Vortex core properties in iron pnictides

The mechanism of unconventional superconductivity in recently discovered Fe-based superconductors has been intensively discussed. A plausible candidate is the superconducting (SC) pairing mediated by antiferromagnetic (AFM) interactions. There are two different approaches predicting the s± pairing state, in which the SC gap shows an s-wave symmetry that changes sign between different Fermi-surface (FS) sheets. The first one is based on the itinerant spin fluctuations promoted by FS nesting, and the second is based on the local AFM exchange couplings. We apply quasiclassical Eilenberger approach to the vortex state to calculate the cutoff parameter, ξh, at different levels of impurity scattering rates and to compare results with experimental data for iron pnictides. The s±-wave pairing symmetry is considered as a presumable state for these materials. Magnetic field dependence of ξh/ξc2 is found to be nonuniversal for s± pairing: depending on the chosen parameter set it can reside both below and above analytical Ginzburg-Landau curve. It is also found that normalized ξ2/ξc2(B/Bc2) dependence is increasing with pair-breaking (interband) impurity scattering, and the intraband scattering results in decreasing of the ξ2/ξc2 value. Here, ξ2 is the vortex core size and ξc2 is the GinzburgLandau coherence length determined from the upper critical field. The ξ2/ξc2(B/Bc2) curve has a minimum at low temperatures and small scattering evolving into monotonously decreasing function at strong scattering and


Introduction
In iron pnictides, superconductivity and magnetism are close neighbors on the phase diagram, and it has been proposed [1,2] that the pairing mechanism is due to AFM spin-fluctuation exchange, similar to high-T c cuprate superconductors. However, the geometry of low-energy states in iron-based pnictides and in the cuprates is different, and in most ferropnicitdes the momentum Q connects low-energy fermionic states near the center and the corner of the Brillouin zone. Thus the SC gap Δ(k) must be symmetric with respect to k x → k y and k x → −k x , but still change the sign under k → k + Q. Such gap is generally called an extended s-wave gap, or s ± gap. Strong-coupling approaches to pairing, based on the AFM local J 1 − J 2 multiple competing exchange model of the magnetism in these systems, are also often used [3]. In that study, the exchange terms have been decoupled in the mean field in the pairing channel. In the region of the general phase diagram with J 2 J 1 , cos k x cos k y was the leading instability for the 2-band Fermi surface, leading to a nodeless s ± state. Experimental evidence for s ± pairing symmetry has been circumstantially reviewed in Ref. [4].
In spite of success of s ± model, there are some indications that a conventional s-wave state without sign reversal (s ++ -wave state) is also a possible candidate for iron a e-mail: ivan.zakharchuk@lut.fi pnictides. It has been proposed that the moderate electronphonon interaction due to Fe-ion oscillation can induce the critical orbital fluctuation, without being prohibited by the Coulomb interaction [5].
Important information about the order parameter symmetry can be obtained from the investigation of the mixed state. The scanning tunneling microscopy (STM) experiments probe the spatial variation of the local density of states [6], whereas μSR is sensitive to the spatial dependence of the local internal magnetic field B(r) [7]. The vortex core size is determined from the μSR measurements by fitting them to a theoretical function for B(r) that includes a cutoff function F(G, ξ h ), where G are the reciprocal lattice vectors and ξ h is the cutoff parameter. The functional form of F(G, ξ h ) depends on the spatial dependence of the superconducting order parameter Δ(r) in the core region. Consequently, cutoff parameter ξ h is generally not the coherence length, but rather a measure of the vortex core size. A definition of the vortex core size is the radius ξ 2 at which the supercurrent density |J(r)| calculated from B(r) reaches its maximum.

Model
Following the microscopical Eilenberger theory, the cutoff parameter, ξ h , can be found from the fitting of the calculated magnetic field distribution h E (r) to the Eilenberger - Hao-Clem (EHC) field distribution h EHC (r) [8] Here, where K 1 (u) is modified Bessel function, u = ξ h G, S is the area of the vortex lattice unit cell and λ(T ) is the penetration depth in the Meissner state. The magnetic-field penetration depth λ(T ) is assumed to be field independent and to have the same value as in the Meissner state. In this approach all field dependent effects are taken into account in ξ h (B) dependence. The London penetration depth can be experimentally measured with great precision and its variation with temperature depends sensitively on the gap structure. For T = T c /3, a conventional isotropic s-wave gap Δ 0 results in an exponential behavior, Δλ(T ) ∝ exp(−Δ 0 /T ), which is preserved even with the addition of non-magnetic impurities. Unconventional pairing states, on the other hand, are susceptible to the presence of non-magnetic impurities, which result in power-law behavior [9].

