Abstract
Iron pnictides are the only known family of unconventional hightemperature superconductors besides cuprates. Until recently, it was widely accepted that superconductivity is driven by spin fluctuations and intimately related to the fermiology, specifically, hole and electron pockets separated by the same wavevector that characterizes the dominant spin fluctuations, and supporting order parameters (OP) of opposite signs^{1,2}. This picture was questioned after the discovery of intercalated or monolayer form of FeSebased systems without hole pockets, which seemingly undermines the basis for spinfluctuation theory and the idea of a signchanging OP^{3,4,5,6,7,8,9,10,11}. Using the recently proposed phasesensitive quasiparticle interference technique, here we show that in LiOHintercalated FeSe compound the OP does change sign, albeit within the electronic pockets. This result unifies the pairing mechanism of ironbased superconductors with or without the hole Fermi pockets and supports the conclusion that spin fluctuations play the key role in electron pairing.
Main
In iron pnictides, it has been widely perceived that superconductivity is driven by spin fluctuations, which supports the sign reversal between order parameters (OP) on the electron and hole pockets^{1,2}. The discovery of superconductivity in intercalated or monolayer FeSe at a critical temperature of the order of 40 K rekindled interest in Febased superconductivity and sent many theorists back to the drawing board^{3,4,5,6,7,8,9,10,11}. Indeed, the simple, transparent and largely accepted idea of spin fluctuations scattering electron pairs between hole and electron pockets was shaken by the absence of hole pockets in K_{x}Fe_{2−y}Se_{2}. The fact that the superconducting phase was formed by filamentary inclusions in a strongly magnetic matrix^{12} spoke against a conventional singlesign swave (‘s^{++}’) pairing^{13}, and model calculations based on the spinfluctuation scenario predicted a dwave state^{14,15}, which, by symmetry, would have generated gap nodes^{16}. On the other hand, later photoemission experiments indicated a nodeless superconducting state^{17,18}. After other materials with similar properties were discovered, including (Li_{1−x}Fe_{x})OHFeSe, which could be synthesized in a singlephase form^{6,7}, as well as FeSe monolayers^{8}, it became increasingly clear that superconductivity at ∼40 K and higher is possible without hole pockets.
We do not discuss here possible mechanisms for this superconductivity, nor even whether it may or may not be similar to the superconductivity in Fepnictides. Instead, we will concentrate on a phenomenological question of utmost importance: is superconductivity in FeSe derivatives (assuming they all belong to the same class) of a constant OP sign, or does it have a signchanging OP? The most natural superconductivity of the latter sort is of x^{2}–y^{2} type (where x and y are the directions of the Fe–Fe bonds). As discussed in refs 5,16, crystallographic symmetrylowering due to the Se positions, and thus doubling of the unit cell, results in this state acquiring gap nodes, although in principle the nodal area may be very small. Moreover, for K_{x}Fe_{2−y}Se_{2}, which exhibits another electron pocket at the zone centre, this should also lead to nodal lines on this pocket, with a typical cosinusoidal angular dependence of the gap. Neither of these effects has been observed^{17}. The other type of (truly gapless) signchanging superconductivity was suggested in ref. 16 and a detailed theory developed by Khodas and Chubukov^{19}. They observed that upon accounting for the spin–orbit interaction, the folded electron ellipses anticross and form two concentric pockets, of which the inner one is mostly of d_{xz}/d_{yz}, and the outer one of d_{xy} orbital character. This ‘bonding–antibonding’ scenario^{5} postulates that the OP on the inner barrel has one sign, and on the outer barrel the other. The goal of this paper is not to distinguish between the d_{x2 − y2} and s^{±} symmetries, but to eliminate another popular hypothesis, namely that all electron pockets have the same sign of the OP^{20}. We emphasize that this question has a principal conceptual importance; it has been widely argued that no hightemperature superconductivity is possible at all, unless the order parameter averages to (nearly) zero over all Fermi surfaces (thus eliminating the effect of the onsite Coulomb repulsion), and it is generally accepted that spinfluctuationdriven superconductivity necessarily requires a signchanging order parameter.
