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 $Δ_0$, 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 $E=Δ_0$ with square-root behavior as $E$ deviates from $Δ_0$. The coefficients of this dependence are different at $E<Δ_0$ and $E>Δ_0$, so the peculiarity is asymmetric.