Abstract
One of the most significant issues for superconductivity is clarifying the momentumdependent superconducting gap Δ(\({\boldsymbol{k}}\)), which is closely related to the pairing mechanism. To elucidate the gap structure, it is essential to investigate Δ(\({\boldsymbol{k}}\)) 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 fluxflow resistivity of ironpnictide superconductors; LiFeAs and BaFe_{2}(As_{1−x}P_{x})_{2} (x = 0.3, 0.45). We evaluated Δ(\({\boldsymbol{k}}\)) by fitting these twoindependent quantities with a twoband model simultaneously. The obtained Δ(\({\boldsymbol{k}}\)) are consistent with the results observed in angleresolved photoemission spectroscopy (ARPES) and scanningtunneling spectroscopy (STS) studies. We believe our approach is a powerful method for investigating Δ(\({\boldsymbol{k}}\)) because it does not require a sample with clean surface unlike ARPES and STS experiments, or a rotational magneticfield system for direct measurements of the angular dependence of thermodynamic quantities.
Introduction
Although conventional superconductors and cuprates possess definitive Δ(k) with \(s\) and \(d\)wave symmetry, iron pnictides have multifarious Δ(k) comprised of combinations of gaps with and without zero points (nodes) reflecting their multipleband nature. An effective method for investigating Δ(k) is to measure physical quantities sensitive to lowenergy quasiparticle excitations. In this paper, we focus on two of such physical quantities. The first quantity is the temperaturedependent superfluid density \({n}_{s}(T)\) at \(T\ll {T}_{{\rm{c}}}\) (\({T}_{{\rm{c}}}\) is the superconductingtransition temperature). For a singlegap superconductor in the clean limit, \({n}_{s}(T)\) behaves as \({n}_{s}\mathrm{(0)}{n}_{s}(T)\propto \exp [\Delta (T)/{k}_{{\rm{B}}}T]\) when Δ(k) has no nodes. Meanwhile, a nodalgap superconductor exhibits \({n}_{s}\mathrm{(0)}{n}_{s}(T)\propto {(T/{T}_{{\rm{c}}})}^{\beta }\) with \(\beta =1\) and 2 when Δ(k) contains linelike and pointlike nodes, respectively. The situation becomes more complicated in multiplegap cases because the characteristics of every gap contribute to \({n}_{s}(T)\). The second quantity addressed herein is the magneticfield dependence of the fluxflow resistivity \({\rho }_{f}(B)\), which is a finite dissipation induced by quasiparticles bound inside the vortex core where \(\Delta \) is suppressed locally^{1,2}. \({\rho }_{f}(B\parallel c)\) behaves as \({\rho }_{f}(B)/{\rho }_{n}=\alpha B/{B}_{c2}\) in the \(B\ll {B}_{{\rm{c}}2}\) region (\({\rho }_{n}\) is the normalstate resistivity and \({B}_{{\rm{c}}2}\) is the upper critical field in the \(B\parallel c\) configuration). The initial slope, \(\alpha \), relates to Δ(k) through vortexcorebound states and increases from unity with increasing anisotropy of Δ(k)^{2,3,4,5,6}. Kopnin and Volovik^{7} successfully reproduced such an empirical relation between \(\alpha \) and Δ(k) in singlegap superconductors as \(\alpha =\langle {\Delta }_{0}^{2}\rangle /\langle {\Delta }^{2}({\boldsymbol{k}})\rangle \), where Δ_{0} is the maximum magnitude of superconducting gap and \(\langle \cdots \rangle =\int \,{\rm{d}}{S}_{{\rm{F}}}(\cdots )/\hslash {{\boldsymbol{v}}}_{{\rm{F}}}\) is the Fermi surface average. Furthermore, systematic investigations on iron pnictides clarified that the KopninVolovik relation also holds in multiplegap superconductors at least in a qualitative manner^{8,9,10,11,12}. These results encouraged us to investigate Δ(k) more quantitatively, based on \({n}_{s}(T)\) and \({\rho }_{f}(B)\) studies.
