Odd and Even Modes of Neutron Spin Resonance in the Bilayer Iron-BasedSuperconductor CaKFe4As4

Tao Xie,1, 2 Yuan Wei,1, 2 Dongliang Gong,1, 2 Tom Fennell,3 Uwe Stuhr,3 Ryoichi

Kajimoto,4 Kazuhiko Ikeuchi,5 Shiliang Li,1, 2, 6 Jiangping Hu,1, 2, 6 and Huiqian Luo1, ∗

1Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

2University of Chinese Academy of Sciences, Beijing 100049, China3Laboratory for Neutron Scattering and Imaging,

Paul Scherrer Institute, CH-5232 Villigen, Switzerland4Materials and Life Science Division, J-PARC Center,

Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan5Neutron Science and Technology Center, Comprehensive Research

Organization for Science and Society, Tokai, Ibaraki 319-1106, Japan6Collaborative Innovation Center of Quantum Matter, Beijing 100190, China

(Dated: July 3, 2018)

We report an inelastic neutron scattering study on the spin resonance in the bilayer iron-based su-perconductor CaKFe4As4. In contrast to its quasi-two-dimensional electron structure, three stronglyL-dependent modes of spin resonance are found below Tc = 35 K. The mode energies are below andlinearly scale with the total superconducting gaps summed on the nesting hole and electron pockets,essentially in agreement with the results in cuprate and heavy fermion superconductors. This obser-vation supports the sign-reversed Cooper pairing mechanism under multiple pairing channels andresolves the long-standing puzzles concerning the broadening and dispersive spin resonance peakin iron pnictides. More importantly, the triple resonant modes can be classified into odd and evensymmetries with respect to the distance of Fe-Fe planes within the Fe-As bilayer unit. Thus, ourresults closely resemble those in the bilayer cuprates with nondegenerate spin excitations, suggestingthat these two high-Tc superconducting families share a common nature.

PACS numbers: 74.70.Xa, 74.20.Rp, 76.50.+g, 78.70.Nx, 75.40.Gb

Understanding the superconducting mechanism in un-conventional superconductors such as copper oxides,heavy fermions, iron pnictides, and iron chalcogenides,is one of the most important topics in modern condensedmatter physics [1, 34, 36]. On cooling below the super-conducting transition temperature (Tc) in these materi-als, the spin excitations form a resonant peak with en-hanced susceptibility at a certain energy and around theantiferromagnetic (AFM) wavevector of the parent com-pounds. This so-called neutron spin resonance, which isargued to be a spin-1 collective mode of particle-hole ex-citations in the superconducting state, gives prominentevidence for the magnetic Cooper pairing in cuprates andheavy fermion superconductors [4–6, 11, 34].

The multiband physics from Fe 3d orbitals in iron-based superconductors opens a new opportunity to ex-plore the unconventional superconductivity [8, 9]. In ironpnictides or chalcogenides, the sign-reversed s-wave (s±)Cooper pairing can be obtained in both weak couplingapproaches [12–15] and strong correlated electron models[14] and has been supported by many experimental evi-dences [15–17, 28, 29]. In the s± superconducting state,a spin resonance is theoretically predicted to arise at thewave vector Q linking between hole-electron or electron-electron pockets, which is experimentally observed inmany systems [20–30, 33, 34, 37, 37, 38, 38, 39, 39, 40, 40].If the resonance is indeed a spin exciton in the supercon-

ducting state, it should be a sharp peak bound at anenergy (ER) below the pair breaking energy, namely, thetotal superconducting gap summed on the two pocketslinked by Q: ∆tot = |∆k|+ |∆k+Q| [13, 15, 34, 41]. Here,∆k and ∆k+Q have opposite signs (probably differentvalues) to yield a finite coherence factor of this process,enhanced by the interband Fermi surface nesting underintraorbital Coulomb repulsion [14, 15]. In contrast, anonresonant broad peak in the magnetic spectrum belowTc is expected above twice the superconducting gap (2∆)in the sign-preserved (s++) state [16–18]. However, com-pared to the resonance mode observed in copper oxides[5, 6], spin resonances in iron pnictides are actually muchbroader in energy distribution and more dispersive bothin plane and along the L direction [29, 30, 33, 39, 40].Lacking of the sharpness in both energy and momentumspaces may be attributed to the complex multi-orbitalnature in iron-based superconductors that can lead tomultiple resonant modes and spin anisotropy [45–49].

To further understand the spin resonance in iron-basedsuperconductors, it is essential to make a full compar-ison to all behaviors of the resonant mode observedin cuprates. In the hole-doped bilayer YBa2Cu3O6+δ

(YBCO) system, the spin resonance exhibits distin-guished odd and even L symmetries due to the non-degenerate interlayer magnetic excitations [5, 50, 51],which is later confirmed in another bilayer system

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Bi2Sr2CaCu2O8+δ (Bi2212) [6, 52]. These two differ-ent modes of spin resonance evolve in a strikingly similardoping dependence in both systems, and their separationin energy is fully determined by a weak AFM interac-tion between Cu-O planes within the bilayer unit, givingstrong evidence for the magnetically mediated supercon-ducting pairing mechanisms. However, this even modehas never been observed in iron-based superconductors,which seems to suggest that the spin resonance may havedifferent origins in these two high-Tc families.

In this Letter, we report an inelastic neutron scat-tering study on the spin excitations of stoichiometriciron-based superconductor CaKFe4As4 (1144 compound)with Fe-As bilayer structure [Fig. 1(a)]. Three spinresonance modes are identified at the wave vector Qfrom Γ to M point [Fig. 1(b)], where the resonanceenergies and the mode intensities are directly propor-tional to the total superconducting gaps summed onthe nesting electron and hole bands. In contrast to itsquasi-two-dimensional (2D) electron structure, the reso-nance intensity for all three modes is highly L dependentwith two opposite harmonic modulations, showing eitherodd symmetry ∼| F (Q) |2 sin2(zπL) or even symmetry∼| F (Q) |2 cos2(zπL) [Figs. 1(c)−1(f)]. Here, F (Q)is the magnetic form factor of Fe2+, and zc = 5.855A (z = 0.4636) is the distance between adjacent Fe-Feplanes within the Fe-As bilayer unit [Fig. 1(a)]. We ar-gue that such phenomenon is essentially similar to thenondegenerate bilayer magnetic excitations in YBCO [5]but under multiband pairing mechanism [8].

We prepared high-quality single crystals of CaKFe4As4using the self-flux method according to the previous re-ports [3, 54–56]. X-ray diffraction, resistivity, and mag-

netization measurements suggest our crystals have a veryhomogenous quality with sharp superconducting transi-tions around 35 K, where the potential impurity phasesfrom CaFe2As2 or KFe2As2 (122 compound) are com-pletely absent. Neutron scattering experiments were car-ried out using thermal triple-axis spectrometer EIGERat SINQ, PSI, Switzerland, with a fixed final energyEf = 14.7 meV and about 2 g (∼200 pieces) of coalignedcrystals [64]. Time-of-flight (TOF) neutron scatteringexperiments were carried out at 4SEASONS spectrom-eter (BL-01) at J-PARC, Tokai, Japan, with incidentenergy Ei = 42 and 23 meV, ki parallel to the c axis,chopper frequency f = 250 Hz, and a total samplemass of about 4.3 g (∼400 pieces) [9, 10, 65]. Thescattering plane was [H, 0, 0] × [0, 0, L], defined usingthe magnetic unit cell with 2-Fe atoms similar to thatof the 122 parent compounds: aM = bM = 5.45 A,c = 12.63 A, in which the wave vector Q at (qx, qy, qz) is(H,K,L) = (qxaM/2π, qybM/2π, qzc/2π) reciprocal lat-tice units (r.l.u.). The collinear (C-type) AFM order sim-ilar to CaFe2As2 (or the noncollinear spin vortex phase[6]), which is expected to form magnetic Bragg peaks atQ = (1, 0, L) [and Q = (0, 1, L)] (L = ±1,±3,±5, ...),does not exist based on our neutron diffraction experi-ments (Supplemental Material [56]). Even so, the spinexcitations still emerge around Q = (1, 0), coincidingwith the Fermi surface nesting vector from Γ to M point[Fig. 1(b)], similar to many other iron pnictides [Fig.1(e) and 1(f)][34, 36].

Figure 2 gives the key results of this paper. After sub-tracting the intensity of spin excitations in the normalstate (T = 40 K) from E = 2 to 22 meV at Q = (1, 0, L)with L in the range 2−3, we can identify three spin reso-nance peaks in the superconducting state (T = 1.5 K) atER = 9.5 ± 0.5, 13 ± 0.5, 18.3 ± 0.5 meV, respectively[Fig. 2(a)]. The intensity distribution of all three peaksseparates into two groups, as clearly shown by the TOFneutron experiments [Fig. 2(b)]. Although the 9.5 and13 meV modes overlap with each other, it seems all threemodes are energy resolution limited and nearly nondis-persive along both the K and L directions. The temper-ature dependence of all three modes confirms their cou-pling to superconductivity: the intensity gain decreaseslike a superconducting order parameter, which ceases atTc = 35 K [Fig. 2(c)]. A spin gapped feature with inten-sity loss below Tc is also found below E = 7 meV. Moreinterestingly, all three modes show strong but different Ldependences with the maximums at L = 3 for the formertwo modes and L = 2 for the latter one [Fig. 2(a)]. Thus,we have further measured the spin excitations over a largerange of Q = (1, 0, L) with L = 0− 6, where those belowL = 2 cannot be measured for E ≥ 16 meV due to thescattering restriction. The results are summarized in Figs.2(d)−2(f). Obviously, two opposite symmetries along Lcan be identified for maximum around L = odd or even,much similar to the cases in bilayer cuprates YBCO and

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FIG. 2: (a) Energy dependence of the spin resonances at Q = (1, 0, L). The solid lines are guides to eyes. (b) 2D slice in Evs K of the spin resonances. (c) Temperature dependence and (d) L dependence of three resonance modes at E = 10, 13, 18meV. The red solid and dashed lines are fitting results by | F (Q) |2 sin2(zπL) [or | F (Q) |2 cos2(zπL)] function. (e), (f) Lmodulation of the odd resonance modes and the even resonance mode at different energy ranges.