Unfortunately, the phasesensitive probes developed for dwave cuprates either fail or are more questionable in Febased materials. Probes based on Josephson loops with πcontacts, instead of providing a qualitative test, offer only a quantitative probe, since all possible junctions have currents arising from various Fermi surface sheets corresponding to both samesign and oppositesign order parameters^{21,22}. Quasiparticle interference (QPI) due to scattering from vortex cores is, in principle, phasesensitive, but the interpretation requires specific models of the superconducting states^{23,24}. The technique of identifying bound states formed at a nonmagnetic impurity by means of scanning tunnelling microscopy (STM) is more promising, and straightforward to measure. However, here the problem is that it is often hard to prove that the investigated impurities are indeed nonmagnetic.
In this paper, we shall report, first, an observation of the abovementioned bound state, which, notwithstanding the reservations above, strongly suggests a signchanging order parameter. Second, we will present a set of QPI measurements, and an analysis that does not rely upon the scattering from vortex cores, a poorly understood process, but upon zerofield integrated quantities, as suggested in ref. 24. As discussed in the following, this analysis involves a minimal number of assumptions and utilizes qualitative differences between the integrated QPI intensities for the signchanging and signconserving QP scatterings. This methodology does not require a separation between intra and interpocket scatterings, and unambiguously identifies the presence of a signchanging OP. However, this universality comes at a cost; it cannot distinguish between a nodeless s^{±} and a nearly nodeless dstate in these systems. This choice has to be made based on separate information.
In this work, we synthesized zincdoped (Li_{1−x}Fe_{x})OHFeSe and measured its material characteristics. The superconducting transition temperature determined using the midpoint of the resistive transition was suppressed from about 37.5 K to about 34.8 K, and the transition width becomes also clearly broadened by Zndoping. More detailed information is presented as Supplementary Note 1. We have measured the scanning tunnelling spectra (STS) at 1.5 K near a single Fesubstituting impurity in (Li_{1−x}Fe_{x})OHFe_{1−y}Zn_{y}Se, and show the results in Fig. 1. The topographic image measured with V_{bias} = 30 mV, I_{t} = 100 pA around the single impurity is shown in Fig. 1a. One can see that the impurity exhibits a typical image of a dumbbell shape with the centre located at the Fe site beneath the top Se layer. This dumbbell shape looks quite similar to the nonmagnetic Cu impurities in NaFe_{0.96}Co_{0.03}Cu_{0.01}As (ref. 25), which suggests that the Zn impurity is the origin of the dumbbell shape. This is also supported by the fact that simple counting of the impurities visible in the field of view (FOV) of the STM agrees well with the analysis of Xray energy dispersive spectrum (EDX). Both yield an approximately 2% Zn/Se atomic ratio in the sample, which is, as expected, less than the nominal 10% concentration. Indeed, we see essentially one type of surface defect with a concentration in the right range, and, while intrinsic defects such as vacancies may or may not occur in a sizeable amount, Zn atoms that exist in the sample must manifest themselves in the STM. Thus, although we cannot definitely identify this impurity as a Zn atom, it seems exceedingly likely. In Fig. 1c, d we show the spatial evolution of the tunnelling spectra along the line shown in Fig. 1a when the magnetic field is zero and 11 T, respectively. One can see that a strong resonant peak appears at about 4.0 meV at the impurity site. When moving away from the impurity, the spectrum recovers a typical shape with two superconducting gaps, similar to the pristine system without Zn (ref. 26). The two gaps Δ_{1} = 14 meV and Δ_{2} = 8.5 meV, determined here from spectra far away from the impurity (see the inset of Fig. 1b) are also quite close to those in the Znfree samples^{26}. To illustrate how the magnetic field affects the resonant impurity states, we show in Fig. 1b two selected spectra measured at the impurity site with and without magnetic field. It is clear that the magnetic field suppresses the resonant peak, but does not shift or split the peak.
