Dispersion of the odd magnetic resonant mode in near-optimally doped Bi Sr CaCu O 2 2 2 8+δ B. Fauqu´e1, Y. Sidis1, L. Capogna2,3, A. Ivanov3, K. Hradil4, C. Ulrich2, A.I. Rykov5, B. Keimer2, and P. Bourges1∗ 1 Laboratoire L´eon Brillouin, CEA-CNRS, CE-Saclay, 91191 Gif sur Yvette, France. 2 Max-Plank-Institute fu¨r Festk¨orperforschung, Heisenbergstr. 1, 70569 Stuttgart, Germany. 7 3 Institut Laue-Langevin, 6 Rue J. Horowitz, 38042 Grenoble cedex 9, France. 0 4 Forschungreaktor Mu¨nchen II, TU Mu¨nchen, Lichtenbergstr, 1 95747 Garching, Germany. 0 5 Department of Applied Chemistry, University of Tokyo, 2 Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan. n We report a neutron scattering study of the spin excitation spectrum in the superconducting a J stateofslightly overdopedBi2Sr2CaCu2O8+δ system(Tc=87K).Wefocuson thedispersion of the resonance peak in the superconducting state that is due to a S=1 collective mode. The measured 3 spinexcitationspectrumbearsastrongsimilarity tothespectrumoftheYBa2Cu3O6+x system for a similar doping level (i.e. x ∼ 0.95−1), which consists of intersecting upward- and downward- ] n dispersing branches. A close comparison of the threshold of the electron-hole spin flip continuum, o deduced from angle resolved photo-emission measurements in the same system, indicates that the c magnetic response in the superconducting state is confined, in both energy and momentum, below - r thegappedStonercontinuum. IncontrasttoYBa2Cu3O6+x,thespinexcitationspectrumisbroader p thantheexperimentalresolution. Intheframeworkofanitinerant-electronmodel,wequantitatively u relate this intrinsic energy width to the superconducting gap distribution observed in scanning s tunnelling microscopy experiments. Our study further suggests a significant in-plane anisotropy of . t themagnetic response. a m - I. INTRODUCTION nance peak. d Recently, the debate became focused on the origin of n o Itisnowwellestablishedthatthespinexcitationspec- the S=1 dispersive collective mode. The theoretical de- c trum in the superconducting (SC) state of many high- scriptionofthemodeisimportant,becauseantiferromag- 1 [ tTrcipsleutpeerxccoitnadtuiocnto1,r2s,3,i4s,5,d6,o7m,8,i9n,1a0t,1e1d,12b.yTahnisuenxucsituaatliosnpiins ntheetisSmCipsagireinnegramlleychbaenliiesvmedintohipghla-yTcacsuipgnraifitecsa2n2t. role in v referred to as the magnetic resonance peak. It is cen- First, based on the spin dynamics data in the stripe 2 tered at the planar antiferromagnetic (AF) wave vector ordered system La Ba CuO , it has been proposed 7/8 1/8 4 5 q =(π/a,π/a) (where a is the lattice spacing) and at that E could be a saddle point in the dispersion, with 0 AF r anenergyE thatscaleswiththeSCcriticaltemperature, spin excitations propagating along a given in-plane di- 1 r ∗ 0 Tc. Further,the resonancepeakintensity vanishesabove rection (say a ) below Er and along the perpendicular 7 Tc andexhibitsatemperaturedependencethatlookslike direction (i.e. b∗) above Er23,24. This gives rise to an 0 an order parameter. A renormalization of its character- X-like shape in twinned crystals where a∗ and b∗ are / istic energy with temperature is not observedwithin the mixed. Saddle points can arise in models with spin and t a experimentalerror5,6. Thisbehaviorhasbeenreportedin chargestripe orderat low temperature. The spin excita- m severalfamilies of copper oxides with maximum SC crit- tion spectrum can then be modelled by a specific bond- - ical temperatures Tmax ≥90 K: in Tl Ba CuO with centered stripe model according to which non-magnetic d c 2 2 6+x uniformly spaced, single CuO layers12, as well as in bi- charge stripes separate a set of weakly coupled two-leg n 2 layersystemssuchasYBa Cu O (YBCO)2,3,4,5,6,7,8,9 spin ladders in the copper oxide layers. The low en- o 2 3 6+x c and Bi2Sr2CaCu2O8+x (Bi2212)10,11. The magnetic res- ergy excitations (below Er) correspond to collective ex- : onance peak indicates the existence of a S=1 collective citations that propagate perpendicular to the ladder di- v mode with a peculiar dispersion. Its downward dis- rection, whereas the high energy part of the spectrum i X persion starting at q was first observed in YBCO13. (above E ) is associated to intra-spin ladder excitations AF r r Complementary measurements provided strong indica- propagating parallel to the lines of charges. This pic- a tions of a second upward dispersion starting from the ture, later on sustained by calculations25,26,27, implies resonance energy in optimally doped YBCO14,15,16. Im- a pronounced in-plane anisotropy of the magnetic spec- portantly, both dispersive branches vanish above T as trum. However,this is not consistent with the spin exci- c does the magnetic resonance peak at q . In strongly tations in SC cuprates, in particular in underdoped and AF underdoped cuprates (T ≤ 62 K), similar dispersive ex- optimally doped YBCO20,21. Indeed, using detwinned c citations have also been reported17,18,19,20,21, but only YBCO samples21, it has been shown that the spin ex- the downward branch vanishes above T whereas the citation spectrum exhibits a 2D geometry both below20 c upward dispersion remains essentially unchanged across and above21 E , inconsistent with a saddle-point disper- r T 21. Therefore, the spin excitation dispersion in YBCO sion. However, recent calculations of the spin dynamics c exhibits a ”hour glass”-like shape centered at the reso- consideringfluctuatingstripesegments28,showsthatthe 2 magnetic spectrum is losing its 1D character for short (ARPES) (see29,40,41 and references therein). Second, charge segments (a few atomic distances) because they Scanning Tunnelling Microscopy (STM) data42,43,44 ev- actually exist in both perpendicular directions. Further, idence a local distribution of the superconducting gap it should be emphasized, once again, that the resonance and then infer the low energy quasi-particle excitations peak intensity in all SC cuprates exhibits systematically and the Fermi surface through Friedel oscillations. Re- a strong temperature dependence in the SC state1. This cently, attempts to compute the spin excitation spec- is in a marked contrastto the data in the stripe-ordered trum in nearly