Direct evidence for a magnetic f-–mediated pairing mechanism of heavy- in CeCoIn5 John S. Van Dykea, Freek Masseeb,c, Milan P. Allanb,c,d, J. C. Séamus Davisb,c,e,f,1, Cedomir Petrovicb, and Dirk K. Morra,1

aDepartment of , University of Illinois at Chicago, Chicago, IL 60607; bCondensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY 11973; cLaboratory of Atomic and Solid State Physics, Department of Physics, Cornell University, Ithaca, NY 14853; dDepartment of Physics, Eidgenössische Technische Hochschule Zürich, CH-8093 Zurich, Switzerland; eSchool of Physics and Astronomy, University of St. Andrews, Fife KY16 9SS, United Kingdom; and fKavli Institute at Cornell, Cornell University, Ithaca, NY 14853

Contributed by J. C. Séamus Davis, May 21, 2014 (sent for review April 23, 2014; reviewed by Zachary Fisk and Andrey Chubukov) To identify the microscopic mechanism of heavy-fermion Cooper electronic fluid as a Fermi liquid but with strong antiferromag- pairing is an unresolved challenge in quantum matter studies; it netic spin fluctuations derived from the f-electron magnetism. may also relate closely to finding the pairing mechanism of high- Moreover, it has been long hypothesized that it is these spin temperature superconductivity. Magnetically mediated Cooper pair- fluctuations that generate the attractive Cooper pairing in- ing has long been the conjectured basis of heavy-fermion supercon- teraction in heavy-fermion materials, specifically, in the d-wave ductivity but no direct verification of this hypothesis was achievable. channel (1–7). Why, in the decades since heavy-fermion super- Here, we use a novel approach based on precision measurements of conductivity was discovered (12), has this been so extremely the heavy-fermion band structure using interference im- difficult to prove? The crux is that identification of the pairing k aging to reveal quantitatively the momentum space ( -space) struc- mechanism in a heavy-fermion compound requires two specific α;β ture of the f-electron magnetic interactions of CeCoIn5. Then, by pieces of information: the heavy-band structures Ek and the solving the superconducting gap equations on the two heavy-fer- α;β α β k-space structure of the superconducting gaps Δk (which encode mion bands E , with these magnetic interactions as mediators PHYSICS k the essentials of the pairing process). However, determination of of the Cooper pairing, we derive a series of quantitative predictions the characteristic heavy-fermion band structure requires pre- about the superconductive state. The agreement found between α;β cision measurement of E both above and below the Fermi these diverse predictions and the measured characteristics of super- k energy (Fig. 1B), making it problematic for angle-resolved pho- conducting CeCoIn5 then provides direct evidence that the heavy-fer- f toemission. Moreover, due to the extreme flatness of the heavy mion Cooper pairing is indeed mediated by -electron magnetism. α;β bands dEk =dk → 0 and a maximum superconducting gap of heavy fermion materials | unconventional superconductivity | typically only a few hundred microelectron volts, no techniques – existed with sufficient combined energy resolution δE ≤ 100 μeV f-electron mediated pairing mechanism α;β α;β and k-space resolution to directly measure Ek and Δk for any uperconductivity of heavy is of abiding interest, heavy-fermion superconductor. Unambiguous identification of Sboth in its own right (1–7) and because it could exemplify the the Cooper pairing mechanism had therefore proven impossible. unconventional Cooper pairing mechanism of high-temperature – Heavy-Fermion Quasiparticle Interference: Experiment superconductors (8 11). Heavy-fermion compounds are inter- and