Both observations are of great importance; the former proves that the peak in question has superconducting origin, and the latter indicates that the impurity is nonmagnetic. Indeed, for a magnetic impurity in the Born limit, the energy of a Bogoliubov–deGennes (BdG) quasiparticle contains three terms: H_{qp} = H_{0} + gμ_{B}S ⋅ H + ∑ _{k, σ, k′, σ′}Jc_{k, σ}^{+}σ_{σ, σ′} ⋅ S c_{k′σ′} (ref. 27). Here the first term H_{0} is just the bare energy of a BdG quasiparticle. The second term arises from the interaction between the magnetic moment S and external field H; g is the Landé factor and μ_{B} is the Bohr magneton. This term shifts the resonant peak position in a magnetic field. The third term is due to the interaction between the BdG quasiparticles and the local magnetic moment, and will lead to a splitting of the resonant peak. J in this term is the exchange coupling parameter between the BdG quasiparticles and the magnetic moment, while σ_{σ, σ′} denotes the Pauli matrix in spin space and c_{k σ}^{+}(c_{k′σ′}) is the quasiparticle creation (annihilation) operation at momentum k(k′) with quasiparticle spin σ(σ′). We have monitored more than ten individual ‘dumbbell’ impurities and have never observed either a shift or splitting of the peak under a magnetic field in the present sample, so we conclude the impurity of this type (presumably Zn atom) is nonmagnetic. According to theory^{28,29,30}, in multiband superconductors, if the pairing gap has a sign reversal, the nonmagnetic impurity with a moderate scattering potential will induce ingap resonant bound states. For a signpreserving gap, such as plain swave (s^{++}), these bound states are not expected. The observation of the bound state peak at the nonmagnetic Zn impurity reflects the signreversal gap of the system. Our conclusion that the bound states are associated with Zn atoms seems at odds with ref. 31, which found no such states for Zn atoms dosed onto the surface. We note, however, that the effective potential for a dosed Zn adatom is significantly weaker than for a substitutional Zn impurity. It is highly likely that the former is simply too weak to create a midgap resonance^{5,28,29,30}. This point is discussed further in Supplementary Note 2.
Earlier angleresolved photoemission^{10,11} and STM measurements^{26} revealed that the hole Fermi surfaces at the Brillouin centre (Γ point) are absent in (Li_{1−x}Fe_{x})OHFeSe, with only electron pockets at the zone corners. In Fig. 2a, we present a sketch of the Fermi surface topology in the folded (2Fe) Brillouin zone. Here is the reciprocal vector of the 2Fe lattice, that is, it is directed along the Fe–Fe square diagonal and equal to , where a is the Fe–Fe distance.
This Fermi surface geometry generates a rather simple pattern in the QPI image. Since all double pockets around the Mpoints are equivalent by symmetry (both in the normal and in the superconducting state), there is one roughly circular spot in QPI around the zone centre q = 0, which extends up to approximately 2k_{F} (where k_{F} is the Fermi vector of the outer pocket). This spot is periodically repeated at each reciprocal lattice vector G. Due to tunnelling matrix elements, one expects the spot around G = 0 to be the strongest, the one around G = (1,0) weaker (labelled as q_{2} in Fig. 2b), and the one around G = (1,1) even weaker (q_{3}). We will see that this is exactly the picture we observe. The q_{2} and q_{3} spots do not carry any additional information, being symmetryequivalent to q_{1}, so in our analysis we concentrated on the latter. The measured QPI results in a large area can be found in Supplementary Note 3. Note that if the spin–orbit coupling induced gap is sufficiently large, one may, in principle, be able to resolve the q_{1} spot into three concentric rings, corresponding to the thresholds of the scattering inside the inner barrel, inside the outer barrel, and the interband scattering. Currently this is beyond attainable resolution, but if in the future, were it to become possible, in the spirit of the discussion below one would be able to distinguish between the bonding–antibonding s^{±} and quasinodeless d states.
In Fig. 2c, we show a spatial map of the measured differential conductivity g(r, E = 8.5 meV) = [dI/dV ](r, E = 8.5 meV) at 1.5 K around the single impurity shown in Fig. 1a. One can clearly see patterns generated by the QPI. A Fourier transformation (FT) of Fig. 2c is shown in Fig. 2d, which can be directly compared with the cartoon in Fig. 2b. This validates the Fermi surface topology assumed in Fig. 2a, although our experimental resolution cannot assess the amplitude of the spin–orbitalcouplinginduced splitting. We emphasize that to obtain dI/dV on a fine energy mesh, we have used 64 × 64 grids in the FOV with a single impurity in the centre^{32}, and measured the STS (− 25 mV to 25 mV) at each grid point, masking the remainder of the observation window. Then we can rearrange the g(r, E) data for all the points to create maps with 64 × 64 pixels at each energy. This new method ensures stability in the following phaserelated analysis on the QPI intensity. For details of the measurements and spatial position corrections we refer the reader to Methods and to Supplementary Note 4.