optimally doped Bi2212 starting from system, where this anomalous temperature dependence ARPES measurements45,46 have been performed using is absent. These inconsistencies cast some doubt about the itinerant-spin approach(excitonic scenario). a similar origin of the resonance peak seen in cuprates In this paper, we present a study of both energy and where superconductivity is well developed1 and the one momentum dependences of the resonantspin excitations reported in the stripe-ordered system23. As a matter in the SC state ofa nearly optimally doped Bi2212. Our of fact, it would be more meaningful to compare the studyrevealsthatthespinexcitationspectruminBi2212 spectrum in non-SC La Ba CuO with the magnetic bears close similarity to the spin excitation spectrum re- 7/8 1/8 4 spectrum in the normal state of other SC cuprates, as it ported in YBCO for the same doping level. We fur- has recently been done in underdoped YBCO21. ther show that the resonant spin collective modes are located below the gapped Stoner continuum computed Second,startingfromthemetallicsideofthephasedi- from ARPES data. The combination of our inelastic agramof high-T superconductors and reducing the hole c neutronscattering(INS)measurementsandARPESand doping, one can try to understand the S=1 collective STMmeasurementsonthesamesystemprovidestheop- mode within an itinerant-electron model. It has been portunitytotesttheoreticalmodelsforthespindynamics proposed that the resonant magnetic collective mode could be described as a spin exciton29,30,31,32,33,34, i.e in the cuprates. a S=1 bound state, pushed below the gapped Stoner continuum in SC state by AF interactions. This type II. EXPERIMENTAL of excitation exists in the SC state only and vanishes when the gap disappears in the normal state. Alterna- tively, when starting from the Mott-insulator side of the For the present study, we used a single crystal of phase diagram and increasing the hole doping, in frame- of slightly overdoped Bi2212 with volume ∼1.5 g and work of localized-spin models the mode can be viewed Tc=87 K . The INS measurements were performed on as the remnant of the magnon observed in the insu- the thermal triple-axis spectrometers IN8 at the Insti- lating AF state35,36. The collective modes of localized tuteLaueLangevininGrenoble(France),2TattheLab- spins on Cu sites may survive in the metallic state, but oratoire L´eon Brillouin at the reactor Orph´ee in Saclay are heavily damped by scattering from charge carriers. (France), and PUMA at the reactor FRM-II in Garch- Long-lifetime collective excitations can then be restored ing (Germany). Measurements were carried out with a intheSCstate,whenscatteringprocessesareeliminated double-focusingPG(002)monochromatorandaPG(002) below the gapped Stoner continuum. When the mode analyzer. The final neutron wave vector was set to energy is close to the gap, it can be viewed as a spin- kf=4.1 ˚A−1, yielding an energy resolution σω ≃6 meV. exciton, as in the itinerant-spin approach35,36. Both ap- A PG filter was inserted into the scattered beam in or- proaches represent two different limits of a dual descrip- der to eliminate higher order contaminations. For high tion of the magnetism of high-T superconductors: lo- energy transfers (h¯ω >55 meV), the PG filter was re- calized and itinerant spins are tigchtly bound and cannot moved and kf set to 5.5 ˚A−1. To cover the full q- be disentangled37,38. It is worth emphasizing that in a dependence of the spin excitations, three different scat- dual approach one can schematically ascribe the upper tering planes wereused for the measurements. The sam- dispersion to the localized character of the magnetic re- plewassuccessivelyorientedsuchthatmomentumtrans- sponse and the lower one to the itinerant one. However, fers Q of the form (H,H,L), (K/3,K,L) and (H,0.3L,L) future quantitativecalculationsarenecessaryto validate were accessible. We use a notation in which Q is in- this picture. dexedinunits ofthe tetragonalreciprocallattice vectors 2π/a=1.64˚A−1 ≡2π/b and 2π/c=0.203˚A−1. In all of these models, the change of the band elec- The magnetic neutron scattering cross section is pro- tronic excitations upon passing through T has an im- c portional to Imχ(Q,ω), the imaginary part of the dy- portant feedback on the spin excitation spectrum in the namical magnetic susceptibility, weighted by the square SC state. The determination of the electron-hole spin of the Cu magnetic form factor, F(Q), and the detailed flip continuum then requires a good knowledge of the balancetemperature factor5,6,8. Fora paramagneticsys- fermionic dispersion relations, which is still in a stage of tem, it reads: rapid development in the YBCO system39. In contrast, the charge excitations in Bi Sr CaCu O have been d2σ r2 1 2 2 2 8+δ = 0|F(Q)|2 Imχ(Q,ω) (1) extensivelystudiedbysurface-sensitivetechniques. First, dΩdω 2π 1−exp(− h¯ω ) theFermisurfaceandthebanddispersionshavebeende- kBT termined by angle resolved photo-emission spectroscopy where Q = (H,K,L) is the full wave vector and q = 3 FIG. 1: Difference between constant-Q scans performed FIG.2: Differencesbetweenconstantenergyscansperformed at 10 K and 100 K. INS measurements were carried out at10Kand100K:a)scansalongthe(110)direction,b)scans on spectrometer 2T. The differential spectra are shown at alongthe(130) direction. Allmeasurementswerecarriedout different wave vectors of the form: a) Q=(H,H,L=14), b) on the spectrometer IN8. The position of the negative back- Q=(K/3,K,L=4.2). In addition to the energy scan data ground used in the analysis of the energy scans, reported in (open circles) are also reported the magnetic intensities Fig 1, isalso shown at theAFwavevector(red circles). The (open squares) and the location of the negative background solids lines correspond to the fit of the data by a single or (crosses), deducedfrom theanalysisofconstant energyscans double Gaussian functional form, on top of a sloping back- (Fig. 2). Those data , obtained on spectrometer IN8, have ground. been rescaled. The lines indicate the energy dependence of the negative background (see text) and the enhancement of themagneticintensityisdescribedbyasetofGaussianfunc- appropriatechoiceoftheLcomponent: L=14forthepla- tions on top of thenegative background. nar wave vector (0.5,0.5) and L=4.7 for the planar wave vector(0.5,1.5). NotethatthesquareoftheCumagnetic formfactorisabout1.4timeslargerfor(0.5,1.5,4.7)than (H,K) is the planar wave vector in the CuO plane. 