Convergence to stationary states of solutions of the transient drift diffusion equations for semiconductor devices with prescribed currents

The transient drift diffusion equations describing the charge transport in semiconductors coupled with Maxwell’s equations are considered. The asymptotic behavior of the solutions is investigated. In particular it is shown that they converge for $t\rightarrow\infty$ to solutions of the stationary drift diffusion equations provided that the currents corresponding to the stationary states are sufficiently small.