Two energy scales in the spin excitations of the high-T superconductor La Sr CuO c 2−x x 4 B. Vignolle,1 S.M. Hayden,1 D.F. McMorrow,2,3 H.M. Rønnow,4 B. Lake,5 C.D. Frost,3 and T.G. Perring3 1H.H. Wills Physics Laboratory, University of Bristol, Tyndall Ave., Bristol, BS8 1TL, UK 2London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, London, WC1E 6BT, UK 7 3ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK 0 4Laboratory for Neutron Scattering, ETH-Zu¨rich and Paul Scherrer Institut, 5232 Villigen, Switzerland 0 5Hahn-Meitner Institut, Berlin D-14109, Germany. 2 n a Theexcitationsresponsibleforproducinghigh- suggestthattherearestronglycoupledelectronicexcita- J temperature superconductivity in the cuprates tions in this energy range [9]. Unfortunately, the high- 8 have not been identified. Two promising candi- energy magnetic excitations in the cuprates have been ] dates are collective spin excitations and phonons most comprehensively studied in YBa2Cu3O6+x, a com- n [1]. A recent argument against spin excitations pound for which ARPES data are scarce. In contrast, o c has been theirinability to explain structures seen La2−xSrxCuO4 is a system for which the magnetic ex- - in electronic spectroscopies such as photoemis- citations may be studied by neutron scattering (because r sion [2, 3, 4, 5] and tunnelling [6]. Here we of the availability of large single crystals) and ARPES p u use inelastic neutron scattering to demonstrate (because a suitable surface may be prepared). s thatcollectivespinexcitationsinoptimallydoped Here we report a high-resolution neutron scattering . at La2−xSrxCuO4 are more structured than previ- study of the magnetic excitations in optimally doped ouslythought. Theexcitationshaveatwocompo- m La2−xSrxCuO4 (x=0.16, Tc=38.5 K). The parent com- nent structure with a low-frequency component - pound La2CuO4 of the La2−xSrxCuO4 superconducting d strongest around 18 meV and a broader compo- series exhibits antiferromagnetic order with an order- n nent strongest near 40-70 meV. The second com- ing wavevector of Q =(1/2,1/2). Doping induces su- o ponent carries most of the spectral weight and its 2D perconductivity and causes low-frequency incommensu- c energy matches structure seen in photoemission [ rate fluctuations [10, 11] to develop with wavevectors and tunnelling spectra [2, 3, 4, 5, 6] in the range 2 50-90 meV. Our results demonstrate that collec- Q2D = (1/2,1/2±δ) and (1/2±δ,1/2). These excita- tions broaden[12] and disperse inwards initially towards v tive spinexcitations canexplainfeatures ofquasi- 1 (1/2,1/2) [13] with increasing energy. The present work particle spectroscopies and are therefore likely to 5 extends the energy range and wavevector resolution of 1 be the strongest coupled excitations. previous studies [10, 11, 12, 13, 14]. 1 0 Since their discovery, considerable progress has been Neutron spectroscopy provides a direct probe of the 7 made in understanding the properties of the high-Tc magnetic response function χ′′(Q,ω). Experiments were 0 cuprate superconductors. For example, we know that performedontheMAPSspectrometerattheISISspalla- / t the superconductivity involves Cooper pairs, as in the tion neutron source of the Rutherford-Appleton Labora- a conventional BCS theory, but the d-wave pairing is dif- tory. Figure 1a-hshowswavevector-dependentimagesof m ferent to the s-wave pairing of conventional supercon- the magneticresponse atvariousenergiesdemonstrating - d ductors. However, a major outstanding issue is the howitevolveswithenergy. Atlowenergies,E =10meV n pairing mechanism. Identifying, the bosonic excitations (panel 1a) we observe the low-energy incommensurate o whichwerestronglycoupledtotheelectronquasiparticles excitations [10, 11, 12, 13]. As the energy is increased, c played an pivotal role in confirming the pairing mecha- E = 18 meV (panel 1b) the response becomes stronger, : v nism in conventional superconductors [7, 8]. In the case the pattern fills in along the line connecting the nearest i X of the copper oxide superconductors, advances in elec- neighbour incommensurate peaks and the incommensu- tronic spectroscopies such as angle resolved photoemis- rability δ decreases. For E = 25 meV (panel 1c), the r a sion (ARPES) and tunnelling have revealed structure in intensity of the pattern is noticeably attenuated. On the low-energy electronic excitations which may reflect further