2 for(0.5,0.5,14). Thisisbecausethemagneticformfactor r = 0.54 10−12 cm is the neutron magnetic scattering 0 is anisotropic in the cuprates49. length. Like the YBCO system, the Bi2212 system con- tains two CuO planes per unit cell. Owing to the in- 2 teractionbetweentheCuO planeswithinabilayer,spin 2 III. INELASTIC NEUTRON SCATTERING excitations with odd (o) and even(e) symmetry with re- MEASUREMENTS spect to the exchange of the layer contribute to the spin susceptibility as in YBCO8,47, so that: Throughout this paper, we focus on the enhancement χ(Q,ω)=sin2(πzL)χ (q,ω)+cos2(πzL)χ (q,ω) (2) of the magnetic response in the SC state, corresponding o e totheresonantspinexcitations. Toextractthisresponse, z=0.109isthereduceddistancebetweentheCuO planes we subtractedscans atT=10 K andT=100K (>T =87 2 c of the bilayer. Using different L values of the c∗ compo- K). This procedure generates a negative background in nent of the momentum transfer, the mode of each sym- the differential spectra, owing to the thermal enhance- metry can be measured. The observation of resonant ment of the nuclear background. The main contribu- modes with both symmetries has recently been reported tion to this negative background comes from the ther- in Bi221248 in two samples and in particular in the sam- malpopulationofphononsgivenbythe detailedbalance ple studied here. In the present study, we focus on the factor. In the energy range of interest for the present oddspinexcitations,thatcanbeselectivelyprobedbyan study, 25-60 meV, the nuclear (phononic) background 4 Refs. Tc(K) Er(meV) σr(meV) σ(meV) 60 Fong et al10 91 43 13 ± 2 12 ± 2 a ) b ) Present study 87 42 13 ± 2 11 ± 2 ) Heet al11 83 38 12 ± 2 10 ± 2 eV 55 Capogna et al48 70 34 8 ± 1 5 ± 1 m ( 50 TABLE I: Characteristic energy and energy width σr gy (FWHM:Full Widthat Half maximum)of themagnetic res- r e onance peak reported until now in Bi2212 samples. σ stands n 45 for the intrinsic FWHM of the resonance peak, after decon- E volution bythe energy resolution σω ≃6 meV. 40 continuouslydecreases. Thus,theenergydependentneg- 35 ative background can be quite well approximated by a -0.4 0 0.4 -0.4 0 0.4 -1 -1 functional form like (a + bω)/(exp(− h¯ω ) − 1), with q-q (Å ) q-q (Å ) kBTo AF AF T =100K.Owing to the weakness of the magnetic signal o ( ≤ 10 % of the nuclear one), one also becomes sensi- FIG. 3: Dispersion of the resonant magnetic excitations de- tive to multiple scattering effects. This is the main rea- duced from constant energy scans: a) along the (130) direc- son why the background in the difference signal remains tion (the full lines are described in the text), b) along the weakly negative even at high energy, where the detailed (110) direction. Horizontal and vertical error bars stand for balance factor has a negligible effect. Difference scans thehalf width at half maximum of thesignal and theenergy performedfarawayfromtheantiferromagneticwavevec- resolution, respectively. toratQ=(0.275,0.275,14)(Fig.1.a)andQ=(0.5,1.8,4.7) (Fig.1.b)showthetypicalenergydependenceoftheneg- ativebackground. Ontopoftheabovedescribednegative (scans at 46 and 50 meV in Fig. 2.b), the magnetic re- background,the enhanced magnetic intensity appears in sponse also broadens, but in contrast to the measure- energy scans at the AF wave vector at Q=(0.5,0.5,14) ments at 38 meV, the magnitude of the signal remains (Fig. 1.a) and Q=(0.5,1.5,4.7) (Fig. 1.b). The resonant comparable with that of the magnetic resonance. While magnetic signal is located at Er=42 meV and exhibits a at 54 meV a magnetic response with a double peak pro- Gaussian profile with an energy width of σr ∼ 13 meV file can be observed, at slightly higher energy, 58 meV, (FWHM). From constant-Q scans (Fig. 1), it appears the magnetic responseseems to be weak andcommensu- that the magnetic intensity at all wavevectors is limited rate,suggestingthatthespinexcitationsmaymoveback atlowenergybyaspin-gap of∼32meV.Belowthespin to the AF wave vector. The broadening of the magnetic gap, the magnetic signal is, at least, less than 1/4 of its response above and below E =42 meV is consistent the r value at Er=42 meV. This value agrees with reports on differential energy scans performed away from the AF YBCO for similar doping levels5,13,50. wave vector at Q=(0.52,1.56,4.7)and Q=(0.54,1.62,4.7) In addition to the constant Q-scans shown in Fig. 1, (Fig. 1.b). Indeed, at these wave vectors, the magnetic systematic constant-energy scans have been performed scattering is widely spread in energy, but shows a slight in each scattering plane (Fig. 2) to characterize the q- dip at 42 meV. The combination of energy scans and dependence of the magnetic intensity. Along the (130) constant-energyscansalongthe(130)directionisconsis- direction, the magnetic resonance peak at 42 meV is tent with the existence of resonant spin excitations dis- centered at the AF wave vector and displays a Gaus- persingupwardanddownward,aspreviouslyobservedin sianlineshape on top of a sloping background(Fig. 2.b). theYBCOsystem. GuidedbythedatainYBCO13,14,we A similar signal can be observed at 40 meV, whereas then model both dispersions of Fig. 3.a by the relations: at slightly lower energy, 38 meV, the magnetic response E± = (Er)2±(α±(q−qAF))2 where (-) and (+) cor- weakens and starts to broaden. Although the spin re- responpd to the downward and upward dispersions, re- sponse is not clearly peakedat an incommensurate wave spectively. One obtains α− =130 meV.˚A and α+ =200 vector, the lineshape of the signal at this energy can be meV.