Nature of Magnetic Excitations in the High-Field Phase of α -RuCl 3
A.N. Ponomaryov, L. Zviagina, J. Wosnitza, P. Lampen-Kelley, A. Banerjee, J.-Q. Yan, C.A. Bridges, D.G. Mandrus, S.E. Nagler, S.A. Zvyagin
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Nature of Magnetic Excitations in the High-Field Phase of α -RuCl A. N. Ponomaryov, ∗ L. Zviagina, J. Wosnitza,
1, 2
P. Lampen-Kelley,
3, 4
A. Banerjee, † J.-Q. Yan, C. A. Bridges, D. G. Mandrus,
4, 3
S. E. Nagler, and S. A. Zvyagin ‡ Dresden High Magnetic Field Laboratory (HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.qmat,Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany Institut f¨ur Festk¨orper- und Materialphysik, TU Dresden, 01062 Dresden, Germany Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37821, USA Department of Materials Science and Engineering,University of Tennessee, Knoxville, TN 37821, USA Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Chemical Science Division, Oak Ridge National Laboratory, Oak Ridge, TN 37821, USA (Dated: July 16, 2020)We present comprehensive electron spin resonance (ESR) studies of in-plane oriented single crys-tals of α -RuCl , a quasi-two-dimensional material with honeycomb structure, focusing on its high-field spin dynamics. The measurements were performed in magnetic fields up to 16 T, applied alongthe [110] and [100] directions. Several ESR modes were detected. Combining our findings withrecent inelastic neutron- and Raman-scattering data, we identify most of the observed excitations.Most importantly, we show that the low-temperature ESR response beyond the boundary of themagnetically ordered region is dominated by single- and two-particle processes with magnons aselementary excitations. The peculiarities of the excitation spectrum in the vicinity of the criticalfield are discussed. Spin systems with honeycomb structures have recentlyattracted a great deal of attention, in particular in con-nection with the Kitaev-Heisenberg model [1]. The modelpredicts a variety of magnetic phases, ranging from theconventional N´eel state to a quantum spin liquid, withthe excitation spectrum formed by spin-flip excitations,fractionalized into gapped flux excitations and gaplessMajorana fermions [2]. α -RuCl has been proposed asone of the prime candidates to test this model [3]. Inthis material, the multiorbital 5 d t g state can be mappedinto a single orbital state with effective pseudospins j eff = 1/2. The spins are arranged into a two-dimensional(2D) honeycomb lattice [Fig. 1(a)] with bond-dependentinteractions, defined by the Kitaev parameter K in theHamiltonian: H = X h ij i (cid:20) JSSS i · SSS j + KS γi S γj + Γ (cid:16) S αi S βj + S βi S αj (cid:17)(cid:21) −− µ B X i B · g · SSS i . (1)Here, S i and S j are spin-1/2 operators at site i and j ,respectively, J is the Heisenberg exchange parameter, Γrepresents a symmetric off-diagonal term, and µ B , B , g correspond to the Bohr magneton, magnetic field, and g tensor, respectively ( α and β are perpendicular to the Ki-taev spin axis γ ). A number of sets of parameters of thegeneralized Kitaev-Heisenberg model for α -RuCl havebeen proposed (for a review see, e.g., Ref. [4]). Below T N ∼ B c [6], followed by apartial polarization of the ground state [7]. In addition,a signature of the second phase (AF2) has been detectedbetween B ∗ c and B c [8–10]. For H k [110], the criticalfields are B ∗ c = 6 . B c = 7 . B c = 7 . H k [100], where theseparation is small, if not zero [9]. FIG. 1. (a) Schematic view of the honeycomb structure, show-ing the [100] and [110] axes relative to the Ru–Ru bonds. Ruions from adjacent zigzag chains are shown by different colors.(b) Schematic temperature-field phase diagram for α -RuCl .AF1 and AF2 correspond to different low-temperature anti-ferromagnetically ordered phases. One striking peculiarity of the spin dynamics in α -RuCl is the presence of a broad excitation continuum,which has been interpreted as a potential signature offractionalized Majorana excitations [11–14]. It can beobserved up to well above 100 K, indicating the ratherhigh-energy scale of magnetic interactions in this com-pound. The continuum remains present well below T N ,when the ground state is magnetically ordered and thelow-energy excitation spectrum is formed by two anti-ferromagnetic resonance (AFMR) modes [15–17]. Basedon that, α -RuCl was proposed to be in close proxim-ity to the predicted Kitaev quantum spin liquid [12]. Inthe field-induced disordered phase the continuum is alsopresent and gapped, with the gap gradually increasingwith the applied magnetic field [8, 18–21].Alternatively, the continuum can be described in termsof incoherent multimagnon processes [22, 23]. In line withthat, recent calculations [4] point toward the physics ofthe strongly