Neutron scattering studies on spin fluctuations in Sr_2RuO_4
K.Jenni, S. Kunkemöller, P. Steffens, Y. Sidid, R. Bewley, Z. Q. Mao, Y. Maeno, M. Braden
NNeutron scattering studies on spin fluctuations in Sr RuO K. Jenni, ∗ S. Kunkem¨oller, P. Steffens, Y. Sidis, R. Bewley, Z. Q. Mao,
5, 6, 7
Y. Maeno, and M. Braden † II . Physikalisches Institut, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, D-50937 K¨oln, Germany Institut Laue Langevin,71 avenue des Martyrs, 38000 Grenoble, France Laboratoire L´eon Brillouin, C.E.A./C.N.R.S., F-91191 Gif-sur-Yvette CEDEX, France ISIS, United Kingdom Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Tulane University, New Orleans, LA 70118, USA Department of Physics, Pennsylvania State University, University Park, PA 16802, USA (Dated: February 15, 2021)The magnetic excitations in Sr RuO are studied by polarized and unpolarized neutron scatteringexperiments as a function of temperature. At the scattering vector of the Fermi-surface nestingwith a half-integer out-of-plane component, there is no evidence for the appearance of a resonanceexcitation in the superconducting phase. The body of existing data indicates weakening of thescattered intensity in the nesting spectrum to occur at very low energies. The nesting signal persistsup to 290 K but is strongly reduced. In contrast, a quasi-ferromagnetic contribution maintains itsstrength and still exhibits a finite width in momentum space. I. INTRODUCTION
A quarter century after the discovery of superconduc-tivity in Sr RuO [1] its character and its pairing mech-anism remain mysterious. Inspired by the ferromagneticorder appearing in the metallic sister compound SrRuO [2] it was initially proposed that ferromagnetic fluctua-tions drive the superconductivity in Sr RuO renderingits superconductivity similar to the A-phase of superfluid He [3, 4]. For a long time chiral p -wave superconductiv-ity with spin-triplet pairing has been considered to bestdescribe the majority of experimental studies [5, 6] al-though the absence of detectable edge currents [7] andthe constant Knight-shift observed for fields perpendicu-lar to the Ru layers [8] were not easily explained in thisscenario [9]. Further insight was gained from experimentsperformed under large uniaxial strain that revealed a con-siderable enhancement of the superconducting transitiontemperature by more than a factor two [10, 11], similarto the enhancement in the eutectic crystals [6]. How-ever, the breaking of the four-fold axis should split thesuperconducting transition of the chiral state in contra-diction with a single anomaly appearing in the specificheat under strain [12]. Furthermore, the strain depen-dence of the transition temperature close to zero strainis flat [10, 13], whereas one expects a linear dependencefor the chiral state.The picture of chiral p -wave superconductivity wasfully shaken, when the two experiments yielding thestrongest support for triplet pairing [14, 15] were revised.The new studies of the Knight shift in NMR [16, 17] andthose of the polarized neutron diffraction [18] reveal anunambiguous drop of the electronic susceptibility that isinconsistent with spin-triplet pairs parallel to Ru layers. ∗ [e-mail: ][email protected] † [e-mail: ][email protected] Since then, numerous proposals for the superconductingstate were made mostly invoking some d -wave state andthe discussion of the superconducting pairing has becomevery active [19–26]. The observations of broken time-reversal symmetry in muon spin relaxation experiments[27, 28] and in measurements of the magneto-optical Kerreffect [29] may require interpretations other than the chi-ral p -wave scenario. Many theories discuss a supercon-ducting state with a complex combination of components[19–25].Assuming a simple boson-mediated pairing followingBCS theory, phonons and magnetic fluctuations or a com-bination of both [30] can be relevant. There are anoma-lies in the phonon dispersion that could be fingerprintsof electron phonon coupling [31, 32]. The phonon modethat describes the rotation of the RuO octahedra aroundthe c axis exhibits an anomalous temperature dependenceand severe broadening [31]. This mode can be associatedwith the structural phase transition and with the shift ofthe van Hove singularity in the γ band through the Fermilevel. Both effects occur upon small Ca substitution[33, 34]. In addition, the Ru-O bond-stretching modesthat exhibit an anomalous downwards dispersion in manyoxides with perovskite-related structure [35] exhibit ananomalous dispersion in Sr RuO as well [32]. Compar-ing the first-principles calculated [36] and measured [32]phonon dispersion in Sr RuO the agreement is worstfor these longitudinal bond-stretching modes, which ex-hibit a flatter dispersion indicating better screening com-pared to the density functional theory (DFT) calcula-tions. Note, however, that perovskite oxides close tocharge ordering exhibit a much stronger renormalizationof the zone-boundary modes with breathing characterthat is frequently labeled overscreening [35, 37].On the other side there is clear evidence for strongmagnetic fluctuations deduced from NMR [38] and inelas-tic neutron scattering (INS) experiments [39–45]. Thedominating magnetic signal is incommensurate and stemsfrom nesting in the one-dimensional bands associated a r X i v : . [ c ond - m a t . s up r- c on ] F e b with d xz and d yz orbitals, see Fig. 1. The relevance ofthis instability towards an incommensurate spin-densitywave (SDW) is underlined by the observation of staticmagnetic order emerging at this q position in recipro-cal space for minor substitution of Ru by Ti [46] or ofSr by Ca [47, 48]. A repulsive impurity potential wasrecently proposed to form the nucleation center for themagnetic ordering that should strongly couple to chargecurrents [49]. Furthermore, the temperature dependenceof these incommensurate magnetic fluctuations in pureSr RuO agrees with a closeness to a quantum criticalpoint [40]. These nesting-induced magnetic fluctuationscan easily be explained by DFT calculations using therandom phase approximations (RPA) [50] but their rel-evance for the superconducting pairing remains contro-versial [51]. Inelastic neutron scattering in the supercon-ducting state can exclude the opening of a large gap forthese nesting-driven [52]. Since magnetic excitations areparticle hole excitations one expects in the most simpleisotropic case a magnetic gap comparable to twice the su-perconducting one, which can be safely excluded. How-ever, the anisotropy of the gap function and interactionscan strongly modify the magnetic response in the super-conducting state. A more recent TOF inelastic neutronscattering experiments confirms the absence of a largegap but reports weak evidence for suppression of spectralweight at very low energies [53]. This experiment alsoclaims the occurrence of a spin resonance mode at thenesting position with a finite perpendicular wave-vectorcomponent, which would point to an essential modulationof the superconducting gap perpendicular to the RuO layers but which is inconsistent with the results of thiswork.In addition to the incommensurate nesting-inducedfluctuations, macroscopic susceptibility [54], NMR [38,55] and also polarized inelastic neutron scattering exper-iments [42, 45] reveal the existence of magnetic fluctua-tions centered at the origin of the Brillouin zone, whichtypically can be associated with ferromagnetism. Futher-more, a small concentration Co doping can lead to staticshort-range ferromagnetic order [56]. All techniques findalmost temperature independent quasi-ferromagnetic ex-citations in pure Sr RuO . This ferromagnetic responsequalitatively agrees with a recent dynamical mean fieldtheory (DMFT) analysis of magnetic fluctuations [57],which finds essentially local magnetic fluctuations super-posed on the well known nesting signal. However, theneutron data disagree with a fully local character as theyshow a finite q dependence [45]. The quasi-ferromagneticfluctuations also disagree with the expectations for anearly ferromagnetic system that exhibits paramagnonscattering [45, 58]. SrRuO clearly exhibits such param-agnon scattering with its well-defined structure in q andenergy space [59].Here we present additional neutron scattering exper-iments on the magnetic fluctuations in Sr RuO , whichfocus on several aspects that are particularly relevant forthe superconducting pairing mechanism involving mag- FIG. 1. Fermi surface of Sr RuO for k z = 0. The bands arebased on LDA+SO calculations from [60] and marked by dif-ferent colors. The black arrow represents the dominant nest-ing vector between the one-dimensional sheets α (red) and β (blue). The incommensurate positions of the in-plane nestingsignal are marked by different symbols. The circles repre-sent the crystallographically equivalent positions ( ± ± ± ± ± ± ± ± netic fluctuations or for the general understanding ofmagnetic excitations in a strongly correlated electron sys-tem. We discuss the possibility of important out-of-planedispersion in the magnetic response in the superconduct-ing and normal states, the shape of nesting scatteringaway from the peak position and the non-local characterof the quasi-ferromagnetic response. II. EXPERIMENTAL
INS experiments were carried out on the ThALES[61, 62] and IN20 [63] triple-axis spectrometers (TAS)at the Institut Laue Langevin and on the LET [64] time-of-flight (TOF) spectrometer at the ISIS Neutron andMuon Source. We used an assembly of 12 Sr RuO crys-tals with a total volume of 2.2 cm in all experiments.At Kyoto University, the crystals were grown using thefloating zone method and similar crystals were studied inmany experiments [5, 6]. The crystal assembly was ori-ented in the [100]/[010] scattering plane (correspondingto a vertical c axis) to study the in-plane physics of theRu layers. Additionally, with the instruments ThALESand LET it was possible to access parts of the q spaceperpendicular to the plane which enables an analysis ofthe out-of-plane dispersion of the magnetic response. Toconduct experiments inside the superconducting phasea dilution refrigerator was used, reaching a temperatureof ∼
200 mK, well below the transition temperature of ∼ ∼ µ eV. The TOF spec-trometer LET records data simultaneously with four dif-ferent values of the incidental energies, E i and resolu-tions, while the energy resolution of the TAS ThALES isdetermined by the chosen final neutron wave vector k f of1.57 ˚ A − combined with the collimations. On ThALESthe best intensity to background ratio was achieved byusing a Si(111) monochromator and PG(002) analyzercombined with a radial collimator in front of the analyzerfor further background reduction. The same configura-tion was also used in an earlier study [52].A polarized neutron scattering experiment was per-formed on the thermal TAS IN20 using Heusler crystalsas monochromator and analyzer. A spin flipper in front ofthe analyzer enabled the polarization analysis. The scanswere performed with a fixed final momentum of k f =4.1 ˚ A − , where the graphite filter in front of the analyzercuts higher order contaminations. Longitudinal polar-ization analysis was performed with a set of Helmholtzcoils. III. RESULTS AND DISCUSSIONA. q dependence of fluctuations associated withnesting The TOF technique enables an imaging of the com-plete Q - E -space, which gives insight on the distributionof scattering intensity in the reciprocal space. Through-out the paper, the scattering vector Q =( H , K , L ) and thepropagation vector in the first Brillouin zone q =( q h , q k , q l )are given in reciprocal lattice units (rlu). We mostlyconsider only the planar wave vector Q =( H , K ) pro-jection. Fig. 2 shows the inelastic scattering plottedagainst the H , K components of the scattering vectorin the superconducting phase. The four different pan-els display sections of the two-dimensional ( H , K ) planefor different incident energies and hence different reso-lutions. The intensities are fully integrated along theenergy transfer (depending on the incident energy) andalong the out-of-plane component of the scattering vec-tor, -0.7 < L < ± ± ± ξ ,0] and [0, ξ ] directions that were firstreported