Results and Discussion
Superfluid density and fluxflow resistivity measurements
We investigated the superfluid density and fluxflow resistivity by measuring the microwave surface impedance in the zerofield limit or under finite magnetic field, respectively. Details of measurements and data analysis are presented in Supplementary Information. The superfluiddensity fraction, \({n}_{s}(T)/{n}_{s}\mathrm{(0)}\), of LiFeAs saturated to unity below ≈0.25T_{c} (Fig. 1a). This behavior suggests that Δ(k) has no nodes, consistent with previous reports^{8,13,14}. As for the fluxflow resistivity, \({\rho }_{f}(B)\) of LiFeAs increased with \(\alpha \) moderately greater than unity (Fig. 1b), indicating that Δ(k) has a finite anisotropy^{8}.
In contrast to that for LiFeAs, \({n}_{s}(T)/{n}_{s}\mathrm{(0)}\) for BaFe_{2}(As_{1−x}P_{x})_{2} (Fig. 1c) did not reach unity, even at \(0.05{T}_{{\rm{c}}}\), and it exhibited quasilinear \(T\) dependence (\({n}_{s}(T)/{n}_{s}(0)\approx 1{T}^{\beta }\)), suggesting the presence of linenodal gaps. The exact values of \(\beta \) obtained from the data below \(0.3{T}_{{\rm{c}}}\) were \(\beta =1.45\pm 0.05\) (\(x=0.30\)) and \(1.65\pm 0.05\) (\(x=0.45\))^{12} (see Supplementary Fig. S2). Possible origins of such fractional \(\beta \) values are (i) a pairbreaking effect due to impurity scattering^{15,16}, (ii) a renormalization of the effective Fermi velocity due to quantum fluctuations^{17,18}, and (iii) multipleband nature of the material^{19}. The first two explanations are unlikely for BaFe_{2}(As_{1−x}P_{x})_{2} because the residual dc resistivity of our samples was small (see Supplementary Fig. S1) and fractional \(\beta \) was also observed in BaFe_{2}(As_{0.55}P_{0.45})_{2}^{12}, which has a composition far from the quantum critical point (\(x\approx 0.3\)). Regarding the flux flow, the value of \(\alpha \) for BaFe_{2}(As_{1−x}P_{x})_{2} was high, exceeding 3, which is greater than that observed in cuprates (\(\alpha \approx 2\)), whose gap is completely anisotropic^{20}. Thus, a very large \(\alpha \) suggests that the multipleband effect plays an important role in addition to the highly anisotropic gap structure.
Summarizing the experimental results, it is expected that LiFeAs has nodeless gaps with moderate anisotropy and that BaFe_{2}(As_{1−x}P_{x})_{2} has highly anisotropic gaps containing at least one linenodal gap. Furthermore, the multipleband effect should play an essential role. To check these hypotheses concerning the Δ(k) of these materials, we evaluated Δ(k) quantitatively by fitting the \({n}_{s}(T)\) and \({\rho }_{f}(B)\) data with a phenomenological model, as described below.
The model we used to fit the data
We considered two sheets of Fermi surface composed of onehole and oneelectron pockets as a minimum model. Such a twoband model has been used elsewhere and is justified when the phase of the wave function on each band does not play a crucial role. Hereafter, we use subscript “h” (“e”) to specify hole (electron)like band component. Although unusual phenomena, such as a timereversalsymmetrybreaking state originating from Josephsontype interband interactions among N(>3)band components^{21}, may affect on superfluid density and fluxflow resistivity, but considering such exotic contributions is beyond our purpose of this manuscript that to demonstrate a new methodology to evaluate anisotropy of superconducting gaps on multipleband superconductors from experimentally obtained data. We hope that our attempt in this manuscript will stimulate and promote more sophisticated theoretical researches in the future.
