Robust upward dispersion of the neutron spin resonance in the heavy fermion superconductor Ce 1 x Yb x CoIn 5

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1 ARTICLE Reeived Fe 26 Aepted 27 Jul 26 Pulished 28 Sep 26 DOI:.38/nomms277 OPEN Roust upwrd dispersion of the neutron spin resonne in the hevy fermion superondutor Ce x Y x CoIn 5 Yu Song, John Vn Dyke 2, I.K. Lum 3,,5, B.D. White,5, Sooyoung Jng 3,,5, Duygu Yzii 3,,5, L. Shu 6,7, A. Shneidewind 8, Petr Čermák 8, Y. Qiu 9, M.B. Mple 3,,5, Dirk K. Morr 2 & Pengheng Di The neutron spin resonne is olletive mgneti exittion tht ppers in the unonventionl opper oxide, iron pnitide nd hevy fermion superondutors. Although the resonne is ommonly ssoited with spin-exiton due to the d(s ± )-wve symmetry of the superonduting order prmeter, it hs lso een proposed to e mgnon-like exittion ppering in the superonduting stte. Here we use inelsti neutron sttering to demonstrte tht the resonne in the hevy fermion superondutor Ce x Y x CoIn 5 with x ¼,.5 nd.3 hs ring-like upwrd dispersion tht is roust ginst Y-doping. By ompring our experimentl dt with rndom phse pproximtion lultion using the eletroni struture nd the momentum dependene of the d x 2 y 2-wve superonduting gp determined from snning tunnelling mirosopy (STM) for CeCoIn 5, we onlude tht the roust upwrd-dispersing resonne mode in Ce x Y x CoIn 5 is inonsistent with the downwrd dispersion predited within the spin-exiton senrio. Deprtment of Physis nd Astronomy, Rie University, Houston, Texs 775, USA. 2 Deprtment of Physis, University of Illinois t Chigo, Chigo, Illinois 667, USA. 3 Mterils Siene nd Engineering Progrm, University of Cliforni, Sn Diego, L Joll, Cliforni 9293, USA. Deprtment of Physis, University of Cliforni, Sn Diego, L Joll, Cliforni 9293, USA. 5 Center for Advned Nnosiene, University of Cliforni, Sn Diego, L Joll, Cliforni 9293, USA. 6 Stte Key Lortory of Surfe Physis, Deprtment of Physis, Fudn University, Shnghi 233, Chin. 7 Collortive Innovtion Center of Advned Mirostrutures, Nnjing 293, Chin. 8 Jülih Center for Neutron Siene JCNS, Forshungszentrum Jülih GmH, Outsttion t MLZ, D-8577 Grhing, Germny. 9 NIST Center for Neutron Reserh, Ntionl Institute of Stndrd nd Tehnology, Githersurg, Mrylnd 2899, USA. Correspondene nd requests for mterils should e ddressed to D.K.M. (emil: dkmorr@ui.edu) or to P.D. (emil: pdi@rie.edu). NATURE COMMUNICATIONS 7:277 DOI:.38/nomms277

2 ARTICLE NATURE COMMUNICATIONS DOI:.38/nomms277 Understnding the origin of unonventionl superondutivity in strongly orrelted eletron mterils ontinues to e t the forefront of modern ondensed mtter physis 5. In opper oxide 6 8, iron pnitide 9, nd hevy fermion,2 superondutors, the pperne of neutron spin resonne elow the superonduting trnsition temperture T suggests tht spin-flutution-medited piring is ommon thred for different fmilies of unonventionl superondutors 2. The neutron spin resonne is olletive mgneti exittion oupled to superondutivity with temperture dependene similr to the superonduting order prmeter 6,7. It is loted ner the ntiferromgneti (AF) ordering wve vetor Q AF of the undoped prent ompound nd its energy E r t Q AF is relted to either T (ref. 3) or the superonduting energy gp D (ref. ). Although it is generlly epted tht the resonne is signture of unonventionl superondutivity 2, there is no onsensus on its mirosopi origin. A ommon interprettion of the resonne is tht it is spin-exiton, rising from prtile-hole exittions involving momentum sttes ner the Fermi surfes tht possess opposite signs of the d (or s ± )-wve superonduting order prmeter 7,2,5. Alterntively, it hs lso een proposed to e mgnon-like exittion 6,7. At present, there is no onsensus on its mirosopi origin 2,7,8,. In hole-doped opper oxide superondutors, the mgneti exittions hve n hourglss dispersion with downwrd dispersion t energies elow E r nd n upwrd mgnon-like dispersion t energies ove E r (ref. 8). The resonne, on the other hnd, otined y sutrting the norml-stte mgneti exittions from those in the superonduting stte, displys predominntly downwrd dispersion 8 2. In the se of Ni-underdoped BFe 2 As 2 with oexisting AF order nd superondutivity 22, the resonne only revels n upwrd mgnon-like dispersion 23. In oth ses, the resonne is well desried y the spin-exiton senrio, the opposite dispersions eing result of d x 2 y 2 or s± symmetry of the superonduting order prmeter 23,2. For the hevy fermion superondutor CeCoIn 5 (T ¼ 2.3 K) (ref. ), the resonne ppers elow T t E r ¼.6±.3 mev nd the ommensurte AF wve vetor Q AF ¼ (/2, /2, /2) in reiprol spe 2. Sine CeCoIn 5 hs superonduting gp with d x2 y2-wve symmetry s determined from snning tunnelling mirosopy (STM) experiments 25,26, the resonne is expeted to show downwrd dispersion similr to the uprtes within the spin-exiton piture 27,28. Alterntively, the resonne, with its three-dimensionl hrter 2, ould e mgnon-like exittion of f eletrons tht eomes visile due to its redued dey rte in the superonduting stte 6,7. In this se, the resonne is not signture of d x 2 y2-wve superondutivity, ut mesure of the hyridiztion etween f eletrons nd ondution eletrons nd its ssoited piring-sensitive Lndu dmping 7. When L is sustituted for Ce in Ce x L x CoIn 5 (refs 29,3), superondutivity nd the energy of the resonne re oth rpidly suppressed, while E r /k B T remins pproximtely onstnt, where k B is the Boltzmnn onstnt. At the sme time, the energy width of the resonne rodens with inresing L-doping 3,32. When Y is doped into CeCoIn 5 to form Ce x Y x CoIn 5, superondutivity is suppressed muh slower 33.Withinresing Y, de Hs-vn Alphen nd ngle-resolved photo-emission spetrosopy studies find hnge in the Fermi-surfe topology etween Y nominl doping levels of x ¼. nd.2 (refs 3,35). In ddition, London penetrtion depth mesurements suggest tht the superonduting gp hnges from nodl to nodeless round similr Y-doping level 36, rising possily from omposite eletron piring in fully gpped superondutor for x.2 (ref. 37). If the resonne in CeCoIn 5 is spin-exiton, it should e drmtilly ffeted y the Y-doping-indued hnges in Fermi surfe topology nd superonduting gp. On the other hnd, if the resonne is mgnon-like exittion, it should e muh less sensitive to Y-doping ross x ¼.2 nd disply upwrd dispersion similr to spin wves in ntiferromgnetilly ordered nonsuperonduting CeRhIn 5 hrteristi of roust effetive nerest-neighour exhnge oupling, regrdless of its itinernt eletron or lol moment origin 7,38,39. Here we use inelsti neutron sttering to demonstrte tht the resonne in the hevy fermion superondutor Ce x Y x CoIn 5 with x ¼,.5 nd.3, nd T E2.3, 2.25 nd.5 K, respetively (Methods setion nd Supplementry Fig. ),2,33, hs dominnt ring-like upwrd dispersion tht is roust ginst Y-doping nd the onomitnt hnges in eletroni struture, feture not present in the spin-exiton senrio. Moreover, downwrd dispersion expeted in the spin-exiton senrio is not oserved. The roust upwrd dispersion of the resonne suggests tht it my hve mgnon-like ontriution 7. Speifilly, we find tht the resonne in Ce.95 Y.5 CoIn 5 displys n upwrd dispersion long [H, H,.5], [.5, K,.5] nd [.5,.5, L] s shown in Fig. d f, respetively. Upon inresing Y-doping to x ¼.3, the energy of the resonne t Q AF dereses orresponding to the redution in T (Supplementry Fig. 2), ut the overll dispersion nd lotion of the mode in reiprol spe remin unhnged. Upwrd dispersions similr to Ce.95 Y.5 CoIn 5 re lso found in undoped CeCoIn 5 nd Ce.7 Y.3 CoIn 5 (Supplementry Figs 3 5). Using the eletroni struture nd the momentum dependene of the d x 2 y2-wve superonduting gp determined from STM for CeCoIn 5 (Fig. g) 28, we lulte the feedk of superondutivity on the mgneti exittions within the spin-exiton senrio (Supplementry Note, Supplementry Figs 6 8). The resulting wve vetor dependene of the spin-exiton long [.5, K] nd [H, H], whih re shown in Fig. h,i, respetively, re inonsistent with the experimentlly determined upwrd dispersion (solid lines). Similr dispersive resonnes in CeCoIn 5 nd Ce.7 Y.3 CoIn 5 (Fig. 3, Supplementry Figs 3 nd nd Fig. 5) re seen in spite of possile hnges in the Fermi surfe nd superonduting gp symmetry on moving from x ¼ to.3 (refs 3 36), lso inonsistent with the expettion tht spin-exiton should depend sensitively on the Fermi surfe. We thus onlude tht the upwrd-dispersing resonne mode in Ce.95 Y.5 CoIn 5 nnot e interpreted s spin-exiton rising from the feedk of unonventionl d-wve superondutivity 2,27,28. On the other hnd, the similrity of the resonne s dispersion long the [H, H,.5] diretion with the spin-wve dispersion in AF-ordered nonsuperonduting CeRhIn 5 long the sme diretion 38,39 (Fig. j) suggests tht the upwrd-dispersing resonne my e mgnon-like. In this senrio, the mgneti resonne rises sine the opening of the superonduting gp leds to strong suppression of Lndu dmping for preexisting mgnon-like exittions, s shown in Fig. k,l (Supplementry Note 2 nd Supplementry Figs 9 ). This is, therefore, the first experimentl oservtion of mgneti resonne in n unonventionl superondutor tht nnot e interpreted s spin-exiton. Results Dispersion of the resonne in Ce.95 Y.5 CoIn 5 long [H, H,.5] nd [.5,.5, L]. Using tetrgonl unit ell with ¼ ¼.6 Å nd ¼ 7.5 Å for Ce.95 Y.5 CoIn 5 (Fig. ), we define the momentum trnsfer Q in three-dimensionl reiprol spe in Å s Q ¼ H þ K þ L, where H, K nd L re Miller indies nd ¼^2p=, ¼^2p= nd ¼^2p=. The experiments re rried out using the [H, H, L] nd [H, K, H] 2 NATURE COMMUNICATIONS 7:277 DOI:.38/nomms277

