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

TL;DR

This paper investigates the nonlinear stability and linear instability of transonic steady-states in a hydrodynamic semiconductor model, revealing conditions under which these states are stable or unstable, with implications for device behavior.

Contribution

It provides new analysis of the structural stability and instability of transonic solutions in Euler-Poisson models, especially considering singularities and the effects of electric field direction.

Findings

C^1-smooth transonic steady-states are structurally stable under small perturbations when the electric field is positive.

Transonic shock steady-states are linearly dynamically unstable when the electric field is negative.

The proofs involve singularity analysis, monotonicity arguments, and transformations of free boundary problems.

Abstract

For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, the existence of steady transonic shock solutions and C-smooth steady transonic solutions for Euler-Poisson Equations were established in [27] and [41], respectively. In this paper we further study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. When the C^1-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. For any relaxation time, by means of elaborate singularity analysis, we first investigate the structural stability of the C^1-smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. Moreover, when the relaxation time is large enough, 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.