We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such
equations arise from the theory of semiconductors and are composed of two continuity equations coupled
with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the
convergence of the scheme and then the existence of solutions to the problem. The key point of the
proof relies on the construction of an approximate gradient of the electric potential which allows us to
deal with coupled terms in the continuity equations. Finally, a numerical example is given to show the
efficiency of the scheme.