Superconducting-Gap Anisotropy of Iron Pnictides Investigated via Combinatorial Microwave Measurements

One of the most significant issues for superconductivity is clarifying the momentum-dependent superconducting gap Δ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{k}}$$\end{document}k), which is closely related to the pairing mechanism. To elucidate the gap structure, it is essential to investigate Δ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{k}}$$\end{document}k) in as many different physical quantities as possible and to crosscheck the results obtained in different methods with each other. In this paper, we report a combinatorial investigation of the superfluid density and the flux-flow resistivity of iron-pnictide superconductors; LiFeAs and BaFe2(As1−xPx)2 (x = 0.3, 0.45). We evaluated Δ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{k}}$$\end{document}k) by fitting these two-independent quantities with a two-band model simultaneously. The obtained Δ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{k}}$$\end{document}k) are consistent with the results observed in angle-resolved photoemission spectroscopy (ARPES) and scanning-tunneling spectroscopy (STS) studies. We believe our approach is a powerful method for investigating Δ(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{k}}$$\end{document}k) because it does not require a sample with clean surface unlike ARPES and STS experiments, or a rotational magnetic-field system for direct measurements of the angular dependence of thermodynamic quantities.

, where G and C are constants determined by the geometry of the sample and the resonator. We assumed the Hagen-Rubens relation, R s = X s = √ µ 0 ωρ dc /2, in the normal state to determine G and C. µ 0 is the vacuum permeability, and ρ dc is the dc resistivity of the sample measured by using a four-probe method (Fig.S1c). For an example, we plotted Z s (B, T ) obtained by this technique for BaFe 2 (As 0.7 P 0. 3  The T -dependent superfluid density, n s (T ) = 1/λ 2 (T ) (λ is the magnetic penetration depth), can be obtained from Z s measured in the zero-field limit (B dc = 0) via the relation λ (T ) = X s (T )/µ 0 ω (Fig.S2a). As clearly seen in Fig.S2b, n s (T ) of BaFe 2 (As 1−x P x ) 2 changed with temperature, with fractional β being between 1 and 2. These results are different from the conventional values for the line-nodal gap (β = 1) and for the gapless state due to impurity scattering (β = 2). We attributed such fractional β to the multiple-band nature of these materials 1 , and our analysis based on two-band model supports this idea (see the main text).
The B-dependence of the flux-flow resistivity was obtained from the data measured under finite fields (B dc > 0) using the Coffey-Clem model 2 . In this model, Z s induced by the vortex motion is calculated considering the effects of normal fluids and of the flux creep due to thermal fluctuations. Because these effects are negligibly small at sufficiently low temperatures, Z s can be described as where ω cr characterizes the crossover from the reactive response (ω ≪ ω cr ) to the resistive response (ω ≫ ω cr ). Although the free-flux-flow state, where vortices move without suffering from pinning, is achieved only at higher frequencies (typically ω > 5ω cr ), we can obtain exact ρ f (B, T ) due to the simultaneous measurement of R s and X s and equation (S2). The flux-flow resistivity of LiFeAs and BaFe 2 (As 1−x P x ) 2 (x = 0.3, 0.45) measured at T /T c ≈ 0.1 are plotted in Fig.S2c. Enlarged plots of these data are presented in the main text as Fig.1.

a c b
Supplementary Figure S2. Measured data. a, The superfluid-density fraction, n s (T )/n s (T ), of LiFeAs and BaFe 2 (As 1−x P x ) 2 (x = 0.3, 0.45) throughout the superconducting phase. b, A double-logarithmic plot of |n s (T )/n s (0) − 1| of BaFe 2 (As 1−x P x ) 2 below 0.3T c . The lines are guides to the eye of n s (T ) = n s (0) − A(T /T c ) β for several β . c, The B-dependent flux-flow resistivity, ρ f (B), of LiFeAs and BaFe 2 (As 1−x P x ) 2 measured at 1.8 K and 2.0 K, respectively.

B) Formulae of the two-band model
For the superfluid-density fraction As a minimum model, the superfluid density of N-band superconductors is expected to be the simple summation of each band contributions, n s (T ) = ∑ N ν=1 n sν (T ). According to the model suggested by Chandrasekhar and Einzel 3 , the νth band component of the in-plane superfluid density, n sν (T ), can be described by the in-plane Fermi velocity, v v v ab Fν , and the kinetic energy of the quasiparticles measured from the Fermi level, ξ , as where (v ab Fν ) 2 = |v v v ab Fν | 2 and ⟨· · ·⟩ ν = ∫ dS Fν (· · · )/|hv v v Fν | is the surface integral of X on the νth sheet of the Fermi surface. The first term of the right-hand side of equation (S3) corresponds to n sν (0) because the second term vanishes at T = 0 K. These formulae give the superfluid-density fraction of N-band superconductors. Thus, the total superfluid-density fraction is given is the weighting factor of the νth-band ingredient. The set of weighting factors satisfies ∑ N ν=1 γ ν = 1. This formula can account for the k k k dependence of the Fermi surface and the superconducting gaps through the Fermi surface integral of v v v F and ∆ 2 ν (k k k). In the manuscript, we used N = 2 and considered hole-like and electron-like sheets of the Fermi surface.

