Multiple-magnon excitations shape the spin spectrum of cuprate parent compounds
Davide Betto, Roberto Fumagalli, Leonardo Martinelli, Matteo Rossi, Riccardo Piombo, Kazuyoshi Yoshimi, Daniele Di Castro, Emiliano Di Gennaro, Alessia Sambri, Doug Bonn, George A. Sawatzky, Lucio Braicovich, Nicholas B. Brookes, Jose Lorenzana, Giacomo Ghiringhelli
MMultiple-magnon excitations shape the spin spectrum of cuprate parent compounds
Davide Betto, ∗ Roberto Fumagalli, Leonardo Martinelli, Matteo Rossi, Riccardo Piombo, KazuyoshiYoshimi, Daniele Di Castro,
5, 6
Emiliano Di Gennaro,
7, 8
Alessia Sambri, Doug Bonn, George A.Sawatzky, Lucio Braicovich, Nicholas B. Brookes, Jos´e Lorenzana, † and Giacomo Ghiringhelli
2, 10, ‡ European Synchrotron Radiation Facility, B.P. 220, 38043 Grenoble, France Dipartimento di Fisica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy ISC-CNR and Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, p.le Aldo Moro 5, 00185 Roma, Italy The Institute for Solid State Physics, The University of Tokyo, Kashiwa-shi, Chiba, 277-8581, Japan Dipartimento di Ingegneria Civile e Ingegneria Informatica,Universit`a di Roma Tor Vergata,, Via del Politecnico 1, 00133 Roma, Italy CNR-SPIN, Universit`a di Roma Tor Vergata, Via del Politecnico 1, 00133 Roma, Italy Dipartimento di Fisica “E. Pancini”, Universit`a degli Studi di Napoli “Federico II”,Complesso Monte Sant’Angelo via Cinthia, 80126 Napoli, Italy CNR-SPIN, Complesso Monte Sant’Angelo via Cinthia, 80126 Napoli, Italy Department of Physics and Astronomy, University of British Columbia,Vancouver, British Columbia V6T 1Z1, Canada CNR-SPIN, Dipartimento di Fisica, Politecnico di Milano,piazza Leonardo da Vinci 32, 20133 Milano, Italy (Dated: February 9, 2021)Thanks to high resolution and polarization analysis, resonant inelastic x-ray scattering (RIXS)magnetic spectra of La CuO , Sr CuO Cl and CaCuO reveal a rich set of properties of the spin / antiferromagnetic square lattice of cuprates. The leading single-magnon peak energy dispersion isin excellent agreement with the corresponding inelastic neutron scattering measurements. However,the RIXS data unveil an asymmetric lineshape possibly due to odd higher order terms. Moreover,a sharp bimagnon feature emerges from the continuum at ( / ,0), coincident in energy with the bi-magnon peak detected in optical spectroscopy. These findings show that the inherently complex spinspectra of cuprates, an exquisite manifestation of quantum magnetism, can be effectively exploredby exploiting the richness of RIXS cross sections. The spin / antiferromagnetic two-dimensional (2D)square lattice is one of the best studied quantum systemsand represents a benchmark for quantum magnetism.Notably, it depicts the spin ground state arrangementin the CuO planes, common to all high-temperature su-perconducting layered cuprates, when no doping charge ispresent and the antiferromagnetic order impedes chargetransport and energetic magnons dominate the spin spec-tra [1, 2]. Upon doping, long range antiferromagnetismis substituted by superconductivity but short-range in-plane spin correlations survive, giving rise to dampedmagnons of comparably high energy [3–6]. Indeed, spinfluctuations are considered to be a main ingredient of theCooper pairing “glue” in these materials [7], as suggestedby the correlation between T c and the exchange interac-tion J in certain cuprate families [8–12]. The spectrum ofmagnetic excitations has been extensively used to exper-imentally determine the coupling parameters with phys-ical importance [1, 2, 10, 13–20], such as J , the hoppingintegral t and the Coulomb repulsion U .The magnon dispersion is traditionally measured byinelastic neutron scattering (INS) and reproduced, withinthe linear spin-wave theory, by an improved Heisenbergmodel that includes higher-order terms [2, 10, 21]. In thelast decade resonant inelastic x-ray scattering (RIXS) [22]has proven to be a valid alternative to INS, in particularfor cuprates [3, 5, 6, 23, 24] and other transition metalcompounds with large superexchange coupling [25–29]. Neutrons interact only with the electrons’ spin, notwith the charge, and with the atomic nuclei, makingthe theoretical treatment of the scattering cross-sectionsrather straightforward [30]. Consequently, once thephononic background is duly subtracted, the interpre-tation of the INS experimental spectra in terms of mag-netic scattering function is in principle simple but, at thesame time, it misses part of the richness of the many-body problem. Instead RIXS allows for a wide energyloss range measured at constant resolving power. Com-pared with INS it can profit from a much larger crosssections and incident flux but requires a more involvedtheoretical analysis [31][32], with less stringent selectionrules. Therefore, RIXS has the potential to provide moreinformation on the problem if one is able to disentanglethe complexity of the spectra by exploiting good energyresolution and the analysis of the scattered x-rays’ polar-ization (polarimetric analysis) [33, 34].These differences in the cross sections stimulated aclose comparison of the two techniques, and some doubtswere raised on the possibility of deriving the actual spindynamical structure factor S ( q , ω ) from RIXS data. In-deed, in the very first RIXS work Braicovich et al. [35]had already shown that in La CuO the single magnonenergy dispersion in INS and RIXS coincide almost per-fectly, a fact that has been recently confirmed more ex-tensively [36]. However, Plumb et al. [21] pointed outsome discrepancies in the magnetic excitation spectrum a r X i v : . [ c ond - m a t . s up r- c on ] F e b (0.25,0.25) (0.5,0) (0,0) (0.25,0.25)00 . . . E n e r g y l o ss ( e V ) INSRIXS (0.5,0) (0,0) (0.25,0.25)SCOC (b)H,K (r . l . u . ) (0.5,0) (0,0) (0.25,0.25)00 . . . . − . − . − . .
51 LCO(0 . ,
0) (d) I n t e n s i t y ( a r b . un i t) − . − . − . . . ,
0) (e)Energy loss (eV) 0 − . − . − . − . − . . ,
0) (f)
FIG. 1. (a-c): Single magnon dispersion determined by RIXS for the three compounds and by INS for SCOC and LCO (Refs. 2and 21, respectively). (d-f): RIXS spectra at (0.4,0) measured with π incident photon polarization (black circles) and theirmain constituents obtained by a phenomenological fitting (red line): Gaussian elastic peak (brown), two resolution limitedphonon contributions (pink), single magnon with Fano lineshape (blue), even order multimagnons (green). of Sr CuO Cl close to q = ( / ,
0) (X-point) of the two-dimensional (2D) Brillouin zone [37], where the RIXS-derived magnon energy exceeds that of INS by ∼
25 meV,i.e., about 10 %. In this Letter, we start from this sin-gle magnon issue and unveil a richer scenario for theRIXS data. We report high-resolution RIXS measure-ments on a La CuO (LCO) thin film ( ∼
100 nm-thick),Sr CuO Cl (SCOC) crystals and a CaCuO (CCO) thinfilm, with special emphasis on the low-energy (magnetic)portion of the spectra and on the comparison with themost recent INS data on the same compounds, whereavailable. A polarimetric analysis at selected momentumpoints, supplemented with theoretical computations, al-lows us to constrain the symmetry of the different con-tributions to the RIXS lineshape.The RIXS spectra have been acquired using the ERIXSspectrometer of the ID32 beamline [38] at the EuropeanSynchrotron ESRF, which includes the polarimeter usedfor the analysis of the polarization of the scattered light[33, 34]. The x-ray energy was tuned to the Cu L edge,at about 931 eV. The incoming x-rays were polarizedeither parallel ( π ) or perpendicular ( σ ) to the scatteringplane (see SM Fig. S1 [39]). The total energy resolutionwas ∼
47 meV for the LCO and CCO, ∼
32 meV for theSCOC and ∼
