In this work, the MOSFET device is considered. The carrier densities in the MOSFET are modeled by the drift-diffusion equation. We manipulate the formulas of the charge density at the equilibrium in order to derive a simple Poisson's or Laplace's equation. To formulate a shape optimization problem, we have defined a cost functional. The existence of an optimal solution is proved. To solve the involved optimization problem, we have designed a numerical approach based on the finite element method combined with the genetic algorithm. Several numerical examples are established to prove the validity of the proposed approach.
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