As discussed, in principle we should have three scattering channels and thus three circles around q = 0, as marked in Fig. 3d. Taking into account the orbital character variation around the Fermi surfaces, we observe that these three rings would roughly correspond to the orbital channels d_{xy} ↔ d_{xz/yz}, d_{xz/yz} ↔ d_{xz/yz}, d_{xy} ↔ d_{xy} for the interband, innerintraband and outerintraband scattering, respectively. Near q = 0 we see a bright spot (Fig. 2b) that arises from the very short qscattering within one single Fermi surface and between the two Fermi surfaces. Due to the existence of significant background QPI intensity in the region close to q = 0 we shall not consider this part as relevant for the analysis.
Some of us have recently proposed^{24} a new methodology for robust determination of possible order parameter sign reversal, using QPI from a single nonmagnetic impurity. The central prediction is that the antisymmetrized intensity of the real part of the FTQPI is proportional to
for a weak scatterer, and the structure of this result is universal in the sense that it does not depend qualitatively on the strength of the scattering and the details of the electronic structure^{24}. Here Δ_{1} and Δ_{2} are the two gaps associated with the two bands, U is a scalar scattering potential, A represents an area in qspace containing gapsignchanging scattering vectors, and ρ(q, E) represents the FTQPI, which is the FT of the spatial map of the differential conductivity g(r, E). The prediction for an s^{±} pairing state is that this quantity, δρ^{−}(E), at an energy between the two gaps will be coherently enhanced and hence does not change sign with E > 0, while for an s^{++} case, this quantity is generally small, with an alternating sign between the two gaps. We have calculated δρ^{−}(E) from our measured data and show it in Fig. 3a. The phase shift has been corrected by taking the impurity as the origin point of the QPI image before the FT, as demonstrated in ref. 32. The details can be found in Supplementary Note 4. To get an enhanced signal/noise ratio, we have integrated the data within the ring defined by 0.13π/a < q < 0.55π/a, with a being the Fe–Fe bond length. This area is determined by simulation, with the requirement that the smallq scattering near q = 0 can be effectively neglected. This is explained in Supplementary Note 5. The energy dependence of δρ^{−}(E) is presented in the main panel of Fig. 3b. One can see a sharp peak at about 4.0 mV, which corresponds to an impurity resonance (see Fig. 1b). This peak is unrelated to the phasedependent analysis of QPI and needs to be removed prior to further analysis. To illustrate the energy dependence of the δρ^{−} image, we present a movie with a series of δρ^{−} images as a function of energy (see Supplementary Movie 1). As one can see, at some low energies, an artificial asymmetry of δρ^{−}(q, E) is observed. Above the smaller gap, this asymmetry becomes weaker and finally vanishes; the energies between the two gaps, namely between 8.5 to 14 meV, which are most relevant to our analysis, are very little affected by this asymmetry.
To eliminate the effect of the bound state, we have masked the central part of the twodimensional (2D) map of dI/dV with a circle of R = 3 pixels radius (29 pixels around the defect) by assuming a parabolic relation δρ(E) = AE^{2} + BE in the energy window from 3 to 5.5 meV. The details of this masking can be found in Supplementary Note 6. The calculated experimental data of δρ^{−}(E) after this masking are shown in the main panel of Fig. 3c. The masking procedures for experimental data and theoretical calculations are given in Supplementary Notes 6 and 7.
