Ferromagnetic Kitaev interaction and the origin of large magnetic anisotropy in α-RuCl_3
J. A. Sears, Li Ern Chern, Subin Kim, P. J. Bereciartua, S. Francoual, Yong Baek Kim, Young-June Kim
FFerromagnetic Kitaev interaction and the origin of large magnetic anisotropy in α -RuCl J. A. Sears, Li Ern Chern, Subin Kim, P. J. Bereciartua, S. Francoual, Yong Baek Kim, and Young-June Kim Deutsches Elektronen-Synchrotron (DESY), 22607 Hamburg, Germany Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario, M5S 1A7, Canada (Dated: October 30, 2019) α -RuCl is drawing much attention as a promis-ing candidate Kitaev quantum spin liquid [1–8].However, despite intensive research efforts, con-troversy remains about the form of the basic in-teractions governing the physics of this material.Even the sign of the Kitaev interaction (the bond-dependent anisotropic interaction responsible forKitaev physics) is still under debate, with con-flicting results from theoretical and experimentalstudies [5, 6, 9–15]. The significance of the sym-metric off-diagonal exchange interaction (referredto as the Γ term) is another contentious question[16–18]. Here, we present resonant elastic x-rayscattering data that provides unambiguous exper-imental constraints to the two leading terms inthe magnetic interaction Hamiltonian. We showthat the Kitaev interaction ( K ) is ferromagnetic,and that the Γ term is antiferromagnetic and com-parable in size to the Kitaev interaction. Ourfindings also provide a natural explanation for thelarge anisotropy of the magnetic susceptibility in α -RuCl as arising from the large Γ term. Wetherefore provide a crucial foundation for under-standing the interactions underpinning the exoticmagnetic behaviours observed in α -RuCl . The magnetic behaviour of the honeycomb material α -RuCl has been the topic of much recent work, followingthe discovery in this material of an unusual continuum ofmagnetic excitations not well explained by spin-wave the-ory [3, 5, 6]. The structural environment and electronicstate of the ruthenium atoms in α -RuCl are such thatthe Kitaev magnetic interaction [19] is expected to be sig-nificant [1, 20]. For this reason, these remarkable findingshave been attributed to fractionalized excitations analo-gous to those found in the spin liquid ground state of theKitaev model [5, 6]. Recent discovery of quantization ofthe thermal Hall signal in the intermediate magnetic fieldphase has further stimulated interest in this material [8].Understanding the salient features of magnetism in α -RuCl requires a good knowledge of the magnetic in-teractions between the ruthenium magnetic moments.The magnetic Hamiltonian relevant to this material in-cludes an isotropic Heisenberg ( J ) term as well as bond-dependent anisotropic Kitaev ( K ), and off-diagonal Γterms [16]. The general Hamiltonian for atoms at ad-jacent sites i and j takes the following form, where α , β , and γ denote the spin components: H ( γ ) ij = J S i · S j + KS γi S γj + Γ( S αi S βj + S βi S αj ) . (1)Often included in this Hamiltonian are further-neighbourisotropic interactions ( J , J , etc.) and additional off-diagonal term Γ (cid:48) due to non-zero trigonal crystal fields[16, 17, 21]. Early ab initio calculations [9] and fitsto inelastic neutron scattering measurements [5, 6] sug-gested an antiferromagnetic Kitaev interaction ( K >
K <
0) [10–15].Although these scenarios can be distinguished by thedirection of the ordered magnetic moment [22], to date,this information has not been experimentally available.The magnetic structural solution from neutron diffrac-tion data [23] suggested two possible structures, withcollinear moments confined to the monoclinic ac plane(See Fig. 1a ). These two magnetic structures, differingonly in the canting angle of the moment direction out ofthe crystallographic ab (honeycomb) plane (Θ), were fitequally well by the neutron data. In the case of
K > − ◦ , whileΘ = +35 ◦ corresponds to a ferromagnetic K ( K < azimuthal dependenceof a magnetic Bragg peak intensity. This measurementis done by rotating the sample around the scattering vec-tor ( (cid:126)q ) as shown in Fig. 2a. With the incident linear po-larization perpendicular to the vertical scattering plane,the diffracted magnetic intensity for electric dipole tran-sitions is proportional to the projection of the orderedmoment onto the scattered beam [24] and shows modula-tion as the sample is rotated about the scattering vector.By modelling this intensity modulation as a function ofthe azimuthal angle, Ψ, one can distinguish between thetwo possible structures suggested by the neutron mea-surement.We have collected REXS data on a single crystal sam-ple of α -RuCl at the known magnetic Bragg peak po-sition expected for zigzag magnetic ordering [2, 5]. Themagnetic diffraction signal in this sample was first char-acterized by measuring its dependence on momentum, a r X i v : . [ c ond - m a t . s t r- e l ] O c t FIG. 1.
