In this paper, we consider an unipolar degenerated drift-diffusion system where the relation between the concentration of the charged species c and the chemical potential h is h(c) = log c 1−c. We design four different finite volume schemes based on four different formulations of the fluxes. We provide a stability analysis and existence results for the four schemes. The convergence proof with respect to the discretization parameters is established for two of them. Numerical experiments illustrate the behaviour of the different schemes.