3 NATURE COMMUNICATIONS DOI:.38/nomms277 ARTICLE Ce/Y L HH In q = Q Q AF Co.5 Q q.5 Q AF.5 HH.5 K [H, H,.5] (r.l.u.) [.5, K,.5] (r.l.u.) [.5,.5, L] (r.l.u.) d e f K MACS K PANDA K PANDA K MACS q (Å ) q (Å ) q (Å ).5 g α h Min i Mx K (r.l.u.) Q AF β α H (r.l.u.).5 k Fit to exp. dt Spin exiton l.5.5 CeRhIn 5, ref. (38) CeRhIn 5, ref. (39) j Mgnon [H, H] (r.l.u.) [.5, K ] (r.l.u.) [H, H ] (r.l.u.).5 Figure Summry of neutron sttering results on Ce.95 Y.5 CoIn 5. () Crystl struture of Ce x Y x CoIn 5.() [H, H, L] sttering plne, where q is mesured from Q AF vi q ¼ Q Q AF. The red nd green rrows represent sns long [.5,.5, L] nd [H, H,.5] entred t Q AF, respetively. () [H, K, H] sttering plne. Here sns long [.5, K,.5] entred t Q AF n e rried out s indited y the lue rrow. (d) Dispersion of the resonne long qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [H, H,.5]. The xis ove the figure is Q in r.l.u., wheres the xis t the ottom is q in Å. An isotropi dispersion E¼ D 2 þ ðjqjþ 2 (D ¼.55() mev, ¼ 3.2() mev Å) is shown s yn solid line, where D represents spin gp nd is the effetive spin wve veloity. The horizontl rs represent experimentlly oserved pek full-width-t-hlf-mximum. The dshed vertil lines indite the ordering wve vetor of the so-lled Q phse t Q ¼ Q AF ±(d, d, ) with d ¼.5 (ref. ). (e,f) re similr to (d), ut re for dispersions long [.5, K,.5] nd [.5,.5, L], respetively. (g) The Fermi surfes of CeCoIn 5, where the lue nd red shding represent the d-wve symmetry of the superondutivity order prmeter. The lk rrow indites Q AF, whih onnets prts of Fermi surfes with sign-reversed superondutivity-order prmeters. (h) Colour-oded lulted intensity long the [.5, K] diretion y onsidering the resonne mode to e spin-exiton. (i) Clulted intensity for the spin-exiton long the [H, H] diretion. (j) Comprison of dispersions of the resonne in Ce.95 Y.5 CoIn 5 (solid yn line) nd spin wves in CeRhIn 5 (dshed purple nd ornge lines) 38,39. (k) Clulted intensity of the resonne long the [.5, K] diretion ssuming it is mgnon-like exittion. Dispersion of the mgnon-like exittions is otined from fits to experimentl dt nd the intensity is ffeted y dmping due to the prtile hole ontinuum. (l) Clulted intensity for the mgnon-like exittion long the [H, H] diretion. NATURE COMMUNICATIONS 7:277 DOI:.38/nomms