˚AingoodagreementwiththedispersionsinYBCO fitted to a double-peak structure. On decreasing the en- obtained for similar doping levels13,14,15. ergy transfer further, no sizeable magnetic response can Along the (110) direction, the evolution of the mo- bedetectedat32meV.Nonetheless,withintheerrorbars mentumdistributionofthemagneticsignalasafunction one cannot exclude a weak magnetic response of magni- of the energy exhibits the same trends as those observed tude at most half of the one measured at 38 meV. It is alongthe(130)direction,withabroadeningatbothhigh worthnotingthatbelow36meV,the largenuclearback- and low energies (46 meV and 40 meV in Fig. 2.a). A ground makes the detection of a magnetic response (if slightdouble-maximumstructurealsoshowsupintheen- any) particularly difficult. The broadening of the mag- ergy scan at Q=(0.425,0.425,14)(Fig. 1.a). But a closer netic response is not observed exclusively below the en- inspection of the data reveals marked differences. First, ergy of the magnetic resonance peak. At higher energy the signalbegins to broadenalready at 40 meV, and be- 5 a ) b ) a) b) FIG.5: Differencesbetweenconstantenergyscansat46meV performed between : a) 10 K and 300K, b) 100 K and 300K. MeasurementswerecarriedoutonthespectrometerIN8. The solidlinecorrespondstothefitofthedatabyadoubleGaus- sian functionalform, on top ofaslopingbackground(dashed line). The background at the AF wave vector is set to 0 to getridofthethevariationofthethermalenhancementofthe nuclearbackgroundandtoallow adirectcomparison ofboth FIG. 4: a) Temperature dependencies at different wave vec- results. tors and energies. b) Differences between constant energy scans at 10 K and 100K. Scans are performed at 40 meV aroundtheAFwavevectoralong3differentdirections: (100), (130),(110). ThelabelΘcorrespondstotheanglebetweenthe tensity.) However,scanswithsimilarresolutionandscat- scanning direction of the (100) direction (see Fig. 6.f). Data teringplane13didrevealincommensuratepeaksbelowE r are normalized so that the intensities at the AF wave vector inYBCO.Therefore,theoriginofthedifferencebetween are thesame. both systems is not purely instrumental, but at least in part intrinsic to Bi2212. It was previously noticed1 that comes almost twice as broad at 38 meV. At this energy, energy scans in Bi2212 exhibit a width larger than the energyresolutionfor alldoping levels, as showninTable the lineshape of the magnetic response is now ratherdif- ferent, with a more complex structure that can be qual- I,whereastheenergywidthoftheoddresonancepeakin YBCO is resolution-limited at optimal doping14,15. The itatively described in terms of a central peak at the AF intrinsic width ofthe mode blurs the details ofthe mode wave vector surroundedby two satellites of similar mag- nitude at Q=(−0.5±δ,−0.5±δ,14) with δ=0.2. As a dispersion in Bi2212. If we nevertheless try to fit the dataawayfromE withadoublepeakstructure,oneob- result of this expansion, a magnetic signal can still be r observedindifferentialenergyscansatQ=(0.35,0.35,14) tains dispersive excitations both below and above Er as shown in Fig. 3, as previously reportedin YBCO13,14,15. (Fig.1.a),i.e ata largedistance inreciprocalspace from The overall energy and momentum dependences of reso- the AF wave vector. nant spin excitations then exhibit the typical hour-glass As emphasized above, one of the hallmarks of the res- lineshape characterizingthe resonantmode dispersionin onant spin excitations in SC cuprates is their peculiar temperature dependence1. In this respect, the behav- YBCO, as evidenced by the color maps of Figs. 6.d- e deduced from our data along the two q-directions of ior of Bi2212 is very similar to the phenomenology re- Fig. 2. At E =42 meV, the magnetic signal shrinks ported in YBCO. The magnetic resonance peak at 42 r aroundq . Further, belowE , additionalmagnetic ex- meV disappears steeply at T , as can be seen in Fig. 4.a AF r c citations appear to develop in a momentum region far at 42 meV. The temperature dependence measured at Q=(0.35,0.35,14)and 40 meV suggests a similar change from qAF. As shown by color maps of Figs. 6.d-e, the spectral weight of these excitations is maximum along at T , indicating that the differential signal at this wave c the diagonal direction, leading to an anisotropy that we vector is also of magnetic resonant type. At the back- ground position, Q=(0.25,0.25,14), the T-dependence discuss in section V. does not show any indication of a magnetic signal. As pointed out at the beginning of this section, we Wenowdiscusssomegeneralaspectsofthespinexcita- focus on the enhancement of the magnetic response in tionsofBi2212. While theoveralllayoutofthemagnetic the SC state. The main reason is that previous INS spectrum of Bi2212 is quite similar to that of YBCO, measurements in optimally doped Bi221210 show that well-defined incommensurate magnetic peaks were not normal state excitations are not measurable around 40 observed in Bi2212. Several reasons can be put forward meV. These early measurements are in agreement with to explain this difference. Obviously, the weak signal- the results in YBCO for the same doping level5,6. In to-noise ratio of the measurements in Bi2212 limits the YBCO7, a systematic study as a function of hole doping possibility to see details of the q-dependence. In partic- indicates that the normal-state AF fluctuations weaken ular, we had to work with a broad q-resolution in order continuously with increasing doping level and fall below to pick up enough intensity. (Attempts with improved the detection limit above optimal doping7. Note that q-resolutionwerenotsuccessfulbecauseofinsufficientin- the fact that the magnetic fluctuations are not sizeable 6 aboveT doesnotnecessarilymeanthattheyareabsent, teractions related to the real part of the electronic self- c but it rather suggeststhat they are muchweakerand/or energy29,40. In principle, v◦ can be determined by the F broader in the normal state than in the SC state. For band structure given by the Local Density Approxima- instance in weakly overdoped YBCO, the typical mag- tion (LDA). In a self-consistent experimental procedure nitude of the spin fluctuations left in the normal state using the Kramers-Kronig transformation to determine at the resonance energy is estimated to be one order of the self-energy41, the energy scale t is deduced from an magnitude weaker that magnetic resonance peak in the estimate of the bare Fermi velocity along the nodal di- SC state5. In our slightly overdoped Bi2212 sample, the rection, v◦ ∼ 4.0 eV.