Theory metallics containing magnetic ions in the 4f- or 5f-electronic state Very recently, however, this situation has changed (13–15). Heavy- within each unit cell. At high temperatures, each f-electron is lo- calized at a magnetic ion (Fig. 1A). At low temperatures, inter- fermion Bogoliubov quasiparticle interference (BQPI) imaging actions between f-electron spins (red arrows Fig. 1A) lead to the f Significance formation of a narrow but the subtly curved f-electron band «k near the chemical potential (red curve, Fig. 1B), and Kondo screening c hybridizes this band with the conventional c-electron band «k of the In heavy-fermion materials, the magnetic moment of an metal (black curve, Fig. 1B). As a result, two new heavy-fermion f-electron atom, such as Ce, is screened via the α;β bands Ek (Fig. 1C) appear within a few millielectron volts of the resulting in the splitting of a conventional light band into two . Their electronic structure is controlled by the hy- heavy bands within few millielectron volts of the Fermi energy. bridization matrix element sk for interconversion of conduction For decades it has been hypothesized that Cooper pairing and c- to f-electrons and vice-versa, such that superconductivity of the resulting heavy electrons are mediated by the f-electron magnetism. By extracting the magnetic inter- v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! u 2 actions of CeCoIn5 from heavy-fermion scattering interference, c f u c f α;β «k + «k t «k − «k and by then predicting quantitatively a variety of characteristics E = ± + s2: [1] k 2 2 k expected for unconventional superconductivity driven by them, we provide direct evidence that the heavy-fermion Cooper pairing in this material is indeed mediated by f-electron magnetism. The momentum structure of the narrow bands of hybridized electron- ic states (Eq. 1 and Fig. 1C, blue curves at left) near the Fermi surface Author contributions: J.C.S.D. and D.K.M. designed research; J.S.V., F.M., M.P.A., C.P., and then directly reflects the form of magnetic interactions encoded with- D.K.M. performed research; C.P. synthesized the samples; J.S.V., F.M., and D.K.M. ana- f in the parent f-electron band «k. It is these interactions that are con- lyzed data; and J.S.V., F.M., J.C.S.D., and D.K.M. wrote the paper. jectured to drive the Cooper pairing (1–5) and thus the opening up Reviewers: Z.F., University of California, Irvine; and A.C., University of Wisconsin. of a superconducting energy gap (Fig. 1C, yellow curves at right). The authors declare no conflict of interest. 1To whom correspondence may be addressed. Email: [email protected] or Heavy Fermions and Cooper Pairing [email protected]. Theoretical studies of the microscopic mechanism sustaining the This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. Cooper pairing of such heavy fermions typically consider the 1073/pnas.1409444111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1409444111 PNAS Early Edition | 1of5 Downloaded by guest on September 25, 2021 A spectroscopic imaging scanning tunneling microscopy system oper- atingdowntoanelectrontemperatureof250mK,weimagethe differential conductance gðr; EÞ with atomic resolution and register, and then determine gðq; EÞ, the power spectral density Fourier transform of each image. Recently, it was demonstrated that this approach can be used to identify elements of heavy-fermion k-space electronic structure (13, 37) because elastic scattering of electrons from −kðEÞ to + kðEÞ generates density-of-states interference patterns occurring as maxima at qðEÞ = kðEÞ − ð−kðEÞÞ = 2kðEÞ in B T>T εc K k gðq; EÞ. The onset of the heavy bands in CeCoIn5 is then detected (15) as a sudden transformation of the slowly changing structure of