interacting and mutually decaying magnons,not to that of the fractionalized excitations.Recent high magnetic field spectroscopy measurementsrevealed a very rich excitation spectrum in the field-induced magnetically disordered phase [15, 18–21], in-cluding several modes below the continuum. The remark-ably large slope of some of them implies the presence oftransitions with ∆ S = 2 (contrary to ∆ S = 1, expectedfor conventional one-particle excitations in S = 1 / α -RuCl has an emergentmultiparticle nature, raising an important question onthe nature of the observed excitations.To understand the complex spin dynamics in α -RuCl ,a comparative analysis of available experimental data isessential. Unfortunately, one critical shortcoming of themajority of magnetic studies of α -RuCl comes from ig-noring its in-plane anisotropy, which makes such a com-parison challenging or even impossible. The anisotropyappears to be rather pronounced, as followed from high-field electron spin resonance [15] and magnetic suscep-tibility [9, 24] measurements, suggesting the presence ofthe Kitaev parameter K and symmetric off-diagonal spinexchange Γ [Eq. (1)] as two key sources of the anisotropy[24, 25].Electron spin resonance (ESR) is traditionally recog-nized as one of the most sensitive high-resolution spec-troscopy tools for studying the spin dynamics in stronglycorrelated electron systems, capable of probing not onlyconventional magnons, but also fractional excitations(such as spinons and solitons [26–29]), the property ofmagnetic materials with quantum spin liquid groundstates. Here, we present results of high-field tunable-frequency ESR studies of α -RuCl , focusing on its spindynamics in the field-induced magnetically disorderedphase.The measurements were performed on high-quality sin-gle crystals from the same batch as reported previously[15]. The platelike samples were prepared using a vapor-transport technique starting from pure RuCl powderand have typical sizes of 3x3x0.5 mm . The experimentswere performed employing a 16 T transmission-type mul-tifrequency ESR spectrometer, similar to that describedin Ref. [30]. A set of backward-wave oscillators, Gunndiodes, and VDI microwave sources (Virginia Diodes Inc,USA) was used, allowing us to study magnetic excitationsin a broad quasicontinuously covered frequency range, Magnetically Disordered A F A F G G*E F r equen cy ( T H z ) A D
H [100]h H (a) B c Zigzag AF1 J Magnetically Disordered A Zigzag AF1 F F r equen cy ( T H z ) Magnetic field (T)
B C
H [100]h H (b)
Magnetically Disordered CD Zigzag AF1 Magnetically Disordered B *c B c H [110]h H
EA(c)
Zigzag AF1 A Magnetic Field (T)
H [110]h H
FCDB(d)
FIG. 2. Frequency-field dependences of magnetic excitationsin α -RuCl for H k [100] (a, b) and H k [110] (c, d) ( h ω k H (a, c), h ω ⊥ H (b, d), where h ω is the magnetic componentof the THz radiation; T = 1 . from 0.05 to 1.2 THz (corresponding to an energy rangeof about 0.2-5 meV). The experiments were performedin the Voigt configuration with magnetic fields H k [100]and H k [110] [i.e., applied parallel and perpendicular toa Ru–Ru bond direction, respectively, Fig. 1(a)]. Fineorientation of the samples was done in situ , employing agoniometer with the rotation axis normal to honeycomblayers. A wire-grid polarizer was installed just before thesample, allowing us to select the polarization of the inci-dent THz radiation with respect to the applied magneticfield and crystallographic axes.The frequency-field dependences of polarized ESR in α -RuCl for H k [100] and H k [110] are shown in Fig. 2.Some examples of polarized-ESR measurement data areshown in the Supplemental Material [31]. Two modeswere observed in the low-field zigzag ordered phase, asreported previously [15]. These excitations (modes A and B ) correspond to conventional relativistic AFMR modesexcited at the Γ point, with the zero-field frequencies0.62 and 0.8 THz [17]. Both modes exhibit pronouncedsoftening in magnetic field. The low-frequency AFMRmode A is dominantly excited when h ω k H , while themode B corresponds to excitations with h ω ⊥ H . At lowfields, mode A was observed also when h ω ⊥ H . Suchan unusual behavior can be explained by a change indomain populations, as suggested by neutron-scatteringstudies [8]. Detailed spin-wave-theory analysis of theAFMR spectrum (including the polarization dependenceof magnetic excitations) was performed by Wu et al. [17],revealing an overall good agreement with the obtainedexperimental data.For both orientations of applied magnetic field, the in-tensity of the AFMR modes decreases significantly whenapproaching the critical region. These changes can beparticularly well seen in unpolarized ESR spectra [Figs. 3and 4]. Remarkably, our ESR measurements revealed thepresence of AFMR