4 ARTICLE NATURE COMMUNICATIONS DOI:.38/nomms277 sttering plnes to study the dispersions of the resonne long [H, H,.5], [.5, K,.5] nd [.5,.5, L] (Fig.,). Figure 2 shows the olour-oded plot of the spin exittions t.6 K otined from fits to the rw dt t energies E ¼.3,.55,.7,.85 nd mev long [H, H,.5] for Ce.95 Y.5 CoIn 5 (Fig. 2). Although the dt show wek ommensurte pek t E ¼.3 mev, we see ler ommensurte resonne t E r E.55 mev nd upwrd-dispersing inommensurte peks for energies E ¼.7,.85 nd mev. Figure 2 shows onstntenergy sns t E ¼.7 mev elow nd ove T.AtT ¼ 2.3 K, we see rod pek entred t the ommensurte AF wve vetor Q AF. Upon ooling to elow T t T ¼.6 K, the ommensurte pek eomes two inommensurte peks, whih disperse outwrd with inresing energy (Fig. 2). Figure 2d shows onstnt- Q sns t Q AF for tempertures T ¼.6,.5 nd 2.3 K. Similr to previous work on pure CeCoIn 5 (ref. 2), the dt revel ler resonne t E r E.55 mev elow T, nd no pek in the norml stte ove T. To further illustrte the dispersive nture of the resonne, we show in Fig. 3 mps of sttering intensities in the [H, H, L] sttering plne of the spin exittions t different energies ove nd elow T otined on the multi-xis rystl spetrometer (MACS) for Ce.95 Y.5 CoIn 5. In the proed reiprol spe, we see ler spin exittions round Q AF, whih disperse outwrd with inresing energy. At n energy (E ¼. mev) elow the resonne, spin exittions re ommensurte elow (Fig. 3) nd ove (Fig. 3) T. The onstnt-energy uts of the dt long the [H, H,.5] diretion onfirm this onlusion (Fig. 3). Figure 3d f shows similr sns t E ¼.55 mev nd indite tht the sttering eomes roder in reiprol spe. Upon moving to E ¼.7 mev (Fig. 3g i),. mev (Fig. 3j l) nd.2 mev (Fig. 3m o), we see ler ring-like sttering dispersing wy from Q AF with inresing energy in the superonduting stte. The norml-stte sttering is ommensurte t ll energies, nd this is most lerly seen in the onstnt-energy uts long the [H, H,.5] diretion. Bsed on the differene of dt t 2. nd K in Fig. 3, one n ompose the dispersions of the resonne long the [H, H,.5] (Fig. d) nd [.5,.5, L] (Fig. f) diretions. By plotting the dispersion in Å wy from Q AF (q s defined in Fig. ), we see tht the resonne disperses lmost isotropilly long these two diretions. Dispersion of the resonne in Ce.95 Y.5 CoIn 5 long [.5, K,.5]. In uprte superondutors suh s YB 2 Cu 3 O 6.5 (ref. 2), YB 2 Cu 3 O 6.6 (ref. ) nd L.875 B.25 CuO (ref. ), spin E =.7 mev K fit to exp. dt.6 K 2.3 K Intensity (ounts per min) 3. mev [H, H,.5] (r.l.u.) Intensity (ounts per min) mev.7 mev.55 mev.3 mev.6 K d Q AF = (.5,.5,.5).6 K.5 K 2.3 K 5 5 Intensity (ounts per min) [H, H,.5] (r.l.u.).5. Figure 2 Neutron sttering results on Ce.95 Y.5 CoIn 5 in the [H, H, L] sttering plne. () Colour-oded intensity of mgneti exittions long [H, H,.5] entred t Q AF t.6 K, otined from fits to dt in (). () Constnt-energy sns long [H, H,.5] entred t Q AF with E ¼.7 mev. The solid symols re dt well elow T (.6 K), where two peks n e resolved wheres open symols re otined ove T (2.3 K) showing single pek entred t Q AF. The solid line is fit to the dt t.6 K with two Gussin funtions, wheres the dshed line is fit to single Gussin funtion for the dt t 2.3 K. Dt t the two tempertures re fit simultneously to hve the sme liner kground. () Constnt-energy sns long [H, H,.5] t.6 K. For lrity, sns with E ¼.55,.75,.75 nd mev re, respetively, shifted upwrds y 5, 3, 8 nd 22. The solid lines re fits to either one or two Gussin funtions with liner kground. (d) Constnt-Q sns t Q AF. The rrows represent energies for whih onstnt-energy sns re shown in (). All vertil error rs in the figure represent sttistil errors of s.d. NATURE COMMUNICATIONS 7:277 DOI:.38/nomms277

5 NATURE COMMUNICATIONS DOI:.38/nomms277 ARTICLE.55 mev. mev.7 mev. mev.2 mev d g j m.3 K HH (r.l.u.) e h k n.3 2. K HH (r.l.u.) f i l o K 2. K [H, H,.5] (r.l.u.) Intensity (ounts) Intensity (ounts) Intensity (ounts) Intensity (ounts) Intensity (ounts) Figure 3 Constnt-energy mps of sttering intensities in the [H, H, L] sttering plne for Ce.95 Y.5 CoIn 5. Constnt-energy mp t E ¼. mev t () K nd () 2. K. A Q -dependent kground hs een sutrted. () Cuts otined from (,) y inning dt with.5rlr.55; solid lines re fits to the dt using either single or two Gussin funtions. Sine kground hs lredy een sutrted in mps in (,), no kground is ssumed in the fits. Similrly, (d f) re for E ¼.55 mev, (g i) re for E ¼.7 mev, (j l) re for E ¼. mev nd (m o) re for E ¼.2 mev. All vertil error rs in the figure represent sttistil errors of s.d. exittions ove the resonne form ring-like upwrd dispersion in the plne slightly softened from the spin wves in their AF-ordered prent ompounds 8. To onlusively determine if the resonne dispersion is lso ring-like in the plne in Ce.95 Y.5 CoIn 5, we ligned the single rystls in the [H,,H] [, K, ]([H, K, H]) sttering plne to mesure the dispersion of the resonne long [.5, K,.5] entred t Q AF. Figure f summrizes the onstnt-energy sns t E ¼.35,.5,.55,.7,.85 nd. mev long [.5, K,.5]. Although the sttering is lerly ommensurte t E ¼.35 nd.5 mev elow the resonne t E r E.55 mev (Fig.,), it eomes inommensurte ove the resonne t E ¼.7,.85 nd. mev with n upwrd dispersion s funtion of inresing energy (Fig. d f). Figure e summrizes the dispersion of the resonne in Å wy from Q AF long [.5, K,.5]. Figure g shows the differene of the onstnt-q sns elow nd ove T t Q AF, gin reveling strong pek t the resonne energy of E r E.55 mev similr to Fig. 2d. Finlly, Fig. h shows the temperture dependene of the sttering t n inommensurte wve vetor (.5,.35,.5) nd E ¼.85 mev, whih revels ler superonduting order-prmeter-like inrese elow T nd indites tht the inommensurte prt of the resonne is lso oupled to superondutivity. Dispersion of the resonne for CeCoIn 5 nd Ce.7 Y.3 CoIn 5. To determine how Y-doping, nd in prtiulr the possile hnges in the Fermi surfe topology nd superonduting gp struture etween Y-doping of x ¼. nd.2, ffets the ehviour of the resonne 3 36, we rried out dditionl inelsti neutron sttering experiments on CeCoIn 5 nd Ce.7 Y.3 CoIn 5 t MACS. Figure 5 shows temperture differenes of onstnt-q sns t Q AF elow nd ove T in Ce.7 Y.3 CoIn 5, whih revels ler resonne t E r E. mev. Figure 5 plots the temperture dependene of the resonne, displying superonduting order-prmeter-like inrese in intensity elow T. From wve vetor sns long the [H, H,.5] nd [.5,.5, L] diretions t different energies elow nd ove T for Ce.7 Y.3 CoIn 5 (Supplementry Fig. 5), we n estlish the dispersions of the resonne long these two diretions s shown in Fig. 5,d, respetively. Similrly, Fig. 5e,f ompres NATURE COMMUNICATIONS 7:277 DOI:.38/nomms