We are now ready to compare the experimental data with the theoretical predictions of ref. 24. To begin with, we have simulated the results of the unmasked processing presented in Fig. 3b, by integrating the simulated δρ^{−}(q, E) over the same ring in qspace. For parameters we used E_{imp} = 4.0 meV, Δ_{2} = 8.5 meV, and Δ_{1} = 14 meV. The calculated result is shown in the inset of Fig. 3b (more details of the calculations are presented in Methods and in Supplementary Note 6). We immediately observe that the s^{±} model reproduces the essential features of the experimental data. One can, however, argue that in the s^{±} case the spectral weight is dominated by the impurity resonance. To address this point, we present in Fig. 3c the δρ^{−}(E) obtained after the removal of the resonance, as discussed above, and, in the inset, the calculated δρ^{−}(E) subjected to the same masking procedure. Again, we see a qualitative difference between the two cases, with only the s^{±} calculation reproducing the experimental spectrum (and, in fact, reproducing it very well). This gives us strong confidence that gaps change sign between two or more Fermi surface sheets, as shown in Fig. 3d. Note that our analysis is not based on a detailed, modeldependent comparison between the measurements and simulations, but upon a very qualitative analysis, and the observed agreement hinges exclusively on the fact that the assumed pairing state has a signchanging order parameter. A new round of measurements has been repeated on the same impurity later, and the resulting energy dependence of the antisymmetrized FTQPI spectra is quite similar to the data presented here. In addition, a new impurity in the same sample was chosen to perform the control experiment, with a very similar result. All these results are described in Supplementary Note 8 and Supplementary Figs 7 and 8. The similar results from the control experiments lend strong support to the analysis and conclusions presented in the paper.
To summarize, we have shown, based on both the observation of an ingap impurity state for the nonmagnetic impurities, and a novel theoretical analysis of quasiparticle interference data, that the order parameter in intercalated FeSe, specifically in (Li_{1−x}Fe_{x})OHFe_{1−y}Zn_{y}Se, alternates sign, either between the Fermi surface sheets, or within individual sheets, as illustrated in Fig. 3d. Furthermore, the order parameter sign must affect a considerable fraction of the scattering processes, meaning that the two opposite signs are roughly balanced. This puts severe restrictions on the available scenarios. Of those proposed so far, two satisfy this experiment: the bonding–antibonding swave state or a nearly nodeless dwave state, as discussed, for instance, in ref. 5. At present, our results do not allow us to distinguish between the two. The latter has an advantage of having a clear candidate for pairing glue, namely spin fluctuations resulting from interpocket nesting in the unfolded zone (note that such fluctuations were observed, at q = {π, π/2} (refs 33,34,35) and calculated, at q = {π, 0.625π} (ref. 14) in K_{x}Fe_{2−y}Se_{2} or other intercalated system, which would have naturally led to a state depicted in Fig. 3d(right). A principal problem with this option, however, is that in the related K_{x}Fe_{2−y}Se_{2} it is incompatible with the nodeless Γpocket gap^{17}, so one has to assume different pairing symmetry in these otherwise fairly similar materials, and the symmetryrequired gap nodes at the points where the red and blue Fermi lines intersect in Fig. 3d(right), must be very steep, that is, dΔ/dk ≫ Δ/k_{F}. The bonding–antibonding s^{±} scenario, on the other hand, has an additional advantage in the sense that in this case the sign of the order parameter naturally follows the orbital character of the bands, with the xy orbitals carrying one sign, and the xz/yz orbitals the other (Fig. 3d). This greatly narrows the selection from various possible scenarios in the related systems^{36}. We must emphasize that our present work is based on a novel theoretical proposal^{24}. It would be interesting and important to do the same experiment and analysis on a system with different gaps having the same sign. A good candidate for this control experiment would be LiFeAs, with its good surface quality and wellseparated hole pockets. But in that case it is not entirely clear whether all hole pockets have the same sign of the order parameter. The optimally doped Ba_{1−x}K_{x}Fe_{2}As_{2} may be another good system to try this idea, supposing that the surface is clean enough. Our observation of signreversing gaps in this electronFermipocketonly system suggests a universality of the pairing mechanism in the Febased superconductors.
Methods
Sample synthesis.
(Li_{1−x}Fe_{x})OHFe_{1−y}Zn_{y}Se single crystals were synthesized by the hydrothermal ionexchange method. Firstly, single crystals with the nominal composition K_{0.8}(Fe_{0.9}Zn_{0.1})_{2−x}Se_{2} were manufactured in advance by the same selfflux method that had been used to synthesize pristine K_{0.8}Fe_{2−x}Se_{2} single crystals. Afterwards, 10 ml deionized water, 5 g LiOH (J&K, 99% purity), 0.6 g iron powder (Aladdin Industrial, 99% purity), and 0.3 g selenourea (Alfa Aesar, 99% purity) were added to a 50 ml Teflonlined stainlesssteel autoclave. After complete mixing, some pieces of K_{0.8}(Fe_{0.9}Zn_{0.1})_{2−x}Se_{2} single crystals were added into the mixture. Then, the autoclave was sealed and heated up to 120 °C, and the temperature was maintained for 40 to 50 h. Finally, the (Li_{1−x}Fe_{x})OHFe_{1−y}Zn_{y}Se single crystals were obtained by cooling the autoclave to room temperature. An Xray energy dispersive spectrum (EDS) analysis using scanning electronic microscopy suggests that the composition ratio of Fe:Se is about 1.2:1, and the ratio of Zn:Se is about 2%, indicating partial substitution of Zn at the Fe sites in the Fe layer. The characteristics of the sample are presented as Supplementary Note 1.