As a model for superfluid density in the \(ab\) plane, \({n}_{s}(T)\), we used twoband extension of the ChandrasekharEinzel model developed for a singlegap superconductors^{16,22}. That is, the superfluiddensity fraction of a twoband superconductor is given by \({n}_{s}(T)/{n}_{s}(0)=\gamma {n}_{s{\rm{h}}}(T)/{n}_{s{\rm{h}}}(0)+(1\gamma ){n}_{s{\rm{e}}}(T)/{n}_{s{\rm{e}}}(0)\), where \(\gamma \) is a weight factor determined by the Fermisurface structure (see Section B in Supplementary Information). \(\nu \)(=h, e)band component of superfluid density is
where \({\mu }_{0}\), \({{\boldsymbol{v}}}_{{\rm{F}}\nu }^{ab}\), and \({\langle \cdots \rangle }_{\nu }=\int \,{\rm{d}}{S}_{{\rm{F}}\nu }(\cdots )/\hslash {{\boldsymbol{v}}}_{{\rm{F}}\nu }\) are the permeability in vacuum, the inplane Fermi velocity and the surface integral on the \(\nu \)th sheet of the Fermi surface. According to Eq. (1), \({n}_{s\nu }(T)\) obviously reflects momentum dependences of superconducting gaps and Fermi surfaces^{22}.
Regarding the fluxflow resistivity, an explanation of \({\rho }_{f}(B)\) in twoband superconductors were attempted in frameworks of twoband extension of timedependent GinzburgLandau (2bandtdGL) theory^{23} and that of a nonequilibrium version of Usadel (2bandKeldyshUsadel) theory^{24}. These calculations showed that various values of initial slope \(\alpha \) can be obtained depending on ratio of diffusion constants on different bands and/or paring interactions. Unfortunately, situations considered in these reports, superconductors in the dirty limit at \(T\simeq {T}_{{\rm{c}}}\) (2bandtdGL theory^{23}) and superconductors in the dirty limit (2bandKeldyshUsadel theory^{24}), are far away from what we measured in this manuscript (fairly clean superconductors at \(T\ll {T}_{{\rm{c}}}\)). In addition, these theories are not applicable to anisotropicgap cases since momentum dependence of gaps is smeared out due to strong impurity scattering in the dirty limit. Therefore, we could not adopt these results to our data. Although an extension of the KeldyshEilenberger theory^{25}, which can treat clean singleband superconductors with anisotropic gap in whole temperature range, to twoband case may give an rigorous description for \({\rho }_{f}(B)\), such a calculation needs heavy analytical and numerical calculations. Hence it does not meet our purpose in this manuscript to demonstrate a new approach to evaluate the gap anisotropy in multipleband superconductors from experimentally obtained superfluid density and fluxflow resistivity.
Instead of deriving a formula for \({\rho }_{f}(B)\) based on rigorous but complicated calculations, we extended a parallelcircuit model for isotropic gaps^{26,27} to anisotropicgap cases by applying the KopninVolovik relation^{7} to each band component. The GoryoMatsukawa model assumes that the interband interaction works to lock the relative phase between two gaps on different bands depending on the sign of interband interaction, which leads the situation that two fractional vortices flow together with the same velocity^{26,27,28}. Such a picture is correct as long as measurements are carried out at low temperature^{26,27} and small driving force^{28}, and microwave measurements reported in this manuscript meet these conditions. At this time, it is clarified that the fluxflow conductivity (\(\mathrm{1/}{\rho }_{f}\)) in twoband superconductors is given by \(\mathrm{1/}{\rho }_{f}(B)=\mathrm{1/}{\rho }_{f{\rm{h}}}(B)+\mathrm{1/}{\rho }_{f{\rm{e}}}(B)\) from the viewpoint of energy minimization. We applied the KopninVolovik relation to \(\nu \)band component in order to take gap anisotropy into account; \({\rho }_{f\nu }(B\parallel c)/{\rho }_{n\nu }={\alpha }_{\nu }B/{B}_{{\rm{c}}2\nu }\) with \({\alpha }_{\nu }(T)={\langle {\Delta }_{\nu 0}^{2}(T)\rangle }_{\nu }/{\langle {\Delta }_{\nu }^{2}(T,{\boldsymbol{k}})\rangle }_{\nu }\), where Δ_{ν0}, \({\rho }_{n\nu }\), and \({B}_{{\rm{c}}2\nu }\) are \(\nu \)(=h, e)band components of the maximum magnitude of superconducting gap, the normalstate resistivity, and the characteristic field (corresponds to the upper critical field with zero interband interaction) in the \(B\parallel c\) configuration. Unfortunately, \({\rho }_{f}(B)\) over the whole \(B\) range for arbitral Δ(k) cannot be obtained theoretically, even in singleband superconductors. Thus, at present, we calculate the initial slope of \({\rho }_{f}(B)\) in twoband superconductors; \(\alpha ={\rm{d}}({\rho }_{f}/{\rho }_{n})/{\rm{d}}(B/{B}_{{\rm{c}}2})\). Obtained expression for \({\rho }_{f}(B)\) in twoband superconductors at \(B\parallel c\ll {B}_{{\rm{c}}2}^{{\rm{\min }}}(T)\) is
where \({B}_{{\rm{c}}2}^{{\rm{\min }}/{\rm{\max }}}\) are the smaller/larger value of \({B}_{{\rm{c}}2{\rm{h}}}\) and \({B}_{{\rm{c}}2{\rm{e}}}\). Obtained \(\alpha \) reflects momentum dependences of Fermi sheets and superconducting gaps through \({\alpha }_{\nu }\) and ratios \({\rho }_{n{\rm{h}}}/{\rho }_{n{\rm{e}}}\) and \({B}_{{\rm{c}}2{\rm{h}}}/{B}_{{\rm{c}}2{\rm{e}}}\). We found that Eq. (2) becomes equivalent to that calculated on the basis of the 2bandKeldyshUsadel theory^{24} with zero interband interaction when we impose isotropic gaps in the dirty limit. This means that Eq. (2) is the simplest extension of \({\rho }_{f}(B)\) in twoband superconductors without interband interactions to anisotropic gaps. In other words, it is difficult to evaluate the strength of interband interaction by present model. We hope that morerigorous evaluation of \({\rho }_{f}(B)\) will be made by a KeldyshEilenberger theory^{25} extended to twoband superconductors with anisotropic gaps in the future. Moredetailed information relating to our model described above are given in the Section B of Supplementary Information.
We assumed BCSlike \(T\) dependence, Δ_{ν}(T) = Δ_{ν} \((0)\tanh (1.785\sqrt{{T}_{{\rm{c}}}/T1})\), instead of solving the gap equation selfconsistently. Such a simplification is justified as long as we focus on the low\(T\) region because Δ\((T\ll {T}_{{\rm{c}}})\) is almost constant. Rather than using oversimplified Fermi cylinders, we used the exact k dependence of the Fermi surface that was evaluated from the ARPES data^{29,30} (Fig. 2). For superconductors for which ARPES measurement cannot be performed, the Fermi surface obtained by band calculations may be used as an alternative. As described above, k dependence of the Fermi surface is important since it is reflected on \({n}_{s}(T)\) and \({\rho }_{f}(B)\) through surface integrals on Fermi sheets. We set Δ_{ν}(k) as