6 ARTICLE NATURE COMMUNICATIONS DOI:.38/nomms277 Intensity (ounts per min) Intensity (ounts per min) Intensity (ounts per min) Intensity (ounts per min) e g.35 mev.5 K.55 mev K.85 mev K [.5, K,.5] (r.l.u.) Q AF = (.5,.5,.5) K [.5, K,.5] (r.l.u.) dispersions of the resonne for CeCoIn 5 (Supplementry Fig. ) nd Ce.95 Y.5 CoIn 5 long the [H, H,.5] nd [.5,.5, L] diretions, respetively. From Fig. 5 f, we see tht the dispersions of the resonne re essentilly Y-doping independent. However, the ottom of the dispersive resonne t Q AF moves down in energy with inresing Y-doping nd E r is proportionl to k B T, similr to L-doped CeCoIn 5 (refs 3,32). d f h.5 mev.5 K.7 mev K. mev K Q = (.5,.35,.5) E =.85 mev K Temperture (K) Figure Neutron sttering results on Ce.95 Y.5 CoIn 5 in the [H, K, H] sttering plne. () Constnt-energy sn long [.5, K,.5] entred t Q AF t.5 K for E ¼.35 mev. The solid line is fit to single Gussin with liner kground. () Similr to (), ut for E ¼.5 mev. () Constnt-energy sn long [.5, K,.5] entred t Q AF, otined y sutrting dt t 2.3 K from dt t.5 K for E ¼.55 mev. The solid line is fit to Gussin funtion with zero kground. (d) Similr to (), ut for E ¼.7 mev, nd the solid line is fit to two Gussin funtions. (e) Similr to (d), ut for E ¼.85 mev. The rrow points to Q ¼ (.5,.35,.5), where mesurement of the temperture dependene ws rried out, shown in (h). (f) Similr to (d,e), ut for E ¼. mev. (g) Constnt-Q sn t Q AF otined y sutrting the 2.3 K dt from the.5 K dt. The solid line is Gussin funtion entred t E ¼.57() mev with zero kground. Arrows represent energies t whih onstnt-energy sns re shown in ( f). (h) Temperture dependene of sttering intensity t Q ¼ (.5,.35,.5) for E ¼.85 mev. The solid line is fit to d-wve superondutivity order prmeter with onstnt kground. The superonduting ritil temperture T otined from the fit is 2.() K. All vertil error rs in the figure represent sttistil errors of s.d. T Intensity (ounts per min) Intensity (ounts per min) Intensity (ounts per min) Intensity (ounts per min) Disussion From the dispersions of the resonne long [H, H,.5] (Fig. d), [.5, K,.5] (Fig. e) nd [.5,.5, L] (Fig. f) for Ce.95 Y.5 CoIn 5, we see tht the mode disperses isotropilly in reiprol spe wy from Q AF, whih is inonsistent with the resonne eing spin-exiton (see Fig. h,i), ut resemles mgnon-like exittion with dispersion similr to spin wves in CeRhIn 5 (Fig. j, Supplementry Note 3 nd Supplementry Fig. 2) tht eomes undmped in the superonduting stte 6,7. However, the ft tht CeCoIn 5 is multind system omplites the identifition of the resonne s origin. Athough we hve ssumed here tht the min ontriution to the resonne rises from the qusi-lolized f-levels identified vi qusi-prtile interferene (QPI) spetrosopy in STM experiment 25,28, it is of ourse possile tht there exist further eletroni nds tht eome superonduting nd ontriute to the resonne (either diretly or through renormliztion of the mgneti intertion) ut were not deteted vi QPI spetrosopy. Clerly, further studies re neessry to investigte this possiility. Moreover, in reent work on undoped CeCoIn 5, it ws suggested tht the resonne in the energy rnge of..7 mev is inommensurte long the [H, H,.5] diretion with wvevetor Q AF ±(d, d, ), where d ¼.2(2) r.l.u. (ref. 2). Sine the inommensurte wve vetors of the resonne pper to e lose to the in-plne mgneti field-indued inommensurte stti mgneti order t Q AF ±(d, d, ) with d ¼.5 (the so-lled Q phse) (see the vertil dshed lines in Fig. d) 3 5, nd sine it ws suggested tht the flututing moment of the resonne is entirely polrized long the -xis similr to the ordered moment of the Q phse 2,2, the resonne hs een desried s dynmil preursor of the Q phse 6. Experimentlly, we did not oserve inommensurte exittions t E ¼.5 mev; nevertheless, our dt suggest smller splitting thn in previous work if the exittions t E ¼.5 re inommensurte (Supplementry Note nd Supplementry Fig. 3). Furthermore, the Q phse preursor interprettion of the resonne is lso inonsistent with the oserved ring-like dispersion t E.7 mev. It is possile tht there re more thn one ontriution to the resonne in CeCoIn 5 given its eletroni omplexity. In the present work, we identify the upwrd-dispersing mgnon-like ontriution s eing dominnt, ut do not rule out finer fetures t lower energies with Eo.6 mev, whih n only e resolved with etter resolution. Our dt nd previous work on CeCoIn 5 (ref. 2) re onsistent with eh other, oth showing no signture of downwrd dispersion. Further insight into the nture of the resonne in CeCoIn 5 n e gined y onsidering its ehviour in n pplied mgneti field. Previous neutron sttering experiments y Stok et l. 7 oserved tht the resonne in the superonduting stte of CeCoIn 5 splits into two modes if mgneti field is pplied long the [,, ] diretion. This splitting into two modes y n in-plne field is rther puzzling, sine for system with Heisenerg spin symmetry splitting into three modes is expeted. Moreover, if the resonne in CeCoIn 5 ws entirely polrized long the -xis 2,2, pplition of n in-plne mgneti field should not split the resonne into the doulet oserved experimentlly 7,8. However, this oservtion n e explined if the system possesses mgneti nisotropy with mgneti esy plne (indited y the green ellipse in Fig. 6) tht is perpendiulr to the diretion of the pplied mgneti field (red rrow in Fig. 6). Sine the mgneti field pplied y Stok et l. 7 lies in the [,, ] diretion, this implies tht the esy plne is spnned y the unit vetors in the [,, ] nd [,, ] diretions. This leds us to suggest tht the resonne in CeCoIn 5 should lso hve omponent long the [,, ] diretion in ddition to the -xis omponent similr to 6 NATURE COMMUNICATIONS 7:277 DOI:.38/nomms277

7 NATURE COMMUNICATIONS DOI:.38/nomms277 ARTICLE Intensity (ounts) Temperture (K) Ce.7 Y.3 CoIn 5 Ce.7 Y.3 CoIn K T E =.5 mev E f = 2.5 mev E f = 3.7 mev Intensity (ounts). [H, H,.5] (r.l.u.) d [.5,.5, L] (r.l.u.) Ce.95 Y.5 CoIn 5 Ce.7 Y.3 CoIn q (Å ) Ce.95 Y.5 CoIn 5 Ce.7 Y.3 CoIn q (Å ).. e [H, H,.5] (r.l.u.) [.5,.5, L] (r.l.u.)..5. f..5.5 Ce.95 Y.5 CoIn 5 Ce.95 Y.5 CoIn 5. CeCoIn 5 CeCoIn q (Å ) q (Å ). Figure 5 Summry of neutron sttering results on Ce.7 Y.3 CoIn 5 nd CeCoIn 5. () Differene of onstnt-q sns t Q AF ¼ (.5,.5,.5) for.3 nd 2 K, displying resonne mode t E r E. mev for Ce.7 Y.3 CoIn 5. Filled symols re otined with fixed sttered neutron energy E f ¼ 3.7 mev nd open symols re for E f ¼ 2.5 mev sled up y times. All of the dt in the rest of figure re otined with E f ¼ 3.7 mev. The solid line is guide to the eye. () Temperture dependene of the resonne mode in Ce.7 Y.3 CoIn 5 for E ¼.5 mev nd Q AF ¼ (.5,.5,.5); the solid line is fit to d-wve superonduting gp, with T ¼.5() K. Dispersion of the resonne long () [H, H,.5] nd (d) [.5,.5,L] for Ce.7 Y.3 CoIn 5. Dispersions of the resonne for CeCoIn 5 long [H, H,.5] nd [.5,.5, L] re showin in (e,f), respetively. The solid yn lines in ( f) re dispersions of the resonne otined for Ce.95 Y.5 CoIn 5. The horizontl rs represent experimentlly oserved pek full-width-t-hlf-mximum. All vertil error rs in the figure represent sttistil errors of s.d. the resonne in eletron-doped iron pnitides 9,5.Suhin-plne spin exittion nisotropy n our due to the presene of spinorit oupling, nd does not rek the four-fold rottionl symmetry of the underlying lttie 5. The present experimentl results do not rule out the presene of suh mode, lthough it is lso hllenging to experimentlly onfirm its presene (Supplementry Note 5 nd Supplementry Figs nd 5). To quntittively understnd the effet of mgneti field on spin exittions, we onsider the Hmiltonin (see Supplementry Eq. in ref. 28) H ¼ X I r;r S r S r þ A X S z 2 r gmb H X S z r ðþ r;r r r with the three terms representing the mgneti intertions etween the f-eletron moments, the mgneti nisotropy of the system nd the intertion with the externl mgneti field, respetively. Here, we define the diretion of the mgneti field long the [,, ] diretion s the z-xis in spin spe. We ssume A, suh tht the system possesses hrd mgneti xis long [,, ] nd n esy plne (green ellipse in Fig. 6) perpendiulr to it. This Hmiltonin implies tht the effetive intertion for the longitudinl, non-spin-flip sttering mode (prllel to the pplied field) is given y I zz (q) ¼ I q þ A, while the intertion for the trnsverse mode is given y I ± (q) ¼ I q, with I q eing the Fourier trnsform of I r,r in Eqution (). In the viinity of the AF wve-vetor Q AF, where I QAF o, we thus otin I zz (Q AF ) o I ± (Q AF ) sine A for n esy plne perpendiulr to the [,, ] diretion. This implies tht the effetive intertion NATURE COMMUNICATIONS 7:277 DOI:.38/nomms