STM/STS measurements.
The STM/STS measurements were carried out in a scanning tunnelling microscope (USM1300, Unisoku) with an ultrahigh vacuum, low temperature and high magnetic field. The samples were cleaved in an ultrahigh vacuum with a base pressure of about 1 × 10^{−10} torr. Tungsten tips were used during all the STM/STS measurements. To raise the signaltonoise ratio, a typical lockin technique was used with an a.c. modulation of 0.4 mV at 987.5 Hz.
Differential conductivity measurements on a dense energy mesh in real space.
To eliminate uncertainties during the QPI analysis, we have measured the full spectrum using a dense energy mesh at each point in the real space. Firstly, we choose an area of 6 nm × 6 nm with a single impurity sitting at the centre of the image. Then, we divide the scanning area into 64 × 64 pixels and measure tunnelling spectra in the voltage window from − 25 mV to 25 mV with the same setpoint at each position. Then the 2D mapping images of the differential conductivity at different energies are derived from the tunnelling spectrum at each point.
Theoretical calculations.
To describe the FTQPI result, we employ a twoband tightbinding parametrization of the two elliptic electron pockets of the Fermi surface, rotated with respect to each other by 90° on the Fermi surface, as described in Supplementary Note 7. The scattering of quasiparticles by a nonmagnetic impurity, measured in the FTQPI, was calculated as a correction to the local density of states (LDOS), using the standard Tmatrix approach describing multiple scattering by a single impurity. In particular, we compute the antisymmetrized correction to the LDOS
Here, G_{μ}^{0}(k, E) = − (iEτ_{0} + ɛ_{μ}(k)τ_{3} + Δ_{μ}τ_{1})/(E^{2} + Δ_{μ}^{2} + ɛ_{μ}^{2}(k)) refers to the Nambu–Gor’kov Green’s function for the band μ, and τ_{i} is the ith Pauli matrix. ɛ_{μ}(k) and Δ_{μ} refer to the quasiparticle energy and superconducting gap, respectively, of the corresponding band. The Tmatrix for the multiple scattering by a single impurity in the band and Nambu–Gor’kov space is defined as
where U_{μμ} = U_{intra}τ_{0} and U_{μν} = U_{inter}τ_{0} are the intra and interband impurity scattering strengths, respectively.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Additional Information
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Acknowledgements
We acknowledge the useful discussions with G. Kotliar, P. Coleman, D.H. Lee and J. Zhao. The work in NJU was supported by National Key R&D Program of China (grant number: 2016YFA0300400), National Natural Science Foundation of China (NSFC) with the projects: A0402/11534005, A0402/11190023, A0402/11374144 and Natural Science Foundation of Jiangsu (grant number: BK20140015). P.J.H. was supported by NSFDMR1407502. I.I.M. was supported by ONR through the NRL Basic Research Program. D.A. and I.E. were supported by the joint DFGANR Project (ER 463/81) and DAAD PPP USA N57316180.
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The lowtemperature STS measurements were performed by Z.D., X.Y., Q.G. and H.Y. Data analysis was done by Z.D., X.Y., Q.G., H.Y. and H.H.W. The samples were grown by H.L. and X.Y.Z. The theoretical calculations were done by D.A. and I.E. All authors contributed to the writing of the paper, with P.H., I.I.M. and H.H.W. responsible for the final text. H.H.W. coordinated the whole work. All authors have discussed the results and the interpretations.
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Du, Z., Yang, X., Altenfeld, D. et al. Sign reversal of the order parameter in (Li_{1−x}Fe_{x})OHFe_{1−y}Zn_{y}Se. Nat. Phys. 14, 134–139 (2018). https://doi.org/10.1038/nphys4299
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DOI: https://doi.org/10.1038/nphys4299
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