  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 $h(c)=\log \frac{c}{1-c}$. 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.