8 ARTICLE NATURE COMMUNICATIONS DOI:.38/nomms277 χ (.u.) gμ B SH = gμ B SH =.5 mev gμ B SH =. mev χ ± χ + H z [ ] Figure 6 Effet of pplied mgneti field on the resonne mode. () Orienttion of the mgneti field H nd tht of the mgneti esy plne in the rystl lttie. The mgneti field is perpendiulr to the mgneti esy plne. () Evolution of the resonne with inresing mgneti field. t Q AF for the longitudinl, non-spin-flip sttering mode (prllel to the pplied field) is smller thn for the two trnsverse, spin-flip sttering modes, whih lie in the esy plne. As result, the longitudinl mode will e loted t energies higher thn the trnsverse modes. In prtiulr, for suffiiently lrge A, the longitudinl mode n e loted ove the onset energy, o (Q AF ), for the prtile hole ontinuum in the superonduting stte, nd thus would not emerge s resonne pek. Hene, only the two trnsverse modes within the esy plne ontriute to the resonne pek. The pplition of mgneti field perpendiulr to the esy plne of the system then splits the two trnsverse modes of the resonne pek in energy (while not ffeting the longitudinl mode), with the energy splitting inresing linerly with the mgneti field, s shown in Fig. 6, thus explining the experimentl oservtion in ref. 7). If spin exittions in CeCoIn 5 re only polrized long the -xis with the existene of n esy xis rther thn n esy plne 2,2, with pplition of mgneti field long the diretion perpendiulr to the esy xis long the [,, ] diretion, the trnsverse mode long the esy xis shifts down with inresing field, ut does not split. Similrly, when field is pplied long the esy xis diretion (-xis field), the two trnsverse modes re loted t higher energies, while the longitudinl mode, whih is loted t lower energies, does not split in the mgneti field. The presene of longitudinl spin exittion long the [,, ] diretion is lso onsistent with the mgneti field effet work of ref. 8, where the resonne is elieved to e omposite exittion, whih ontins three exittion hnnels involving oth trnsverse nd longitudinl modes. While unonventionl superondutivity in opper oxide, iron pnitide nd hevy fermion superondutors ppers with the suppression of the stti AF order in their prent ompounds, dispersive mgnon-like exittions persist in the doped superondutors 8,,5. Our disovery tht the resonne itself in Ce x Y x CoIn 5 shows roust ring-like upwrds dispersion suggests tht, insted of eing spin-exiton in d- wve superondutor 2,7, the resonne my e mgnon-like exittion reveled in the superonduting stte 7. Sine the presene of propgting spin resonne is hrteristi of nery AF stte, we propose tht the mgnon-like resonne mode in Ce x Y x CoIn 5 is the strong-oupling nlogue of wek oupling spin-exiton. This would imply tht the nture of the mgneti resonne spin-exiton versus mgnon-like exittion represents new riterion to distinguish etween more wekly nd more strongly oupled unonventionl superondutors. Methods Smple preprtion. Single rystls of Ce x Y x CoIn 5 (x ¼,.5 nd.3) were prepred y the indium self-flux method. Detils of smple preprtion nd hrteriztions hve een previously reported; lttie prmeters for Ce x Y x CoIn 5 remin similr to pure CeCoIn 5 for ll reported doping levels 33. We use the nominl doping throughout the pper to e onsistent with erlier work 33, lthough the tul doping is B/3 of the nominl doping 52. Supplementry Fig. shows the out-of-phse AC mgneti suseptiility (5.9 Hz) mesured on Ce x Y x CoIn 5 smples with x ¼.5 nd.3 from the sme growth thes used for neutron sttering experiments. Bulk superondutivity ppers t T ¼ 2.25 K nd T ¼.5 K, respetively, wheres T ¼ 2.3 K in pure CeCoIn 5 (ref. 33). Hundreds of Ce x Y x CoIn 5 single rystls with totl msses of.8, 2.5 nd. g, respetively, for x ¼,.5 nd.3 were o-ligned on severl luminium pltes using CYTOP s hydrogen-free glue (Supplementry Fig. ). The pltes re then mounted in either the [H, H, ] [,, L]([H, H, L]) (Supplementry Fig. ) or the [H,,H] [, K, ]([H, K, H]) sttering plne (Supplementry Fig. d). The totl thikness of smples on o-ligned pltes is 2 mm, minimizing neutron sorption due to indium. Asorption eomes most signifint when the inident or the sttered neutron em eomes perpendiulr to [,, ], whih does not our for reiprol spe regions shown in this work. Experiment detils nd nlysis. Neutron sttering experiments were rried out on the PANDA old triple-xes spetrometer 53 t Heinz Mier-Leinitz Zentrum nd the MACS instrument t the NIST Center for Neutron Reserh. The experiments on PANDA used Be filter 8 mm in length fter the smple, whih is highly effetive in removing ontmintion from higher-order neutrons; oth the nlyser nd the monohromtor re douly foused to mximize neutron flux t the smple. Vertil fousing of the nlyser is fixed, wheres horizontl fousing is vrile. Both the horizontl nd vertil fousing of the monohromtor re vrile. The vrile fousings re djusted depending on the neutron wvelength, whih is sed on empirilly optimized vlues. The PANDA experiment in the [H, H, L] sttering plne used fixed k f of.3 Å (E f E3.5 mev) nd the experiment in the [H, K, H] sttering plne used fixed k f of.57 Å (E f E5. mev). The MACS experiments in the [H, H, L] sttering plne used Be filters oth efore nd fter the smple with fixed E f ¼ 3.7 mev. MACS onsists of 2 spetrosopi detetors, eh seprted y 8. By rotting the smple nd shifting ll of the detetors to ridge the 8 gps, mp in terms of smple rottion ngle nd sttering ngle t fixed energy trnsfer n e effiiently onstruted. A signifint portion of the reiprol spe in the sttering plne n e overed, whih further llows uts long the high-symmetry diretions. Ninety-degree ollimtors re used etween the smple nd eh individul nlysers. The nlysers re vertilly foused, while the monohromtor is douly foused. For the neutron sttering results on PANDA, liner kground is ssumed for ll mesured onstnt-energy sns, while no kground is used for sns otined y sutrting dt ove T from those otined elow T. The onstntenergy sns re then simply fit to either one or two Gussin peks. For the neutron sttering results otined on MACS, mps of lrge portions of the sttering plne for severl energy trnsfers were olleted oth elow nd ove T.A Q -dependent kground is otined y msking the signl ner (.5,.5,.5) nd is then fit to polynomil. The signl with Q o.5 Å is msked throughout the nlysis. The fit kground is then sutrted from the mp nd the dt re folded into the first qudrnt of the sttering plne to improve sttistis. The results for Ce.95 Y.5 CoIn 5 re shown in Fig. 3 nd Supplementry Fig. 3. Cuts long [H, H,.5] re otined y inning dt with.5rlr.55 nd fit with single or two Gussin peks. Cuts long [.5,.5, L] re otined y inning dt with.5rhr.55 nd fit y sum of 8 NATURE COMMUNICATIONS 7:277 DOI:.38/nomms277