increasing to E = 41 meV (panel 1d), the re- coupling to bosonic excitations. ARPES measurements sponse “recovers”, becoming more intense again, but is on Bi2Sr2CaCu2O8, Bi2Sr2CuO6 and La2−xSrxCuO4 now peaked at the commensurate wavevector (1/2,1/2). have shown that there are rapid changes or “kinks” in At higher energy E = 90 meV (panel 1e), the struc- the quasiparticle dispersion E(k) in the nodal direction ture resembles a square box with the corners pointing [k (1/2,1/2)] for energies in the range 50–80 meV along the (110) type directions, i.e. towards the Bril- k [2, 3, 4, 5]. The origin of these kinks has been dis- louin zone center. Thus the square pattern is rotated ◦ cussed in terms of coupling to collective spin excitations 45 with respect to the low energy response (e.g. panel and phonons. Optical conductivity measurements also 1b). A similar high-energy response has been observed 2 FIG.1: Images ofthe magneticexcitations inLa1.84Sr0.16CuO4 for various energiesat T=12K.a-hThemeasured χ′′(Q,ω) is plotted in units of µ2B eV−1 f.u.−1 as a function of wavevector. a-c show the emergence and disappearance of the componentat(1/2±δ,1/2) and(1/2,1/2±δ)whichismostintenseatlowerenergies. d-hshowthehigherenergycomponent which emerges around 41 meV and disperses outwards with increasing energy. i-p Model fits to the images of the magnetic excitations shown in a-h. The phenomenological model [Eq. (1)] provides a good description of the experimentally measured magnetic excitations and can therefore be used to parameterize the data. At higher energies (panels n-p) the data are best fitted with themodel χ′′(Q,ω) rotated 45◦ in theh−k plane. Wavevectors are labelled by their positions in reciprocal space Q=ha∗+kb∗+lc∗. in underdoped YBa2Cu3O6+x [15, 16] and the stripe or- where κ(ω) is an inverse correlation length, the position dered composition La1.875Ba0.125CuO4 [17]. Thus a ro- of the four peaks is specified by δ, and λ controls the tated continuum appearsto be a universalfeature of the shape of the pattern (λ =4 corresponds to four distinct cuprates. peaks and λ=0 corresponds to a pattern with circular In order to make our analysis quantitative, we fitted symmetry). a modified Lorentzian cross-section previously used to Fig. 1i-p shows plots of the fitted model response for describedthecupratesandothersystems[18]tothedata: the same energies as Fig. 1a-h. Another way of display- κ4(ω) ing the results is to take constant energy cuts through ′′ χ (Q,ω)=χδ(ω)[κ2(ω)+R(Q)]2 (1) our data set. Fig. 2 shows cuts along the dashed trajec- toryinFig.1forvariousenergiestogetherwithfitsofour with model response (Eq. 1) convolvedwith the experimental (cid:2)(h 1)2+(k 1)2 δ2(cid:3)2+λ(h 1)2(k 1)2 resolution. The good agreement between the data and R(Q)= − 2 − 2 −4δ2 − 2 − 2 fits allows us to use the parameters derived from the fits 3 La Sr CuO (T=10 K) 200 1.84 0.16 4 c 3 a E=10 meV b E=18 meV3 150 2.5 2.5 ) V 2 2 e m100 -1-1V f.u.)1.315 c E=41 meV d E=50 meV 11.5 wh ( 50 e m 3 1 2.5 0 -sr 2 0.3 0.2 0.1 0 0.1 0.2 0.3 n 2.5 |d | (r.l.u.) ar1.5 b 0.5 m 1 2 sity (1.1 e E=90 meV f E=130 meV0.45 0.4 b n e nt 1 0.4 0.3 I ) u. 0.9 0.35 r.l.0.2 ( 0.8 0.3 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 k 0.1 h in (h,1/2) 0 TFI=G1.22:KM. aa-gfnsehtoiwc tehxecivtaartiaiotinosn ionf tLhae1s.8c4aSttre0r.1e6dCiuntOen4siatyt 3.5 a ) 23 1300 0KK jawe-icbtthosrwhyaovwoefvttehhceetoicrnucftoormisvmasrehinoosuwusnrafitbxeyedlotehwxe-cfridteaaqstuhioeenndceylninecreogmyin.poTFnhiegen.tt1raoa-f. -1-1V f.u.)2.523 cw''(010 1h0 w (2m0eV3)0 the response. The high-frequencycomponent is strongest for e 40–50 meV (c-f). Two distinct peaks are seen at higher en- 2 B1.5 ergies (e-f), these disperse away from (1/2,1/2) in a similar m) ( 1 mannertothespin-waveexcitations intheparentcompound w(0.5 La2CuO4. c'' 0 0 50 100 150 200 h w (meV) (Fig. 3) to interpret our results. In order to distinguish betweenmagneticandphononscatteringthe experiment FIG. 3: Magnetic excitation spectrum and evolution was performed at a number of incident energies. This of the form of the magnetic response with energy. ′′ meansthatthe samein-planemomentum(h,k)couldbe The susceptibility wavevector-averaged susceptibility χ (ω) (a) shows a “peak-dip-hump” structure suggesting that the probed with a variety of l (momentum perpendicular to magneticresponsehastwocomponents. Theemergenceofthe plane)valuesandstrongphononsisolated. Thefactthat