where \(\varphi \) is the azimuth angle measured from the ΓM direction in the Brillouin zone (Fig. 2), and \({\Delta }_{\nu }^{{\rm{\max }}}\) (\({\Delta }_{\nu }^{{\rm{\min }}}\)) is the maximum (minimum) value of Δ_{ν}(k). By considering the Fermisurface symmetry and referring to the ARPES data^{31}, we selected prefactors \(({p}_{{\rm{h}}},{q}_{{\rm{h}}},{r}_{{\rm{h}}},{p}_{{\rm{e}}},{q}_{{\rm{e}}},{r}_{{\rm{e}}})\) of LiFeAs as \((\,\,1,4,0,1,4,0)\), which leads to an inplane fourfold anisotropy. In contrast, \((1,0,1,1,2,1)\) was used for BaFe_{2}(As_{1−x}P_{x})_{2}, which has a possibility of appearance of horizontalnodal lines in Δ_{h}^{30,32} and/or loopnodal lines in Δ_{e}^{33,34}. These prefactors are selected so that the superconducting gaps reflect the symmetry of hole and electronlike Fermi surface. For example, in the case of BaFe_{2}(As_{1−x}P_{x})_{2}, \(({p}_{{\rm{h}}},{q}_{{\rm{h}}},{r}_{{\rm{h}}})=(1,0,1)\) expects an isotropy in the \({k}_{x}\)\({k}_{y}\) plane and a twofold symmetry in the \({k}_{z}\) direction but \(({p}_{{\rm{e}}},{q}_{{\rm{e}}},{r}_{{\rm{e}}})=(1,2,1)\) gives twohold symmetries in the \({k}_{x}\)\({k}_{y}\) plane and along \({k}_{z}\) direction. These symmetry is the same to those of Fermi sheets of hole and electron bands. Calculations with other prefactors are given in the Section C of Supplementary Information. Consequently, we can evaluate \({\Delta }_{\nu }^{{\rm{\max }}}\) and \({\Delta }_{\nu }^{{\rm{\min }}}\) as fit parameters through the simultaneous fitting of \({n}_{s}(T)\) and \({\rho }_{f}(B)\) with the model for the exact Fermisurface structure.
Fitted results
The results of the twoband model analysis are plotted as solid curves in Fig. 1, exhibiting good agreements with the measured data of \({n}_{s}(T\ll {T}_{{\rm{c}}})\) and \({\rho }_{f}(B\ll {B}_{{\rm{c}}2})\). The obtained fit parameters and its resolution are presented in Table 1 and Supplementary Table S1, respectively. The obtained gap anisotropies were reasonable compared to those measured by other probes, as mentioned below, validating our approach to Δ(k) determination.
Δ_{h}(k) and Δ_{e}(k) of LiFeAs were finite over the entire hole and electronlike sheets of the Fermi surface (Fig. 3a,b), and their minima were located in the ΓM and MM directions, respectively. A barometer of gap modulation defined by \({M}_{\nu }\equiv 1{\Delta }_{\nu }^{{\rm{\min }}}/{\Delta }_{\nu }^{{\rm{\max }}}\) was \({M}_{{\rm{h}}}\approx 36 \% \) and \({M}_{{\rm{e}}}\approx 29 \% \). Such moderately anisotropic gaps and these k dependences are consistent with those reported in refs. ^{31,35}.
Next, we consider BaFe_{2}(As_{1−x}P_{x})_{2} with \(x=0.3\). The results for \(x=0.45\) was similar to the case of \(x=0.3\) qualitatively. Δ_{h} of BaFe_{2}(As_{0.7}P_{0.3})_{2} is finite over the entire holelike sheet similar to LiFeAs, but its anisotropy was remarkably high up to \({M}_{{\rm{h}}}\approx \mathrm{87 \% }\) (Fig. 3c). \({\Delta }_{{\rm{h}}}^{{\rm{\min }}}\) appeared around point X, where horizontal nodes were reported via ARPES measurement^{30}, and its magnitude (≈\(1\,{\rm{meV}}\)) was smaller than the energy resolution of ref. ^{30}. Thus, the Δ_{h} based on our analysis is consistent with that of ref. ^{30} if we assume that the horizontal nodes in Δ_{h} reported in ref. ^{30} are actually small minima. Regarding the electronlike band, our result suggested that Δ_{e} has looplike nodes at the flat parts of the Fermi surface (Fig. 3d). The emergence of loop nodes is consistent with results of the angleresolved thermalconductivity study^{33} and the ARPES study^{34}. In particular, note that the location where the loop nodes appear is almost the same as that suggested in ref. ^{33}.
