On the One-Dimensional Current Driven Semiconductor Equations - Kerkhoven, Thomas (SIAM journal on applied mathematics, 1991)
The steady state semiconductor equations in one dimension are examined subject to a given current through the device, supplemented by Dirichlet boundary conditions on one end of the device. Solutions to this current driven model are identified as the fixed points of a mapping T, and uniqueness of solutions for the unipolar model is shown for arbitrary currents, by demonstrating that T is unconditionally a contraction. For both the uni- and the bipolar model existence of solutions close to thermodynamic equilibrium is shown. This means that for the one-dimensional unipolar current driven model the situation is complementary to that for the potential driven model where it is possible to show existence of solutions for arbitrary applied potentials, and uniqueness close to thermodynamic equilibrium only. For the unipolar case, iteration with the mapping T defines an algorithm for the solution of this model which is unconditionally convergent to the unique solution if it exists.