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

Abstract

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.