Characterization of magnetic scattering. a.
Crystal structure and ordered moment directions of α -RuCl proposed in Ref. [23]. b. Temperature dependence of magnetic diffraction intensity at (0,-1,1.43), showing an orderingtemperature of 12K. At each temperature the magnetic peak was measured by simultaneously scanning the sample and detectorangles. Inset: intensity dependence on L reciprocal space direction (perpendicular to the honeycomb plane) showing a broadpeak at the L=1.5 position. At each L value, the magnetic peak was measured by scanning along the momentum space Kdirection. The integrated intensities for all scans were found by fitting the scans with a Gaussian peak shape. Error barsshown are the square root of covariance value from the fit. c. Energy dependence of the magnetic diffraction intensity at(0,-1,1.43), showing resonance at the Ru L resonant energy of 2837.5 eV. Integrated intensities and error bars were calculatedfrom combined scans of the sample and detector angles, as in b. d. Comparison of the magnetic signals obtained with theincident photon energy at the Ru L edge (2837.5 eV) and the Ru L edge (2970 eV). Scans were collected by simultaneouslyscanning sample and detector angles. Error bars shown are the square root of the number of photons detected. A constantbackground was subtracted, and the overall photon counts normalized to the monitor recording incident beam intensity. temperature, and incident photon energy. The momen-tum dependence of the magnetic signal showed an ex-tended rod in the out-of-plane direction, with a broadpeak at the position expected for ABAB type layer stack-ing as shown in Fig. 1b (inset). This stacking order haspreviously been reported in neutron diffraction measure-ments with an ordering temperature of 14 K, as opposedto the 7 K ordering temperature observed for the three-layer stacking [5]. We measured an ordering temperatureof 12 K for this sample, as shown in Fig. 1b. We notethat diffraction measurements at this x-ray energy will behighly surface-sensitive, since the beam penetrates thesample to a depth of only a few hundred nm, and theobserved 2-layer stacking may not reflect the bulk crys-tal structure. As the magnetic interactions are stronglytwo-dimensional, we do not anticipate that stacking hasa large effect on the moment direction. Scans along theL direction in reciprocal space were also collected at sev-eral different azimuthal positions, to ensure that the az-imuthal dependence does not depend on the L positionselected. The position L=1.43 was selected to maximizeintensity while maintaining an accessible position for thediffractometer.The azimuthal dependent measurement was collectedat an incident photon energy of 2837.5 eV (correspondingto the Ru L edge), where the intensity is at a maximum. The dependence of the peak intensity on the incidentphoton energy was measured both to find the optimalenergy for measurement, and to confirm the resonant na-ture of the magnetic peak. This energy dependence isplotted in Fig. 1c. Following the measurements at the L edge, the magnetic peak intensity at the same recipro-cal space position was also measured at the ruthenium L edge (incident photon energy 2970 eV). The integratedintensity was substantially weaker at the L edge (com-parison is shown in Fig. 1d), and we calculate a ratioof approximately 20 for the intensities at the two photonenergies.The detailed azimuthal dependence of the magneticscattering measured at the (0,-1,1.43) reciprocal latticeposition is shown in Fig. 2b, which exhibits substantialvariation in intensity as the sample is rotated about (cid:126)q .The zero position of the azimuthal angle Ψ correspondsto the orientation with the in-plane direction (-2,0,0.65)pointing along the outgoing beam. The maxima in inten-sity correspond to the positions when the moment lies inthe scattering plane, while the minima are at positionswhere the magnetic moment lies approximately orthogo-nal to the scattering plane. The difference in intensity ofthe two maxima is directly related to the degree of out-of-plane canting, with the largest peak directly indicatingwhich way the moment is canted out of the honeycomb FIG. 2.
Azimuthal dependence. a.