Vem 02 Vem ygrenE gðq; EÞ, which appears at E ≈ 4 meV, followed by a rapid evolution EF of the maximum intensity features toward a smaller jqj-radius, f jqj ε < then by an abrupt jump to a larger -radius, and then by a k second rapid diminution of interference pattern jqj-radii. Thus, heavy-fermion QPI directly reveals the momentum structure of two -π/a momentum π/a heavy bands in the energy range −4 meV < E < 12 meV, and the 0 0 formation of their hybridization gap. Fig. 2 A and B shows typical examples of our CeCoIn5 heavy-fermion QPI data and shows the C T

r

enE EF and consists of a small hole-like Fermi surface arising from the heavy β-band and two larger electron-like Fermi surfaces ðα1; α2Þ resulting from the heavy α-band (15) (SI Text, section 1). Eq. 1 β Ek

-π/a0 momentum π/a0 (2π,0) Fig. 1. Effects of f-electron magnetism in a heavy-fermion material. (A) A B

The magnetic subsystem of CeCoIn5 consists of almost localized magnetic f-electrons (red arrows) with a weak hopping matrix element yielding a very narrow band with strong magnetic interactions between the f-electron spins. (B) The heavy f-electron band is shown schematically in red and the light c-electron band in black. (C) On the left, schematic of the result of hybridizing the c- and f-electrons in B into new composite electronic states referred to as heavy fermions (blue). On the right, the opening of a super- -2.1 meV -2.1 meV conducting energy gap is schematically shown by back-bending bands near the chemical potential. The microscopic interactions driving Cooper pairing of C these states, and thus of heavy-fermion superconductivity, have not been iden- tified unambiguously for any heavy-fermion compound. (π,π)

α2 implemented with δE ≈ 75 μeV at T ≤ 250 mK allowed detailed α k Δα;β 1 measurements of the -space energy gap structure k for the ar- β (π,0) chetypical heavy-fermion superconductor CeCoIn5 (16). Its normal- state properties are somewhat unconventional (17, 18) and seem to reflect the presence of strong antiferromagnetic spin fluctuations (19) [whether these fluctuations are associated with the existence of ky – a true (20 22)ispresentlyunclear].The kx compound is electronically quasi-2D (23) and its Tc = 2:3 Kis among the highest of the heavy-fermion superconductors (16). The Δα;β Cooper pairs are spin singlets (24, 25) and therefore k must D gnilpuocgnilpuoc I(r)

exhibit even parity. Application of the recently developed heavy- MFA (1,1) fermion QPI imaging technique (13, 15) to CeCoIn5 reveals the expected development with falling temperature of the heavy bands (1,0)(0,0) (2,0) (3,0) (26) in agreement with angle-resolved photoemission (27, 28). Ev- (0,0)(1,0) idence for a spin fluctuation-driven pairing mechanism in related

M heavy fermion compounds was adduced by comparing the change in F the magnetic exchange energy to the condensation energy (29). At (0,0) (1,1) (2,2) (3,3) lower temperatures, there is clear evidence that CeCoIn5 is an unconventional superconductor with a nodal energy gap (30–33) Fig. 2. Heavy-fermion band-structure determination for CeCoIn5.(A)Typical example of measured gðq,EÞ within the heavy bands. (B) Typical example of possibly with dx2−y2 order parameter symmetry (15, 34) [although this has not been verified directly using a phase-sensitive method predicted gðq,EÞ for our parameterization of the heavy-band structure. They are (35)]. The microscopic Cooper pairing mechanism of CeCoIn has, in good detailed agreement, as are the equivalent pairs of measured and pre- 5 dicted gðq,EÞ (SI Text,section1). (C) The Fermi surface of our heavy-band struc- however, not been established (7,16, 25, 36). r ∼ ture model (SI Text,section1). (D) -space structure of the magnetic interaction Heavy QPI imaging and BQPI studies of CeCoIn5 at 250 mK strength, IðrÞ,asobtainedfromEq. S2b.ThisformofIðrÞ reflects the existence can now yield accurate knowledge of the k-space structure α;β α;β of strong antiferromagnetic (AFM) correlations between adjacent localized of both Ek and Δk (15). Using a He-3 refrigerator-based moments and ferromagnetic (FM) correlations between next nearest neighbors.

2of5 | www.pnas.org/cgi/doi/10.1073/pnas.1409444111 Van Dyke et al. Downloaded by guest on September 25, 2021 A (π,2π) (2π,2π) B (0, π) (π, π) C + 0.6 θα - α + 2 0.4 α1 - 0.2 - α α (π,π) 1 2 k β qy 65.9 y + 600 0.0

)

Ve β - θβ -0.2

m(

cs - -0.4

V -41.7 -600+ -0.6 (0,0) (-π,-π) k 0 30 60 90 120 150 180 qx x θα , θβ (degrees)