For the flux-flow resistivity
Recently, the magnetic field dependence of flux-flow resistivity, ρ f (B), in two-band superconductors has been studied on the basis of two-band version of the Keldysh-Usadel equation 4 and time-dependent Ginzburg-Landau (tdGL) equations 5 . These studies show that various kinds of initial slope, Fν τ ν /3, can be obtained, suggesting the importance of multiple-band effects. Unfortunately, these reports assumed two isotropic gaps and considered situation is far from that in our measurement (fairy clean superconductors, T ≪ T c ) since the Keldysh-Usadel equation is applicable to dirty superconductors and the tdGL equation is adequate for limited regions in the B-T phase diagram where superconducting order parameter is suppressed. An extension of a non-equilibrium quasiclassical (Keldysh-Eilenberger) theory 6 , which can treat clean superconductors, to multiple bands with anisotropic gaps may give rigorous description for ρ f (B) in multiple-band system. However, it needs heavy numerical calculations, and hence it does not meet the purpose of this manuscript to demonstrate a new approach to evaluate the anisotropy of superconducting gaps in multiple-band materials from measured superfluid density and flux-flow resistivity by a microwave technique. Instead of such a rigorous calculation based on microscopic framework, we used the Goryo-Matsukawa model 7,8 , which is the simplest application of the Bardeen-Stephen model to two-band systems and succeeds in explaining ρ f (B) in MgB 2 9 and Y 2 C 3 10 , as a minimal model to describe the flux-flow resistivity in two-band superconductors.
The Ginzburg-Landau free energy for two-band superconductors 7,8,11,12 is given by where A A A(r r r) is a vector potential. The fourth term is a Josephson-type interaction g 12 , where θ ν is a phase of superconducting wave function of νth-band component and g 12 = g 21 is assumed. Regarding the interband Josephson coupling, dislocation of one magnetic vortex into two vortices with fractional flux quanta due to large driving force 13 or entropic term at high temperatures 14 has been suggested theoretically. However, the dislocation of fractional vortices is difficult to occur in our case since the current (and the driving force) introduced by the microwave perturbation technique is sufficiently small and we carried out flux-flow measurements at low temperatures around 2 K. Thus, the inter-band Josephson interaction is expected to result in a locking effect of phase difference θ 1 − θ 2 = 0 (g 12 < 0) or π (g 12 > 0) 7,8,13,14 . This is the situation considered in Goryo-Matsukawa 7, 8 and small driving force regions in Lin-Bulaevskii 13 , and two fractional vortices flow with the same velocity. In this case, current applied to the superconductor is shared by two bands, J J J = J J J 1 + J J J 2 , and energy dissipations caused by flux motion 15 in the unit volume become W = ρ f 1 J J J 2 1 + ρ f 2 J J J 2 2 . These expression are the same to that in currents flowing through a parallel circuit of two resistors; the net flux-flow resistivity is given by .
(S6) ρ f ν (B, T ) reaches the normal-state resistivity, ρ nν (B, T ), when the applied field equals to "the upper critical field" of νth band in the zero inter-band coupling limit, B c2ν (T ) 7,8 . Hence, ρ f (B, T ) equals to the net normal-state resistivity at the highest upper critical field; In the zero inter-band interaction case, coefficients in the GL free energy for two-band superconductors (equation (S5)) in the clean limit become and a tensor product of Fermi velocity. Consequently, the νth-band component of in-plane upper critical field in the zero inter-band interaction limit is In the case of single-band superconductors, the flux-flow resistivity behaves as ρ f (B, T )/ρ n (B, T ) = α(T )B/B c2 (T ) at B ≪ B c2 (T ) 15 , and the slope α(T ) in the case of anisotropic gap can be described by the Kopnin-Volovik relation 16 in fairly clean superconductors with an anisotropic gap is not understood even theoretically. To address the k k k-dependent superconducting gaps in two-band superconductors, we applied the Kopnin-Volovik relation to each band component; As a result, we can evaluate the initial slope in two-band superconductors, α, as where B min c2 (T ) = min{B c21 (T ), B c22 (T )}. By considering that the scattering rate τ −1 is proportional to the density of states at the Fermi level N(E F ), the normal-state conductivity can be approximated as Let us consider isotropic gaps (α 1 = α 2 = 1) in the dirty limit, the initial slope of ρ f (B) in equation (S9) can be transformed as . This expression matches to equation (55) in  , where ρ f (B) in two-band superconductors with V 12 = V 21 = 0 in the dirty limit was calculated on the basis of the Keldysh-Usadel thepry, with small difference in factor of 1/β 0 = 1/0.9 17 . Thus, equation (S9) is the simplest extension of ρ f (B) in two-band materials with zero inter-band interaction to anisotropic gaps.
In summary, we can calculate the initial slope α by taking the k k k dependence of the superconducting gap and the Fermi surface into account. We would like to note about further extension of our model given above. In cases of N(≥ 3)-band superconductor, it has been proposed that an emergence of time-reversal symmetry-broken state caused by the cross term, ∑ N ν=1 ∑ N µ̸ =ν g µν ∆ ν ∆ * µ (e.g. Silaev-Babaev 18 ). Such a unusual state may affects ρ f (B) in multiple-band superconductors. In addition, as mention before, an application of Keldysh-Eilenberger model to multiple bands with anisotropic gaps may provide the rigorous description of ρ f (B) in iron-based materials. We hope that further modifications of our minimal model based on the Keldysh-Eilenberger theory will give more precise picture of ρ f (B) in multiple-band (N > 3) superconductors in the clean limit with anisotropic superconducting gaps with inter-band coupling in the future.