65 meV for the polarimetric spectra. Wemapped the magnon dispersion for the three compoundswith π polarization along the ( / , / ) → ( / , → (0 , → ( / , / ) path in reciprocal space.Fig. 1 shows the low energy-loss portion of selectedspectra (see SM Fig. S2 for the complete set of data [39]).Each spectrum has been decomposed by phenomenolog- ical multi-peak fitting into an elastic line at zero energyloss, a phonon contribution and its overtone, a Fano line-shape (comprising a leading single magnon peak and amultimagnon tail) and an additional multimagnon peak,in order of increasing energy loss, as shown in panels (d-f) of Fig. 1. The elastic and the multimagnon peaks weremodeled using Gaussian functions, while for the phononpeaks we employed a Lorentzian shape convoluted withthe experimental resolution. For the additional multi-magnon contribution, the choice of lineshape is not cru-cial since the spectrum is very broad energy-wise. For thesingle magnon peak and its tail, we found that the Fanoasymmetric function gives the best results in the fittingprocedure and its implications will be discussed below.We emphasize that the peak is not resolution limited for q > q = (0 ,
0) are notshown because the elastic component is too intense thereand hinders the determination of the loss features. Allthree compounds show the same asymmetric lineshape ofthe main peak, indicating that this is a common featureof 2D spin- / lattices and of cuprates.The extracted single magnon peak positions are shownin Fig. 1 (a-c) and compared to the INS results takenfrom literature when available [2, 21]. The LCO dis-persion is almost superimposed to the INS data over thewhole Brillouin zone in agreement with previous litera-ture [3, 24, 36]. In SCOC, the agreement is very good . , π pol. (a) πππ πσ (0 . , σ pol. (b) σσσ σπ . , . π pol. (c) I n t e n s i t y ( a r b . un i t) (0 . , . σ pol. (d)-0.6 -0.4 -0.2 -002468 (0 . , π pol. (e)Energy loss (eV)-0.6 -0.4 -0.2 -0(0 . , σ pol. (f) FIG. 2. Polarimetric spectra for SCOC at different q . Thered and blue solid lines are the result of a 3-points adjacent-averaging of the experimental data points (squares). π (cid:48) and σ (cid:48) indicate the scattered polarizations. everywhere except close to the X-point. However, thebetter resolution of our RIXS data with respect to thoseof Ref. 41, previously used in Ref. 21 for the comparison,allows us to better model the lineshape and to reducedthe energy difference to ∼
15 meV. The small differenceat ( / ,
0) is mainly due to the inadequacy of a singlepeak to reproduce the actual spectral shape, whose de-termination is more challenging for INS than RIXS inthis region of the momentum space where the scatter-ing intensity is particularly low. Indeed, the X-pointis characterized by a series of very interesting anoma-lies (weakening and broadening of the leading magnonpeak, emergence of high energy tail) both in theoreti-cal studies[42, 43], and in physical realizations of spin- / square-lattice antiferromagnetic systems irrespective ofthe actual exchange energy scale [2, 44, 45]. We can thusconclude that no significant discrepancy between INS andRIXS single magnon dispersion remains if the data aremeasured with adequate energy resolution and statisticalquality and are analysed with the proper lineshape.The richness of the RIXS spectra invites to go beyondthe traditional analysis made on INS data and to bet-ter exploit the complexity of the RIXS cross sections.In that spirit, we acquired RIXS spectra of SCOC withanalysis of the scattered light polarization at three dif-ferent q values (Fig. 2). Measurement methods, analysis and spectral assignments were made as in Refs. 33 and34. Figure 2(a) confirms that the main peak used todraw the single magnon dispersion of Fig. 1 has crossedpolarization character πσ (cid:48) (the prime indicates the scat-tered x-rays polarization). The rotation of the photonpolarization after the scattering process, which impliesa transfer of angular momentum, is needed for an oddnumber of magnons to be excited, i.e., for ∆ S = 1 spinflip process. Conversely the parallel polarization scatter-ing channels must correspond to ∆ S = 0 spin conservingexcitations, i.e., to an even number of magnons simulta-neously excited [46]. With this in mind, the polarimetricdata appear immediately of non-trivial complexity: theydisprove the simplistic assignment of the high energy tailto two-magnons only and they reveal that parallel polar-ization spectra are different when π or σ incident polar-ization is used. The latter is particularly evident by com-paring the blue curves of Fig. 2 in panels of the same row,and is very striking at the X-point. It is often assumedthat, in σπ (cid:48) or πσ (cid:48) polarization (∆ S = 1 excitations), thescattering involves a single on-site spin-flip operator ˆ S ± r ,leading to a RIXS spectrum proportional to the trans-verse magnetic structure factor S ⊥ ( q , ω ), irrespective ofthe relative orientation between electric field and lattice .From theoretical studies the latter is known to consist ofa single magnon peak and a continuum of odd numberof magnons developing above it [18, 42, 47, 48], with thelatter having maximum relative spectral weight (40 %)at ( / ,0) [42, 43] and around 21 % weight on average inthe whole Brillouin zone [18]. The crossed polarizationlineshapes at ( / , / ) and ( / ,
0) [red in Fig. 2(c,d,e,f)are consistent with these predictions. Furthermore, the πσ (cid:48) polarization at ( / ,
0) shows relatively more weightin the continuum as expected from the theory.Conversely, the σπ (cid:48) polarization at ( / ,
0) does not fitthis scenario. Indeed, one might expect σπ (cid:48) and πσ (cid:48) (redlines) to be proportional to each other and, eventually,to S ⊥ (( / , , ω ), which is clearly not the case. This im-plies that the scattering operator in RIXS, in addition tothe standard on-site spin-flip process ˆ S ± ( r ) includes non-local contributions that can be sensitive to the electricfield (i.e., photon polarization) orientation. A possibleexplanation is a generalization of the three spin operatorproposed in Ref. [53], ˆ S ± r ˆ S r · ˆ S r + δ with a matrix elementdepending on the projection of the electric field on thebond direction δ . Interference between these two chan-nels allows to rationalize the need for a Fano line shapefor the fitting, and the different spectral shapes betweenRIXS and INS and between the πσ (cid:48) and σπ (cid:48) configura-tions. Multi-magnon scattering in RIXS was discussedbefore [48] as part of the standard S ⊥ ( q , ω ). Here wepropose that the weight of these excitations can be mod-ulated by the photon polarization (see SM[39]).We now turn to the ∆ S = 0 excitations, which canbe probed in RIXS as well as Raman and infrared (IR)spectroscopy. In all these cases the scattering opera- (cid:15) × IR Exp.IR ISW ππ (0.5,0) σσ , − . − . − . σσ ππ − . − . − . − . . . . . ,
0) (b)Energy loss (eV) R I X S i n t e n s i t y ( a r b . un i t) RIXS ππ Theory ππ FIG. 3. (a) Blue shows the IR line shape from Ref. 49 plot-ted as imaginary part of the dielectric function (assuming adielectric constant (cid:15) = 5 and a phonon frequency shift of0.61 meV). The other curves are the interacting spin-wavetheory with J = 0.108 eV, see Refs. 14–16, and 50 for details.(b) The ππ (cid:48) experimental RIXS lineshape and the theoreti-cal two-magnon theory with the same energy position usedto fit the IR spectra and an experimental Gaussian broad-ening FWHM = 65 meV. The inset shows the multimagnonresponse using exact diagonalization in the Heisenberg modelin a 32 site cluster as implemented in Ref. 51 and 52. tor involves two spin operators B δ r ≡ ˆ S r · ˆ S r + δ , andcan access excitations with an even number of magnons.In RIXS the scattering operator is usually derived as-suming that the main effect of the intermediate 2 p d state is to transiently eliminate one magnetic site. Thislocal approximation [35, 53] yields a polarization inde-pendent lineshape of the even order multi-magnon spec-trum. However, also in this case, we find that the line-shape at the X-point is strongly dependent on the po-larization (blue lines in Fig 2), which again calls for non-local effects of the core hole. Here the polarization ef-fects can be incorporated already at the leading two-spin operator channel, which facilitates an explicit com-putation of the spectral shape. Versions of such po-larization dependent operators have already been pro-posed for RIXS [50, 54]. We adopt the following form, A nl ( q ) = f ( q ) (cid:80) µ (ˆ e i · δ )( δ · ˆ e o ) B δ ( q ). Here ˆ e i,o are thepolarization vectors of the incoming and outgoing pho-tons and B δ ( q ) the Fourier transform of B δ r with f ( q ) apolarization independent form factor. The resulting line-shape is closely related to the theory of phonon-assistedmultimagnon excitation, where the same associated spec-tral function appears but momentum integrated with adifferent form factor [14–16].Figure 3(a) shows the IR experiment [49, 55] togetherwith an interacting spin-wave theory (ISWT) computa-tion restricted to two magnons[14, 15]. This explains the leading peak in terms of the momentum-integratedtwo-magnon response but misses substantial weight inhigher order side-bands, which was thus assigned to four-and higher multi-magnon processes [14–16, 49, 55]. Wealso show the two-magnon spectral function at specific re-ciprocal space points, corresponding to the RIXS (greenand pink) or Raman (brown) lineshape. Upon momen-tum integration, the IR lineshape is dominated by a two-magnon resonance dubbed the bimagnon , which corre-sponds to the proposed RIXS lineshape in the ππ (cid:48) chan-nel at the X-point. Indeed, the theoretical bimagnon line-shape, whose energy is assigned by the IR experiment, ex-plains fairly well the leading observed RIXS peak (panelb). It may appear surprising that the bimagnon hasnearly the same energy as the single magnon (Fig. 2a).This is explained by the attractive magnon-magnon in-teraction and by the fact that the bimagnon has contri-butions from low-energy magnons whose individual mo-mentum is away from the zone boundary. Strikingly, itis clear from Fig. 3 that both RIXS and IR leave a sim-ilar fraction of spectral weight in higher multi-magnonprocesses further supporting a common explanation.In Fig. 3 (a) we show also the σσ (cid:48) RIXS two-magnonISWT prediction, which gives a broad and very weakpeak (pink, multiplied by 5 to make the curves visible)in agreement with the absence of the ≈ . σσ (cid:48) spectrum shown in bluein Fig. 2(b). The structure at ≈ . σσ (cid:48) and ππ (cid:48) ) is qualitatively reproduced if instead of restrictingto two-magnons we perform an exact computation in asmall cluster. This treatment, however, underestimatesthe relative weight of high energy side bands. The sameproblem arises for the IR lineshape and was explained asdue to finite size effects and a substantial four-ring ex-change term in the Hamiltonian [16], which was omittedhere for simplicity.Taking full advantage of the additional informationcontained in the RIXS lineshape (both in parallel andcrossed polarization channels) requires computationswhich include non-local effects in the scattering cross sec-tion and resonant effects in the matrix elements and se-lection rules, which we hope our work will stimulate. Inthe future, the present findings can be extended to othermagnetic systems and doped cuprate compounds. In IRexperiments a remarkably different situation was foundfor spin-1 2D antiferromagnetic square lattice, wherefour-magnon and higher order processes remain negli-gibly weak [15] and the two-magnon theory suffices toreproduce the experimental lineshape[56]. We can under-stand this drastic difference by noting that these compu-tations are based on a 1 /S expansion, which might poseconvergence problems for S <
1. Our results are fur-ther evidence that S = / systems belong to a differentclass and are characterized by proximity to more exoticground states [57], as also proposed earlier by analyzingoptical[58] and INS studies [2, 44, 45].The experimental data were collected at the beam lineID32 of the European Synchrotron (ESRF) in Grenoble(France) using the ERIXS spectrometer designed jointlyby the ESRF and the Politecnico di Milano. This workwas supported by ERC-P-ReXS Project No. 2016-0790of the Fondazione CARIPLO, Regione Lombardia andby MIUR Italian Ministry for Research through projectPIK Polarix and PRIN Project No. 2017Z8TS5B. J.L.acknowledges financial support from Regione Lazio (L.R. 13/08) under project SIMAP. 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