Fig. 3. Predicted gap structure for CeCoIn5 if f-electron magnetism mediates Cooper pairing. (A) Magnetically mediated pairing potential, VSC ðqÞ = −IðqÞ=2, of CeCoIn5. The arrow represents the momentum Q = ð ± 1, ± 1Þπ=a0, where the pairing potential is large and repulsive. (B) Angular dependence of the α,β predicted superconducting gaps Δk in the α- and β-bands (note that the angles θα and θβ are measured around q = ðπ,πÞ and q = ð0,0Þ, respectively). The thickness and color of the Fermi surface encode the superconducting energy gap size and sign, respectively. These results were obtained at T = 0 using a Debye frequency of ωD = 0:66 meV. The arrow represents the momentum Q = ð ± 1, ± 1Þπ=a0 which connects momentum states on the Fermi surfaces where the superconducting gaps possess different signs. (C) Predicted values of the superconducting energy gaps as a function of the Fermi surface angle. The largest gap is located on the α1-Fermi surface, whereas a small gap exists on the central β–Fermi surface and the outer α2-Fermi surface. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α;β α;β α;β 2 α;β 2 shows how the precise dispersions Ek of the two heavy-fermion Here Ωp = ðEp Þ + ðΔp Þ are the two pairs of Bogoliubov f bands are fixed by «k, which is itself generated by the Heisenberg bands, wk and xk, are the coherence factors of the heavy- PHYSICS interaction energy Iðr − r′ÞSr · Sr′ between spins Sr and Sr′ at f- fermion hybridization process, and the primed sum runs only α;β electron sites r and r′ (38) (Eq. S1 and SI Text,section1). over those momentum states p whose energies Ep lie within ω Therefore, the real space (r-space) form of the magnetic in- the interaction cutoff energy, D, of the Fermi energy. Our sol- teraction potential, Iðr − r′Þ, can be determined directly from utions to Eq. 2 (SI Text,section2) predict that the α-andβ-bands α;β α;β Δ 2− 2 the measured Ek (Eq. S2b and SI Text, section 1). Carrying out of CeCoIn5 possess superconducting gaps k of nodal dx y this procedure reveals quantitatively the form of IðrÞ for the symmetry as shown in Fig. 3 B and C. This symmetry is dictated Q = ð ± ; ± Þπ= f-electron magnetic interactions of CeCoIn (Fig. 2D) and by the large repulsive pairing potential near 1 1 a0 5 (arrow in Fig. 3A), which requires that the superconducting gap therefore that strong antiferromagnetic interactions occur be- Q tween adjacent f-electron moments (SI Text, section 1). changes sign between Fermi surface points connected by ,as shown in Fig. 3B. We predict that the maximum gap value occurs α Δα;β Solution of Gap Equations with a Magnetic f-Electron on the 1-Fermi surface, with k being well approximated by Interaction Kernel Δα È à  à α 0 These interactions are hypothesized to give rise to an effective Δk = cosðkxÞ − cosðkyÞ + α1 cosð2kxÞ − cosð2kyÞ ðr − r′Þ = − ðr − r′Þ= S14 2 electron-pairing potential VSC I 2(Eq. and SI  ÃÉ ðr − r′Þ Text,section2.1), with the opposite sign to I because anti- + α2 cosð3kxÞ − cosð3kyÞ [3] parallel spins at sites r and r′ (for I > 0) experience an attractive ðV < 0Þ pairing potential; we are assuming spin-singlet pairing Δβ à SC β 0 3 ðrÞ ðqÞ Δk = cosðkxÞ − cosðkyÞ : throughout. Fourier transformation of VSC yields VSC as 2 shown in Fig. 3A, thereby revealing that the putative pairing ðqÞ q = ð ± ; ± Þπ= Δα = : α = − : α = − : Δβ = − : potential VSC is strongly repulsive at 1 1 a0 and Here 0 0 49 meV, 1 0 61, 2 0 08, and 1 04 meV 0Δα;β 3 attractive at q = ð ± 1; 0Þπ=a0; ð0; ± 1Þπ=a0. It is this strong re- represent the quantitative predictions for the k in Eq. 2 ω = : pulsion at q = ð ± 1; ± 1Þπ=a0 which is the long-anticipated (1–11) derived from the hypothesis of Eq. with D 0 66 meV constrain- requirement for unconventional Cooper pairing to be mediated ing the maximum possible energy gap to be Δmax ≤ 600 μeV. The by antiferromagnetic interactions. Finally, inserting this hypoth- solution of Eq. 2 (without any further adjustable parameters) = : = : esized pairing potential VSCðqÞ into the coupled superconducting then also predicts Tc 2 96 K, which is reduced to Tc 2 55 K α;β once one accounts for the experimentally observed mean free gap equations for both heavy bands Ek of CeCoIn5 yields path of l = 81 nm (Eq. S21). Overall, the predicted gap structure α;β 2 X Δk (Eq. 3) and Tc are in striking quantitative agreement with Δα = −xk ′ ð p − kÞ k VSC the measured ΔiðkÞ (15) and Tc = 2:3 K (16) for CeCoIn5. N p These results raise several interesting questions. First, al- "    # α α β β though ωD appears above as a phenomenological parameter, the Δp Ωp Δp Ωp 3 x2 tanh + w2 tanh question naturally arises of how such a cross-over scale arises from p Ωα p Ωβ 2 p 2kBT 2 p 2kBT the interplay between the form of the spin excitation spectrum, [2] the strength of the coupling between f-electrons and spin fluc- 2 X β wk ′ tuations, and the flatness of the f-electron bands. To address this Δk = − VSCð p − kÞ N p question, it will be necessary to extend the above method to a strong coupling, Eliashberg-type approach. Second, although our "    # Δα Ωα Δβ Ωβ approach assumes that the formation of coherent (screened) Kondo 2 p p 2 p p 3 xp α tanh + wp tanh : lattice is concluded prior to the onset of superconductivity, it has 2Ω 2k T Ωβ 2k T p B 2 p B been suggested (39) that singlet formation might continue into the