in [43, 44]. The arc visible in Fig. 2(d) connect-ing (-0.3,0.3) and (0.3,0.3) is a spurious signal; it doesnot appear for the other incidental energies.Neglecting electronic dispersion perpendicular to theplanes and assuming an idealized scheme of flat one-dimensional bands originating from the d xz and d yz or-bitals, one expects nesting induced magnetic excitations FIG. 2. In-plane scattering in the superconducting phase (T= 0.2 K). The TOF data at four different incidental ener-gies display the magnetic scattering distribution in the ab plane. The intense signal at the incommensurate positions(0.3,0.3), (0.7,0.7), and (0.3,0.7) is visible for all E i . Addition-ally there is magnetic scattering between the incommensuratepositions in [ ξ ,0] and [0, ξ ] directions, respectively. To increasethe statistics the data is integrated over the maximum L range of [-0.7,0.7] and full E range depending on the inciden-tal energy (1.75 < E <
10 for E i = 14 . < E < E i = 8 . < E < E i = 5 . < E < E i = 4 . for any two-dimensional vector Q =(0.3, ξ ) and ( ξ ,0.3)and accordingly a peak at (0.3,0.3) [50]. The peaksclearly dominate but the ridges are also detectable -mostly for the positions connecting the nesting peaks,i.e. 0.3 < ξ < ξ ,0] directioncalculated from the data taken with E i =14.13 meV (Fig.2(a)). By subtracting the background obtained fromthe average of ( ξ ,0.15) and ( ξ ,0.45), shown in Fig. 3(b)and (c) respectively, we isolate the signal along the line( ξ ,0.3) shown in Fig. 3(d). The ridge scattering ismainly detectable between the peaks at the incommensu-rate positions, as it is visible in the two one-dimensionalcuts representing the background parallel to the ridgeon both sides (Fig. 3(b) and (c)). While the ( ξ ,0.15)cut exhibits only a weak signal around (-0.3,0.15) the( ξ ,0.45) cut shows clearly two peaks at the (-0.7,0.45)and (-0.3,0.45) positions representing the ridges in [0, ξ ]direction. The rounding of the one-dimensional Fermi-surface sheets suppresses the susceptibility at (0.3, ξ ) with ξ lower than 0.3, but this suppression is not abrupt. Be-sides the ridge scattering we may also confirm the pro-nounced asymmetry of the nesting peak with a shouldernear (0.25,0.3) and equivalent positions. This shoulderwas reported in [40] and was also found in the full RPAcalculations.The asymmetry of the nesting peaks and the ridge scat-tering between the incommensurate positions can also beseen in the data of lower incident energies (see Fig. 3(e)and (f)). The one-dimensional cuts for different K val-ues confirm the asymmetric shape of the nesting peaks.A thorough analysis of the pure magnetic signal as in thecase of E i = 14 .
13 meV is not possible due to uncertaintyin the background. Furthermore, the ridge scattering isless pronounced in the data obtained with lower incidentenergies, which indicates a higher characteristic energy ofthe ridge scattering. This further explains why the muchweaker scattering in the ridges has not been detected inearly TAS studies [39–41].
B. Search for gap opening or a resonance modebelow T c The opening of a superconductivity-induced gap in thespectrum of magnetic fluctuations would have strong im-pact on the discussion of the superconducting characterin Sr RuO . Previous INS experiments using a TAS re-vealed the clear absence of a large gap at the nesting po-sition [52], whereas a recent TOF experiment reports atiny gap although the statistics remained very poor [53].Studying the magnetic response of Sr RuO in its su-perconducting phase by INS is challenging, because one FIG. 3. Magnetic scattering along the connection of the in-commensurate positions. (a)-(c) show one-dimensional cutsfrom Fig. 2(a) along the ( ξ , K ) paths for K = 0.15 (a), 0.3(b), and 0.45 (c). The background at both sides of the incom-mensurate positions is displayed in (a) and (c) (represented bythe same colored rectangles in 3(a)). An averaged backgroundis formed from both (gray open circles) and fitted with a lin-ear contribution and two Gaussians (black solid line). This iscompared to the incommensurate signal in (b). In (d) the lin-ear background contribution (black dashed line) is subtractedand the signal along the [ ξ ,0] direction is fitted with two skewGaussians for the incommensurate signal and a broad Gaus-sian fixed at ξ = 0 . K and two different incident energies 8.78 meV and5.64 meV taken from Fig. 2(b) and (c). The integration rangein [0, ξ ] direction is ± .
025 around the K value and the scansare shifted vertically for better visibility. needs to focus on small energies of the order of 0.2 to0.5 meV. At these energies the signal in the normal stateis at least one order of magnitude below its maximumstrength at 6 meV, and the required high-energy reso-lution further suppresses statistics. Fig. 4 presents theTOF data obtained with E i = 3 meV by calculating theenergy dependence at the nesting position integrated overall L values. The full L integration is needed to enhancethe statistics. In Fig. 4 (a) and (b) we compare the rawdata for both temperatures with the background signal.In (c) the background subtracted magnetic response inthe superconducting phase is compared to that in the FIG. 4. Low energy dependence of incommensurate signalbelow and above the superconducting transition extractedfrom TOF data. (a) and (b) display the energy scans atat q =(0.3,0.3) below (T = 0.2 K) and above (T = 2 K) thesuperconducting phase transition. The background in bothpanels is derived from the constant Q cut at (0.09,0.41) forboth temperatures ( | Q IC | = | Q bg | ). To increase statistics theTOF data with an incidental energy of 3 meV is fully inte-grated over L (range [-0.7,0.7]) and symmetrized by foldingin q space at (0.3,0.7) along the (1,-1,0) plane. The H and Kcomponent is integrated with the range [0.25,0.35] (c) Back-ground subtraction and Bose factor correction yields the puremagnetic response at low energies which is compared insideand outside the superconducting phase. normal phase. There is no evidence for the opening of agap within the statistics of this TOF experiment. Also aresonance at a finite energy cannot be detected. Admit-tedly the statistics of this TOF data is too poor to detectsmall signals or their suppression.Following the claim of Iida et al. [53] the TOF datais also analyzed in terms of a possible resonance modeappearing at a finite value of the L component, i.e. at(0.3,0.3,0.5). Therefore, the L dependence of the mag-netic signal at (0.3,0.3, L ) is determined by backgroundsubtraction and compared for the two temperatures (seeFig. 