Bimodal distribution of delay times and splitting of the zero-bias conductance peak in a double-barrier normal-superconductor junction

We formulate a scattering theory of the proximity effect in a weakly disordered SININ junction (S = superconductor, I = insulating barrier, N = normal metal). This allows to relate the conductance and density of states of the junction to the scattering times $\tau$ (eigenvalues of the Wigner-Smith time-delay matrix). The probability density $P(\tau )$ has two peaks, at a short time ${\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and a late time ${\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}$. The density of states at the Fermi level is the geometric mean of the two times. The splitting of the zero-bias conductance peak is given by $\mathrm{\hslash }\mathrm{/}{\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}$.