Van Dyke et al. PNAS Early Edition | 3of5 Downloaded by guest on September 25, 2021 (π,π) 1000 1000 A D E q1 100 100 q 2 + 10 -1 10

(s ) (q,E)

1 th

1/T Δg

(arb. units) 1 1 E=-0.5 meV - 1

(π,π) 1/T 0.1 0.1 B 0.01 0.01 0.01 0.1 1 10 0.01 0.1 1 10 + T (K) T (K)

normal state T=1.3 K Δg(q,E) F G SC state T=3 K E=-0.5 meV - 30 Q=(π,π) 30 (π,π) C q 20 B 20 2 + (arb. units) -

(Q,E)

χ 10 10 - q’1 χ″(q,ω) (μ² /meV)

Im

0 0 + 0.40.0 1.20.8 0.80.40.0 1.2 E (meV) E (meV)

Fig. 4. Comparison of predicted phenomenology of CeCoIn5 if f-electron magnetism mediates Cooper pairing, with experimental data. (A) Predicted PQPI Δα,β Δ ðq Þ = ðq Þ − ðq Þ = − : Δ ðq Þ scattering pattern for the predicted k with dx2 −y2 symmetry: g ,E,B g ,E,B g ,E,0 for E 0 5 meV. g ,E,B is negative (red) for scattering q vectors connecting parts of the Fermi surface with opposite signs in the order parameter, such as 1 (C), whereas sign-preserving scattering leads to a positive Δ ðq Þ q Δ ðq Þ = ðq Þ − ðq Þ = − : ðq Þ ðq Þ g ,E,B (blue) such as 2 (C). See SI Text, section 2.3 for full details. (B) Measured PQPI g ,E,B g ,E,B g ,E,0 for E 0 5 meV. g ,E,B and g ,E,0 are measured in the identical field of view using identical measurement parameters at B = 0 and B = 3T. Δgðq,E,BÞ exhibits the same enhancement and q suppression for 1,2 as in A. The good correspondence, especially for the relevant scattering vectors between regions whose gaps are predicted to have opposite signs between theoretically predicted Δgðq,E,BÞ and measurements thereof is a phase-sensitive verification of a d-wave gap symmetry in CeCoIn5. See SI Text, section 2.3 for additional energies and details. (C) Equal energy contour (EEC) for E = −0:5 meV used in A and B (i.e., momentum points with = −Ωαβ, ðq Þ q ðq Þ E k ) in the superconducting state. Scattering processes yielding the dominant contribution to g 1,2,E , with 1 2 connecting points on the EEC with + − q′ = ð π πÞ − q opposite phases (the same phase) of the superconducting gap are shown (the phases are indicated by / ). For simplicity, we show 1 2 ,2 1,the q Umklapp vector to 1. Note that the coordinate system is rotated by 45° with respect to A and B.(D) Predicted temperature dependence of the nuclear α,β relaxation rate 1=T1 for the superconducting state of CeCoIn5 based on combining the f-electron magnetic interactions IðqÞ and the predicted Δk .(E) Measured temperature dependence of 1=T1 for the superconducting state of CeCoIn5 (data from ref. 25). (F) Predicted imaginary part of the dynamical spin susceptibility χðQ,EÞ at Q = ð ± 1, ± 1Þπ=a0 in the superconducting state of CeCoIn5 based on combining the f-electron magnetic interactions IðqÞ and α,β the predicted Δk . A strong resonance is predicted below Tc at E ≈ 0:6 meV. (G) Measured imaginary part of the dynamical spin susceptibility χðQ,EÞ at Q ≈ ð ± 1, ± 1Þπ=a0 in the superconducting state of CeCoIn5 (modified from ref. 36). There is a strong quantitative correspondence to the model prediction.