Bi2212 [5, 6, 50–52]. In the metallic YBCO, both odd andeven modes of spin resonance are found corresponding tothe acoustic and optical spin waves in the AFM insulatingphase [33]. Although there is no evidence for any opticalbranch from antiphase spin excitations in the paramag-netic CaKFe4As4, by simply considering the symmetricand antisymmetric combinations from the contribution ofmagnetically decoupled Fe-As bilayers similar to metal-lic YBCO [35, 56], we can obtain both odd and even Lsymmetries of the spin excitations. Here, the intensity oftwo spin resonances at ER = 9.5 and 13 meV follows theL modulation | F (Q) |2 sin2(zπL) (so-called odd mode),and the one at high energy (ER = 18.3 meV) can bedescribed by | F (Q) |2 cos2(zπL) (so-called even mode)instead, with respect to the distance (zc) between theadjacent Fe-Fe planes within the Fe-As bilayer unit [Fig.1(a)] [56]. The data points agree very well with such sine-squared (cosine-squared) modulation, as shown in Fig.2(d). Here, we have z = 0.4636 for the unique bilayerstructure due to the shift of the intermediate FeAs layerout of their high-symmetry positions (z = 0.5) [3, 54, 55].Consequently, the peak positions actually shift to nonin-tegral L in comparison with the high-symmetric struc-ture, such as L = 1.08, 3.24, and 5.39 (odd mode) or L= 2.16, 4.31, and 6.47 (even mode), etc [Fig. 1(c)−(f)].Unlike the weak L modulation of spin resonance in 122-type iron pnictides [29, 30, 40], the minimum intensity at

each valley here is near zero [56].

Figure 3 summarizes the intensity distribution of thespin resonances and spin gap within [H,K] plane. Allthree resonance modes and the spin gap follow Gaussianline shapes around Q = (1, 0, L). While both the in-tensity loss at 3 meV and the intensity gain at 10 meVlook like ellipses elongated along the H direction, similarto the hole-doped Ba1−xKxFe2As2 [30], the 13 and 18meV resonant modes are more like circles in the [H,K]plane. The peak width at half maximum of the intensityis determined by the dispersion of the paramagnetic ex-citation, and the relative intensity depends on the energytransfer coupled with the L position in the TOF neutronscattering experiment when ki ‖ c.

The triple modes of spin resonance in CaKFe4As4can be naturally explained by multiple pairing chan-nels. Although the density functional theory calcula-tions predict ten Fermi pockets (six hole bands and fourelectron bands) [5], the angle-resolved-photoemission-spectroscopy measurements reveal three hole pockets(α, β, γ) around the Γ point and one electron pocket (δ)around the M point, with large diversity of the supercon-ducting gaps and 2D shapes of each pocket [Fig. 1(b)][4]. The observation of several full gaps and matchedsizes of electron and hole pockets is consistent with thes± pairing scenario under Fermi surface nesting. Thus,three different values of the total superconducting gaps

4

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L = 0.9 L = 1.5

5 K - 40 K

5 K - 40 K 5 K - 40 K

L = 1.9 L = 2.6

FIG. 3: Constant-energy scans along the H direction for (a)the spin gap at 3 meV and (b)−(d) three resonance modes at10, 13, 18 meV with intensity differences below and above Tc.The red solid lines are Gaussian fits to the data. (Insets) 2Dslices at half maximum of the intensity with identical energytransfer but different Ls [56].

(∆tot) summed on the nesting hole and electron pocketsshould yield three modes of spin resonance at differentenergies [56]. It turns out that the resonance energy andthe maximum intensity gain for each mode [∆S(Q,ω)]show contrary linear dependence with ∆tot [4, 56], asshown in Fig.4 (a) and (b). In fact, a universal relation-ship ER/2∆ = 0.64 was proposed among copper oxide,heavy fermion, and iron pnictide superconductors [11],where 2∆ is twice the superconducting gap in the sin-gle band case. We then summarize the reported resultsabout ER and ∆tot in Fig. 4(d) for iron-based supercon-ductors [4, 16, 21–25, 28–30, 37, 38, 40, 73–86]. The samelinear scaling with ER = 0.64∆tot can also describe thesedata together with our results of CaKFe4As4. Anotherwell-known scaling behavior with ER = 4.9kBTc maybe still applicable in this new compound [37, 40], onlyif considering the average resonance energy ER = 12.5meV determined on a powder sample [Fig. 4(c)][87]. Itshould be noticed that all pockets are fully gapped in thesuperconducting state, and there is no evidence for gapmodulation along kz or gap nodes in the spectroscopicinvestigations [4, 30–32]. This agrees with the nondisper-sive feature of all three resonant modes and rules out thesign-changed gaps within a single Fermi pocket. More-over, the orbital selective pairing could generate dou-ble resonant modes and possibly even L modulation, asshown in the NaFe1−xCoxAs system [25, 34, 49]. Unfor-tunately, further analysis on the orbital selection of pair-ing in CaKFe4As4 would be very difficult, given the equaloccupation of Fe orbitals, including dxz, dyz, dx2−y2 , anddz2 [5].

More importantly, the CaKFe4As4 compound actuallyis a bilayer system where the two Fe-As layers linked by

(a)

(b)

(c)

(d)

FIG. 4: (a), (b) The linear relationship of ER and ∆S(Q,ω)vs ∆tot. (c) ER vs Tc for CaKFe4As4 single crystal and pow-der samples under the linear scaling: ER = 4.9kBTc. (d) Thelinear scaling between ∆tot and ER for iron-based supercon-ductors [4, 16, 21–25, 28–30, 37, 38, 40, 73–86]. The dashedline marks ER = ∆tot, and the solid line is ER = 0.64∆tot

[11].

Ca have a shorter distance along the c axis than thoselinked by K for their different ionic radius [3, 4, 54, 91, 92][Fig. 1(a)], thus the magnetic coupling within the bilayerunit is much stronger than the interbilayer interaction.We also notice that the interlayer exchange coupling SJcin CaFe2As2 is much larger (about 5.5 meV) [93] thanthat in BaFe2As2 (about 0.22 meV) [94], and almost zeroin KFe2As2 [95, 96], accompanied by the stretched Fe-As interlayer spacing with 0.5c = 5.84, 6.51, and 6.94A [97–100], respectively. The distance of the Fe-Fe planewithin one Fe-As bilayer of CaKFe4As4 is 0.4636c = 5.855A , almost the same as the Fe-As interlayer spacing inCaFe2As2. Moreover, the energy difference between theodd (13 meV) and even (18.3 meV) spin resonance peaksis 5.3 meV, similar to SJc in CaFe2As2. All these factsclosely resemble those in the metallic YBCO with strongintrabilayer coupling J⊥ ∼ 10 meV in the magneticallydecoupled bilayers, where two spin resonance modes arefound at 41 and 53 meV following the odd and even Lsymmetries, respectively [35, 51]. The existence of twospin resonance modes in YBCO and Bi2212 indicatesthat there is still an AFM coupling between Cu-O planeseven in the superconducting state, which probably drivesthe bilayer systems to higher Tc than the monolayer sys-tems [5, 6]; whereas the multiband nature of CaKFe4As4induces further splitting of the odd modes, thus generat-ing triple peaks of spin resonance.

It should be noticed that the even mode of spin reso-nance in cuprates always has weaker intensity and higherenergy than the odd mode. This is attributed to thepresence of a threshold of the electron-hole spin flip con-tinuum slightly below 2∆, which supports the spin ex-

5

citon scenario [6]. Although the dichotomy of theoret-ical descriptions of magnetism is still an unresolved is-sue in iron-based superconductors, the nearly isotropicspin resonance in most compounds basically agrees withthe spin-1 exciton picture [34, 36, 37, 40]. The multi-ple resonant modes remind us to recall the broadeningand asymmetric spin resonance peak in many other iron-based superconducting systems, which are more likely in-duced by several overlapped odd and even modes due tosmall SJc [21–25, 28–30, 37–40]. If in analogy to the caseof CaKFe4As4, the low-energy part of the resonance peakis probably filled with odd modes, while the even modesmostly contribute to the high-energy part. When chang-ing L from odd to even within one Brillouin zone, theoverall peak position will naturally shift to higher energy[56]. This makes the resonant mode in appearance witha dispersion along the L direction [25, 28–30, 33, 34, 37–40]. Finally, compared with our recent discovery of 2Dspin resonance under three-dimensional Fermi surfacesin a 112-type iron-based superconductor [40], it suggeststhat the resonance intensity is much more sensitive to thelocal magnetic couplings rather than the kz dependenceof fermiology, even though the resonant energy is mostlycoupled to superconducting gaps from itinerant electronsnear the Fermi surfaces.

In summary, we have discovered strongly L-dependenttriple spin resonance modes in the new iron-based super-conductor CaKFe4As4 under multiple pairing channels.The resonance energies are below and proportional tothe total superconducting gaps, consistent with the s±pairing mechanism. Both odd and even L symmetries ofthe resonance intensity are found, which are attributedto the nondegenerate spin excitations from the Fe-As bi-layer similar to the cuprate superconductors with the Cu-O bilayer. Our results suggest that the spin resonancein iron-based superconductors has an intrinsic commonnature with cuprate superconductors, and the high-Tcsuperconductivity in both families is strongly associatedwith local magnetic interactions coupled with itinerantelectrons.

This work is supported by the National Key Researchand Development Program of China (2017YFA0302903,2017YFA0303103, 2016YFA0300502, 2015CB921300,2017YFA0303100), the National Natural Science Founda-tion of China (11374011, 11374346, 11674406, 11334012,and 11674372), the Strategic Priority Research Pro-gram (B) of the Chinese Academy of Sciences (CAS)(XDB07020300), and the Key Research Program of theCAS (XDPB01). H. L. is grateful for the support fromthe Youth Innovation Promotion Association of CAS(2016004). This work is based on experiments performedat the Swiss Spallation Neutron Source (SINQ), PaulScherrer Institute, Villigen, Switzerland. The neutronexperiment at the Materials and Life Science Experimen-tal Facility of J-PARC was performed under a user pro-gram (Proposal No. 2017A0051).