9 NATURE COMMUNICATIONS DOI:.38/nomms277 ARTICLE Lorentzin peks, ounting for the Ce 3 þ mgneti form ftor f(q) nd the polriztion ftor ssuming exittions re dominntly polrized long the -xis similr to previous work 2. The possile presene of exittions polrized long the [,, ] diretion is disussed in Supplementry Note 5. The funtion used to fit sns long [.5,.5, L] n e written s IðQÞ / f ðqþ 2 2 X ^Q ^ n¼ Fnþ ð LÞ ð2þ where F(L) is either single Lorentizn pek entred t L ¼.5 or two Lorentzin peks eqully displed from L ¼.5. The peks long [.5,.5, L] re signifintly roder ompred to those long [H, H,.5], nd remin non-zero even for L ¼ (Supplementry Fig. 3). This ontrsts with similr sns long [H, H,.5] in Fig. 3, where the intensity drops to zero wy from Q AF. MACS dt of CeCoIn 5 nd Ce.7 Y.3 CoIn 5 with the orresponding mps nd uts re shown in Supplementry Figs nd 5. Similr to Ce.95 Y.5 CoIn 5, the resonne mode lerly disperses upwrd with inresing energy. Dt vilility. The dt tht support the findings of this study re ville from the orresponding uthor upon request. Referenes. Monthoux, P., Pines, D. & Lonzrih, G. G. Superondutivity without phonons. Nture 5, (27). 2. Slpino, D. J. A ommon thred: the piring intertion for unonventionl superondutors. Rev. Mod. Phys. 8, 383 (22). 3. Keimer, B., Kivelson, S. A., Normn, M. R., Uhid, S. & Znen, J. From quntum mtter to high-temperture superondutivity in opper oxides. Nture 58, (25).. Thompson, J. D. & Fisk, Z. Progress in hevy-fermion superondutivity: Ce5 nd relted mterils. J. Phys. So. Jpn 8, 2 (22). 5. White, B. D., Thompson, J. D. & Mple, M. B. Unonventionl superondutivity in hevy-fermion ompounds. Physi C 5, (25). 6. Rosst-Mignod, J. et l. Neutron sttering study of the YB 2 Cu 3 O 6 þ x system. Physi C 85, (99). 7. Eshrig, M. The effet of olletive spin- exittions on eletroni spetr in high-t superondutors. Adv. Phys. 55, 7 83 (26). 8. Trnqud, J. M., Xu, G. & Zliznyk, I. A. Superondutivity, ntiferromgnetism, nd neutron sttering. J. Mg. Mg. Mter 35, 8 6 (2). 9. Christinson, A. D. et l. Resonnt spin exittion in the high temperture superondutor B.6 K. Fe 2 As 2. Nture 56, (28).. Di, P. C. Antiferromgneti order nd spin dynmis in iron-sed superondutors. Rev. Mod. Phys. 87, 855 (25).. Sto, N. K. et l. Strong oupling etween lol moments nd superonduting hevy eletrons in UPd 2 Al 3. Nture, 3 33 (2). 2. Stok, C., Broholm, C., Hudis, J., Kng, H. J. & Petrovi, C. Spin resonne in the d-wve superondutor CeCoIn 5. Phys. Rev. Lett., 87 (28). 3. Inosov, D. S. et l. Crossover from wek to strong piring in unonventionl superondutors. Phys. Rev. B 83, 252 (2).. Yu, G., Li, Y., Motoym, E. M. & Greven, M. A universl reltionship etween mgneti resonne nd superonduting gp in unonventionl superondutors. Nt. Phys. 5, (29). 5. Hirshfeld, P. J., Korshunov, M. M. & Mzin, I. I. Gp symmetry nd struture of Fe-sed superondutors. Rep. Prog. Phys. 7, 258 (2). 6. Morr, D. K. & Pines, D. The resonne pek in uprte superondutors. Phys. Rev. Lett. 8, 86 (998). 7. Chuukov, A. V. & Gor kov, L. P. Spin resonne in three-dimensionl superondutors: the se of CeCoIn 5. Phys. Rev. Lett., 7 (28). 8. Bourges, P. et l. The spin exittion spetrum in superonduting YB 2 Cu 3 O Siene 288, (2). 9. Di, P. C., Mook, H. A., Hunt, R. D. & Doğn, F., Evolution of the resonne nd inommensurte spin flututions in superonduting YB 2 Cu 3 O 6 þ x. Phys. Rev. B 63, 5525 (2). 2. Reznik, D. et l. Dispersion of mgneti exittions in optimlly doped superondutor YB 2 Cu 3 O Phys. Rev. Lett. 93, 273 (2). 2. Stok, C. et l. From inommensurte to dispersive spin-flututions: the highenergy inelsti spetrum in superonduting YB 2 Cu 3 O 6.5. Phys. Rev. B 7, 2522 (25). 22. Lu, X. Y. et l. Avoided quntum ritility nd mgnetoelsti oupling in BFe 2 x Ni x As 2. Phys. Rev. Lett., 257 (23). 23. Kim, M. G. et l. Mgnonlike dispersion of spin resonne in Ni-doped BFe 2 As 2. Phys. Rev. Lett., 772 (23). 2. Eremin, I., Morr, D. K., Chuukov, A. V., Bennemnn, K. H. & Normn, M. R. Novel neutron resonne mode in d x 2 y2 -wve superondutors. Phys. Rev. Lett. 9, 7 (25). 25. Alln, M. P. et l. Imging Cooper piring of hevy fermions in CeCoIn 5. Nt. Phys. 9, (23). 26. Zhou, B. B. et l. Visulizing nodl hevy fermion superondutivity in CeCoIn 5. Nt. Phys. 9, 7 79 (23). 27. Eremin, I., Zwikngl, G., Thlmeier, P. & Fulde, P. Feedk spin resonne in superonduting CeCu 2 Si 2 nd CeCoIn 5. Phys. Rev. Lett., 87 (28). 28. Vn Dyke, J. et l. Diret evidene for mgneti f-eletronmedited piring mehnism of hevy-fermion superondutivity in CeCoIn 5. PNAS, (2). 29. Petrovi, C., Bud ko, S. L., Kogn, V. G. & Cnfield, P. C. Effets of L sustitution on the superonduting stte of CeCoIn 5. Phys. Rev. B 66, 553 (22). 3. Tntr, M. A. et l. Unpired eletrons in the hevy-fermion superondutor CeCoIn 5. Phys. Rev. Lett. 95, 672 (25). 3. Pnrin, J., Rymond, S., Lpertot, G., Flouquet, J. & Mignot, J.-M. Effets of nonmgneti L impurities on the spin resonne of Ce x L x CoIn 5 single rystls s seen vi inelsti neutron sttering. Phys. Rev. B 8, 5255 (2). 32. Rymond, S., Pnrin, S., Lpertot, G. & Flouquet, J. Evolution of the spin resonne of CeCoIn 5 s funtion of mgneti field nd L sustitution. J. Phys. So. Jpn 8, SB23 (2). 33. Shu, L. et l. Correlted eletron stte in Ce x Y x CoIn 5 stilized y oopertive vlene flututions. Phys. Rev. Lett. 6, 563 (2). 3. Polykov, A. et l. Fermi-surfe evolution in Y-sustituted CeCoIn 5. Phys. Rev. B 85, 259 (22). 35. Dudy, L. et l. Y vlene hnge in Ce x Y x CoIn 5 from spetrosopy nd ulk properties. Phys. Rev. B 88, 658 (23). 36. Kim, H. et l. Nodl to nodeless superonduting energy-gp struture hnge onomitnt with Fermi-surfe reonstrution in the hevy-fermion ompound CeCoIn 5. Phys. Rev. Lett., 273 (25). 37. Erten, O., Flint, R. & Colemn, P. Moleulr piring nd fully gpped superondutivity in Y-doped CeCoIn 5. Phys. Rev. Lett., 272 (25). 38. Ds, P. et l. Mgnitude of the mgneti exhnge intertion in the hevy-fermion ntiferromgnet CeRhIn 5. Phys. Rev. Lett. 3, 26 (2). 39. Stok, C. et l. Single to multiqusiprtile exittions in the itinernt helil mgnet CeRhIn 5. Phys. Rev. Lett., 275 (25).. Hyden, S. M., Mook, H. A., Di, P. C., Perring, T. G. & Doğn, F. The struture of the high-energy spin exittions in high-trnsition temperture superondutor. Nture 29, (2).. Trnqud, J. M. et l. Quntum mgneti exittions from stripes in opper oxide superondutors. Nture 29, (2). 2. Rymond, S. & Lpertot, G. Ising inommensurte spin resonne of CeCoIn 5 : dynmil preursor of the Q phse. Phys. Rev. Lett. 5, 37 (25). 3. Kenzelmnn, M. et l. Coupled superonduting nd mgneti order in CeCoIn 5. Siene 32, (28).. Kenzelmnn, M. et l. Evidene for mgnetilly driven superonduting Q phse of CeCoIn 5. Phys. Rev. Lett., 27 (2). 5. Gerer, S. et l. Swithing of mgneti domins revels sptilly inhomogeneous superondutivity. Nt. Phys., (2). 6. Mihl, V. P. & Mineev, V. P. Field-indued spin-exiton ondenstion in the d x2 y 2 -wve superondutor CeCoIn 5. Phys. Rev. B 8, 5258 (2). 7. Stok, C. et l. Mgneti field splitting of the spin resonne in CeCoIn 5. Phys. Rev. Lett. 9, 6727 (22). 8. Rymond, S., Kneko, K., Hiess, A., Steffens, P. & Lpertot, G. Evidene for three flutution hnnels in the spin resonne of the unonventionl superondutor CeCoIn 5. Phys. Rev. Lett. 9, 2372 (22). 9. Steffens, P. et l. Splitting of resonne exittions in nerly optimlly doped B(Fe.9 Co.6 ) 2 As 2 : n inelsti neutron sttering study with polriztion nlysis. Phys. Rev. Lett., 37 (23). 5. Luo, H. Q. et l. Spin exittion nisotropy s proe of oritl ordering in the prmgneti tetrgonl phse of superonduting BFe.9 Ni.96 As 2. Phys. Rev. Lett., 76 (23). 5. Stokert, O. et l. Mgnetilly driven superondutivity in CeCu 2 Si 2. Nt. Phys. 7, 9 2 (2). 52. Jng, S. et l. Resolution of the disrepny etween the vrition of the physil properties of Ce x Y x CoIn 5 single rystls nd thin films with Y omposition. Philos. Mg. 9, (2). 53. Heinz, Mier-Leinitz Zentrum et l. PANDA: Cold three xes spetrometer. J. Lrge-Sle Reserh Filities, A2 (25). Aknowledgements We thnk Qimio Si, S. Rymond nd C. Stok for helpful disussions. We lso thnk S. Rymond for shring with us his unpulished dt on CeCoIn 5. We knowledge help from Mengshu Liu, Xingye Lu nd Wenling Zhng for ssistne with smple olignment, nd Sott Crr, Weiyi Wng nd Jose Rodriguez for preliminry mesurements on Ce.7 Y.3 CoIn 5. The neutron sttering work t Rie is supported y the U.S. DOE, BES, under Grnt No. DE-SC23 (P.D.). Prt of the mteril hrteriztion efforts t Rie is supported y the Roert A. Welh Foundtion Grnt No. C-839 (P.D.). The reserh t UCSD ws supported y the U.S. DOE, BES, under Grnt No. DE-FG2- ER65 (smple synthesis), nd the U.S. NSF, under Grnt No. DMR (smple hrteriztion). The work y JVD nd DKM ws supported y the U.S. DOE, BES, under Grnt No. DE-FG2-5ER6225. The reserh t Ntionl Institute of NATURE COMMUNICATIONS 7:277 DOI:.38/nomms