superconducting state. New theoretical/experimental approaches altering the nature of the scattering potential provide direct in- Δ Δ will be required to determine if this type of composite pairing might formation on the relative phase difference between k1 and k2 . be detectable in the temperature dependence of physical proper- This phenomenon has been beautifully demonstrated by con- ties for T K Tc. Notwithstanding these questions, our first focus sidering magnetic field-induced changes in the conductance ratio is now to evaluate the predictive utility of the relatively simple in Bi2Sr2CaCu2O8, a single-band cuprate superconductor with Δα;β approach that was proposed originally (1–4) and has been dx2−y2 symmetry (35). Although the predicted symmetry of k in implemented here. CeCoIn5 is similarly dx2−y2 (Fig. 3), the detailed predictions for magnetic field-induced changes in gðq; EÞ are quite complex Phase-Sensitive QPI α;β because of the multiple superconducting gaps and intricate band Given this detailed new understanding of Δk (Eq. 3 and Fig. 3 B geometry (SI Text,section2.3). In Fig. 4A we show our pre- and C), a variety of other testable predictions for superconducting dicted values of the field-induced QPI changes Δgðq; E; BÞ = characteristics of CeCoIn5 become possible. In particular, the gðq; E; BÞ − gðq; E; 0Þ for a typical energy, E = −0:5 meV below phase of the predicted dx2−y2 symmetry gaps is directly reflected the gap maximum (SI Text, section 2.3). For comparison, in Fig. in the magnitude of gðq; EÞ. The scattering of Bogoliubov qua- 4B we show the measured Δgðq; E; BÞ at the same energy (SI siparticles between momentum points k1 and k2 near the Fermi Text,section2.3). Not only is the agreement between them as surface leads to a contribution to gðq = k1 − k2; EÞ that is directly to which regions of q-space have enhanced or diminished scattering Δ Δ Δ ðq; ; Þ proportional to the product k1 k2 . For time-reversal invariant intensity evident, but they also demonstrate that g E B is ðq = k − k ; Þ q ðq Þ scalar-potential scattering, this contribution enters g 1 2 E positive (negative) for scattering vectors, 1 2 connecting parts with a sign that is opposite to that for time-reversal violating of the Fermi surface with the same sign (different signs) of the magnetic scattering. As a result, changes in gðq; EÞ generated by superconducting gap (Fig. 4C). In contrast, the equivalent