∗ Electronic address: hqluo@iphy.ac.cn[1] J. M. Tranquada, G. Xu, and I. A. Zaliznyak, J. Magn.

Magn. Mater. 350, 148 (2014).[2] D. S. Inosov, C. R. Phys. 17, 60 (2016).[3] P. Dai, Rev. Mod. Phys. 87, 855 (2015).[4] O. Stockert, J. Arndt, E. Faulhaber, C. Geibel, H. S.

Jeevan, S. Kirchner, M. Loewenhaupt, K. Schmalzl, W.Schmidt, Q. Si, and F. Steglich, Nat.Phys. 7, 119 (2011).

[5] M. Eschrig, Adv. Phys. 55, 47 (2006).[6] Y. Sidis, S. Pailhes, V. Hinkov, B. Fauque, C. Ulrich, L.

Capogna, A. Ivanov, L.-P. Regnault, B. Keimer, and P.Bourges, C. R. Phys. 8, 745 (2007).

[7] G. Yu, Y. Li, E. M. Motoyama and M. Greven, Nat.Phys. 5, 873 (2009).

[8] Q. Si, R. Yu, and E. Abrahams, Nat. Rev. Mater. 1,16017(2016).

[9] P. Richard, T. Sato, K. Nakayama, T. Takahashi, and H.Ding, Rep. Prog. Phys. 74, 124512 (2011).

[10] M. M. Korshunov and I. Eremin, Phys. Rev. B 78,140509(R) (2008).

[11] A. V. Chubukov, D. V. Efremov, and I. Eremin, Phys.Rev. B 78, 134512 (2008).

[12] T. A. Maier, S. Graser, D. J. Scalapino, and P. Hirschfeld,Phys. Rev. B 79, 134520 (2009).

[13] I. Mazin and J. Schmalian, Physica (Amsterdam) 469C,614 (2009).

[14] K.J. Seo, B. A. Bernevig, and J. Hu, Phys. Rev. Lett.101, 206404 (2008).

[15] T. Hanaguri, S. Niitaka, K. Kuroki, and H. Takagi, Sci-ence 328, 474 (2010).

[16] H. Yang, Z. Wang, D. Fang, Q. Deng, Q. Wang, Y. Xiang,Y. Yang, and H. Wen, Nat. Commun. 4, 2749 (2013).

[17] A. A. Kalenyuk, A. Pagliero, E.A. Borodianskyi, A. A.Kordyuk, and V. M. Krasnov, Phys. Rev. Lett. 120,067001 (2018).

[18] Z. Du, X. Yang, H. Lin, D. Fang, G. Du, J. Xing, H. Yang,X. Zhu, and H.-H. Wen, Nat. Commun. 7, 10565(2016).

[19] Z. Du, X. Yang, D. Altenfeld, Q. Gu, H. Yang, I. Eremin,P. J. Hirschfeld, I. I. Mazin, H. Lin, X. Zhu, and H. -H.Wen, Nat. Phys. 14, 134 (2018).

[20] A. D. Christianson, E. A. Goremychkin, R. Osborn,S. Rosenkranz, M. D. Lumsden, C. D. Malliakas, I. S.Todorov, H. Claus, D. Y. Chung, M. G. Kanatzidis, R. I.Bewley, and T. Guidi, Nature (London) 456, 930 (2008).

[21] Y. Qiu, W. Bao, Y. Zhao, C. Broholm, V. Stanev, Z.Tesanovic, Y. C. Gasparovic, S. Chang, J. Hu, B. Qian,M. Fang, and Z. Mao, Phys. Rev. Lett. 103, 067008(2009).

[22] D. S. Inosov, J. T. Park, P. Bourges, D. L. Sun, Y.Sidis, A. Schneidewind, K. Hradil, D. Haug, C. T. Lin,B. Keimer, and V. Hinkov, Nat.Phys. 6, 178 (2010).

[23] N. Qureshi, P. Steffens, Y. Drees, A. C. Komarek, D.Lamago, Y. Sidis, L. Harnagea, H.-J. Grafe, S. Wurmehl,B. Buchner, and M. Braden, Phys. Rev. Lett. 108,117001 (2012).

[24] S. Wakimoto, K. Kodama, M. Ishikado, M. Matsuda, R.Kajimoto, M. Arai, K. Kakurai, F. Esaka, A. Iyo, H.Kito, H. Eisaki, and S. Shamoto, J. Phys. Soc. Jpn. 79,074715 (2010).

[25] C. Zhang, R. Yu, Y. Su, Y. Song, M. Wang, G. Tan, T.Egami, J. A. Fernandez-Baca, E. Faulhaber, Q. Si, and

6

P. Dai, Phys. Rev. Lett. 111, 207002 (2013).[26] P. D. Johnson, G. Xu, and W. -G. Yin, Iron-Based Su-

perconductivity(Springer, New York, 2015), pp 165−169.[27] M. Wang, H. Luo, J. Zhao, C. Zhang, M. Wang, K.

Marty, S. Chi, J. W. Lynn, A. Schneidewind, S. Li,andP. Dai, Phys. Rev. B 81, 174524 (2010).

[28] H. Luo, X. Lu, R. Zhang, M. Wang, E. A. Goremychkin,D. T. Adroja, S. Danilkin, G. Deng, Z. Yamani, and P.Dai, Phys. Rev. B. 88, 144516 (2013).

[29] S. Chi, A. Schneidewind, J. Zhao, L. W. Harriger, L. Li,Y. Luo, G. Cao, Z. Xu, M. Loewenhaupt, J. Hu, and P.Dai, Phys. Rev. Lett. 102, 107006 (2009).

[30] C. Zhang, M. Wang, H. Luo, M. Wang, M. Liu, J. Zhao,D. L. Abernathy, T. A. Maier, K. Marty, M. D. Lumsden,S. Chi, S. Chang, J. A. Rodriguez-Rivera, J. W. Lynn,T. Xiang, J. Hu, and P. Dai, Sci. Rep. 1, 115 (2011).

[31] C. H. Lee, P. Steffens, N. Qureshi, M. Nakajima, K. Ki-hou, A. Iyo, H. Eisaki, and M. Braden, Phys. Rev. Lett.111, 167002 (2013).

[32] J. Zhao, C. R. Rotundu, K. Marty, M. Matsuda, Y. Zhao,C. Setty, E. Bourret-Courchesne, J. Hu, and R. J. Birge-neau, Phys. Rev. Lett. 110, 147003 (2013).

[33] M. G. Kim, G. S. Tucker, D. K. Pratt, S. Ran, A. Thaler,A. D. Christianson, K. Marty, S. Calder, A. Podlesnyak,S. L. Budko, P. C. Canfield, A. Kreyssig, A. I. Gold-man, and R. J. McQueeney, Phys. Rev. Lett. 110, 177002(2013).

[34] C. Zhang, H.-F. Li, Y. Song, Y. Su, G. Tan, T. Netherton,C. Redding, S. V. Carr, O. Sobolev, A. Schneidewind, E.Faulhaber, L. W. Harriger, S. Li, X. Lu, D. X. Yao, T.Das, A. V. Balatsky, T. Bruckel, J. W. Lynn, and P. Dai,Phys. Rev. B 88, 064504 (2013).

[35] J. T. Park, G. Friemel, Y. Li, J. -H. Kim, V. Tsurkan,J. Deisenhofer, H. -A. Krug von Nidda, A. Loidl, A.Ivanov, B. Keimer, and D. S. Inosov, Phys. Rev. Lett.107, 177005 (2011).

[36] G. Friemel, W. P. Liu, E. A. Goremychkin, Y. Liu, J.T. Park, O. Sobolev, C. T. Lin, B. Keimer, and D. S.Inosov, Europhys. Lett. 99, 67004 (2012).

[37] M. A. Surmach, F. Bruckner, S. Kamusella, R. Sarkar,P. Y. Portnichenko, J. T. Park, G. Ghambashidze, H.Luetkens, P. K. Biswas, W. J. Choi, Y. I. Seo, Y. S.Kwon, H.-H. Klauss, and D. S. Inosov, Phys. Rev. B 91,104515 (2015).

[38] Q. Wang, Y. Shen, B. Pan, Y. Hao, M. Ma, F. Zhou,P. Steffens, K. Schmalzl, T. R. Forrest, M. Abdel-Hafiez,X. Chen, D. A. Chareev, A. N. Vasiliev, P. Bourges, Y.Sidis, H. Cao, and J. Zhao, Nat. Mater. 15, 159 (2016).

[39] M. Ma, L. Wang, P. Bourges, Y. Sidis, S. Danilkin, andY. Li, Phys. Rev. B 95, 100504(R) (2017).

[40] T. Xie, D. Gong, H. Ghosh, A. Ghosh, M. Soda, T. Ma-suda, S. Itoh, F. Bourdarot, L.-P. Regnault, S. Danilkin,S. Li, and H. Luo, Phys. Rev. Lett. 120, 137001 (2018).

[41] L. W. Harriger, O. J. Lipscombe, C. Zhang, H. Luo, M.Wang, K. Marty, M. D. Lumsden, and P. Dai, Phys. Rev.B 85, 054511 (2012).

[42] S. Onari, H. Kontani, and M. Sato, Phys. Rev. B 81,060504(R) (2010).

[43] S. Onari and H. Kontani, Phys. Rev. Lett. 109, 137001(2012).

[44] Q. Wang, J. T. Park, Y. Feng, Y. Shen, Y. Hao, B. Pan,J. W. Lynn, A. Ivanov, S. Chi, M. Matsuda, H. Cao, R.J. Birgeneau, D. V. Efremov, and J. Zhao, Phys. Rev.Lett. 116, 197004 (2016).