˚A41, yielding t=397 meV, in good F upper limit on the magnitude of the magnetic signal left agreement with LDA predictions. An alternative ap- inthe normalstate extractedfromourdatais about1/4 proachto analysethe ARPESdataessentiallydefinesan of the signal in the SC state at 46 meV (Fig. 5). This effective band structure describing only low energy elec- is consistent with a previous report at 43 meV in an op- tronic excitations (see e.g.53). The hopping parameter timally doped Bi2212 sample10. determined in this way is about a factor of two smaller: t ∼ 200 meV. This does not significantly change the lineshape of the gapped Stoner continuum and can be ignored in this section. However, it is important for a IV. FINGERPRINTS OF THE quantitativedescriptionoftheS=1modeorigin(seesec- ELECTRON-HOLE SPIN FLIP CONTINUUM tion VI). Figures 6d-e show color maps of the magnetic inten- In order to elucidate the relationship between the res- sity deduced from fits to the constant energy scans of onant spin excitations in the SC state and the gapped Fig. 2. The threshold of the gapped Stoner continuum Stoner continuum, we have computed the threshold of is superimposed on the magnetic signal. For the sake of the continuum in the odd channel for the two main di- simplicity, one can define two distinct areas below the rectionsalongwhichmostoftheINSmeasurementswere continuum (Fig. 6.b-c). In area I, the threshold of the carried out (see Fig. 6.b-c): (130) and (110). One has continuum decreases away from the AF wave vector, ex- to pay attention to the bilayer structure that affects hibiting a dome-like shape. Area II corresponds to the both spin and charge properties. The motion of elec- remainingpartofthe gappedportionofthephasespace. tronbetweenthe twolayersin abilayerunit leadsto the Along the (130) direction, the magnetic response in the formation of anti-bonding (a) and bonding (b) states, SC state is confined to area I and appears to asymp- and consequently to a splitting of the Fermi surface, as totically approach the threshold of the continuum. The shown in Fig. 6.a. The odd neutron scattering chan- signal vanishes below the detection limit in areas where nel originates from electronic interband spin flip excita- continuum excitations are possible according to the cal- tions, and the threshold of the Stoner continuum in the SC state is defined as the minimum [Ea+Eb ], where culation. k k+q Along the (110) direction, the situation is more com- Ea,b = ξa,b2+∆2 is the electronic dispersion relation k q k k plex. As shown in Fig. 6c,e, area I is smaller than along in the SC state. ∆k = ∆m(coskx − cosky)/2 stands (130), so that its spread in momentum becomes simi- for the d-waveSC gapand the bareelectronic dispersion lar to the q-width of the experimental resolution ellip- relation is described using a simplified tight-binding ex- soid (Fig. 6.c). Moreover,the intrinsic energy width dis- pression: ξka,b = 2t(coskx +cosky)−4t′coskxcosky + cussed above scrambles the observed magnetic response, 2t”(cos2kx−cos2ky)± 14t⊥(coskx−cosky)2−µ. For a limiting an accurate determination of the exact number nearlyoptimallydopedBi2212samplewithaTc of87K, of branches of the magnetic dispersion and their pre- weusedthefollowingparametersdeterminedbyARPES: ciselocation. Someinterestingobservationscanbemade ∆m=35 meV51 and {0.219,0.108,0.207,−0.95} for the nonetheless. In particular, along (110) area II extends parameters {t′,t”,t⊥,µ} in units of t41. The chemical uptoenergiesintherange35-48meV,higherthanalong potential corresponds to a hole doping level determined (130). The extra magnetic signal along the (110) at 38 fromtheSCtransitiontemperatureaccordingtothephe- meV may hence be attributable to secondary spin exci- nomenological relationship of Tallon et al52. tations in area II, in addition to those observed in area Before proceeding to the comparison of the locations I. In slightly underdoped YBCO , such a secondary 6.85 ofthegapintheStonercontinuumandtheobservedres- magnetic contribution has been observed in area II, al- onant spin excitations, one needs to mention that there beit at higher energies around ∼ 54 meV14. There it aretwowaystodeterminetheabsolutescaleofelectronic was also shown that the continuum gives rise to almost parameters. The value of these parameters in absolute vertical silent bands14, where the intensity of collective units (meV) strongly relies on the estimate of the value magnetic modes is suddenly suppressed presumably due of the nearest-neighbor hopping parameter t, which in to the decay into elementary electron-hole spin flip ex- turn can be extracted from the Fermi velocity v mea- citations. This scenario was further confirmed theoret- F suredinARPESexperiments. Alongthenodaldirection, ically within the spin exciton model34. Interestingly in v ≃ 2.0 eV˚A40 is related to the bare Fermi velocity by Bi2212 the scan along (110) at 38 meV (Fig. 2.a) dis- F v = v◦/(1 + λ) where λ describes the electronic in- plays a slight minima at the planar wave vectors q≃(- F F 7 Q QQQ excitations directly measured by ARPES for the same a ) f ) doping level41, without adjustable parameters. This in- terpretationisalsoqualitativelyconsistentwith the con- versecomputationoftheimaginarypartofthedynamical X susceptibility fromARPESdata in the frameworkof the X spin exciton scenario45,46. These computations indeed X indicate the existence of two magnetic resonant modes, located in area I and area II respectively. The detailed momentum shape of the magnetic spectra (Figs. 6.d,e) further suggests that the resonant spin excitation spec- b ) d ) traexhibitahourglasslineshapemainly confinedto area I. Overall, these results are in good agreement with pre- vious studies carried out in YBCO system14,18,19,21, al- thoughthe intrinsicenergywidthofthemagneticmodes in Bi2212 obscures some of the features. II IIIIII I III IIII IIII V. ANISOTROPY OF THE MAGNETIC RESPONSE Another interesting experimental feature is the in- e ) plane anisotropy of the magnetic response. Figure 4.b c ) shows scans performed at 40 meV around the AF wave vector along 3 different directions: (100), (130), and (110). The data are labelled