4of5 | www.pnas.org/cgi/doi/10.1073/pnas.1409444111 Van Dyke et al. Downloaded by guest on September 25, 2021 predictions for phase-sensitive Δgðq; E; BÞ if the gap symmetry of the spin resonance, as well as the form of the spin excitation is a nodal s-wave bear little discernible relationship to the ex- spectrum both below and above Tc is remarkable. Moreover, the perimental data (SI Text,section2.3). Thus, the application in value of ωD used in Eq. 2 is now seen to be quite consistent with the CeCoIn5 of the phase-sensitive QPI (PQPI) technique (35) energy scale of the spin fluctuation spectrum (SI Text,section4). reveals the predicted effects of sign changes in d 2− 2 symmetry α;β x y gaps of structure Δk . Conclusions To summarize, the r- and q-space structure of magnetic inter- Spin Excitations actions between f-electrons, IðrÞ and IðqÞ, in the heavy-fermion Lastly, by combining the f-electron magnetic interactions IðrÞ (Fig. α;β state of CeCoIn5 are determined quantitatively (Figs. 2D and 2D) and the predicted Δ (Fig. 3), we can investigate the spin α;β k 3A) from our measured heavy-band dispersions Ek (SI Text,sec- dynamics of CeCoIn5 in the superconducting state (SI Text,section tion 1). The coupled superconducting gap equations (Eq. 2)are 3). One test for the nodal character of the predicted dx2−y2 sym- then solved using the hypothesis that it is these magnetic inter- metry superconducting gap is the temperature dependence of the actions that mediate Cooper pairing with V ðqÞ = −IðqÞ=2. This spin–lattice nuclear relaxation rate 1=T : Its theoretically predicted SC 1 allows a series of quantitative predictions regarding the physical formisshowninFig.4D.Thisisingoodagreementwiththatof properties of the superconducting state of CeCoIn5. These include the measured 1=T1 reproduced in Fig. 4E from ref. 25. The the superconducting critical temperature Tc,thek-space structure theoretically predicted and experimentally observed power-law α;β α of the two energy gaps Δk (Fig. 3), the phase-sensitive BQPI sig- dependence 1=T1 ≈ T with α ≈ 2:5 (straight lines in Fig. 4 D and E) shows an exponent that is reduced from the expected α = 3 for nature arising from the predicted dx2−y2 symmetry of the super- conducting gaps (Fig. 4A), the temperature dependence of the spin– a single-band d 2− 2 wave superconductor. We demonstrate that x y α;β = this effect is actually due to fine details of the Δ multigap lattice relaxation rate 1 T1 (Fig. 4D), and the existence and struc- k q = ð ± ; ± Þπ= structure on the α- and β-Fermi surfaces (Fig. 3C). Finally, our ture of a magnetic spin resonance near 1 1 a0 (Fig. theoretically predicted dynamic spin susceptibility in the super- 4F). The demonstrated quantitative agreement between all these α;β conducting state, using the extracted IðrÞ and computed Δk with predictions (themselves based on measured input parameters) no further adjustable parameters, exhibits a spin resonance peak and the disparate experimental characteristics of superconducting at q = ð ± 1; ± 1Þπ=a0 and at E ≈ 0:6 meV (SI Text, section 3)as CeCoIn5 (15, 16, 25, 29, 31, 34, 36) provide direct evidence that

shown in Fig. 4F. Such spin resonances are a direct signature of its Cooper pairing is indeed mediated by the residual f-electron PHYSICS an unconventional superconducting order parameter: They arise magnetism (Eqs. 2 and 3). from spin–flip transitions that involve states k1 and k2 with op- posite signs of the superconducting gap. The experimental data ACKNOWLEDGMENTS. We acknowledge and thank M. Aprili, P. Coleman, M. H. Fischer, R. Flint, P. Fulde, M. Hamidian, E.-A. Kim, D. H. Lee, A. P. Mackenzie, on spin excitations in superconducting CeCoIn5 from inelastic neutron scattering (reproduced from ref. 36) are shown in Fig. 4G, M. R. Norman, J. P. Reid, and J. Thompson for helpful discussions and com- ≈ : munications. This work was supported by the US Department of Energy un- and exhibit a strong resonance located at E 0 6 meV. The der Contract DEAC02-98CH10886 (to J.C.S.D. and C.P.) and Award DE-FG02- quantitative agreement between the predicted and measured energy 05ER46225 (to D.K.M. and J.S.V.).

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