The Kosztin-Leggett Nonlocal Effect on the Magnetic Penetration Depth in Superconductors with Nodes in the Gap

Penetration depth measurements have played a very fundamental role in the study of the gap symmetry of all unconventional superconductors. For instance, from magnetic penetration depth measurements it was first suggested the dwave symmetry for the energy gap in high critical temperature superconductors (HTSC) [1]. It is expected that, for superconducting energy gaps with nodes on the Fermi surface, the deviation of the magnetic penetration depth λ(T ) from its zero temperature value goes as a power law in temperature, ∆λ(T ) = λ(T )−λ(0) ∝ T, at temperatures well below the critical temperature Tc. The exponent α depends on gap geometry, dimensionality and the rate at which the gap vanishes near the zeroes. For line nodes in three dimensional systems, like those with a d-wave symmetry, ∆λ(T ) ∝ T (the same conclusion holds for point nodes in two dimensional systems). In these systems, however, the penetration depth has been found in some cases to cross over from linear to quadratic temperature dependence at a temperature T ∗ << Tc. Such a crossover has been explained theoretically as due to impurities [2] or nonlocal electrodynamics [3]. The impurity model considers that a dilute concentration of strong scatterers in the unitary limit in unconventional superconductors with line nodes causes a finite residual density of states near the nodes and negligible suppression of Tc. Thus, as T → 0 a ”gapless” temperature behavior [4] ∆λ(T ) ∝ T 2 should be expected for such superconductors, and the crossover in ∆λ(T ) is predicted to occur at T ∗ ∼ Γ∆0. Here, Γ is the scattering rate and ∆0 is the energy gap maximum. The nonlocal model supposes that the local condition λ(0)/ξ0 >> 1, for which the spatial variation of the electromagnetic field over the extent of the Cooper pair is neglected and the pair is taken as a point object, is not satisfied at temperatures well below Tc in most unconventional superconductors. Here ξ0 ≡ ~νF /π∆0 is the BCS coherence length. The nonlocal model is based on the fact that in superconductors with an anisotropic energy gap with nodes on the Fermi surface the coherence length ξ0(k) = ~νF /π∆(k). Kosztin and Leggett (KL) [3] suggested for the local condition