mode A not only below B ∗ c , but alsobetween B ∗ c and B c [Fig. 2(c)]. It has been recently pro-posed, that the antiferromagnetic interlayer coupling in α -RuCl results in a triple-layer structure modulation inthe direction perpendicular to the honeycomb direction(corresponding to the magnetic order 3f-zz), while theferromagnetic interaction would lead to zigzag orderedstate with a unit cell of six layers (6f-zz); the latter islikely realized in α -RuCl between B ∗ c and B c [32]. Basedon this assumption, the observation of the mode A in theAF2 phase suggests the coexistence of the 3f-zz and 6f-zz magnetic structures in this narrow intermediate fieldrange [Fig. 1(b)]. More details of high-field magneticstructure studies of α -RuCl will be reported elsewhere[33]. c B c B c A 144 GHz 366 GHz AA AA 276 GHz C 177 GHz 96 GHz
Magnetic Field (T)
440 GHz B FFA AAB F 680 GHz CCCC 491 GHz 598 GHz 529 GHz
Magnetic Field (T)
FA CB T r an s m i t an c e ( a r b . un i t s ) D 1182 GHz C CC CDE FG 877 GHz 976 GHz 1084 GHz
Magnetic Field (T)
G, EG DG, EE
FIG. 3. Examples of unpolarized ESR spectra in α -RuCl for H k [100] at various frequencies; T = 1 . Several magnetic resonance modes were observedabove B c . The frequency-field diagrams of these modesfor different polarizations of the incident THz radiationare shown in Fig. 2.Recent neutron-scattering measurements of α -RuCl revealed a sharp magnon mode at the lower bound of astrong continuum [10]. This mode has a measurable dis-persion in the direction perpendicular to the honeycombplanes, suggesting the presence of non-negligible inter- A C 335 GHz A C 370 GHz A C 392 GHz B *c B c B c B *c B c B *c A 109.6 GHz AC 104 GHz 128 GHz A 208 GHz 280 GHz AA 313 GHz A CA A F 440 GHz CCCD 572 GHz 500 GHz
Magnetic Field (T)
D CFD FFAA BB FF CC CCCD E D 615 GHz 646 GHz 707 GHz 752 GHz
Magnetic Field (T)
FDBE D T r an s m i t an c e ( a r b . un i t s ) G 801 GHz CCCCCDEGG 881 GHz 929 GHz 1084 GHz 1181 GHz
Magnetic Field (T)
FE DE DEDE D
FIG. 4. Examples of unpolarized ESR spectra in α -RuCl for H k [110] and T = 1 . plane interactions (the dispersion perpendicular to theplane was seen also in the magnetically ordered phasebelow B c , but it is much weaker than the in-plane dis-persion). The corresponding neutron-scattering data at(0, 0, 3.3) and (0, 0, 4.3) are shown in Fig. 2(d) by closedsquares and triangles, respectively. The dispersion pe-riodicity along the (0, 0, L ) direction suggests that theexcitation energy at the Γ point (maximum of the ex-citation dispersion) and at the magnon zone boundary(dispersion minimum) are approximately the same as for(0, 0, 3.3) and (0, 0, 4.3), respectively. Based on that,the excitations C and F are identified as relativistic andexchange modes of magnetic resonance [Fig. 5(a)] (thesame interpretation is given in Ref. [4]); similar excita-tions were observed, e.g., in the field-induced polarizedphase in the triangular-lattice antiferromagnet Cs CuCl [34]. The mode C is the most intensive resonance (Figs. 3and 4), having maximal intensity for the polarization h ω ⊥ H . This mode was observed also by means of far-infrared [18, 20] and Raman-scattering [21] spectroscopy[the latter is denoted as M F has a polarization h ω ⊥ H . The corresponding transi-tions (modes C and F ) are shown in Fig. 5(a) by the solidred and blue arrows, respectively. The observation of theexchange mode F (which is, as expected, much weakerthan the mode C ) becomes possible due to the staggeredDzyaloshinskii-Moriya interaction [35, 36], which is al-lowed in α -RuCl due to the absence of an inversion sym-metry center between the Ru ions in adjacent layers. Ourscenario is supported by recent calculations for a three-dimensional exchange model [32]. The distance betweenthe C and F modes gets larger with increasing field, indi-cating that spin correlations in the system are becomingless 2D in high fields. Similar behavior was observed byinelastic neutron-scattering experiments [10]. E2F2C C E ne r g y ( a r b . un i t s ) F q -2 c* (a) A B 2C (b)
E 2FC M2’ F r equen cy ( T H z ) F Magnetic field (T)
M3’M1
