Surface density of states in superconductors with inhomogeneous pairing constant: Analytical results

We consider a superconductor with surface suppression of the BCS pairing constant λ (x ) . We analytically find the gap in the surface density of states (DOS), behavior of the DOS ν (E ) above the gap, a "vertical" peculiarity of the DOS around an energy equal to the bulk order parameter Δ₀, and a perturbative correction to the DOS at higher energies. The surface gap in the DOS is parametrically different from the surface value of the order parameter due to a difference between the spatial scale r_c at which λ (x ) is suppressed and the coherence length. The vertical peculiarity implies an infinite-derivative inflection point of the DOS curve at E =Δ₀ with square-root behavior as E deviates from Δ₀. The coefficients of this dependence are different at E <Δ₀ and E >Δ₀ , so the peculiarity is asymmetric.