Δ_{ν} between our calculations and results of other experiments have slight difference in a quantitative manner, and there are two possibilities for this difference; the first one is the difference in sample properties and second origin is additional effects caused by interband coupling not included in our model. LiFeAs is sensitive to air and impurities, and characteristics in BaFe_{2}(As_{1−x}P_{x})_{2} are sensitive to phosphorus contents. Slight difference in such sample conditions might affect on the value of superconducting gaps. As for the second possibility, our model is the simplest extension of twoband superconductors in the zero interband interaction limit to anisotropicgaps cases. Exotic phenomena, such as the timereversalsymmetrybreaking state expected in N(>3)band superconductors^{21}, and/or nontrivial effects originating from multipleband components, which were not included in our model, may influence on actual \({n}_{s}(T)\) and \({\rho }_{f}(B)\). We hope that further developments of our model will give more precise description of \({n}_{s}(T)\) and \({\rho }_{f}(B)\) in multipleband superconductors and allow for more accurate evaluation of gap anisotropies in the future.
Finally, we refer to the comparison between BaFe_{2}(As_{0.7}P_{0.3})_{2} and BaFe_{2}(As_{0.55}P_{0.45})_{2}, where \(\alpha \) were found to be almost the same but the \({n}_{s}(T)\) values clearly differed. Examining the formulae of the twoband model (Section B in Supplementary Information), \({n}_{s}(T\ll {T}_{{\rm{c}}})\) is expected to be sensitive to smaller parts of Δ(k) while the fluxflow resistivity reflects the square of Δ(k) averaged over the Fermi surface. Obtained parameters listed in Table 1 show that \({\Delta }_{{\rm{h}}}^{{\rm{\min }}}/{k}_{{\rm{B}}}{T}_{{\rm{c}}}\) and \({\rm{d}}{\Delta }_{{\rm{e}}}({\boldsymbol{k}})/{\rm{d}}{\boldsymbol{k}}{}_{{\boldsymbol{k}}\to {{\boldsymbol{k}}}_{{\rm{F}}}}\) (the slope of Δ_{e}(k) approaching to gap nodes at the Fermi surface) of BaFe_{2}(As_{0.55}P_{0.45})_{2} were larger than those of BaFe_{2}(As_{0.7}P_{0.3})_{2}. These values are consistent with the fact that \({n}_{s}(T\ll {T}_{{\rm{c}}})\) of BaFe_{2}(As_{0.55}P_{0.45})_{2} changed slowly in comparison with that of BaFe_{2}(As_{0.7}P_{0.3})_{2}. On the other hand, \({M}_{{\rm{h}},{\rm{e}}}\) of BaFe_{2}(As_{0.7}P_{0.3})_{2} and BaFe_{2}(As_{0.55}P_{0.45})_{2} were close to each other and differences in anisotropy of the Fermi surface between these two compounds are not so remarkable. These characteristics lead each of \({\alpha }_{{\rm{h}}}\) and \({\alpha }_{{\rm{e}}}\) of BaFe_{2}(As_{0.7}P_{0.3})_{2} and BaFe_{2}(As_{0.55}P_{0.45})_{2} to be similar values. Therefore, observed \({n}_{s}(T\ll {T}_{{\rm{c}}})\) and \({\rho }_{f}(B\ll {B}_{{\rm{c}}2})\) can be understood by the difference in sensitivity of superconductinggap structure; \({n}_{s}(T\ll {T}_{{\rm{c}}})\) is sensitive to smaller parts of Δ(k) and \({\rho }_{f}(B\ll {B}_{{\rm{c}}2})\) is sensitive to anisotropy of Δ(k). In other words, these results suggest that we can evaluate Δ(k) from two independent physical quantities of \({n}_{s}(T\ll {T}_{{\rm{c}}})\) and \({\rho }_{f}(B\ll {B}_{{\rm{c}}2})\) by using different gap sensitivities.
