Title:Stationary solutions to the hydrodynamic model of semiconductors
The hydrodynamic model of semiconductors is actually a compressible Euler-Poisson system with relaxation. It describes the transport of electrons or holes in semiconductor devices. In our recent project, we aim to clarify the whole dynamics of this hydrodynamic model. At the first step, we focus on the classification of its stationary solutions with appropriate boundary conditions in 1 D (here we choose sonic boundary condition). Mathematically, the sonic boundary condition is equivalent to the degeneracy of the system at boundary, which makes the project interesting and also challenging. We find that the structure of the solutions depends on the positions of doping profile and the strength of relaxation. More precisely, we show that, when the doping profile is subsonic, the steady-state equations possess a unique interior subsonic solution, and at least one interior supersonic solution; and if further the relaxation time is large, then the equations admit infinitely many interior transonic solutions of shock type; while if the relaxation time is small, then the system has no transonic solutions of shock type but has infinitely many smooth transonic solutions. When the doping profile is supersonic, the system does not hold any subsonic solution; furthermore, it doesn't admit any supersonic solution or any transonic solution if the supersonic doping profile is small or the relaxation time is small, but it has at least one supersonic solution and infinitely many transonic solutions if the doping profile is close to the sonic line and the relaxation time is large. This explains the physical phenomenon that pure semiconductor device does not work well. We also study the asymptotic stability of the steady subsonic solutions under suitable initial-boundary-value conditions, by overcoming the non-flatness of the solutions and the degeneracy with different strength at two boundary points.