Spin-orbit scattering in d-wave superconductors

– When non-magnetic impurities are introduced in a d-wave superconductor, both thermodynamic and spectral properties are strongly affected if the impurity potential is close to the strong resonance limit. In addition to the scalar impurity potential, the charge carriers are also spin-orbit coupled to the impurities. Here it is shown that (i) close to the unitarity limit for the impurity scattering, the spin-orbit contribution is of the same order of magnitude than the scalar scattering and cannot be neglected, (ii) the spin-orbit scattering is pair-breaking and (iii) induces a small idxy component to the off-diagonal part of the self-energy. In high-Tc superconductors, disorder has important effects on both thermodynamic and spectral properties. The critical temperature Tc and the superfluid density ρs are lowered by non-magnetic impurity substitution [1, 2, 3, 4] and disorder induced by irradiation [5, 6]. Recent ARPES data show clearly how disorder leads to a redistribution of spectral intensity by adding new states at the Fermi level [6]. The basic elements of the current theory have been inspired by previous studies on heavy fermion superconductors and are given by the anisotropy of the order parameter and the strong resonance limit for the impurity potential [7, 8]. These elements, adjusted to describe condensates with a d-wave symmetry of the order parameter, are able to account for most of the features observed by experiments on high-Tc d-wave superconductors [9, 10, 11]. However, discrepancies still exist, like the overestimation of the Tc-suppression [12]. In order to correct this situation, and to provide a more realistic picture, several improvements of the theory have been proposed [13, 14, 15], and, recently, the effect of spatial variation of the order parameter has been taken into account [12, 16, 17]. In addition to the scalar impurity potential, the charge carriers are also spin-orbit coupled to the impurities. So far, this additional scattering channel has not been considered because the spin-orbit interaction is believed to provide, if any, only negligible effects (at least in the absence of a Zeeman magnetic field). This argument is based on the observation that the spin-orbit potential is of order vso ∼ ∆g v where v is the impurity potential and ∆g is the shift of the g-factor [18]. The value of ∆g depends on the specific impurity, however its order of magnitude is roughly ∆g ≃ 0.1. From this estimate, it is expected therefore that the spin-orbit scattering rate 1/τso ≃ N0v 2 so, where N0 is the charge carriers density of Typeset using EURO-LaTEX 2 EUROPHYSICS LETTERS states, should be at most of order 1/τso ∼ 10 /τimp, therefore negligible with respect to the scalar impurity scattering rate 1/τimp [19]. Although such an estimate is correct in the Born approximation (weak scattering) nevertheless it underestimates the effect by orders of magnitude in the strong resonance limit, believed to be valid for high-Tc superconductors. To illustrate such a substantial discrepancy between the Born and the unitarity limit, let us first consider the self-consistent t-matrix solution of the impurity problem for ∆g = 0. In this case the electron (hole) propagator isG(k, iωn) = iω̃n−ρ3ǫ(k)−ρ1∆(k), where ∆(k) = ∆cos(2φ) is the d-wave order parameter and φ is the polar angle, ǫ(k) is the electron dispersion for an half-filled band and ρ1, ρ3 are Pauli matrices. The renormalized Matsubara frequency iω̃n satisfies the following equation [8, 11]: iω̃n = iωn + Γ g0(iωn) c2 − g0(iωn) , (1) where g0(iωn) = 〈iω̃n/[ω̃ 2 n + ∆(k) ]〉 and 〈· · ·〉 is the average over the polar angle φ. In equation (1), Γ = ni/(πN0) and c = 1/(πN0v) where ni is the impurity concentration. For c ≫ 1, eq.