A semi-linear elliptic integro-differential equation subject to homogeneous Neumann boundary conditions for the equilibrium potential in an insulated semiconductor device is considered. A variational formulation gives existence and uniqueness. The limit as the scaled Debye length tends to zero is analysed. Two different cases occur. If the number of free electrons and holes is sufficiently high, local charge neutrality prevails throughout the device. Otherwise, depletion regions occur, and the limiting potential is the solution of a free boundary problem.
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Luis A. Caffarelli, Peter A. Markowich, Christian Schmeiser, Jean Dolbeault, On Maxwellian equilibria of insulated semiconductors. Interfaces Free Bound. 2 (2000), no. 3, pp. 331–339