Schematic diagram showing the geometry of the REXS experiment. b. Azimuthaldependence of the magnetic diffraction signal at (0,-1,1.43). The azimuthal dependence is fit best with a magnetic momentangle of θ = +32 ◦ . The modeled intensities for Θ = +25 ◦ , +45 ◦ and − ◦ are shown for comparison. Ψ = 0 correspondsto the position with the in-plane direction (-2,0,0.65) pointing along the scattered beam. The magnetic peak was measuredby scanning the sample angle. Integrated intensities were found by fitting the scans with a Gaussian peak shape. Error barsshown are the square root of the covariance value from the fit. plane. This can be seen in the modeled intensity for thetwo proposed moment directions (Fig. 2b), which showopposite behaviour in this respect. The measured az-imuthal dependence collected for α -RuCl is clearly fitbest by the model with the moment direction pointingtowards the RuCl octahedral face, indicating that themoment direction is along the face-centered direction ex-pected in the case of a ferromagnetic Kitaev term.We also allow the angle within the ac plane (Θ) to varyas shown in dashed lines in Fig. 2b. The best fit is ob-tained when the moment is confined to the ac plane, withΘ = 32 ◦ ± ◦ . This result is consistent with one of thetwo models proposed by the neutron diffraction result,and also provides insight into the form of the magneticHamiltonian. In the case of a ferromagnetic K term,Chaloupka and Khaliullin showed that a substantial an-tiferromagnetic Γ interaction term is required to keep themoment in the ac plane. Specifically they showed thatwith increasing Γ, the moment rotates away from the lo-cal octahedral xy plane (Θ ∼ ◦ ) and slowly approachesΘ = 32 ◦ from the positive side. According to Ref. [22],in order to have Θ ∼ ◦ the magnitude of Γ must bea significant fraction of, or even exceed the magnitudeof K . We note that Θ ∼ ◦ was obtained for anotherKitaev material Na IrO [25], which would suggest thatthe Γ term is much smaller in Na IrO .Our REXS results provide a clue for solving one of theremaining questions regarding the magnetic properties of α -RuCl : its large magnetic anisotropy. As reported bymany groups [2, 4, 26, 27], the in-plane magnetic suscep-tibility measured by applying magnetic field along the di- rection in the ab plane is significantly larger than the out-of-plane susceptibility. A conventional way to explainthis would be resorting to the g -factor anisotropy. How-ever, experimental data suggest that g -factor anisotropycannot be very large, certainly not large enough to ac-count for the anisotropic susceptibility [28, 29]. Anotherroute to obtain a large magnetic anisotropy is via a largeΓ term as suggested in Ref. [30]. Physically, the effect ofthe Γ interaction is to force the moments towards the abplane, which accentuates magnetic anisotropy.We demonstrate that a large Γ is sufficient to explainthe observed magnetic anisotropy by comparing the ex-perimental data with theoretical calculation results. Thelow-field magnetization data for fields applied in-planeand out-of-plane are plotted in Fig. 3, which shows thatthe susceptibility (slope) anisotropy is about χ ab /χ c ∼ JK Γ model (Eq. (1)),where the model parameters are chosen to be consis-tent with the magnetic moment direction determined byREXS. Either a small Γ (cid:48) or J term was added to en-sure the zigzag ground state of the model (details aboutthe calculation are provided in the Supplementary Infor-mation). The data can be fitted for several parameterchoices with ferromagnetic K and antiferromagnetic Γ ofsimilar magnitude, demonstrating that the magnetiza-tion data can be explained without resorting to g -factoranisotropy. We note that in [30] it was shown that a ratioof | Γ /K | ∼ FIG. 3. Fitting the experimental data through simulatedannealing calculations on the classical spin model. Theexperimental measurements were carried out using a com-mercial SQUID magnetometer at 2 K. The two represen-tative parametrizations theory-I and theory-II correspondto (
J, K, Γ , Γ (cid:48) ) = ( − . , − , . , − .