Conclusion
We measured \({n}_{s}(T)\) and \({\rho }_{f}(B)\) by using a microwave technique and fitted the data with a phenomenological model developed for twoband systems that considered the Fermisurface structure. As a result, we found that LiFeAs has nodeless gaps with moderate anisotropy. In contrast, the data for BaFe_{2}(As_{1−x}P_{x})_{2} (\(x=0.3,0.45\)) can be reproduced by a highly anisotropic nodeless gap on the holelike sheet and another gap with looplike nodes on the electronlike sheet. These results are consistent with those for Δ(k) obtained using other probes and reasonable in quantitatively, thereby validating our combinatorial investigation of \({n}_{s}(T)\) and \({\rho }_{f}(B)\).
Our approach has several advantages over other probes for investigating Δ(k); neither \({n}_{s}(T)\) nor \({\rho }_{f}(B)\) measurements require (i) a clean and uncharged sample surface, unlike ARPES and STS investigations, and (ii) a rotational magneticfield system for angleresolved measurements of thermodynamic quantities. Furthermore, these measurements can be performed in a lower\(T\) region than the typical ARPES measurement, meaning that our approach can be applied to a broader range of superconductors. Thus, we believe that our approach is a novel and powerful method for investigating superconductinggap structures.
As for further advance of our model, it is expected that an extension of the KeldyshEilenberger theory^{25} to multipleband superconductors with anisotropic gaps including effects caused by interband interactions will give more precise expression of fluxflow resistivity in multipleband superconductors. We hope that our approach for gapanisotropy evaluation based on our phenomenological twoband model for \({n}_{s}(T)\) and \({\rho }_{f}(B)\) stimulates more sophisticated theoretical studies on it in the future.
Methods
Single crystals of LiFeAs (\({T}_{{\rm{c}}}=18\,{\rm{K}}\)), BaFe_{2}(As_{0.7}P_{0.3})_{2} (\({T}_{{\rm{c}}}=29.5\,{\rm{K}}\)), and BaFe_{2}(As_{0.55}P_{0.45})_{2} (\({T}_{{\rm{c}}}=22.5\,{\rm{K}}\)) were synthesized by selfflux methods^{36,37}. These samples exhibited the residual dc resistivity of less than 35 μΩcm (see Supplementary Fig. S1), evidencing the high quality of the samples. \({n}_{s}(T)\) and \({\rho }_{f}(B)\) were obtained from microwave surface impedance measured by using a cavity perturbation technique in the zerofield limit and under finite magnetic fields of up to 8 T, respectively. Detailed information on these procedures is presented in the Section A of Supplementary Information.
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Acknowledgements
T.O. and A.M. thank Prof. Takami Toyama and Prof. Yusuke Kato for their valuable comments and fruitful discussions. This research was partially supported by the Strategic International Collaborative Research Program (SICORP) of the Japan Science and Technology Agency, and also supported by JSPS KAKENHI (GrantinAid for JSPS Fellows: 15J09645 and EarlyCareer Scientists: 18K13783).
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T.O. performed the superfluid density and fluxflow resistivity measurements of LiFeAs and BaFe_{2}(As_{1−x}P_{x})_{2}, analytical and numerical calculations, and wrote the manuscript. Y.I. performed some of the superfluid density measurements of LiFeAs. K.K., K.M. and Y.U. synthesized the LiFeAs single crystals and performed transport measurements. M.N., A.I. and H.E. synthesized the BaFe_{2}(As_{1−x}P_{x})_{2} single crystals and performed transport measurements. A.M. wrote the manuscript. All authors discussed the results and commented on the manuscript at all stages of development.
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Okada, T., Imai, Y., Kitagawa, K. et al. SuperconductingGap Anisotropy of Iron Pnictides Investigated via Combinatorial Microwave Measurements. Sci Rep 10, 7064 (2020). https://doi.org/10.1038/s41598020633040
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