Tail states in superconductors with weak magnetic impurities

We analyse the behavior of the density of states in a singlet s-wave superconductor with weak magnetic impurities in the clean limit by using the method of optimal fluctuation. We show that the density of states varies as lnN(E) ∝ −|E −∆0| near the mean field gap edge ∆0 in a d-dimensional superconductor. The optimal fluctuation in d > 1 is strongly anisotropic. We compare the density of states with that obtained in other recent approaches. Studies of spectral properties of disordered superconductors remain an active area of research as they help advance our understanding of the competition between disorder and interactions. Recent years witnessed a renewed interest in the behavior of the density of states (DOS) in singlet s-wave superconductors with magnetic impurities. Such impurities are pairbreaking, and are characterized, in the weak scattering limit, by the spin flip scattering time, τs. In the self-consistent Born approximation (SCB) [1], the dimensionless parameter controlling the suppression of the single particle spectral gap, ∆0, is ∆τs, where ∆ is the amplitude of the superconducting order parameter. In this paper we consider the clean limit, ∆τs ≫ 1, where the SCB approach yields a finite spectral gap, ∆0 ≈ ∆, with the DOS N(E) = 0 at energies E < ∆0. It was argued in Ref.[2] that rare regions where local impurity concentration is high enough to locally destroy superconductivity lead to a finite density of states at the Fermi level. The argument was similar to the method of optimal fluctuation (OF), well known from studies of doped semiconductors [3]: in averaging over all the realizations of the impurity distribution, the probability of finding the realization which locally destroys the gap determines the residual DOS. Later, Lamacraft and Simons [4] considered in detail the energy dependence of the DOS below the mean field gap 1 Corresponding author. E-mail: [email protected] in a dirty superconductor, where the scattering rate due to potential scattering greatly exceeds the spin-flip pairbreaking scattering rate, 1/τs. Very recently we analysed the subgap DOS in a clean s-wave superconductor, where the spin-flip scattering is dominant [5]; we argued that at least in some cases this limit is relevant experimentally. Here we briefly review the results of Ref.[5] and then present a more detailed comparison of the DOS obtained in Refs.[2,4,5]. We consider a mean field hamiltonian Ĥ = ξ̂τ3 +∆(r)τ1σ2 + Û , (1) where ξ̂ = −∇/(2m)−μ, μ is the chemical potential, τi and σi are the Pauli matrices in the particle-hole and the spin space respectively. The potential due to magnetic impurities Û = U(r) · s, where s is the electron spin operator, U(r) = ∑ i JSiδ(r − ri), J is the exchange constant, and Si is the impurity spin at a site i. For an energy, E < ∆0, OF is the most probable configuration of impurities that creates a state at E, and therefore contributes the most to the DOS [3]. OF provides nonperturbative corrections to the DOS determined in the framework of SCB. In essentially all the energy range below the gap the size of the OF is significantly greater than the distance between impurities, so that the exact impurity potential can be replaced by a smooth function, and its probability density is well approximated by a Gaussian [3] with a width U 0 = Preprint submitted to LT23 Proceedings 1 February 2008 nimpJ S(S+1)/3, so that τ s = 2πN0U 2 0 , where nimp is the impurity concentration, and N0 is the normal state DOS. The DOS is then given by ln [N(E)/N0] ≈ −S[Uopt], where S[Uopt] is obtained by minimizing S[U] = 1 2U 0 ∫ drU(r) + λ ( E[U]− E )