[45] H. Luo, M. Wang, C. Zhang, X. Lu, L.-P. Regnault, R.Zhang, S. Li, J. Hu, and P. Dai, Phys. Rev. Lett. 111,107006 (2013).

[46] P. Steffens, C. H. Lee, N. Qureshi, K. Kihou, A. Iyo, H.Eisaki, and M. Braden, Phys. Rev. Lett. 110, 137001(2013).

[47] M. Ma, P. Bourges, Y. Sidis, Y. Xu, S. Li, B. Hu, J. Li,F. Wang, and Y. Li, Phys. Rev. X 7, 021025 (2017).

[48] D. Hu, W. Zhang, Y. Wei, B. Roessli, M. Skoulatos, L.-P.Regnault, G. Chen, Y. Song, H. Luo, S. Li, and P. Dai,Phys. Rev. B 96, 180503(R) (2017).

[49] W. Wang, J. T. Park, R. Yu, Y. Li, Y. Song, Z. Zhang,A. Ivanov, J. Kulda, and P. Dai, Phys. Rev. B 95, 094519(2017).

[50] S. Pailhes, Y. Sidis, P. Bourges, C. Ulrich, V. Hinkov, L.-P. Regnault, A. Ivanov, B. Liang, C. Lin, C. Bernhard,and B. Keimer, Phys. Rev. Lett. 91, 237002 (2003).

[51] S. Pailhes, Y. Sidis, P. Bourges, V. Hinkov, A. Ivanov, C.Ulrich, L.-P. Regnault, and B. Keimer, Phys. Rev. Lett.93, 167001 (2004).

[52] L. Capogna, B. Fauque, Y. Sidis, C. Ulrich, P. Bourges,S. Pailhes, A. Ivanov, J. L. Tallon, B. Liang, C. T. Lin,A. I. Rykov, and B. Keimer, Phys. Rev. B 75, 060502(R)(2007).

[53] A. Iyo, K. Kawashima, T. Kinjo, T. Nishio, S. Ishida, H.Fujihisa, Y. Gotoh, K. Kihou, H. Eisaki, and Y. Yoshida,J. Am. Chem. Soc. 138, 3410 (2016).

[54] W. R. Meier, T. Kong, U. S. Kaluarachchi, V. Taufour,N. H. Jo, G. Drachuck, A. E. Bohmer, S. M. Saunders,A. Sapkota, A. Kreyssig, M. A. Tanatar, R. Prozorov, A.I. Goldman, F. F. Balakirev, A. Gurevich, S. L. Bud’ko,and P. C. Canfield, Phys. Rev. B 94, 064501 (2016).

[55] W. R. Meier, T. Kong, S. L. Bud’ko, and P. C. Canfield,Phys. Rev. Mater. 1, 013401 (2017).

[56] See Supplemental Material for the sample characteriza-tion and raw data of neutron scattering experiments,which includes Refs. [57−63].

[57] Q. Fan, W. H. Zhang, X. Liu, Y. J. Yan, M. Q. Ren, R.Peng, H. C. Xu, B. P. Xie, J. P. Hu, T. Zhang, and D.L. Feng, Nat. Phys. 11,946 (2015).

[58] D. Liu et al., Nat. Commun. 3, 931(2012).[59] S. He et al., Nat. Mater. 12, 605(2013).[60] S. Tan, Y. Zhang, M. Xia, Z. Ye, F. Chen, X. Xie, R.

Peng, D. Xu, Q. Fan, H. Xu, J. Jiang, T. Zhang, X. Lai,T. Xiang, J. Hu, B. Xie, and Donglai Feng, Nat. Mater.12, 634(2013).

[61] J. He et al., Proc. Natl. Acad. Sci. USA 111, 18501(2014).

[62] L. Zhao et al., Nat. Commun. 7, 10608 (2016).[63] X. Liu et al., Nat. Commun. 5, 5047(2014).[64] U. Stuhr, B. Roessli, S. Gvasaliya, H. M. Rønnow, U.

Filges, D. Graf, A. Bollhalder, D. Hohl , R. Burge,M. Schild, L. Holitzner, C. Kaegi, P. Keller, and T.Muhlebach, Nucl. Instrum. Methods Phys. Res., Sect. A853, 16 (2017).

[65] M. Nakamura, R. Kajimoto, Y. Inamura, F. Mizuno, M.Fujita, T. Yokoo, and M. Arai, J. Phys. Soc. Jpn. 78,093002 (2009).

[66] R. Kajimoto et al., J. Phys. Soc. Jpn. 80, SB025 (2011).[67] Y. Inamura, T. Nakatani, J. Suzuki, and T. Otomo, J.

Phys. Soc. Jpn. 82, SA031 (2013).[68] W. R. Meier, Q.-P. Ding, A. Kreyssig, S. L. Budko, A.

Sapkota, K. Kothapalli, V. Borisov, R. Valent, C. D.Batista, P. P. Orth, R. M. Fernandes, A. I. Goldman,

7

Y. Furukawa, A. E. Bohmer, and P. C. Canfield, npjQuantum. Mater. 3, 5 (2018).

[69] D. Reznik, P. Bourges, H. F. Fong, L. P. Regnault, J.Bossy, C. Vettier, D. L. Milius, I. A. Aksay, and B.Keimer, Phys. Rev. B 53, R14741(R)(1996).

[70] H. F. Fong, P. Bourges, Y. Sidis, L. P. Regnault, J. Bossy,A. Ivanov, D. L. Milius, I. A. Aksay, and B. Keimer,Phys. Rev. B 61, 14773 (2000).

[71] F. Lochner, F. Ahn, T. Hickel, and Ilya Eremin, Phys.Rev. B 96, 094521 (2017).

[72] D. Mou, T. Kong, W. R. Meier, F. Lochner, L. L. Wang,Q. Lin, Y. Wu, S. L. Budko, I. Eremin, D. D. Johnson,P. C. Canfield, and A. Kaminski, Phys. Rev. Lett. 117,277001 (2016).

[73] K. Nakayama, T. Sato, P. Richard, Y.-M. Xu, Y. Sekiba,S. Souma, G. F. Chen, J. L. Luo, N. L. Wang, H. Ding,and T. Takahashi, Europhys. Lett. 85, 67002 (2009).

[74] K. Nakayama, T. Sato, P. Richard, Y.-M. Xu, T. Kawa-hara, K. Umezawa, T. Qian, M. Neupane, G. F. Chen,H. Ding, and T. Takahashi, Phys. Rev. B 83, 020501(R)(2011).

[75] C. H. Lee, K. Kihou, J. T. Park, K. Horigane, K. Fujita,F. Waßer, N. Qureshi, Y. Sidis, J. Akimitsu, and M.Braden, Sci. Rep. 6, 23424 (2016).

[76] K. Terashima, Y. Sekiba, J. H. Bowen, K. Nakayama, T.Kawahara, T. Sato, P. Richard, Y.-M. Xu, L. J. Li, G. H.Cao, Z.-A. Xu, H. Ding, and T. Takahashi, Proc. Natl.Acad. Sci. USA 106, 7330 (2009).

[77] M. Wang, M. Yi, H. L. Sun, P. Valdivia, M. G. Kim,Z. J. Xu, T. Berlijn, A. D. Christianson, S. Chi, M.Hashimoto, D. H. Lu, X. D. Li, E. Bourret-Courchesne,P. Dai, D. H. Lee, T. A. Maier, and R. J. Birgeneau,Phys. Rev. B 93, 205149 (2016).

[78] Y. Zhang, Z. Ye, Q. Ge, F. Chen, J. Jiang, M. Xu, B.Xie, and D. Feng, Nat. Phys. 8, 371 (2012).

[79] J. Maletz, V. B. Zabolotnyy, D. V. Evtushinsky, S. Thiru-pathaiah, A. U. B. Wolter, L. Harnagea, A. N. Yaresko,A. N. Vasiliev, D. A. Chareev, A. E. Bohmer, F. Hardy,T. Wolf, C. Meingast, E. D. L. Rienks, B. Buchner, andS. V. Borisenko, Phys. Rev. B 89, 220506(R)(2014).

[80] H. Miao, P. Richard, Y. Tanaka, K. Nakayama, T. Qian,K. Umezawa, T. Sato, Y.-M. Xu, Y. B. Shi, N. Xu, X.-P.Wang, P. Zhang, H.-B. Yang, Z.-J. Xu, J. S. Wen, G.-D.Gu, X. Dai, J.-P. Hu, T. Takahashi, and H. Ding, Phys.Rev. B 85, 094506 (2012).

[81] Q. Q. Ge, Z. R. Ye, M. Xu, Y. Zhang, J. Jiang, B. P.Xie, Y. Song, C. L. Zhang, P. Dai, and D. L. Feng, Phys.Rev. X 3, 011020 (2013).

[82] Z.-S. Wang, Z.-Y. Wang, H.-Q. Luo, X.-Y. Lu, J. Zhu,C.-H. Li, L. Shan, H. Yang, H.-H. Wen, and C. Ren,Phys. Rev. B 86, 060508(R) (2012).

[83] Y. V. Pustovit, and A. A. Kordyuk, J. Low Temp. Phys.42, 995 (2016).

[84] J. Zhu, Z. Wang, Z. Wang, X. Hou, H. Luo, X. Lu, C.Li, L. Shan, H. Wen, and C. Ren, Chin. Phys. Lett. 32,077401 (2015).

[85] Y. Zhang, L. X. Yang, M. Xu, Z. R. Ye, F. Chen, C. He,H. C. Xu, J. Jiang, B. P. Xie, J. J. Ying, X. F. Wang, X.H. Chen, J. P. Hu, M. Matsunami, S. Kimura, and D. L.Feng, Nat. Mater. 10, 273 (2011).

