combining the Green's function method. Here, the results obtained perfectly develop the existing studies.