(1) reduces to the Born limit while for c ≪ 1 it leads to the unitarity, or strong resonant, limit. Now, let us suppose that the spin-orbit impurity scattering leads to a renormalization contribution of the same form of eq.(1). Since vso/v ∼ ∆g, the renormalization induced by both the impurity and the spin-orbit scatterings can be estimated by: iω̃n = iωn + Γ g0(iωn) c2 − g0(iωn) + Γ g0(iωn) (c/∆g)2 − g0(iωn) . (2) For c ≫ 1, iω̃m ≃ iωn + Γg0(iωn)c (1 + ∆g) and, as expected in the Born limit, the contribution of spin-orbit scattering is ∆g times smaller than that of scalar impurity scattering. On the other hand, for strong scattering, c/∆g can be very small and eventually it vanishes in the unitarity limit c → 0. As a result, in this limit the spin-orbit scattering leads to a renormalization of the same order of the impurity scattering, namely of order Γ/∆. To provide more solid grounds to the above simple picture, it is necessary to treat the impurity and spin-orbit interactions on the same level by generalizing the usual t-matrix approach for the impurity scattering also to the spin-orbit contribution. To this end, let us start by considering the spin-orbit impurity potential. Two-dimensionality is often assumed in describing the main electronic excitations at least for some of the high-Tc superconductors. In the present context, the reduced dimensionality has the following implication. If the charge carriers are confined to move in the x-y plane and the impurity potential is V (r) = v ∑ i,k exp[ik · (r − Ri)], where Ri denotes the impurity positions, the spin-orbit interaction assumes the following form: Vso(r) = iηsov ∑ i,k eik·(r−Ri)[k × p] · σ = i∆g v ∑ i,k eik·(r−Ri) [k× p]z k2 F σz, (3) where p = −i∇r is the momentum operator and the spin-orbit coupling has been parametrized by ηsov = ∆g v/k 2 F where kF is the Fermi momentum. The effect of two-dimensionality is therefore to couple the electron spin only along the z direction. Hence, by choosing the z axis as the direction of spin quantization, the spin-orbit interaction (3) does not mix the spin components. The generalized Green’s function in the particle-hole spin space is: G(k, iωn) −1 = G0(k, iωn) −1 − Σ(k, iωn), (4) where G0(k, iωn) −1 = iωn − ρ3ǫ(k) − ρ2τ2∆(k) and the Pauli matrices ρi and τj act on the particle-hole and spin subspaces, respectively [20]. In the self-consistent t-matrix approximaC. GRIMALDI: SPIN-ORBIT SCATTERING IN ETC. 3 tion the self-energy is Σ(k, iωn) = niTtot(k,k, iωn) where the t-matrix satisfies the following equation: Ttot(k,k ′, iωn) = u(k,k ′) + ∑ k′′ u(k,k′′)G(k′′, iωn)Ttot(k ′′,k′, iωn), (5) where u(k,k) = ρ3v + iτ3∆g v[k̂ × k̂ ′]z is the impurity potential including the spin-orbit contribution. Because of the angular dependence of the spin-orbit interaction, it can be shown that the t-matrix (5) reduces to Ttot(k,k ′iωn) = T (iωn) + Tso(k,k , iωn) where T (iωn) = ρ3v + ρ3v ∑ k G(k, iωn)T (iωn) (6) is the usual momentum-independent contribution from non-magnetic impurities [8] and Tso(k,k ′, iωn) = iτ3∆g v[k̂ × k̂ ′]z + i∆gv ∑ k′′ [k̂ × k̂′′]zτ3G(k ′′, iωn)Tso(k ′′,k′, iωn), (7) is the t-matrix for the spin-orbit interaction. Before proceeding with the complete solution of eq.(7), it is useful to analyze the lowest order contributions in ∆g. By replacing G(k, iωn) with G0(k, iωn), the expansion of eq.(7) up to the third order in ∆g leads to a spin-orbit part of the self-energy of the form Σso(k, iωn) = Σ (2) so (k, iωn) + Σ (3) so (k, iωn), where the first term is the usual Born contribution Σ (2) so (k, iωn) = ni∆g v ∑ k′ |k̂ × k̂ |τ3G0(k , iωn)τ3 which renormalizes both the frequency and, contrary to the normal impurity scattering, the gap function. The term proportional to ∆g is instead: Σ so (k, iωn) = −ini(∆g v) 3 ∑ k′,k′′ [k̂ × k̂′]z[k̂ ′ × k̂′′]z[k̂ ′′ × k̂]zτ3G0(k ′, iωn)τ3G0(k ′′, iωn)τ3 = −i2 sin(2φ)Γ ∆g c3 〈