[86] X. H. Niu, S. D. Chen, J. Jiang, Z. R. Ye, T. L. Yu, D.F. Xu, M. Xu, Y. Feng, Y. J. Yan, B. P. Xie, J. Zhao,D. C. Gu, L. L. Sun, Q. Mao, H. Wang, M. Fang, C. J.Zhang, J. P. Hu, Z. Sun, and D. L. Feng, Phys. Rev. B

93, 054516 (2016).[87] K. Iida, M. Ishikado, Y. Nagai, H. Yoshida, A. D. Chris-

tianson, N. Murai, K. Kawashima, Y. Yoshida, H. Eisaki,and A. Iyo, J. Phys. Soc. Jpn. 86, 093703 (2017).

[88] R. Yang, Y. Dai, B. Xu, W. Zhang, Z. Qiu, Q. Sui, C. C.Homes, and X. Qiu, Phys. Rev. B 95, 064506 (2017).

[89] P. K. Biswas, A. Iyo, Y. Yoshida, H. Eisaki, K.Kawashima, and A. D. Hillier, Phys. Rev. B 95,140505(R) (2017).

[90] K. Cho, A. Fente, S. Teknowijoyo, M. A. Tanatar, K. R.Joshi, N. M. Nusran, T. Kong, W. R. Meier, U. Kalu-arachchi, I. Guillamon, H. Suderow, S. L. Bud’ko, P. C.Canfield, and R. Prozorov, Phys. Rev. B 95, 100502(R)(2017).

[91] J. Cui, Q.-P. Ding, W. R. Meier, A. E. Bohmer, T. Kong,V. Borisov, Y. Lee, S. L. Bud’ko, R. Valentı, P. C. Can-field, and Y. Furukawa, Phys. Rev. B 96, 104512 (2017).

[92] Q. -P. Ding, W. R. Meier, A. E. Bohmer, S. L. Bud’ko,P. C. Canfield, and Y. Furukawa, Phys. Rev. B 96,220510(R) (2017).

[93] J. Zhao, D. T. Adroja, D. -X. Yao, R. Bewley, S. Li, X.F.Wang, G. Wu, X. H. Chen, J. Hu, and P. Dai, Nat.Phys. 5, 555 (2009).

[94] J. T. Park, G. Friemel, T. Loew, V. Hinkov, Y. Li, B.H. Min, D. L. Sun, A. Ivanov, A. Piovano, C. T. Lin, B.Keimer, Y. S. Kwon, and D. S. Inosov, Phys. Rev. B 86,024437 (2012).

[95] K. Horigane, K. Kihou, K. Fujita, R. Kajimoto, K.Ikeuchi, S. Ji, J. Akimitsu and C. H. Lee, Sci. Rep. 6,33303 (2016).

[96] M. Wang, C. Zhang, X. Lu, G. Tan, H. Luo, Y. Song,M. Wang, X. Zhang, E.A. Goremychkin, T.G. Perring,T.A. Maier, Z. Yin, K. Haule, G. Kotliar and P. Dai,Nat. Commun. 4, 2874(2013).

[97] N. Ni, S. Nandi, A. Kreyssig, A. I. Goldman, E. D. Mun,S. L. Bud’ko, and P. C. Canfield, Phys. Rev. B 78, 014523(2008).

[98] M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W.Hermes, and R. Pottgen, Phys. Rev. B 78, 020503(R)(2008).

[99] H. Luo, Z. Wang, H. Yang, P. Cheng, X. Zhu, and H. -H.Wen, Supercond. Sci. Technol. 21, 125014 (2008).

[100] K. Kihou, T. Saito, S. Ishida, M. Nakajima, Y. Tomioka,H. Fukazawa, Y. Kohori, T. Ito, S. Uchida, A. Iyo, C.-H.Lee, and H. Eisaki, J. Phys. Soc. Jpn. 79, 124713 (2010).

8

Supplementary Materials

A. SAMPLE CHARACTERIZATION

We prepared high quality single crystals of CaKFe4As4using self-flux method as previous reports [1, 2]. Thesample photos and results of characterization are shownin Fig. S1 and Fig. S2. We co-aligned the crys-tals by a X-ray Laue camera (Photonic Sciences) inbackscattering mode with incident beam along c-axis onthin aluminium plates using CYTOP hydrogen-free gluein [H, 0, 0] × [0, 0, L] scattering plane (Fig. S1). Thecrystalline quality was examined by single crystal X-raydiffraction (XRD) on a SmartLab 9 kW high resolutiondiffraction system with Cu Kα radiation(λ = 1.5406 A)at room temperature ranged from 5 to 90 in reflectionmode. Comparing to the CaFe2As2 system, the inequiv-alent position of Ca and K atoms in CaKFe4As4 leadsto two different Fe-As distances below and above Fe-Feplane and changes the space group from I4/mmm toP4/mmm with Fe-As bilayer structure (Fig.S1(e)) [1, 3–5], thus all odd and even Bragg peaks along c-directionare observed in XRD measurements due to the noncen-trosymmetric structure. The sharp (0 0 L) peaks of X-raydiffraction in Fig. S1(b) indicate high c-axis orientationof our samples. The elastic neutron scattering results areshown in Fig. S1(c) and (d). There is no magnetic signalaround Q = (1, 0, 3) in this stoichiometric compound dueto nearly zero intensity difference between 1.5 K and 70K, suggesting no collinear (C-type) antiferromagnetismlike CaFe2As2 or non-collinear spin vortex (hedgehog)structure like the Ni/Co doped case [6], where both ofthem have antiferromagnetic interlayer coupling along c-axes.

Figure S2(a) shows the normalized resistivity of ourCaKFe4As4 crystals. The sharp superconducting transi-tion width (less than 0.3 K), uniform Tc and nearly iden-tical normal state behaviors among 18 randomly selectedsamples indicate high quality and homogeneity of oursamples. The normal state resistivity with high residual-resistivity-ratio RRR ≈ 15 also suggests high purity ofour samples. Figure S2(b) shows the temperature de-pendence of the DC magnetic susceptibility for 4 typi-cal crystals picked up from the resistivity measurements.The superconducting transitions of these samples are allvery sharp with transition width less than 1 K. All thesesamples have full Meissner shielding volume (4πχ ≈ −1)at based temperature (T = 2 K). From the results ofresistivity and DC magnetic susceptibility, we also canconclude that our CaKFe4As4 sample is purely homo-geneous without any impurity phases from CaFe2As2 orKFe2As2, which may appear in the crystal growth pro-cess. For example, CaFe2As2 will induce a jump in theresistivity curve around 160 K for its structure transi-tion, and KFe2As2 will result in another superconductingtransition around 4 K in the magnetic susceptibility data

[1, 2].

B. TRIPLE-AXIS NEUTRON SCATTERING

Inelastic neutron scattering experiments were carriedout using thermal triple-axis spectrometer EIGER atSINQ, PSI, Switzerland, with fixed final energy Ef =14.7 meV. The total sample mass is about 2 grams(about 200 pieces). Figure S3 shows energy scans atQ = (1, 0, L) (L = 2, 2.5, 2.8, 3). After subtractingthe intensity of spin excitations at the normal state (T= 40 K) from E = 2 to 22 meV at Q = (1, 0, L) withL = 2 ∼ 3, we can identify three spin resonance modesat the superconducting state (T = 1.5 K) with centerenergies E = 9.5 ± 0.5, 13 ± 0.5, 18.3 ± 0.5 meV, and aspin gap below 7 meV, respectively. The intensity gainfor three resonant modes can be fitted by Gaussian peakswith modulated intensity and different width determinedby energy resolution (δE = 2.3, 2.8, 2.9 meV). The max-imums of the former two modes are at L = 3 and thelatter one is at L = 2. It should be noticed that thestrong peak above 16 meV in the raw data mainly comesfrom the phonon scattering of aluminum sample holdersfor both 1.5 K and 40 K. Since the phonon is almost un-changed for this temperature range (1.5 ∼ 40 K), we canidentify the resonance mode around 18 meV by compar-ing the intensity below and above Tc.

Figure S4 summarizes constant-energy scans along Hdirection for the gap energy 3 meV and the resonantmodes at E = 10, 13, 18 meV. There are some spuriousscattering signal in the raw data from aluminium phononand high order neutron scattering. After subtracting the1.5 K (40 K for 3 meV) data by the 40 K (1.5 K for 3meV) data, we have well-defined Gaussian peaks for allmeasured energies.

In order to figure out the L−dependence of the spinexcitations, we have measured the spin excitations forthe energies over a wide range of Q = [1, 0, L] withL = 0 ∼ 6. The L dependence of the results for thegapped and resonant energies at E = 3, 10, 13, 18 meVwere presented in Fig. S5. Periodic modulations canbe identified in the raw data of these L scans, which ismore clear in the the subtracted data between 1.5 K and40 K. The modulation of the resonance around 18 meVhas an opposite behavior with 10 and 13 meV intensi-ties. This is also consistent with the results in Fig. S3(b,d, f, h). Figure S6 summarizes the intensity differencesbetween 1.5 K and 40 K (resonance intensity) for ener-gies from 7 to 21 meV, which can be clearly separatedto two different groups fitted by harmonic functions. Wedefine the “odd mode”described by | F (Q) |2 sin2(zπL),and “even mode”described by | F (Q) |2 cos2(zπL), re-spectively, where F (Q) is the magnetic form factor ofFe2+, and zc = 5.855 A (z =0.4636, c = 12.63 A) isthe distance between adjacent Fe-Fe planes within thecrystalline block of Fe-As bilayer (Fig.S1(e))[7, 8]. It

9

should be mentioned that z =0.4636 is not a free fit-ting parameter in our case, but obtained from the struc-ture refinement of the CaKFe4As4 samples [3, 4]. Be-cause the translation symmetry along c-axes is broken inthis compound, the intermediate FeAs-layers shift out oftheir high-symmetry positions, forming the bilayer struc-ture with z <0.5 and two different Fe-As distances belowand above the Fe-Fe plane (Fig.S1). The non-integralL−positions of maximum intensity of spin resonance donot mean it is incommensurate along L direction, sincethe intensity modulations spread in the entire Brillouinzone (Fig.S5,Fig.S6).

