In this talk, I will present a series of studies on the structure of steady-states to  Euler-Poisson equations for semiconductors  subjected to the sonic boundary, a critical case of boundary setting. Based on the subsonic/supersonic/transonic properties of the doping profile, we give the criteria on the structure of steady-states: in which cases there exist subsonic solutions, supersonic solutions, shock transonic solutions, smooth transonic solutions, and the number of these solutions, and in which cases there are no physical solutions.