higherfrequencycomponentaboveabout40meVcorresponds datacollectedwithdifferentincidentenergiesyields sim- toabroadeningofresponseinwavevectorasdemonstratedby ilarresultsconfirmsthevalidityofouranalysis. Wehave the rapid increase in the κ (b). There is a strong dispersion expressedthestrengthofthespinfluctuationsintermsof of the peak positions in constant excitation-energy cuts as the local or wavevector-averaged susceptibility χ′′(ω) = shown by the energy dependence of the incommensurability R χ′′(Q,ω)d3Q/R d3Q determined from the fitted cross- δ(ω)(c). Thehigh-energydispersionindicatesthepersistence ′′ of residual antiferromagnetic interactions. Symbols indicate section. The local susceptibility χ (ω) is a measure of differentincidentenergies: Ei =30(♦),55((cid:3)),90((cid:4)),160( ), the density of magnetic excitations for a given energy. • 240 meV ( ). ◦ Figure 3a illustrates one of the key results of this work: the magnetic response of La1.84Sr0.16CuO4 has a two component structure. The lower-energy peak corre- gies probed (E=155 meV) in this experiment. The ob- spondstotheincommensuratestructurewhichisrapidly servedtwocomponentstructurediffersfromtheresponse attenuated above 20 meV. The higher-energy structure observed in the magnetically ordered but weakly su- is peaked at (1/2,1/2) for E 40–50 meV and broad- perconductingcompoundLa1.875Ba0.125CuO4 [17]which ≈ ′′ ens out with increasing energy above this. Although the does not show the lower peak in χ (ω). Although wavevector-averaged susceptibility χ′′(ω) of the higher La1.875Ba0.125CuO4 does a similar dispersion in δ . | | frequency component is peaked around 50 meV, it has Giventhemarkedlydifferentcharacteristicsofthe two a long tail with a measurable response at highest ener- components which make up the magnetic response in 4 La1.84Sr0.16CuO4,itislikelythattheyhavedifferentori- ponents. At low energies, the excitations are incommen- gins. One possible scenario is that the lower-energy in- surate and disperse with increasing energy towards the commensuratestructureisduetoquasiparticle(electron- commensurate antiferromagnetic ordering wavevector of hole)paircreationwhichmightbecalculatedfromanun- La CuO ,peakinginintensityaround20meV.Athigher 2 4 derlying band structure [19, 20], while the higher-energy energies a second component develops which displays a structureisduetotheresidualantiferromagneticinterac- broad, commensurate peak around 50 meV. It then dis- tions. It is interesting to compare the magnetic response plays a spin-wave-likedispersionat higher energies. The atoptimaldopingwiththatoftheantiferromagneticpar- high-frequencyexcitationsaremostnaturallyinterpreted ′′ ent compound La CuO [21]. In La CuO , χ (ω) is ap- as being due to residual antiferromagnetic interactions. 2 4 2 4 proximately constant over the energy range probed here Comparison of the present data with electron photoe- (0–160 meV) with χ′′(ω) 1.7 µ2 eV−1 f.u.−1. Thus missionand tunnelling data suggeststhat it is this high- B ≈ the effect of doping is to suppress the high-energy re- frequencycomponentwhichaffectstheelectronquasipar- sponse(~ω >50meV)andenhancetheresponseatlower ticles moststrongly. As the excitationsthat couplemost frequencies, creating a double peak structure. Fig. 3c effectivelytothequasiparticlesaremostlikelytoplayan showsthatthehighenergypartoftheresponsedisperses important part in the superconducting pairing, our re- with increasing energy. Constant energy cuts through sults support the notion that high-Tc superconductivity the data yield two peaks (see Fig. 2) which are reminis- is magnetically mediated. centofspinwaveinthe parentcompoundLa CuO . We 2 4 may use the high-energy dispersion to estimate an effec- tiveHeisenbergexchangeconstantJ whichquantifiesthe strength of the coupling between the copper spins. Us- [1] Chubukov,A.V.,Pines,D.&Schmalian,J. Aspinfluc- ing the fitted values of δ in Fig. 3c for E >40 meV, we estimate the gradient to| |be dE/dδ =510 50 meV˚A−1. tuation model for d-wave superconductivity. In K. H. ± Bennemann & J. B. Ketterson, eds., The Physics of This may then be compared with the standard expres- Superconductors, vol. 2, pp. 495–590 (Springer, Berlin, sionfor the spinwavevelocityinasquarelattice antifer- 2003). romagnet, ~vs = Zc√8SJa, where Zc, S, and a are the [2] Bogdanov, P. V. et al. 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