C. TIME-OF-FLIGHT NEUTRONSCATTERING

Time-of-flight (TOF) neutron scattering experimentswere carried out at 4SEASONS spectrometer (BL-01) atJ-PARC, Tokai, Japan[9, 10]. The incident energies Ei= 42 and 23 meV with ki in parallel to c-axis, chopperfrequency f = 250 Hz. Thus the energy transfer E ismostly in coupled with L for the quasi-2D lattice struc-ture. The total mass of co-aligned samples is about 4.3grams (about 400 pieces, see Fig. S1(a)). Figure S7 givesthe 2D slices of E vs. K for T = 5 K and 40 K measuredby TOF experiments with Ei = 42 meV, where both the| Q | −dependent background from phonon scatteringand constant background from incoherent scattering aresubtracted. The spin excitations significantly increasebelow Tc around E = 10 and 18 meV. By directly sub-tracting the raw data at 40 K in the normal state fromthe 5 K data, we get the 2D slice in E vs. K of the spinresonances at 5 K (superconducting state)(Fig. 2(b)).

In our TOF experiments, the scattering plane is[H, 0, 0]× [0, 0, L] under ki ‖ c, thus energy transfer E ismostly in coupled with L for the quasi-2D lattice struc-ture. To show the 2D constant-energy slices in [H,K]plane for the spin gap at 3 meV and three modes of spinresonance at 10, 13, 18 meV (Fig.S8), we integrated thesignal in a narrow energy window E ± ∆E as shown ineach panel, which is corresponding to a specific L indi-cated by the inserts of Fig.3. The two bright specks inFig.S8(a) are spurious from the contamination of (1, 1, 1)Bragg peaks accidentally hitting on the detector, where isactually no signal due to the full spin gap. From these 2Dcolor maps we can get the clear information of lineshapeof the spin gap and the spin resonances. Four ellipseselongated along radial direction are found for the para-magnetic excitations. The four-fold slices are shown inFig.3 with normalized intensity cutoff at the half maxi-mum. The elongated ellipses along H direction at E =10meV agree with the mismatch between γ band and δband.

D. SUPERCONDUCTING GAP AND SPINRESONANCE ENERGY

In the single-band system such as cuprates, the spinresonance is believed to be a collective mode of spin-1singlet-triplet excitations in the superconducting state,thus the spin resonance energy ER is below the pair-breaking energy 2∆ (twice the superconducting gap) dueto creation of particle-hole pairs [11]. However, the iron-based superconductors are multi-band systems, differentgaps (probably with different sign) are observed on dif-ferent pockets of Fermi surface. In the itinerant picture,the sign-reversed s−wave (s±) superconductivity estab-lishes via the Fermi surface nesting between the holepocket and electron pocket (Fig.S9(a)) [12–14]. Here,the pair-breaking energy is the sum of superconduct-ing gaps on the nesting pockets ∆tot = |∆k| + |∆k+Q|,or ∆tot = |∆h| + |∆p|, where Q is the nesting vector,∆k or ∆h is the gap on hole sheet, ∆k+Q or ∆p isthe gap on electron sheet. The spin resonance modeis an in-gap bound state determined by the coherencefactor [1 − ∆k∆k+Q/EkEk+Q]/2, where Ek (Ek+Q) isthe quasiparticle energy [15, 16]. At the Fermi level,Ek =

√ε2k + ∆2

k = |∆k|, the gap sign should be re-versed (∆k∆k+Q/|∆k||∆k+Q| = −1) to produce a fi-nite intensity of the resonance with ER < ∆tot. Al-ternatively, a non-resonance broad peak above 2∆ (here∆k = ∆k+Q = ∆) may also emerge under the con-ventional sign-preserved (s++) pairing picture [17, 18],due to the redistribution of the magnetic spectral weightwhen cooling down to the superconducting state. In someiron chalcogenide superconductors (e.g. KxFe2−ySe2,(Li1−xFex)OHFeSe, mono-layered FeSe thin film), thereare only electron pockets [19–25]. The spin resonancemode is then observed at the nesting wave vectorQ linkedby two electron pockets [26, 27], and a sign change ofthe gap between the inner and outer electron pocket isproposed from the results of quasi-particle interferencemeasurements (Fig.S9(b)) [28, 29]. In this case, the pair-breaking energy is still valid to represent by the totalsuperconducting gaps ∆tot = |∆k|+ |∆k+Q| summed onthe two nesting electron pockets. In both cases with s±pairing, the spin excitations form a sharp resonant peakat energy ER below ∆tot.

For our case in CaKFe4As4, ∆tot is the total super-conducting gaps summed on the nesting pair of holeband (α, β, γ) and electron band (δ), which is deter-mined by Angle-Resolved-Photoemission-Spectroscopy(ARPES) experiments [4]. We list them below and com-pare with the resonance energies in Tabel.S1. The ratioER/∆tot of three resonant modes is around 0.6, which isconsistent with the s± pairing mechanism. We also no-tice that the best nesting condition (β to δ) results in thelargest superconducting gaps and the highest resonanceenergy, and the elongated ellipses in the 2D slices of spinresonance in [H,K] plane at E = 10 meV agree with themismatch between γ band and δ band (Fig.3). Besidesthe conventional s± state with sign change between holeand electron pockets, a recent theory also predicts a C-

10

Table S1. Superconducitng gaps and resonance energies inCaKFe4As4.

hole ∆h ∆p(δ) ∆tot ER ER/∆tot

pocket (meV) (meV) (meV) (meV)

α 10.5 22.5 13 0.58

β 13 12 25 18.3 0.73

γ 8 20 9.5 0.48

state pairing symmetry with an additional sign changewithin hole or electron pockets in CaKFe4As4 [5]. Whenthe Coulomb repulsion U is weak, some interband inter-actions may change sign and become weakly attractive.Thus the C-state could induce a weak enhancement of thespin fluctuation below Tc around relatively high energyabove ∆tot and a near-nodal behavior of the quasiparticleexcitations at some electron pockets. However, it turnsnot exactly the case in our neutron experiments, since thehigh energy even mode of spin resonance has resolution-limited peak width and ER < ∆tot and no gap nodes areobserved in the spectroscopic investigations [4, 30–32].

E. L−SYMMETRY OF THE SPINRESONANCE

In the insulating bilayer cuprate YBa2Cu3O6 (YBCO),the acoustic magnons are defined as the low energy sec-tor of the spin excitation spectrum which evolves out ofin-phase precession modes of spins in directly adjacentlayers, while the optical magnons are the higher energysector evolving out of antiphase spin excitations [33]. Theformer have odd symmetry and the latter have even sym-metry under exchange of the two layers. In the metallicphase of YBa2Cu3O6+δ, such acoustic and optical spinwaves will develop into odd and even spin excitationswith two corresponding spin resonance modes below Tc,so called odd and even modes of spin resonance.

Except for some magnetically ordered iron chalco-genides, so far there is no evidence for optical magnonsin parent compounds of iron-based superconductors[34]. However, regardless the acoustic- or optical-likemagnons, we can still deduce the odd and even symme-tries from the non-degenerate inter-layer magnetic exci-tations within decoupled bilayer similar to YBCO [35].We assume the eigenstate of the Fe-As layer n as | n〉(n = 1, 2):

z | 1〉 = +d/2 | 1〉,

z | 2〉 = −d/2 | 2〉,(1)

where d = zc is the distance between two adjacent Fe-Felayers. We can then define symmetric and antisymmetric

combinations of the states centered on the two layers:| s〉 = (| 1〉+ | 2〉)/

√2,

| a〉 = (| 1〉− | 2〉)/√

2.

(2)

With momentum transfer Q along c−axis, the spin exci-tations characterized by the transitions between | s〉 and| a〉 states are given by the following superposed states:

〈s | eiQz | s〉 = 〈a | eiQz | a〉 = cos(Qd/2) (even),

〈s | eiQz | a〉 = 〈a | eiQz | s〉 = i sin(Qd/2) (odd).(3)

Here Qd/2 = 2πL/c × zc/2 = zπL . Thus the dynamicspin susceptibility of odd and even excitations can bedescribed by:

χ′′odd(Q,ω) ∼ sin2(zπL),

χ′′even(Q,ω) ∼ cos2(zπL).

(4)

The spin-spin correlation function S(Q,ω) is re-lated to the local susceptibility: S(Q,ω) = (1 +n(ω))χ′′(Q,ω)/πg2µ2

B , where 1 + n(ω) is the Bose fac-tor. The cross section d2σ/(dΩdE) should be furthernormalized by the square of Fe2+ magnetic form factor| F (Q) |2 and Debye-Waller exponent exp(−2W (Q)).

F. COMPARISON WITH OTHER PNICTIDES

The presence of both even and odd modes of spin res-onance are seen specifically in CaKFe4As4 for the follow-ing reasons. Firstly, this compound has a stoichiomet-ric superconductivity without any magnetic order or anydisorder from the dopants which may induce complexityin the pairing process. Secondly, the Fermi surfaces arenearly 2D with a range of diameters, resulting in multiplenesting conditions. Thirdly, the superconducting gapsare divergent from each Fermi sheet, therefore differentmodes of the spin resonance from the multiple nestingcan be separately identified at different energies. Finally,the inequivalent position of Ca and K atoms leads to abilayer symmetry similar to YBCO. The split betweenodd and even resonant modes (about 5.3 meV) are de-termined by the strong intra-bilayer coupling (SJc = 5.5meV in CaFe2As2).