as a function of the an- gle Θ between the direction of the scans and the (100) direction (see Fig. 6.f). Furthermore, the data are plot- II IIIIII I III II IIIIII tedasafunction ofthe reduceddistance tothe AFwave vector (in units of ˚A−1), so that the q-widths of the sig- nals are directly comparable. In all directions, we fit the magnetic peak to a single Gaussian. From Θ = 0◦ FIG. 6: (color online) a) Fermi surfaces in nearly optimally where ∆ = 0.4 ˚A−1 (Full Width at Half Maximum) to doped Bi2212 deduced from ARPES measurements41,51. Θ = 18◦qwhere ∆ = 0.23 ˚A−1, the magnetic signal ex- (red=bonding band, black=anti-bounding band). Threshold q hibits a Gaussian profile, with a reduction of its q-width oftheelectron-holespinflipcontinuumintheSCstatealong at Θ = 18◦. On the contrary at Θ = 45◦, the q-width differentdirections: b)(130),c)(110). Belowthethresholdof of the signal more than doubles, ∆ = 0.9 ˚A−1, indicat- the continuum, one distinguishes two distinct areas labelled q respectively I and II as in14 (see text). The vertical dashed ing a net anisotropy of the magnetic response along the linesindicatethelocationoftheenergyscanswhichwereper- (110) direction. Along that direction and at the reso- formed. The projection of theresolution ellipsoid at 40 meV nanceenergy42meV,oneobtains≃0.45˚A−1 forthein- is also represented. d-e) Color maps showing the enhanced trinsicq-widthofthemagneticsignalafterdeconvolution magnetic response in the SC state as deduced from the fit from the resolution function in agreement with previous of the constant energy scan, in addition to the threshold of reports10,11. thecontinuum. f)Areaoccupiedbythecontinuumat40meV Concerning the origin of the observed anisotropy of (black). Theredpointsindicatetheextensionofthemagnetic the magnetic response along the diagonals, one needs to asdeducedfromFig.4.b(seetext)andthewhitecrossesshow keep in mind that the structure of Bi2212 is not simply where the temperature dependencies of the scattering inten- tetragonal. Indeed, Bi2212 is actually an orthorhombic sity were measured. systemwithanincommensuratedistortionlikelyofcom- positetype54,becausethelatticeparametersoftheCuO 2 planes and those of the BiO planes do not match. The 2 0.4,-0.4) and q≃(-0.6,-0.6), which could be indicative of orthorhombic axes are along the diagonals of the square silentbands(seeFig.6.c)inalocationthatcoincideswith lattice: the sample growth direction, i.e. a∗ , was the one expected basedon the Fermi surface topology41. ortho kept perpendicular to the scattering plane. The direc- The comparison between the location of the observed tion (110) in the tetragonal notation we have adopted magneticexcitationspectraandthemomentumshapeof in this article correspond to b∗ . The incommensu- ortho ∗ ∗ theStonercontinuumofthed-wavesuperconductorthus rate modulation is given by 0.21b +c . Because of ortho showsthatspinexcitationsareonlypresentinthegapped thismodulation,theBi2212samplesarenottwinned,al- regionsofthecontinuum. Weemphasizethatthelocusof though the in-plane lattice parameters are not very dif- the Stoner continuum was computed from the electronic ferent. Itisthereforeconceivablethattheobservedmag- 8 netic anisotropy along b∗ is related to this 1D struc- found in optimally doped YBCO typically corresponds ortho tural anisotropy. It is interesting to remark here that in to only area I, and extra scattering in area II occurs at the YBCO system, the magnetic response also exhibits larger energy14. In optimally doped Bi2212, this extra a 1D-like anisotropy of the magnetic intensity, which is magnetic scattering in area II is found around the same maximalalonga particularin-plane direction, the (100) energy as the resonance peak, yielding a much broader directionperpendiculartotheCuOchains20. AstheCuO peakinq-space. Therefore,thepresentstudyshowsthat chains in YBCO, the BiO planes in the Bi2212 system the assumption of an isotropic magnetic response was 2 play the role of charge reservoir. We therefore cannot notcorrect,leadingtoanoverestimationofthelocalspin exclude that the diagonalanisotropyis a feedback ofthe susceptibility. Thisimpliesthattheq-integratedspectral structural distortion on the magnetic properties. How- weight is likely quite similar in both systems. ever, there are differences in the magnetic anisotropy of bothsystems. Inthe YBCO system,the anisotropynear optimaldopingismostlyrelatedtotheintensity20. Here, VI. ORIGIN OF THE S =1 COLLECTIVE the q-extensionof the magnetic signalappears to be dif- MODE: RPA DESCRIPTION ferentalongthetwodirections. FurtherINSexperiments inwhichthe twoin-planedirectionsa∗ andb∗ are Basedontheenergyandmomentumdistributionofthe ortho ortho studied under the same resolution conditions, following resonantspinexcitationsinthe SC state,wehaveshown prior work in YBCO20, will be needed to shed light on above that our INS data exhibit fingerprints of the d- the origin of the anisotropy of the magnetic response in wave gapped Stoner continuum. This puts constraints Bi2212. on the origin of the S = 1 collective mode. However, WereportinFig.6.fthemomentumdependenceofthe as far as the electron-hole spin flip continuum is taken threshold of the continuum at 40 meV. The area I cen- into account, both itinerant and localized spin models tered at the AF wave vector exhibits a diamond shape, can, in principle, describe the data. Since the charge whereastheareaIIsplitsinto8segments(4alongthedi- excitation spectrum and the SC gap are well-known in agonalsand2alonga∗ andb∗). Theredpointsinthefig- theBi2212familyfromaconsiderableamountofARPES ure indicate the total momentum expansion of the mag- data,this systemis the rightcandidate to test these sce- neticresponses(roughlytwicethefullwidthofthesignal narios quantitatively. Here, we apply the spin-exciton at half maximum), as deduced from Fig. 4.b. The data model29,30,31,32,33,34 using the measured band structure have been symmetrized to account for the fourfold sym- parametersandSCgap. Furthermore,withinthatframe- metry ofthe CuO plane although,as we haveremarked work, one can relate the energy width of the resonance 2 above, it is not known whether the magnetic signal