5). The different panels represent the energy rangesfrom reference 53, where a resonance appearing at 0.56meV is proposed for L =0.5. In our data shown in Fig.5(b), there is no difference visible between superconduct-ing and normal phase at L = ± L dependencein more detail and with better statistics the TAS is bet-ter suited since measurements can be focused to single Q ,E points. Using ThALES and its high flux and en-ergy resolution constant Q scans at the incommensurateposition (0.3,0.7, L ) with L =0, 0.25, and 0.5 were mea- FIG. 5. L dependence of incommensurate signal at low en-ergies extracted from TOF data. Constant E cuts with anintegration width of 0.2 meV at the incommensurate position(0.3,0.3, L ) were adjusted for the measured background at thesame energy at (0.09,0.41, L ) and corrected for the Bose factor.The L dependence of the magnetic response in the supercon-ducting phase (blue) is compared to the normal phase (red).Additionally, the square of the Ru form factor is depictedin each panel (black dashed line). There is no evidence for apeak at L =0.5. sured to investigate the L dependence of the low energyresponse (see Fig. 6). This incommensurate position waschosen due to a better signal-to-noise ratio compared to(0.3,0.3, L ) and because the larger | Q | value allows oneto reach finite L values by tilting the cryostat. Similarto Fig. 4 (a) and (b) the raw data for two tempera-tures is shown in Fig. 6 (a)-(c). The background wasmeasured by rotating ω by 20 degrees for each L valueand then combining all three backgrounds to an aver-age. For all L values the intensity of the incommen-surate signal increases approximately linearly for smallenergies, following the established single relaxor behav-ior. Comparing the two temperatures there is no differ-ence noticable for any L value down to the energy res-olution. Especially around 0.56 meV where Iida et al.[53] propose a resonance at the incommensurate position(0.3,0.3,0.5) the two temperatures yield comparable sig-nals. It should be noted here that while the incommen-surate positions (0.3,0.3,0) and (0.3,0.7,0) are crystallo-graphically not equivalent both positions become equiva-lent with the L component 0.5, see Fig. 1. Therefore, thedata taken at (0.3,0.3,0.5) and (0.7,0.3,0.5) can be com-pared. To emphasize the absence of a resonance modearound 0.56 meV the data from Fig. 6 is plotted witha larger energy binning to further increase the statistics(see Fig. 7, which also indicates the broad energy integra-tion used in [53]). There is no significant deviation fromthe general linear behavior for any L value at low tem-peratures detectable. Iida et al. [53] report an increase ofsignal of ∼
60% for L =0.5 in the superconducting phase,which clearly is incompatible with our data that offerhigher statistics.Since no L dependence of the magnetic low energyresponse can be established (Fig. 6 and 7) we mergethe data and compare it with the former published low-energy dependence of the incommensurate signal [52] (seeFig. 8). The new experiments below T c fully confirm thatthe nesting excitations in Sr RuO do not exhibit a largegap, i.e. a magnetic gap comparable to twice the su-perconducting one. Combining all the previous and newdata there is, however, some weak evidence for the sup-pression of magnetic scattering at very low energies below0.25 meV. With the neutron instrumentation of today itseems very difficult to further characterize the suppres-sion of the small signal at such low energy.For the previously assumed superconducting state de-tailed theoretical analyzes of the magnetic response werereported [51], but concerning the more recently proposedsuperconducting symmetries [19–26] such investigationslack. The d x − y state deduced from quasiparticle in-terference imaging [26] exhibits nodes at Fermi-surfacepositions that are connected through the nesting vector.This implies that even at very low energies, the nest-ing induced excitations are not fully suppressed in such d x − y superconducting state, in agreement with the ex-perimental absence of a large gap in the nesting spectrum[52]. Within the d x − y superconducting state the nest-ing vector also connects Fermi-surface regions with maxi-mum and minimum gap values and it connects either tworegions of the β sheet or one β region with an α region.Therefore the conditions for a spin-resonance mode aremore complex and less favorable than in the case of theFeAs-based superconductors, where the s + − supercon-ducting symmetry and the nesting magnetic fluctuationsperfectly match each other [65]. C. Shape of the quasi-ferromagnetic fluctuations
The polarization analysis of inelastic neutron scatter-ing provides the separation of the magnetic from anyother scattering contribution. It is therefore possibleto identify a tiny magnetic response that is little struc-tured in q space. This technique was used to detect FIG. 6. L dependence of incommensurate signal at low en-ergies extracted from TAS data. The constant q scans wereconducted at the incommensurate positions (0.3,0.7, L ) with L = 0, 0.25, and 0.5 in the superconducting and normal phase.The background for each L is measured after ω rotation of20 deg, thus keeping | q | constant, and later averaged for allscans, yielding the presented background (black circles) andits fit (gray). The intensity is normalized with 1980000 mon-itor counts which corresponds to a measuring time of about15 min per point.FIG. 7. Comparison of the background-free incommensu-rate signal Q=(0.3,0.7, L ) for different L values and temper-atures. The compared data originates from the constant q scans shown in Fig. 6. The binning