9) and (
J, K, Γ , J ) =( − . , − , . , . . ◦ andΘ = 34 . ◦ between the moments and the honeycomb planeat zero field, respectively. Energy is in units of meV. Themagnetization curves of these two parameterizations are verysimilar such that they overlap each other. We fix g = 2 . S = 1 / mined that the ordered moment direction in α -RuCl points toward the octahedral face, rather than towardsone of the cubic axes of the RuCl octahedra. This resultestablishes that the Kitaev interaction is ferromagneticin this material. In addition, we show that a substan-tial antiferromagnetic Γ interaction is essential for un-derstanding magnetism of α -RuCl . In particular, thepresence of large Γ interaction could reconcile the largemagnetic anisotropy observed experimentally with the al-most isotropic g -factors expected in this material. Thefindings of our REXS measurement provide new exper-imental constraints on the magnetic Hamiltonian of α -RuCl , indicating that it lies within the ferromagneticK, antiferromagnetic Γ regime [22]. This result is inagreement with the findings of a number of ab-initio cal-culations, and can inform future investigations into theunusual magnetic behavior of α -RuCl . METHODS
REXS measurements were carried out at the beam-line P09 at PETRA III at DESY (see [31] for details)at the ruthenium L and L edges (2838 and 2967 keVrespectively). Most of the measurements, including mo- mentum, temperature, and azimuthal dependence werecollected at the L edge. The magnetic intensity wasalso measured at the L edge to determine the branchingratio. The monochromator was detuned to minimize thepresence of higher harmonics in the beam, and the mea-surements were made with a sodium iodide scintillationdetector. An all-in-vacuum path was used to minimizex-ray absorption by air. α -RuCl single crystals weregrown by vacuum sublimation in sealed quartz tubes us-ing commercial RuCl powder. The single crystal usedfor this measurement was a flat plate with largest dimen-sion ∼ µ m.The orientation of the crystal used for this measure-ment was checked at room temperature with the crystalin the known monoclinic structural phase, by checkingstructural Bragg peaks using higher energy (third har-monic) photons. The crystal was also checked to ensurethat it did not possess a twin rotated by 180 ◦ , whichwould affect the result of the azimuthal measurement.This was done by searching for structural peaks at thepositions expected for the rotated structure. No intensitywas found at the peak positions expected for the rotatedcrystal structure. The azimuthal dependence data wascorrected for beam absorption (as described in [32]), andthe beam footprint on the sample. Beam footprint on thesample was calculated from the angle between the samplesurface and the incoming beam, and depended only onthe ratio of the beam height and the smallest dimensionof the sample. This ratio of beam height to sample size,and the magnetic moment angle Θ were the only param-eters refined in the fit of the azimuthal dependence data.More detailed information about the fitting procedurecan be found in the Supplementary Information. ACKNOWLEDGEMENTS
We would like to thank Joel Bertinshaw and HakutoSuzuki for their help with the experiment. We ac-knowledge DESY (Hamburg, Germany), a member ofthe Helmholtz Association HGF, for the provision of ex-perimental facilities. Parts of this research were carriedout at PETRA III. Work at the University of Torontowas supported by the Natural Science and EngineeringResearch Council (NSERC) of Canada, Canadian Foun-dation for Innovation, Ontario Innovation Trust, andthe Center for Quantum Materials at the University ofToronto. Y.B.K. is also supported by the Killam Re-search Fellowship from the Canada Council for the Arts.This work was performed in part at Aspen Center forPhysics, which is supported by National Science Founda-tion grant PHY-1607611. [1] Plumb, K. W. et al. α -RuCl : A spin-orbit assistedMott insulator on a honeycomb lattice. Phys. Rev. B , 041112(R) (2014).[2] Sears, J. A. et al. Magnetic order in α -RuCl : ahoneycomb-lattice quantum magnet with strong spin-orbit coupling. Phys. Rev. B α -RuCl . Phys. Rev. Lett. α -RuCl and the zigzag antiferromagnetic ground state. Phys. Rev. B
Nat. Mat. α -RuCl . Science α -RuCl . Nat. Phys.
Nature , 228 (2018).[9] Kim, H.-S., Shankar, V. V., Catuneanu, A. & Kee, H.-Y.Kitaev magnetism in honeycomb RuCl with interme-diate spin-orbit coupling. Phys. Rev. B α -RuCl : an ab initio study. Phys. Rev. B
Phys. Rev. B α -RuCl . Sci. Rep. Phys. Rev. B α -RuCl . Phys.Rev. B α -RuCl from first prin-ciples. Preprint at https://arxiv.org/abs/1904.01523 (2019).[16] Rau, J. G., Lee, E. K.-H. & Kee, H.-Y. Generic spinmodel for the honeycomb iridates beyond the Kitaev limit. Phys. Rev. Lett. IrO are large andferromagnetic: insights from ab initio quantum chemistrycalculations. New J. Phys. IrO . Phys. Rev. B
Ann. Phys.
Phys. Rev. Lett. IrO in the presence of strong spin-orbit inter-action and electron correlations. Phys. Rev. Lett. IrO and RuCl . Phys.Rev. B α -RuCl . Phys. Rev. B
Acta Crystallogr. A IrO . Nature Physics , 462466 (2015).[26] Kubota, Y. et al. Successive magnetic phase transitionsin α -RuCl : XY-like frustrated magnet on the honey-comb lattice Phys. Rev. B , 094422 (2015).[27] Majumder, M. et al. Anisotropic Ru magnetism inthe α -RuCl honeycomb system: Susceptibility, specificheat, and zero-field NMR Phys. Rev. B , 180401(R)(2015).[28] Agrestini, S. et al. Electronically highly cubic conditionsfor Ru in α -RuCl . Phys. Rev. B α -RuCl . Phys. Rev. B
Phys. Rev. B
J. Synchrotron Rad.
Eur. Phys. J. B19,