In other iron-based superconducting systems, the spinresonance peak is thus likely including several overlappedodd and even modes due to small SJc [34]. Since themaximum intensity of spin resonant mode (∆S) de-creases with increasing resonance energy (ER) (Fig.4(b)),if neighboring odd and even modes are overlapped witheach other due to small interlayer coupling, an asymmet-ric broad peak with a long tail at high energy part then isexpected, instead of several individual peaks. In fact, this

11

feature has been already observed in many iron-based su-perconductors [34, 36, 37]. Here we take three systems forexample: BaFe1.925Ni0.075As2, Ba(Fe1−xRux)2As2 andBaFe2(As1−xPx)2 (Fig.S10, Fig.S11 and Fig.S12) [38–40]. In the underdoped BaFe1.925Ni0.075As2 with stripe-type magnetic order, a clear odd L−modulation of thespin excitations is observed. The spin resonance peak isindeed asymmetric with a long tail at high energy part,and the peak center shifts to lower energy when increas-ing L from 0 to 1 [38]. Similar behaviors exist in theBa(Fe1−xRux)2As2 system, no matter in the magneti-cally ordered state (underdoped regime) or paramagneticstate (optimally doped level) [39]. If we suppose the evenmodes are hidden in the high energy part above the peakcenter with weak intensity, when L increases from 0 to1, the intensity at high energy part (even modulation)decreases, while the intensity at low energy part (oddmodulation) increases. Then it looks like the resonancepeak center shifts to lower energy, resulting in a disper-sion of the resonance mode. By digging out the data ofBaFe2(As1−xPx)2 in Fig.S12 (a), we indeed find oppo-site L dependence of resonance intensity at 9 meV and12 meV. To clarify this issue, further neutron scatteringexperiments studied on the spin resonance over a largerange of L need to be done, the polarized neutron analy-sis may also help to identify the overlapped modes withdifferent orbital character.

∗ Electronic address: hqluo@iphy.ac.cn[1] W. R. Meier, T. Kong, U. S. Kaluarachchi, V. Taufour,

N. H. Jo, G. Drachuck, A. E. Bohmer, S. M. Saunders,A. Sapkota, A. Kreyssig, M. A. Tanatar, R. Prozorov, A.I. Goldman, F. F. Balakirev, A. Gurevich, S. L. Bud’ko,and P. C. Canfield, Phys. Rev. B 94, 064501 (2016).

[2] W. R. Meier, T. Kong, S. L. Budko, and P. C. Canfield,Phys. Rev. Materials 1, 013401 (2017).

[3] A. Iyo, K. Kawashima, T. Kinjo, T. Nishio, S. Ishida, H.Fujihisa, Y. Gotoh, K. Kihou, H. Eisaki, and Y. Yoshida,J. Am. Chem. Soc. 138, 3410 (2016).

[4] D. Mou, T. Kong, W. R. Meier, F. Lochner, L. Wang,Q. Lin, Y. Wu, S. L. Budko, I. Eremin, D. D. Johnson,P. C. Canfield, and A. Kaminski, Phys. Rev. Lett. 117,277001 (2016).

[5] F. Lochner, F. Ahn, T. Hickel, and Ilya Eremin, Phys.Rev. B 96, 094521 (2017).

[6] W. R. Meier, Q.-P. Ding, A. Kreyssig, S. L. Budko, A.Sapkota, K. Kothapalli, V. Borisov, R. Valent, C. D.Batista, P. P. Orth, R. M. Fernandes, A. I. Goldman, Y.Furukawa, A. E. Bohmer, and P. C. Canfield, npj Quan.Mat. 3, 5 (2018).

[7] S. Pailhes, Y. Sidis, P. Bourges, C. Ulrich, V. Hinkov, L.-P. Regnault, A. Ivanov, B. Liang, C. Lin, C. Bernhard,and B. Keimer, Phys. Rev. Lett. 91, 237002 (2003).

[8] S. Pailhes, Y. Sidis, P. Bourges, V. Hinkov, A. Ivanov, C.Ulrich, L.-P. Regnault, and B. Keimer, Phys. Rev. Lett.93, 167001 (2004).

[9] R. Kajimoto, M. Nakamura, Y. Inamura, F. Mizuno, K.

Nakajima, S. Ohira-Kawamura, T. Yokoo, T. Nakatani,R. Maruyama, K. Soyama, K. Shibata, K. Suzuya, S.Sato, K. Aizawa, M. Arai, S. Wakimoto, M. Ishikado, S.Shamoto, M. Fujita, H. Hiraka, K. Ohoyama, K. Yamada,and C.-H. Lee, J. Phys. Soc. Jpn. 80, SB025 (2011).

[10] Y. Inamura, T. Nakatani, J. Suzuki, and T. Otomo, J.Phys. Soc. Jpn. 82, SA031 (2013).

[11] G. Yu, Y. Li, E. M. Motoyama and M. Greven, Nat.Phys. 5, 873 (2009).

[12] M. M. Korshunov and I. Eremin, Phys. Rev. B 78,140509(R) (2008).

[13] A. V. Chubukov, D. V. Efremov, and I. Eremin, Phys.Rev. B 78, 134512 (2008).

[14] I. Mazin and J. Schmalian, Physica C 469, 614 (2009).[15] T. A. Maier, S. Graser, D. J. Scalapino, and P. Hirschfeld,

Phys. Rev. B 79, 134520 (2009).[16] Q. Wang, J. T. Park, Y. Feng, Y. Shen, Y. Hao, B. Pan,

J. W. Lynn, A. Ivanov, S. Chi, M. Matsuda, H. Cao, R.J. Birgeneau, D. V. Efremov, and J. Zhao, Phys. Rev.Lett. 116, 197004 (2016).

[17] S. Onari, H. Kontani, and M. Sato, Phys. Rev. B 81,060504(R) (2010).

[18] S. Onari and H. Kontani, Phys. Rev. Lett. 109, 137001(2012).

[19] Q. Fan, W. H. Zhang, X. Liu, Y. J. Yan, M. Q. Ren, R.Peng, H. C. Xu, B. P. Xie, J. P. Hu, T. Zhang, and D.L. Feng,Nat. Phys. 11,946 (2015).

[20] D. Liu, W. Zhang, D. Mou, J. He, Y.-B. Ou, Q.-Y. Wang,Z. Li, L. Wang, L. Zhao, S. He, Y. Peng, X. Liu, C. Chen,L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen, Z. Xu, J. Hu,X. Chen, X. Ma, Q. Xue, and X.J. Zhou, Nat. Commun.3, 931(2012).

[21] S. He, J. He, W. Zhang, L. Zhao, D. Liu, X. Liu, D. Mou,Y.-B. Ou, Q.-Y. Wang, Z. Li, L. Wang, Y. Peng, Y. Liu,C. Chen, L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen,Z. Xu, X. Chen, X. Ma, Q. Xue, and X. J. Zhou, Nat.Mater. 12, 605(2013).

[22] S. Tan, Y. Zhang, M. Xia, Z. Ye, F. Chen, X. Xie, R.Peng, D. Xu, Q. Fan, H. Xu, J. Jiang, T. Zhang, X. Lai,T. Xiang, J. Hu, B. Xie, and Donglai Feng, Nat. Mater.12, 634(2013).

[23] J. He, X. Liu, W. Zhang, L. Zhao, D. Liu, S. He, D. Mou,F. Li, C. Tang, Z. Li, L. Wang, Y. Peng, Y. Liu, C. Chen,L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen, Z. Xu, X.Chen, X. Ma, Q. Xue, and X. J. Zhou, Proc. Natl. Acad.Sci. USA 111, 18501 (2014).

[24] L. Zhao, A. Liang, D. Yuan, Y. Hu, D. Liu, J. Huang,S. He, B. Shen, Y. Xu, X. Liu, L. Yu, G. Liu, H. Zhou,Y. Huang, X. Dong, F. Zhou, K. Liu, Z. Lu, Z. Zhao, C.Chen, Z. Xu, and X. J. Zhou, Nat. Commun. 7, 10608(2016).

[25] X. Liu, D. Liu, W. Zhang, J. He, L. Zhao, S. He, D. Mou,F. Li, C. Tang, Z. Li, L. Wang, Y. Peng, Y. Liu, C. Chen,L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen, Z. Xu, X.Chen, X. Ma, Q. Xue, and X. J. Zhou, Nat. Commun. 5,5047(2014).

[26] J. T. Park, G. Friemel, Y. Li, J. -H. Kim, V. Tsurkan,J. Deisenhofer, H. -A. Krug von Nidda, A. Loidl, A.Ivanov, B. Keimer, and D. S. Inosov, Phys. Rev. Lett.107, 177005 (2011).

[27] G. Friemel, W. P. Liu, E. A. Goremychkin, Y. Liu, J.T. Park, O. Sobolev, C. T. Lin, B. Keimer, and D. S.Inosov, Europhys. Lett. 99, 67004 (2012).

[28] Z. Du, X. Yang, H. Lin, D. Fang, G. Du, J. Xing, H. Yang,

12

X. Zhu, and H.-H. Wen, Nat. Commun. 7, 10565(2016).[29] Z. Du, X. Yang, D. Altenfeld, Q. Gu, H. Yang, I. Eremin,

P. J. Hirschfeld, I. I. Mazin, H. Lin, X. Zhu and H. -H.Wen, Nat. Phys. 14, 134 (2018).

[30] R. Yang, Y. Dai, B. Xu, W. Zhang, Z. Qiu, Q. Sui, C. C.Homes, and X. Qiu, Phys. Rev. B 95, 064506 (2017).

[31] P. K. Biswas, A. Iyo, Y. Yoshida, H. Eisaki, K.Kawashima, A. D. Hillier, Phys. Rev. B 95, 140505(R)(2017).

[32] K. Cho, A. Fente, S. Teknowijoyo, M. A. Tanatar, K. R.Joshi, N. M. Nusran, T. Kong, W. R. Meier, U. Kalu-arachchi, I. Guillamon, H. Suderow, S. L. Bud’ko, P. C.Canfield, and R. Prozorov, Phys. Rev. B 95, 100502(R)(2017).

[33] D. Reznik, P. Bourges, H. F. Fong, L. P. Regnault, J.Bossy, C. Vettier, D. L. Milius, I. A. Aksay, and B.Keimer, Phys. Rev. B 53, R14741(R)(1996).

[34] P. Dai, Rev. Mod. Phys. 87, 855 (2015).[35] H. F. Fong, P. Bourges, Y. Sidis, L. P. Regnault, J. Bossy,

A. Ivanov, D. L. Milius, I. A. Aksay, and B. Keimer,Phys. Rev. B 61, 14773 (2000).

[36] D. S. Inosov, C. R. Physique 17, 60 (2016).[37] P. D. Johnson, G. Xu, W. -G. Yin Iron-Based Supercon-

ductivity, Springer, P165-P169 (2015).[38] M. Wang, H. Luo, J. Zhao, C. Zhang, M. Wang, K.