10 ARTICLE NATURE COMMUNICATIONS DOI:.38/nomms277 Stndrds nd Tehnology is in prt supported y U.S. NSF, under Agreement No. DMR The reserh t Fudn University is in prt supported y the NSFC, under Grnt No. 76. Author ontriutions The smples were prepred y I.K.L., B.D.W., S.J., D.Y., L.S. nd M.B.M. Neutron sttering experiments were rried out y Y.S., A.S., P.C., Y.Q., nd P.D. Dt nlysis ws done y Y.S. Theoretil lultions were done y J.V. nd D.K.M. The pper ws written y P.D., D.K.M., nd Y.S. with input from ll o-uthors. Additionl informtion Supplementry Informtion ompnies this pper t ntureommunitions Competing finnil interests: The uthors delre no ompeting finnil interests. Reprints nd permission informtion is ville online t reprintsndpermissions/ How to ite this rtile: Song, Y. et l. Roust upwrd dispersion of the neutron spin resonne in the hevy fermion superondutor Ce x Y x CoIn 5. Nt. Commun. 7:277 doi:.38/nomms277 (26). This work is liensed under Cretive Commons Attriution. Interntionl Liense. The imges or other third prty mteril in this rtile re inluded in the rtile s Cretive Commons liense, unless indited otherwise in the redit line; if the mteril is not inluded under the Cretive Commons liense, users will need to otin permission from the liense holder to reprodue the mteril. To view opy of this liense, visit r The Author(s) 26 NATURE COMMUNICATIONS 7:277 DOI:.38/nomms277

11 Supplementry Informtion: [,,] [,,] χ'' (Ar. Units) T ~.5K x =.3 T ~ 2.25K x =.5 [,,] Temperture (K) [,-,] d [,,] [,,] [,,] [,,] 3 [,,] [,,] [H,H,L] sttering plne [H,K,H] sttering plne Supplementry Figure : Mesurement of T nd o-lignment of Ce xy xcoin 5 single rystls. () AC mgneti suseptiility mesured on Ce xy xcoin 5 (x =.5 nd.3), with T =2.25 K nd.5 K. () Severl luminum pltes with hundreds of o-ligned Ce xy xcoin 5 (x =.5) single rystls. The rystllogrphi xes re mrked y red rrows. () Co-ligned pltes in the [H,H,L] sttering plne. (d) Co-ligned pltes in the [H,K,H] sttering plne. The ngle etween [,,] nd [,,] is 3. 3 E r /k B T 2 Ce -x Y x CoIn Y onentrtion (nominl) Supplementry Figure 2: E r/k BT in Ce xy xcoin 5. The result for CeCoIn 5 is otined from previous work [] nd the results for x =.5 nd x =.3 re from this work. The vertil error rs re estimtes of the unertinty of E r y rrying out onstnt-q sns t Q AF = (.5,.5,.5).