is increased to ∆ E =0.1 meV which yields better statistics. A linear fit (red line)provides a guide to the eye. The energy range of the proposedspin resonance [53] is indicated by the red box. FIG. 8. Comparison of the energy dependence of the in-commensurate signal with former published data from [52](labelled: Kunkem¨oller PRL). Background corrected datarecorded at (0.3, 0.3, 0) (circles) is given in panel (a); thebackground free signal at (0.3, 0.7) is averaged over all L values for both temperatures (diamonds), panel (b), and theincommensurate signal reported in [52] (triangles) is shownin (c). Data were corrected for the Bose and magnetic formfactors. quasi-ferromagnetic fluctuations and to determine theirstrength in comparison to the incommensurate fluctua-tions in Sr RuO [45]. We wished to extend this studyfocusing on the q dependence of the magnetic quasi-ferromagnetic response. Recent DMFT calculations [57]find evidence for local fluctuations superposing the well-established nesting excitations, which qualitatively agreewith the experimental quasi-ferromagnetic signal. How-ever, while the neutron experiments indicate a finite sup-pression of the quasi-ferromagnetic response towards theboundaries of the Brillouin zone, the DMFT calculationobtains an essentially local feature without such q de-pendence.The polarized neutron study was performed on thethermal TAS IN20 and the results are shown in Fig. 9.An example of the raw data with different spin channelsthat are needed for the polarization analysis, is given FIG. 9. Polarized neutron analysis of the scattering along thediagonal of the 1 st Brillouin zone. (a) Example of constantenergy scans for all three spin flip channels displays the in-creased scattering at the incommensurate position (0.7,0.3,0)at 8 meV and 1.6 K. The polarization analysis of all channelsyields the purely magnetic scattering signal displayed in (b)for different energies and temperatures. The black circles rep-resent data of the previously reported polarization analysis,taken from [45]. This data set was as well measured at 8 meVand 1.6 K. (c) The magnetic signal at 290 K can be describedby the susceptibility model used in [45] (light red line). Theintensity in (a) is normalized with 7800000 monitor countswhich corresponds to a measuring time of about 20 minutesper point. in Fig. 9(a) where a diagonal constant energy scan at8 meV, reaching from the zone boundary (0.5,0.5) overthe incommensurate position (0.7,0.3) to the zone center(1,0), is shown. The x , y , z indices refer to the commoncoordinate system used in neutron polarization analysisin respect to the scattering vector Q [45]. The threespin flip channels SF x , SF y , and SF z clearly exhibit amaximum at the incommensurate position. While SF y and SF z exhibit comparable amplitudes SF x carries thedoubled intensity as it senses both magnetic componentsperpendicular to the scattering vector. There is an en-hancement of magnetic excitations polarized along the c direction, that can be seen in the stronger SF y and thatwas studied in reference 42. Assuming a polarization in-dependent background, 2I(SF x )-I(SF y )-I(SF z ) yields thebackground free magnetic signal, see discussion in refer-ence 45.Fig. 9(b) displays the magnetic signal, corrected forBose factor, i.e. the imaginary part of the susceptibil-ity, for different energies and temperatures. The datawell agrees with the results for 8 meV and 1.6 K pre- FIG. 10. Comparison of magnetic scattering (T=1.6 K) atprominent points in k-space with L =0. The magnetic signalwas extracted using the polarization analysis (2I(SF x )-I(SF y )-I(SF z )) and is displayed for the points in the Brillouin zoneand different energies: Γ point, the incommensurate position,and the different zone boundaries X and M. The inset mag-nifies the intensity region around zero. sented in [45]. Additionally, the data at 290 K indi-cates a significant drop of the incommensurate nestingsignal, which however still is finite and clearly observ-able. The temperature dependence of the incommensu-rate signal was first discussed in reference 39, where theneutron scattering results are compared to the NMR re-sults from reference 38. The incommensurate signal wasfound to strongly decrease with increasing temperatureup to room temperature while the ferromagnetic com-ponent of the NMR is nearly temperature independent.Also the previous polarized neutron experiment foundthe quasi-ferromagnetic contribution to be almost iden-tical at 1.6 and 160 K [45]. As indicated in Fig. 9(b)the quasi-ferromagnetic signal does not change up to290 K, so that the peak heights of incommensurate andquasi-ferromagnetic contributions are comparable at am-bient temperature. Taking the much broader q shape ofthe quasi-ferromagnetic excitations into account, the q -integrated spectral weight of the latter clearly dominates.Around room temperature the quasi-ferromagnetic fluc-tuations possess thus a larger impact on any integrat-ing processes such as electron scattering. The quasi- ferromagnetic fluctuations at 290 K, however, do not ex-hibit a local character as the signal is significantly re-duced at the antiferromagnetic zone boundary (0.5,0.5,0)(Fig. 9(c)). This confirms the conclusion of Steffens et al.[45] that the quasi-ferromagnetic fluctuations are sharperin q space than expected from the calculations. Also atthe other zone boundary (0.5,0,0) there is no significantmagnetic signal detectable (see Fig. 10). IV. CONCLUSION
Polarized and unpolarized neutron scattering experi-ments were performed to study several aspects of themagnetic fluctuations in Sr RuO that are particularlyrelevant for a possible superconducting pairing scenario.The TOF instrument LET yields full mapping of theexcitations and reveals the well-studied incommensuratefluctuations at (0.3,0.3) in the two-dimensional recipro-cal space. There is also ridge scattering at (0.3, ξ ) re-flecting the one-dimensional character of the d xz and d yz bands, as first reported in references 43 and 44. Theseridges are stronger between the four peaks surrounding(0.5,0.5), i.e. for ξ> .
3, but the suppression of the sig-nal at smaller ξ is gradual. The TOF data confirm thepronounced asymmetry of the nesting peaks. Concerningthe study of the nesting fluctuations at very low energyin the superconducting phase, TAS experiments yieldhigher statistics due to the possibility to focus the ex-periment on the particular position in Q ,E space. Datataken at different out-of-plane components of the scatter-ing vector exclude a sizeable resonance mode emerging at L =0.5 in the superconducting phase. Only by combiningthe results of several experiments one can obtain someevidence for the suppression of spectral weight at verylow energies.With neutron polarization analysis the magnetic exci-tations were further characterized at 290 K. The incom-mensurate nesting signal is strongly reduced but still vis-ible, while the quasi-ferromagnetic contribution is almostunchanged. At this temperature there is a suppression ofthis quasiferromagnetic scattering at the Brillouin-zoneboundaries, which underlines that this response is notfully local.We acknowledge stimulating discussions with IlyaEremin. This work was funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) - Project number 277146847 - CRC 1238, projectB04, the JSPS KAKENHI Nos. JP15H05852 andJP17H06136, and the JSPS core-to-core program. [1] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki,T. Fujita, J. G. Bednorz, and F. Lichtenberg, Supercon-ductivity in a layered perovskite without copper, Nature , 532 (1994).[2] G. Koster, L. Klein, W. Siemons, G. Rijnders, J. S.Dodge, C. B. Eom, D. H. Blank, and M. R. Beasley,Structure, physical properties, and applications of SrRuO thin films, Rev. Mod. Phys. , 253 (2012).[3] G. Baskaran, Why is Sr RuO not a high T c super-conductor? Electron correlation, Hund’s coupling andp-wave instability, Phys. B Condens. Matter , 490(1996).[4] T. M. Rice and M. Sigrist, Sr RuO : an electronic ana-logue of He?, J. Phys. Cond. Matter , L643 (1995). [5] A. P. Mackenzie and Y. Maeno, The superconductivityof Sr RuO and the physics of spin-triplet pairing, Rev.Mod. Phys. , 657 (2003).[6] Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, andK. Ishida, Evaluation of Spin-Triplet Superconductivityin Sr RuO , Journal of the Physical Society of Japan ,011009 (2012).[7] C. W. Hicks, J. R. Kirtley, T. M. Lippman, N. C.Koshnick, M. E. Huber, Y. Maeno, W. M. Yuhasz, M. B.Maple, and K. A. Moler, Limits on superconductivity-related magnetization in Sr RuO and PrOs Sb fromscanning SQUID microscopy, Phys. Rev. B , 214501(2010).[8] H. Murakawa, K. Ishida, K. Kitagawa, Z. Q. Mao, andY. Maeno, Measurement of the Ru-Knight Shift ofSuperconducting Sr RuO in a Parallel Magnetic Field,Phys. Rev. Lett. , 167004 (2004).[9] C. Kallin, Chiral p-wave order in Sr RuO , Reports Prog.Phys. , 42501 (2012).[10] C. W. Hicks, D. O. Brodsky, E. A. Yelland, A. S. Gibbs,J. A. N. Bruin, M. E. Barber, S. D. Edkins, K. Nishimura,S. Yonezawa, Y. Maeno, and A. P. Mackenzie, Strong In-crease of Tc of Sr RuO Under Both Tensile and Com-pressive Strain, Science , 283 (2014).[11] A. Steppke, L. Zhao, M. E. Barber, T. Scaffidi,F. Jerzembeck, H. Rosner, A. S. Gibbs, Y. Maeno, S. H.Simon, A. P. Mackenzie, and C. W. Hicks, Strong peakin Tc of Sr RuO under uniaxial pressure, Science ,6321 (2017).[12] Y. S. Li, N. Kikugawa, D. A. Sokolov, F. Jerzembeck,A. S. Gibbs, Y. Maeno, C. W. Hicks, M. Nicklas, andA. P. Mackenzie, High sensitivity heat capacity mea-surements on Sr RuO under uniaxial pressure (2020),arXiv:1906.07597 [cond-mat.supr-con].[13] M. E. Barber, F. Lechermann, S. V. Streltsov, S. L. Sko-rnyakov, S. Ghosh, B. J. Ramshaw, N. Kikugawa, D. A.Sokolov, A. P. Mackenzie, C. W. Hicks, and I. I. Mazin,Role of correlations in determining the Van Hove strainin Sr RuO , Phys. Rev. B , 245139 (2019).[14] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q.Mao, Y. Mori, and Y. Maeno, Spin-triplet superconduc-tivity in Sr RuO identified by 17O Knight shift, Nat. , 658 (1998).[15] J. A. Duffy, S. M. Hayden, Y. Maeno, Z. Mao, J. Kulda,and G. J. McIntyre, Polarized-Neutron Scattering Studyof the Cooper-Pair Moment in Sr RuO , Phys. Rev. Lett. , 5412 (2000).[16] A. Pustogow, Y. Luo, A. Chronister, Y. S. Su, D. A.Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks,N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown,Constraints on the superconducting order parameter inSr RuO from oxygen-17 nuclear magnetic resonance,Nature , 72 (2019).[17] K. Ishida, M. Manago, K. Kinjo, and Y. Maeno, Re-duction of the 17O Knight Shift in the Superconduct-ing State and the Heat-up Effect by NMR Pulses onSr RuO , Journal of the Physical Society of Japan ,034712 (2020).[18] A. N. Petsch, M. Zhu, M. Enderle, Z. Q. Mao, Y. Maeno,I. I. Mazin, and S. M. Hayden, Reduction of the spin sus-ceptibility in the superconducting state of Sr RuO ob-served by polarized neutron scattering, Phys. Rev. Lett. , 217004 (2020).