Marty, S. Chi, J. W. Lynn, A. Schneidewind, S. Li,andP. Dai, Phys. Rev. B 81, 174524 (2010).

[39] J. Zhao, C. R. Rotundu, K. Marty, M. Matsuda, Y. Zhao,C. Setty, E. Bourret-Courchesne, J. Hu, and R. J. Birge-neau, Phys. Rev. Lett. 110, 147003 (2013).

[40] C. H. Lee, P. Steffens, N. Qureshi, M. Nakajima, K. Ki-hou, A. Iyo, H. Eisaki, and M. Braden, Phys. Rev. Lett.111, 167002 (2013).

13

(a)

[110]

[100]

(b) (c)

(d)

(e)

K

As(1)

As(2)

Fe

Ca

FIG. S1: (a) Photos of the co-aligned CaKFe4As4 crystals for neutron scattering experiments. (b) X-ray diffraction patternof CaKFe4As4 single crystal at room temperature, the inset is a Laue photo of CaKFe4As4 single crystal, the high symmetrydirections [100] and [110] are indicated by green arrows. (c, d) Q and temperature dependence of the elastic neutron scatteringat Q = (1, 0, 3) . (e) The crystalline blocks with FeAs bilayers.

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0- 1 . 0

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0 . 0

0 . 0 0

0 . 0 2

0 . 0 4

0 . 0 6

0 . 0 8 3 5 . 0 3 5 . 2 3 5 . 4 3 5 . 6 3 5 . 8

ρ / ρ 30

0 K

T ( K )

# 1 # 2 # 3 # 4 # 5 # 6 # 7 # 8 # 9 # 1 0 # 1 1 # 1 2 # 1 3 # 1 4 # 1 5 # 1 6 # 1 7 # 1 8

( a ) ( b )

4πχ

T ( K )

# 1 F C # 1 Z F C

# 2 Z F C # 3 Z F C # 4 Z F C

H = 1 0 O eH / / a b p l a n eT ( K )

ρ / ρ 30

0 K

FIG. S2: Superconductivity of CaKFe4As4 single crystals: (a) Temperature dependence of the resistivity, all the data isnormalized by the resistivity at 300 K; (b) Temperature dependence of DC magnetic susceptibility.

14

FIG. S3: Energy scans for scattering below and above Tc, and their differences at Q = (1, 0, 2), (1, 0, 2.5), (1, 0, 2.8), (1, 0,3). The solid lines are guides to the eyes. The shadow area are gaussian fits for the intensity gain of each resonant modes andthe intensity loss of the spin gap. The dashed lines indicate three resonant modes around E = 9.5 meV, 13 meV, 18.3 meV,respectively.

15

0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6

4 0

8 0

1 2 0

1 6 0

2 0 0

0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6

0

4 0

8 0

1 2 0

1 6 0

1 0 0

2 0 0

3 0 0

4 0 0

0

1 0 0

2 0 0

3 0 0

5 0

1 0 0

1 5 0

2 0 0

0

2 0

4 0

6 0

0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 61 0 01 5 02 0 02 5 03 0 03 5 0

0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6

0

4 0

8 0

1 2 0

3 m e V

H ( r . l . u . )

1 . 5 K4 0 K

Coun

ts / 1

0 mins L = 3

( a ) ( b )4 0 K - 1 . 5 K

Inten

sity (a

.u.)

H ( r . l . u . )

L = 33 m e V

( c )

1 . 5 K 4 0 K

Coun

ts / 1

0 mins

1 0 m e VL = 3

( d )

Inten

sity (a

.u.)

1 . 5 K - 4 0 K 1 0 m e VL = 3

( e ) 1 . 5 K 4 0 K

Coun

ts / 1

0 mins 1 3 m e V

L = 2 . 5

( f )

Inten

sity (a

.u.)

1 . 5 K - 4 0 KL = 2 . 51 3 m e V

( g ) 1 . 5 K 4 0 K

Coun

ts / 1

0 mins

H ( r . l . u . )

L = 21 8 m e V

( h )1 . 5 K - 4 0 K

Inten

sity (a

.u.)

H ( r . l . u . )

1 8 m e VL = 2

FIG. S4: Constant-energy scans along H direction for the spin gap at E = 3 meV and three resonant modes at E = 10, 13, 18meV with L = 2 ∼ 3. The red solid lines are Gaussian fits.

16

0 1 2 3 4 5 6

1 0 0

2 0 0

3 0 0

4 0 00 1 2 3 4 5 6

0

5 0

1 0 0

1 5 0

2 0 0

1 0 02 0 03 0 04 0 05 0 06 0 0

05 01 0 01 5 02 0 02 5 03 0 03 5 0

1 0 02 0 03 0 04 0 05 0 06 0 0

05 01 0 01 5 02 0 0

0 1 2 3 4 5 6

2 0 04 0 06 0 08 0 0

1 0 0 01 2 0 0

0 1 2 3 4 5 6

0

5 0

1 0 0

( a ) 3 m e V

[ 1 , 0 , L ] ( r . l . u . )

1 . 5 K 4 0 K

Coun

ts / 10

mins

( b )

[ 1 , 0 , L ] ( r . l . u . )

Inten

sity (a

.u.)

4 0 K - 1 . 5 K3 m e V

( c )

1 . 5 K 4 0 K

Coun

ts / 10

mins

1 0 m e V ( d )

Inten

sity (a

.u.)

1 . 5 K - 4 0 K1 0 m e V

( e )

1 . 5 K 4 0 K

Coun

ts / 10

mins

1 3 m e V ( f )

Inten

sity (a

.u.)

1 . 5 K - 4 0 K1 3 m e V

( g )

1 . 5 K 4 0 K

Coun

ts / 10

mins

[ 1 , 0 , L ] ( r . l . u . )

1 8 m e V ( h )

Inten

sity (a

.u.)

1 . 5 K - 4 0 K1 8 m e V

[ 1 , 0 , L ] ( r . l . u . )FIG. S5: Constant-energy scans along L direction for the spin gap at E = 3 meV and three resonant modes at E = 10, 13, 18meV. The solid lines in (b, d, f) are fitting results by | F (Q) |2 sin2(zπL) (odd mode), while the red line in (h) is fitting resultby | F (Q) |2 cos2(zπL) (even mode).

17

FIG. S6: (a) and (b)Intensity differences of the scans along L direction between 1.5 K and 40 K with E = 7 ∼ 21 meV . Thesolid lines are fitting results by | F (Q) |2 sin2(zπL) for 8 ∼ 16 meV , and | F (Q) |2 cos2(zπL) for 17 ∼ 21 meV, respectively.Each curve is shifted upward for clarity with the horizontal dashed lines indicating the zero intensity. The vertical dashed linesindicate the non-integral L positions [L = 1.08, 3.24, 5.39 for (a), L = 2.16, 4.31 for (b) ] . (c) and (d) 2D mapping of the spinresonance intensity corresponding to (a) and (b).

- 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 85

1 0

1 5

2 0

- 0 . 8 - 0 . 4 0 . 0 0 . 4 0 . 85

1 0

1 5

2 0

5 K

0 . 1 5

E (me

V)

K ( r . l . u . )0

( a )E i = 4 2 m e V E i = 4 2 m e V

4 0 K

( b )

E (me

V)

K ( r . l . u . )

Inten

sity (a

.u.)

FIG. S7: 2D slices of E versus K at T = 5 K and 40 K measured by TOF experiments.

18

FIG. S8: 2D constant-energy slices for the spin gap at E = 3 meV and three resonance modes at E = 10, 13, 18 meV, for T =5 K and 40 K, and their differences.

19

M

Q

k

k+Q

M

M M

M

Q

k k+Q

M

MM

(a) (b)

FIG. S9: Two cases of Fermi surface nesting in iron pnictides and iron chalcogenides. The sign-reversed superconducting gapsare marked by different colors.

2 0 0

4 0 0

6 0 0

8 0 0

- 4 - 2 0 2 42 0 0

4 0 0

6 0 0

8 0 0

- 2 0 0

- 1 0 0

0

1 0 0

2 0 0

0 4 8 1 2 1 6- 2 0 0

- 1 0 0

0

1 0 0

E = 5 . 5 m e V

2 K 2 0 K

Coun

ts / 5

minu

tes

( a ) B a F e 1 . 9 2 5 N i 0 . 0 7 5 A s 2

( b )

2 K 2 0 K

Coun

ts / 5

minu

tes

[ 1 , 0 , L ] ( r . l . u . )

E = 7 m e V

( c )

Coun

ts / 1

0 minu

tes

Q = ( 1 , 0 , 1 )

( d ) Q = ( 1 , 0 , 0 )

Coun

ts / 1

0 minu

tes

E n e r g y ( m e V )

FIG. S10: (a)(b) Strong L−modulation of spin excitations in underdoped BaFe1.925Ni0.075As2; (c)(d) Broadening of spinresonance peak and L−dependent resonance energy [38].

20

Energy (meV) Energy (meV)

-50

-50

50

50

0

0

-50

-50

50

50

0

0

20 4 6 8 10 12 2 4 6 8 10 12

2 K - 25 K 2 K - 25 K

2 K - 25 K 2 K - 25 K

Q = (1, 0, 1) Q = (1, 0, 1)

Q = (1, 0, 0) Q = (1, 0, 0)

(a)

(b)

(c)

(d)

FIG. S11: Strong L−dependent spin resonance in Ba(Fe1−xRux)2As2 system with x = 0.25 (underdoped) and x = 0.35(optimally doped) [39].

FIG. S12: (a)L−dispersion of spin resonance peak in optimally doped BaFe2(As1−xPx)2. (b)(c) Opposite L−dependence ofspin resonance intensity at 9 meV and 12 meV [40].