12 2 K-2.K.7meV.55meV.meV e i K-2.K K d 2.K f g h j k l K-2.K meV m 2 n o p 2.2meV q r s t HH (r.l.u.) [H,H,.5] (r.l.u.) [.5,.5,L] (r.l.u.) [.5,.5,L] (r.l.u.) Supplementry Figure 3: Additionl neutron sttering dt for Ce.95Y.5CoIn 5. () Constnt-energy mp for Ce.95Y.5CoIn 5 t E =. mev nd K fter sutrting dt from 2. K. () Cut long [H,H,.5] for the mp in () t E =. mev, the solid line is fit ssuming zero kground. () Cuts long [.5,.5,L] t oth K nd 2. K otined from mps in Figure 3() nd () in the min text. (d) Cut long [.5,.5,L] for the mp in () t K fter sutrting dt from 2. K. The solid lines in () nd (d) re sums of Lorentzin peks polrized long ĉ. Similrly (e), (f), (g) nd (h) re for E =.55 mev, (i), (j), (k) nd (l) re for E =.7 mev, (m), (n), (o) nd (p) re for E =. mev nd (q), (r), (s) nd (t) re for E =.2 mev. For uts long [.5,.5,L] t 2.3 K nd E =. nd.55 mev, F(L) is single Lorentzin pek entered t L =.5. For K nd K-2. K uts with E =.7,. nd.2 mev, F(L) is two Lorentzin peks eqully displed from L =.5. All vertil error rs in the Figure represent sttistil errors of stndrd devition.

13 3 CeCoIn 5. K K.5 mev mev d 3 2 e f mev g h i 5 5. mev j HH (r.l.u.) 6 2 k [H,H,.5] (r.l.u.) l [.5,.5,L] (r.l.u.) Supplementry Figure : Neutron sttering dt for CeCoIn 5. () Constnt-energy mp for CeCoIn 5 t E =.5 mev nd. K fter sutrting dt from 2.5 K. () Cut long [H,H,.5] for the mp in (), the solid line is fit ssuming zero kground. () Cut long [.5,.5,L] for the mp in (), the solid line is sum of Lorentzin peks polrized long ĉ. Similrly (d), (e) nd (f)re for E =.6 mev, (g), (h) nd (i) re for E =.8 mev nd (j), (k) nd (l) re for E =. mev. For uts long [.5,.5,L] for E =.5 nd.6 mev, F(L) is single Lorentzin pek entered t L =.5. For E =.8 nd. mev, F(L) is two Lorentzin peks eqully displed from L =.5. All vertil error rs in the Figure represent sttistil errors of stndrd devition.

14 Ce.7 Y.3 CoIn 5.3 K - 2. K.3 mev. mev d 2 2 e f mev g h i 2.7 mev j 8 k l.85 mev m 6 2 n o. mev p 3 2 q h HH (r.l.u.) [H,H,.5] (r.l.u.) [.5,.5,L] (r.l.u.) Supplementry Figure 5: Neutron sttering dt for Ce.7Y.3CoIn 5. () Constnt-energy mp for Ce.7Y.3CoIn 5 t E =.3 mev nd.3 K fter sutrting dt from 2. K. () Cut long [H,H,.5] for the mp in (), the solid line is fit ssuming zero kground. () Cut long [.5,.5,L] for the mp in (), the solid line is sum of Lorentzin peks polrized long ĉ. Similrly (d), (e) nd (f)re for E =. mev, (g), (h) nd (i) re for E =.55 mev, (j), (k) nd (l) re for E =.7 mev, (m), (n) nd (o) re for E =.85 mev nd (p), (q) nd (r) re for E =. mev. For uts long [.5,.5,L] for E =.3,. nd.55 mev, F(L) is single Lorentzin pek entered t L =.5. For E =.7,.85 nd. mev, F(L) is two Lorentzin peks eqully displed from L =.5. All vertil error rs in the Figure represent sttistil errors of stndrd devition.

15 5 χ [/mev] Re χ SC Im χ SC - -/ I (Q) Re χ norm. Im χ norm. ω () Im χ [/mev] π ky π π () Q=(π,π) π kx norml SC stte Frequeny [mev] Frequeny [mev] Supplementry Figure 6: χ in the spin-exiton senrio t Q AF. () Rel nd imginry prts of χ t Q AF in the norml (dshed lines) nd superonduting (solid lines) stte of CeCoIn 5. The onset energy of the prtile-hole ontinuum in the superonduting stte is shown y green rrow. The lue dotted line indites the vlue of /Ī(Q), suh tht its intersetion with Re χ in the superonduting stte for ω < yields the position of the spin exiton. () Full χ in the norml nd superonduting stte t Q AF. Inset: Fermi surfe nd sttering vetor Q AF etween momentum sttes on the Fermi surfe. 2π () (3) () () Im χ [/mev] () (2) (3) π α β α2 () Energy [mev] (3) (2) () Frequeny [mev] () (2) π π π 2π qx [π] long (H,H) Supplementry Figure 7: χ in the spin-exiton senrio t.88q AF. () Imginry prts of χ t q =.88Q AF in the superonduting stte of CeCoIn 5. The onset energies (i) of the prtile-hole ontinuum in the superonduting stte re shown y green rrows. () Fermi surfes in the extended Brillouin zone with sttering vetors orresponding to (i) in (). () Momentum dependene of the onset energies (i) long the [,,] diretion. The dshed line orresponds to the momentum q =.88Q AF for whih Im χ is shown in (), with the indited (i) eing the sme s in ().

16 6 3 (3) (2) () (2) (H,H) (r.l.u.) Supplementry Figure 8: Momentum dependene of the resonne s spin-exiton with some of the onset energies of Fig. 7() overlin s solid lines. Im Π [/mev] norml stte SC stte ω Im χ [/mev] π ky π π QAF π kx norm stte SC stte ω Frequeny [mev] Frequeny [mev] Supplementry Figure 9: The resonne s mgnon-like exittion t Q AF. () Imginry prts of Π t Q AF in the norml (lk line) nd superonduting (red line) stte of CeCoIn 5. The onset energy of the prtile-hole ontinuum in the superonduting stte is shown y green rrow. () Full χ in the norml (lk line) nd superonduting (red line) stte Q AF. The resonne ours t ω = sw elow (see green rrow). Inset: Sttering proess ontriuting to Im Π.

17 7 Im Π [/mev] () (2) (3) () 2π π β (3) () () α (2) α2 Energy [mev] () (3) (2) () Frequeny [mev] π π π 2π qx [π] long (H,H) Supplementry Figure : The resonne s mgnon-like exittion t.95q AF. () Imginry prt of Π in the superonduting stte of CeCoIn 5 t q =.95Q AF. The onset energies (i) for prtile-hole sttering in the superonduting stte re shown y green rrows. () Fermi surfes in the extended Brillouin zone with sttering vetors orresponding to (i) in (). () Momentum dependene of (i) long q x = q y. The dshed line orresponds to the momentum q =.95Q AF for whih Im Π is shown in (), with the indited (i) eing the sme s in (). 8 (3) (2) (H,H) (r.l.u.) Supplementry Figure : Momentum dependene of the resonne s mgnon-like exittion with some the onset energies of SFig. () overlin s solid lines.