[19] H. S. Røising, T. Scaffidi, F. Flicker, G. F. Lange, and S. H. Simon, Superconducting order of Sr RuO froma three-dimensional microscopic model, Phys. Rev. Re-search , 033108 (2019).[20] A. T. Rømer, D. D. Scherer, I. M. Eremin, P. J.Hirschfeld, and B. M. Andersen, Knight Shift and Lead-ing Superconducting Instability from Spin Fluctuationsin Sr RuO , Phys. Rev. Lett. , 247001 (2019).[21] W.-S. Wang, C.-C. Zhang, F.-C. Zhang, and Q.-H. Wang,Theory of Chiral p -Wave Superconductivity with NearNodes for Sr RuO , Phys. Rev. Lett. , 027002 (2019).[22] Z. Wang, X. Wang, and C. Kallin, Spin-orbit couplingand spin-triplet pairing symmetry in Sr RuO , Phys.Rev. B , 064507 (2020).[23] H. G. Suh, H. Menke, P. M. R. Brydon, C. Timm,A. Ramires, and D. F. Agterberg, Stabilizing even-paritychiral superconductivity in Sr RuO , Phys. Rev. Re-search , 032023 (2020).[24] A. T. Rømer, A. Kreisel, M. A. M¨uller, P. J. Hirschfeld,I. M. Eremin, and B. M. Andersen, Theory of strain-induced magnetic order and splitting of T c and T TRSB inSr RuO , Phys. Rev. B , 054506 (2020).[25] S. A. Kivelson, A. C. Yuan, B. Ramshaw, andR. Thomale, A proposal for reconciling diverse exper-iments on the superconducting state in Sr RuO , npjQuantum Materials , 43 (2020).[26] R. Sharma, S. D. Edkins, Z. Wang, A. Kostin, C. Sow,Y. Maeno, A. P. Mackenzie, J. C. S. Davis, and V. Mad-havan, Momentum-resolved superconducting energy gapsof Sr RuO from quasiparticle interference imaging, Pro-ceedings of the National Academy of Sciences , 5222(2020).[27] G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin,J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno,Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist,Time-reversal symmetry-breaking superconductivity inSr RuO , Nature , 558 (1998).[28] V. Grinenko, R. Sarkar, K. Kihou, C. H. Lee, I. Morozov,S. Aswartham, B. B¨uchner, P. Chekhonin, W. Skrotzki,K. Nenkov, R. H¨uhne, K. Nielsch, S. L. Drechsler,V. L. Vadimov, M. A. Silaev, P. A. Volkov, I. Eremin,H. Luetkens, and H. H. Klauss, Superconductivity withbroken time-reversal symmetry inside a superconductings-wave state, Nat. Phys. , 789 (2020).[29] J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer,and A. Kapitulnik, High Resolution Polar Kerr EffectMeasurements of Sr RuO : Evidence for Broken Time-Reversal Symmetry in the Superconducting State, Phys.Rev. Lett. , 167002 (2006).[30] I. Schnell, I. I. Mazin, and A. Y. Liu, Unconventionalsuperconducting pairing symmetry induced by phonons,Phys. Rev. B , 184503 (2006).[31] M. Braden, A. H. Moudden, S. Nishizaki, Y. Maeno, andT. Fujita, Structural analysis of Sr RuO , Phys. C Su-percond. , 248 (1997).[32] M. Braden, W. Reichardt, Y. Sidis, Z. Mao, andY. Maeno, Lattice dynamics and electron-phonon cou-pling in Sr RuO : Inelastic neutron scattering and shell-model calculations, Phys. Rev. B , 014505 (2007).[33] Z. Fang and K. Terakura, Magnetic phase diagram ofCa − x Sr x RuO governed by structural distortions, Phys.Rev. B , 10.1103/PhysRevB.64.020509 (2001).[34] O. Friedt, M. Braden, G. Andr´e, P. Adelmann, S. Nakat-suji, and Y. Maeno, Structural and magnetic aspects ofthe metal-insulator transition in Ca − x Sr x RuO , Phys. Rev. B , 174432 (2001).[35] M. Braden, W. Reichardt, S. Shiryaev, and S. Barilo,Giant phonon anomalies in the bond-stretching modesin doped BaBiO : comparison to cuprates manganitesand nickelates, Physica C: Superconductivity ,89 (2002).[36] Y. Wang, J. J. Wang, J. E. Saal, S. L. Shang, L.-Q. Chen,and Z.-K. Liu, Phonon dispersion in Sr RuO studiedby a first-principles cumulative force-constant approach,Phys. Rev. B , 172503 (2010).[37] M. Braden, L. Pintschovius, T. Uefuji, and K. Ya-mada, Dispersion of the high-energy phonon modes inNd . Ce . CuO , Phys. Rev. B , 184517 (2005).[38] T. Imai, A. W. Hunt, K. R. Thurber, and F. C. Chou, O NMR Evidence for Orbital Dependent Ferromag-netic Correlations in Sr RuO , Phys. Re , 3006 (1998).[39] Y. Sidis, M. Braden, P. Bourges, B. Hennion,S. Nishizaki, Y. Maeno, and Y. Mori, Evidence for In-commensurate Spin Fluctuations in Sr RuO , Phys. Rev.Lett. , 3320 (1999).[40] M. Braden, Y. Sidis, P. Bourges, P. Pfeuty, J. Kulda,Z. Mao, and Y. Maeno, Inelastic neutron scattering studyof magnetic excitations in Sr RuO , Phys. Rev. B ,064522 (2002).[41] F. Servant, B. F˚ak, S. Raymond, J. P. Brison, P. Lejay,and J. Flouquet, Magnetic excitations in the normal andsuperconducting states of Sr RuO , Phys. Rev. B ,184511 (2002).[42] M. Braden, P. Steffens, Y. Sidis, J. Kulda, P. Bourges,S. Hayden, N. Kikugawa, and Y. Maeno, Anisotropy ofthe Incommensurate Fluctuations in Sr RuO : A Studywith Polarized Neutrons, Phys. Rev. Lett. , 097402(2004).