Results
Using the similarity to the model of spin-flip superconductors, B c2 (T ) for two-dimensional s ± pairing was determined in Ref. [11]. Dashed line in Fig. 1 (a) demonstrates the result of analytical Ginzburg-Landau (AGL) model for ξ v [7] ξ Here, we use the notation of the AGL theory for the cutoff parameter, ξ v instead of ξ h , and b = B/B c2 . This dependence with ξ c2 as a fitting parameter is often used for the description of the experimental μSR results [7]. As can be seen from Fig. 1 (a), the magnetic field dependence of ξ h /ξ c2 is nonuniversal because it depends not only on B/B c2 (as in the AGL theory, dashed line in Fig. 1  (a)), but also on interband and intraband impurity scattering parameters. In the clean case where Γ 0 = Γ π = 0, the results are similar for s ± and s ++ pairing symmetries. We indicated that this curve is "clean" one. In this figure, the case Γ 0 Γ π is considered and the value of ξ h is reduced considerably in comparison with the clean case. One can compare the observed behavior with that in s ++ pairing model. In s ++ pairing symmetry the intraband and interband scattering rates act in a similar way and ξ h /ξ c2 decreases always with impurity scattering. In contrast, in s ± model ξ h /ξ c2 (B/B c2 ) curves show different forms of behavior with Γ π . A crossing point appears in the ξ h /ξ c2 (B/B c2 ) dependences for s ± and s ++ pairings. In case of Γ * = Γ 0 = 0.5 and Γ π = 0.04, ξ h /ξ c2 increases regarding s ++ values at B/B c2 < 0.8 and decreases at higher fields becoming more flattened. A similar effect is visible for Γ * = Γ 0 = 1 with the crossing point at B/B c2 ∼ 0.55. We also calculated the magnetic field dependence of mean square deviation of h EHC distribution of the magnetic field from the Eilenberger distribution normalized by the variance of the Eilenberger distribution where · · · is the average over a unit vortex cell. It can be seen in the inset to Fig. 1 (a) that the accuracy of effective London model is deteriorating as the magnetic field increases; however, in superconductors with impurity scattering the error is below 6% even when it is close to the second critical field.
In Fig. 1 (b), the interband scattering Γ π dependences of ξ h are presented in low fields for the s ± pairing at different temperatures T/T c0 . As can be seen ξ h /ξ c2 increases with the interband scattering rate Γ π . Strong decreasing of ξ h /ξ c2 with a decrease in the temperature can be explained by the Kramer-Pesch effect. Recently, the image of the vortices in a wide field range from 0.1 T to 11 T by mapping the tunneling conductance at the Fermi energy in LiFeAs compound was obtained [12]. It was found that the vortex radius shrank with decreasing temperature and became smaller than the coherence length estimated from the upper critical field. This effect was considered as a direct evidence of the Kramer-Pesch effect expected in a clean superconductor [12]. It should be noted that the normalization constant ξ c2 increases with Γ π because Γ π suppress T c similar to superconductors with spin-flip scattering (violation of the Anderson theorem). Thus, rising of ξ h /ξ c2 implies more strong growth of ξ h than ξ c2 (from GL theory one would expect ξ h /ξ c2 = Const). Qualitatively, it can be explained by the strong temperature dependence of ξ h (B, T/T c ), which is connected to the Kramer-Pesch effect. Increasing Γ π results in suppression of T c , i.e. effective increasing of T and ξ h (T/T c ). The ξ c2 (T/T c ) does not have such strong T c dependence, thus leading to the increase of ξ h /ξ c2 ratio with Γ π . A small value of the cutoff parameter (ξ h /ξ c2 ∼ 0.4 at T/T c = 0.18) was observed in iron pnictide BaFe 1.82 Co 0.18 As [13], which is comparable with our theoretical prediction, Fig. 1 (b). In Ref. [13] the magnetic field distribution shape of the sample was explained by effects of field-induced magnetic order and vortex-lattice disorder. Our consideration shows the im-portance of impurity scattering even in the triangular lattice giving second possible explanation for the experimental results.

Conclusions
In conclusion, Eilenberger equations have been solved for superconductors with s ± and s ++ pairing symmetries in the mixed state. It is found that normalized values of ξ h /ξ c2 and ξ 2 /ξ c2 decrease with temperature due to Kramer-Pesch effect. In unconventional superconductors, ξ 2 /ξ c2 increases with pair-breaking impurity scattering.
The intraband scattering results in decreasing of ξ 2 /ξ c2 value. The transformation from diminishing to growing of ξ 2 /ξ c2 (B/B c2 ) dependence with lowering of the temperature is obtained for s ++ -wave symmetry. A reasonable agreement between calculated ξ h /ξ c2 and ξ 2 /ξ c2 and those obtained experimentally in Ba(Fe 1−x Co x ) 2 As 2 and LiFeAs is found. This work was supported by the Finnish Cultural Foundation (Etelä-Karjalan rahasto).