re- peaktothespatialdistributionoftheSCgapasreported byScanningTunnellingMicroscopy(STM)43,44. Wethen spects the square lattice symmetry. Anyway,one notices investigate the magnetic excitation dispersion along the in Fig. 6.f that the magnetic signal matches the area I two directions studied by INS in order to put this com- along (100)but extends into areaII along the diagonals. parison on a quantitative footing. However, the intensity of the magnetic signal in area II In the spin-exciton scenario, the spin susceptibility along the (110) direction is of the same order as the er- takes an RPA-like form29 : ror bars in the measurement performed along the (100) direction. We therefore cannot rule out the existence of χ0(q,ω) a weakresponsein areaII alsoalongthe (100)direction, χ(q,ω)= (3) 1−V χ0(q,ω) but this extra magnetic response (if any) vanishes along q the (130) direction where area II is absent. χ0(q,ω) stands for the standard non-interacting spin Finally, whatever the origin of the anisotropy, its ob- susceptibility of a superconductor29,30,31,32,33,34. The in- servation allows us to solve an old puzzling issue. In the teraction V that enhances the magnetic response can q initialworkofFongetal. inoptimallydopedBi221210,it be interpreted as: (i) the on-site Coulomb repulsion on was found that the energy-integrated spectral weight at copper U in weak coupling models (U ≤ 8t, 8t being q of the magnetic resonance peak was similar in opti- of the order of the band width), (ii) the AF superex- AF mally doped Bi2212 and YBCO (∼ 2µ2/f.u), whereas changecoupling−J(q)=−2J(cosq +cosq )instrongly B x y the local spin susceptibility (integrated in momentum correlated models (U ≥ 8t, such as in the t-t’-J model, space) was actually four times larger in Bi2212 than with J = 4t2/U ∼ 120 meV)31,32, (iii) an effective spin- YBCO. This discrepancy came from the estimation of fermion coupling g(q) in phenomenological models30,34. local spin susceptibility based on the assumption that As before, we limit our calculations to the odd excita- themomentumdistributionofthemagneticresponsewas tions. Inthischannelχ0(q,ω)involvestwo-particleexci- isotropic in Bi2212, as it is the case at q in YBCO. tationsbetweenbonding(b)andanti-bonding(a)states: AF Since the measurement of the momentum width of the χ0 = (χ0 + χ0 )/2. To calculate χ0(q,ω), we use a ab ba magnetic signal along the (110) direction of Bi2212 was SC gap with dx2+y2 symmetry and a normal-state tight foundtobetwiceaslargeasinYBCO,thelocalsuscepti- binding dispersion previously defined in section IV. In bility wasestimatedto be fourtimes larger. Considering order to facilitate the numerical calculations, we used a Fig. 6.f, the momentum shape at the resonance energy smalldampingparameterof2meV,whichissmallerthan 9 0.075 ) n 3mi bility 0.05 / ba nts p pro c a 0.025 ( G y t si n e 0 15 20 25 30 35 40 45 50 55 t Gap n I FIG. 8: Spatial gap distribution considered for the calcula- tion of χ(q,ω). Using a Gaussian fit of this distribution, one obtains a spatial average gap of 35 meV and a FWHM of σ∆ =15 meV corresponding to thesample with Tc =87 K. Energy (meV) FIG.7: Resonantmagneticmodesinnearlyoptimallydoped the convolution product of this δ-function by the Gaus- Bi2Sr2CaCu2O8+δ (TC=87K)measuredatQ=(0.5,0.5,14) sianresolutionfunctionoftheinstrument: itshouldtake (as in Fig. 1.a). The solid lines correspond to computed the form, Imχ(q ,ω)=W exp(−4ln2(ω−Ωr)2) where imaginary part of the susceptibility convoluted with the en- AF r σω2 σ ≃ 6 meV is the energy resolution (as shown by the ergy resolution function: the green line was obtained using a ω green line (RPA 1) in Fig. 7). Clearly, the energy width asingled-waveSCgap(RPA1)andtheredonewithadistri- butionofd-waveSCgaps(RPA2). Inthelattercase,theSC of the resonant mode is significantly broader than the gapaverageis∆m=35meVwithafullwidthatmaximumof expected spectrum, showing that there is an additional 14 meV. contribution to the broadening. This discrepancy is not specific to our sample. Table I shows the corresponding energy position and energy width of all Bi2212 samples the energy width of the INS resolution. Finally, we use investigated so far by neutron scattering. For all doping the following phenomenological form for the interaction levels, the measured resonant mode is broader than the Vq = Ueff −2Jeff(cosqx +cosqy)34. This allows us to calculated resolution-limited RPA peak. For a long time capture the different physical origins of the interaction. now, it has been argued1 that this broadening might be For any given wave vector q, a sharp magnetic ex- relatedtotheelectronicinhomogeneityoftheBi2212sys- citation shows up at an energy Ω when the conditions tem,whichalsomanifestsitselfintheSCgapdistribution r 1−V Reχ0(q,Ω )=0andImχ0(q,Ω )=0arefulfilled. observed in the real space by STM42,43,44. Interestingly, q r r For a givenmagnitude ofthe SC gapamplitude ∆ , the when going into the overdoped regime, the broadening m measuredenergypositionofthe oddmagnetic resonance of the INS magnetic resonance peak48 and the SC gap peakatq =(π,π)constrainsthesumoftheinteraction distribution44 arereducedsimultaneously(seeTableIfor AF parameters V = U +4J , whereas the shape of the resonance peak). qAF eff eff the dispersion is controlled by their ratio Jeff/Ueff. Basedonthisobservation,wehavetestedtheeffectofa Let us start by studying the odd magnetic resonance spatialdistributionofSCgaponthespinexcitationspec- peak at the planar AF wave vector qAF. In Fig. 7, we trumwithin the spinexcitonscenario. Spatialvariations comparethemeasuredresonantpeakenergyscanatqAF of the SC gap at the surface of Bi2212 were reported by and the spin excitation spectrum computed with Eq. 3 severalgroups42,43,44. Thespatiallyaveragedvalueofthe and convoluted with the energy resolution of the spec- superconducting gap, ∆ , measured by STM is consis- m trometer. ForaslightlyoverdopedsamplewithTc=87K, tentwithotherspectroscopictechniquessuchasARPES. ∆m is set to 35 meV, yielding a threshold of the Stoner The FWHM of the gap distribution, σ∆, decreases with continuum of ωc=62 meV at qAF. To obtain a reso- increasing doping level (Fig. 9), from around 15 