[43] K. Iida, M. Kofu, N. Katayama, J. Lee, R. Kajimoto,Y. Inamura, M. Nakamura, M. Arai, Y. Yoshida, M. Fu-jita, K. Yamada, and S.-H. Lee, Inelastic neutron scatter-ing study of the magnetic fluctuations in Sr RuO , Phys.Rev. B , 060402 (2011).[44] K. Iida, J. Lee, M. B. Stone, M. Kofu, Y. Yoshida,and S. Lee, Two-Dimensional Incommensurate MagneticFluctuations in Sr (Ru . Ti . )O , Journal of the Phys-ical Society of Japan , 124710 (2012).[45] P. Steffens, Y. Sidis, J. Kulda, Z. Q. Mao, Y. Maeno, I. I.Mazin, and M. Braden, Spin Fluctuations in Sr RuO from Polarized Neutron Scattering: Implications for Su-perconductivity, Phys. Rev. Lett. , 047004 (2019).[46] M. Braden, O. Friedt, Y. Sidis, P. Bourges, M. Minakata,and Y. Maeno, Incommensurate Magnetic Ordering inSr Ru − x Ti x O , Phys. Rev. Lett. , 197002 (2002).[47] J. P. Carlo, T. Goko, I. M. Gat-Malureanu, P. L. Russo,A. T. Savici, A. A. Aczel, G. J. MacDougall, J. A.Rodriguez, T. J. Williams, G. M. Luke, C. R. Wiebe,Y. Yoshida, S. Nakatsuji, Y. Maeno, T. Taniguchi,and Y. J. Uemura, New magnetic phase diagram of(Sr,Ca) RuO , Nature Materials , 323 (2012).[48] S. Kunkem¨oller, A. A. Nugroho, Y. Sidis, and M. Braden,Spin-density-wave ordering in Ca . Sr . RuO studied byneutron scattering, Phys. Rev. B , 045119 (2014).[49] B. Zinkl and M. Sigrist, Impurity induced double transi-tions for accidentally degenerate unconventional pairingstates (2020), arXiv:2009.10089 [cond-mat.supr-con].[50] I. I. Mazin and D. J. Singh, Competitions in LayeredRuthenates: Ferromagnetism versus Antiferromagnetismand Triplet versus Singlet Pairing, Phys. Rev. Lett. , 4324 (1999).[51] J.-W. Huo, T. M. Rice, and F.-C. Zhang, Spin densitywave fluctuations and p -wave pairing in Sr RuO , Phys.Rev. Lett. , 167003 (2013).[52] S. Kunkem¨oller, P. Steffens, P. Link, Y. Sidis, Z. Q.Mao, Y. Maeno, and M. Braden, Absence of a LargeSuperconductivity-Induced Gap in Magnetic Fluctua-tions of Sr RuO , Phys. Rev. Lett. , 147002 (2017).[53] K. Iida, M. Kofu, K. Suzuki, N. Murai, S. Ohira-Kawamura, R. Kajimoto, Y. Inamura, M. Ishikado,S. Hasegawa, T. Masuda, Y. Yoshida, K. Kakurai,K. Machida, and S. Lee, Horizontal Line Nodes inSr RuO Proved by Spin Resonance, J. Phys. Soc. Japan , 053702 (2020).[54] Y. Maeno, K. Yoshida, H. Hashimoto, S. Nishizaki, S.-i.Ikeda, M. Nohara, T. Fujita, A. Mackenzie, N. Hussey,J. Bednorz, and F. Lichtenberg, Two-Dimensional FermiLiquid Behavior of the Superconductor Sr RuO , Journalof the Physical Society of Japan , 1405 (1997).[55] H. Mukuda, K. Ishida, Y. Kitaoka, K. Asayama, Z. Q.Mao, Y. Mori, and Y. Maeno, 17O NMR probe of spinfluctuations in triplet superconductor Sr RuO , Phys. BCondens. Matter , 944 (1999).[56] J. E. Ortmann, J. Y. Liu, J. Hu, M. Zhu, J. Peng,M. Matsuda, X. Ke, and Z. Q. Mao, CompetitionBetween Antiferromagnetism and Ferromagnetism inSr RuO Probed by Mn and Co Doping, Scientific Re-ports , 2950 (2013).[57] H. U. R. Strand, M. Zingl, N. Wentzell, O. Parcollet, andA. Georges, Magnetic response of Sr RuO : Quasi-localspin fluctuations due to Hund’s coupling, Phys. Rev. B , 125120 (2019).[58] T. Moriya, Spin Fluctuations in Itinerant Electron Mag-netism (Springer-Verlag Berlin Heidelberg, 1985).[59] K. Jenni, S. Kunkem¨oller, D. Br¨uning, T. Lorenz,Y. Sidis, A. Schneidewind, A. A. Nugroho, A. Rosch,D. I. Khomskii, and M. Braden, Interplay of Electronicand Spin Degrees in Ferromagnetic SrRuO : AnomalousSoftening of the Magnon Gap and Stiffness, Phys. Rev.Lett. , 017202 (2019).[60] C. N. Veenstra, Z. H. Zhu, B. Ludbrook, M. Capsoni,G. Levy, A. Nicolaou, J. A. Rosen, R. Comin, S. Kit-taka, Y. Maeno, I. S. Elfimov, and A. Damascelli, De-termining the surface-to-bulk progression in the normal-state electronic structure of Sr RuO by angle-resolvedphotoemission and density functional theory, Phys. Rev.Lett. , 1 (2013).[61] K. Jenni, M. Braden, P. Steffens, and Y. Sidis, In-terplay of magnetic excitations and superconductiv-ity in Sr RuO (2018), Institut Laue-Langevin (ILL),DOI:10.5291/ILL-DATA.4-02-537.[62] K. Jenni, M. Braden, P. Steffens, and Y. Sidis, pi,pi res-onance mode in the superconducting state of Sr RuO (2020), Institut Laue-Langevin (ILL), DOI:10.5291/ILL-DATA.4-02-586.[63] K. Jenni, M. Braden, P. Steffens, and Y. Sidis, Elec-tronic interaction in the unconventional superconduc-tor Sr RuO (2020), Institut Laue-Langevin (ILL),DOI:10.5291/ILL-DATA.4-02-565.[64] M. Braden, K. Jenni, S. Kunkem¨oller, P. Steffens,and Y. Sidis, Low-energy magnetic excitations in thesuperconducting state of Sr RuO : opening of agap? (2017), STFC ISIS Neutron and Muon Source,DOI:10.5286/ISIS.E.RB1710381. [65] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H.Du, Unconventional superconductivity with a sign rever- sal in the order parameter of lafeaso − x f x , Phys. Rev.Lett.101