Peculiarities of the density of states in SN junctions

We study the density of states (DoS) ν(E) in a normal-metallic (N) film contacted by a bulk superconductor (S). We assume that the system is diffusive and the SN interface is transparent. In the limit of thin N layer (compared to the coherence length), we analytically find three different types of the DoS peculiarity at energy equal to the bulk superconducting order parameter Δ0. (i) In the absence of the inverse proximity effect, the peculiarity has the check-mark form with ν(Δ0)=0 as long as the thickness of the N layer is smaller than a critical value. (ii) When the inverse proximity effect comes into play, the check-mark is immediately elevated so that ν(Δ0)>0. (iii) Upon further increasing of the inverse proximity effect, ν(E) gradually evolves to the vertical peculiarity (with an infinite-derivative inflection point at E=Δ0). This crossover is controlled by a materials-matching parameter which depends on the relative degree of disorder in the S and N materials.