meV nance energyatthe properenergy,Ωr=42meV, wellbe- at optimum doping to nearly 7 meV for an overdoped low ωc, one needs to adjust the interaction VqAF=1070 samplewithagapaverageof∆m=23meV44. Inourcal- meV, avalue significantlyexceeding the hopping param- culation, we use the gap distribution found by STM at etert=397meV.Asystematicinvestigationyieldsaratio optimal doping, and we further assume that the inter- VqAF/t=2.7,independentoftintherange200-400meV, action Vq is independent of the local SC gap. Since the in agreement with Ref. 34. local gap amplitude varies, the resonance peak position At q , the computed spin excitation spectrum is is different in different patches of the sample. We com- AF dominated by the spin exciton, whose contribution to pute the susceptibility χ(q,ω) in each patch character- the imaginary part of the dynamical spin susceptibility izedbyadifferentSCgapvalue,labellednowχ(q,ω,∆). reads: Imχ(q ,ω) = W δ(ω − Ω ) for ω > 0. The χ(q,ω,∆) is computed for seventeen different SC gaps AF r r measured magnetic resonance peak should be given by with ∆ from 20 to 50 meV. The gap distribution is im- 10 30 obtain the best agreement between the experimental q- McElroy dependence(Figs. 6.d,e)andthecomputedmapsofFigs. Kapitulnik 10. This corresponds to the case when J /U <<1. 25 eff eff v) Lee We compute the maps of Fig.10 with Jeff/Ueff =0.025 me Neutron for both Q directions. As for the energy width at the ( 20 AF wave vector, the model does not describe the results n in the absence of a SC gap distribution (Fig. 10.a and o ti d). One obtains a rather sharp mode dispersion down- u 15 b wardasobservedinYBCO13. Incontrast(Fig. 10.band ri st e), the addition of a SC gap distribution (correspond- di 10 ing to the STM data) allows us to describe the main p a features of the measured magnetic excitation spectrum G (Fig. 10.c and f). For the (130) direction, the experi- 5 mental and computed mappings are quite similar. For the (110) direction, both mappings are consistent, even 0 ifbelow42meVthe computedspectrumfailstodescribe 0.1 0.15 0.2 0.25 the q-broadening of the signal. We found that it is not p possible to obtain a signal around 38-40 meV in area II, FIG.9: Width(FWHM)oftheSCgapdistributionmeasured whatever the q-dependence of the interaction used. by STM42,43,44 and width of the magnetic resonance peak in To summarize, using the band structure and the SC INS data10,11,48. gapknownfromARPESmeasurements41,51,thecalcula- tion of the magnetic excitations in an itinerant spin ap- plemented using the histogram shown in Fig. 8, which proach can reproduce most of the characteristic features approximatesaGaussiandistributionoftheSCgapwith of the INS spectrum measured the spin excitation spec- a width σ . The full susceptibility, χ(q,ω), is given by trum in the SC state of Bi2212. Further, the observed ∆ the sum over all local susceptibilities χ(q,ω,∆): energy width of the resonant spin excitations in Bi2212 is naturally explained within this model as a result of the gap distribution reported by STM42,43,44. The mea- (∆−∆ )2 χ(q,ω)= χ(q,ω,∆)exp(−4ln2 m )d∆ (4) sured q-dependence of the magnetic excitations can be Z σ∆2 also captured by a broadening of dispersive excitations due to the SC gap distribution. The totalspinsusceptibility is further convolutedby the Gaussian resolution function in order to fit the data. Finally, we commentona few points about the nature Using this procedure, we can simulate the broadening of the interaction, that should provide an indication on of the magnetic resonance peak by adjusting the width the Hamiltonian needed to obtain a spin exciton. of the Gaussian gap distribution, σ (Eq. 4). Figure 7 First, one finds a ratio V /t = 2.7 whatever the ∆ qAF (fit: RPA 2) shows that the enhancement of the mag- used band structure. We consider the energy range for neticresponseintheSC stateatthe AFwavevectorcan t which is typically 200-400 meV. Two opposite limits be wellaccountedfor by anintrinsic SC gapdistribution can then be discussed: (i) a weak coupling approach, with σ =14 meV. For an overdopedsample with T =70 where the band structure is not far from the LDA one, ∆ c K48, the same analysis gives σ =5 meV. It is striking i.e. t∼400 meV, then the interaction is an effective on- ∆ that the gap distribution deduced from the fit of the site Coulomb repulsion, (ii) a strong coupling approach, magnetic resonance peak matches the one reported by where one chooses an effective band width, i.e. t ∼ 200 STM for both doping levels (Fig. 9). In the framework meV,reducedfromtheLDAvalueduetothestrongelec- of spin exciton scenario, one can thus give a clear ex- tronic correlations and with an interaction given by the planation of the energy width of the magnetic resonance AF superexchange interaction J (in principle measured excitation. Note that such energy broadening is absent intheAFinsulatingstate). Inthefirstlimit,usingt=395 in the YBCO family compounds. This shows that the meV, the effective interaction V = 1070 meV can be qAF electronic inhomogeneity is not generic to the cuprates. directlycomparedtothebandwidth, W =8t. Onefinds SimilarconclusionsaboutabetterhomogeneityinYBCO V /W ≈ 0.35, which clearly belongs to the weak cou- qAF were reached by measuring the quasi-particle lifetimes56 plingapproach60withaneffectiveon-siteCoulombinter- or the 89Y nuclear magnetic resonance linewidths57. actiononcopper. Intheotherlimit,tisaround200meV Next, we address the consequences of the SC gap dis- andthenV =540meV.Thisvaluecanthenbereduced qAF tribution for the q-dependence of magnetic excitations, to ∼ 4J as expected in the t−t′ −J model32, yielding still within the spin-exciton model. We performed the J = 135 meV. In the insulating state, J is usually re- calculation in the q − ω range covered by our experi- lated to the on-site Coulomb interaction as J = 4t2/U, ments, i.e. along both the (110) and (130) directions yielding U=1185 meV. Surprisingly, the ratio between and between 30 and 60 meV. Using the same set of pa- the interaction and the band width is found to be quite rameters, we additionally adjust the ratio J /U to smallas U/W ≈ 0.74,i.e. in anintermediate regime not eff eff