C) Two-band model analysis Fitted results with other gap structures
In the main text, we presented the fitted results with prefactors of (p h , q h , r h , p e , q e , r e ) = (1, 0, 1, 2, 1), which provide the possibility of horizontal-nodal lines in ∆ h (k k k) and loop-like nodal lines in ∆ e (k k k). These prefactors are selected so that the superconducting gaps reflect the symmetry of hole-and electron sheets of Fermi surface. The obtained fitting parameters indicated that the measured data can be reproduced by the combination of a highly anisotropic nodeless gap on the hole-like sheet and a gap with loop-like nodes on the electron-like sheet. To show the validity of these gap structures, we refer to the fitted results with other prefactors leading other gap structures. Fig.S3a-d show the calculated results with (p h , q h , r h , p e , q e , r e ) = (1, 0, 1, 1, 4, 0). These prefactors have four-fold in-plane anisotropy and will give eight vertical-nodal lines in ∆ e (k k k) if ∆ max e ∆ min e < 0. The authors of Ref. 19 discussed a similar gap structure and concluded that such vertical nodes are inadequate for explaining the angle-dependence of the thermal conductivity. Meanwhile, Fig.S3e-h show the results with prefactors of (1, 0, 1, 1, 2, 1) and with the restrictions of ∆ max h ∆ min h < 0 and ∆ max e ∆ min e > 0, leading to horizontal-nodal lines in ∆ h (k k k) and no nodes in ∆ e (k k k). As shown in Fig.S3, the measured data for n s (T ) and ρ f (B) were also reproduced by these gap structures with the fit parameters enumerated in Table S1. However, the 5/9 residual sum of squares (RSS) of ∆ h,e compared with the ARPES data 20 was the smallest in the case of highly anisotropic nodeless ∆ h (k k k) and ∆ e (k k k) with loop-like nodes. Thus, we considered that such a gap structure is realized in BaFe 2 (As 0.7 P 0.3 ) 2 .  1, 0, 1, 1, 4, 0). e-h, Results of the fitting with prefactors of (1, 0, 1, 1, 2, 1) and with the condition of ∆ max h ∆ min h < 0, leading to horizontal nodal lines in ∆ h . a,b,e, and f, Schematic images of ∆(k k k) evaluated by the two-band model fitting. c,e,g, and h, Calculated results of n s (T )/n s (0) (c,e) and ρ f (B)/ρ n (g,h).

Resolution of the gap anisotropy
To show the resolution of the two-band model analysis, we present the parameter dependence of the calculated results for LiFeAs and for BaFe 2 (As 0.7 P 0.3 ) 2 in Figs.S4 and S5, respectively. The black curves, which are the same as the thick curves presented in Fig.1 in the main text, gave the best fit. The red (blue) curves are the results calculated after changing one parameter by +20% (−20%) from the best-fitted value and fixing the other parameters. As an example, we focus on the parameter dependence of the calculated results for LiFeAs (Fig.S4). It is clear that variations of |∆ max h | or |∆ min h | by ±20% changed n s (T ) drastically, but had nearly no effect on ρ f (B). In contrast, ρ f (B) was sensitive to |∆ max e | and |∆ min e |, whereas n s (T ) was insensitive to them. By comparing these results, we can determine (∆ max h , ∆ min h , ∆ max e , ∆ min e ) with errors of less than 10%.