Negative differential resistance in normal narrow bands - superconducting junctions with Andreev reflection

We have calculated current-voltage characteristic curves for normal-superconducting junctions with Andreev reflections and different types of electronic bands. We found that when the normal band is narrow, of the order of the superconducting energy gap, a negative differential resistance appears at a voltage of the order of the band width plus the gap value. In case two bands contribute to the total current the conductance can be smaller than unity at voltages above the gap value. Our simulations may provide an answer to different experimental data of the literature that were not yet understood.