Bc H II [100] J FIG. 5. (a) Proposed schematic energy diagram for α -RuCl in an arbitrary magnetic field above B c . The modes C and F are single-magnon excitations, while the modes 2 C and 2 F correspond to two-magnon excitations. The mode E corre-sponds to an excitation of a two-magnon bound state. (b)Frequency-field dependences of selected ESR modes [fromFigs. 2(a) 2(b)] and the color contour plot of the high-fieldRaman-scattering intensity [21] ( H k [100]). Raman scattering is known as a very powerful tool toprobe two-particle processes in strongly correlated spinsystems. Such two-magnon excitations, the modes M ′ and M ′ , were observed in α -RuCl in the field-inducedphase [21, 37] (Fig. 5(b); for comparison we show simu-lated modes 2 C and 2 F with the excitation energy twicelarger than that for the modes C and F, respectively).The continuum is spread well above 2 C , suggesting con-tributions of multiple-particle processes in the entire Bril-louin zone [32]. Based on the proposed scenario, onewould expect the presence of higher-energy excitations(such as modes G and G ∗ in Fig. 2), involving multipar-ticle processes with different wave numbers.The ESR mode E, with excitation energy slightly largerthan that for the mode 2F (but smaller than that for themode 2C), was observed for h ω k H and can be ten-tatively interpreted as an excitation of a two-magnonbound state [dashed green arrow in Fig. 5(a)].In the vicinity of B c the modes G and E are super-imposed (Fig. 3). To obtain more details about thiscritical range, we refer to our polarized ESR measure-ment data (Supplemental Material [31]). Surprisingly, atabout B c our experiments revealed a broad dip denotedas J , whose field position is almost independent on thefrequency. The dip was observed in the ∼ −
900 GHzfrequency range with the polarization of the incident THzradiation h ω k H ( H k [100], Fig. 1(a), SupplementalMaterial) [38]. The position of the dip J is shown in Figs. 2(a) and 5(b). Remarkably, this frequency range islocated between the excitation energies for modes 2 F and2 C , corresponding to the lower and upper boundaries ofthe two-magnon continuum, respectively (Fig. 5). Thisstrongly suggests that the field-induced transition fromthe magnetically ordered to disordered phase strongly af-fects not only the ground state properties, but also theexcitations spectrum, including multiparticles processes.We hope that our observation will stimulate further the-oretical studies of the unconventional spin dynamics in α -RuCl , in particular, in the critical regime in the vicin-ity of B c .The ESR mode D is relatively weak at low frequen-cies, gaining intensity at higher frequencies and fields.This mode is excited for both polarizations of incidentTHz radiation, h ω k H and h ω ⊥ H (Fig. 2). Similar toother high-field modes, the resonance field for the modeD exhibits a 60 ◦ periodicity [15]. On the other hand, theangular dependence of this mode is significantly differentfrom the others (e.g., for modes C , E , F ), demonstrat-ing a shift of 30 ◦ . The observed very peculiar angulardependence of mode D might provide a potential hintfor identifying the nature of this excitation.Very recently, a plateau in the thermal Hall effect hasbeen observed over a finite field range [39–41]. This hasbeen interpreted as a signature of fractional non-Abelianexcitations, possibly the Majorana fermions of the Ki-taev model on a honeycomb lattice. The presence of aplateau over a limited range of applied fields (approxi-mately between 9.7 and 11.5 T for H k [110] [40]) wouldsuggest the presence of additional phase transitions atthe fields corresponding to the upper and lower boundsof the plateau. Possible evidence for that has been seenin magnetocaloric [10] and magnetostriction [42] exper-iments, while another thermodynamic study (magneticGr¨uneisen parameter and specific heat) detected no signof such transitions [43]. Our high-field ESR measure-ments show magnon modes, characteristic of a partiallypolarized state emerging right above B c , not revealingany evidence for additional high-field phases or phasetransitions in magnetic fields up to 16 T. The question ofsuch a coexistence (the nontrivial topological excitations,if any, and conventional bulk magnons, observed by us in α -RuCl ) remains open, demanding more systematic ex-perimental and theoretical investigations.In conclusion, we have reported on the high-resolutionhigh-field THz ESR spectroscopy studies of in-plane ori-ented single crystals of α -RuCl in magnetic fields upto B c and beyond, applied parallel and perpendicularto Ru–Ru bond directions. We have confirmed therather anisotropic ESR response, highlighting the signifi-cant role of anisotropic in-plane interactions in α -RuCl .Complemented by the results of recent inelastic neutron-and Raman-scattering measurements, we have arguedthat the high-field spin dynamics in this material is dom-inated by one- and two-particle excitations identified asmagnons. We hope that our observations will stimulatefurther theoretical studies of the unconventional spin dy-namics in α -RuCl , in particular, in the critical regimein the vicinity of B c .This work was supported by the Deutsche Forschungs-gemeinschaft through Garnt No.ZV 6/2-2, the excellencecluster ct.qmat (EXC2147, Project-id No 390858490),and SFB 1143, as well as by the HLD at HZDR, memberof the European Magnetic Field Laboratory. A.B. andS.E.N. were supported by the Division of Scientific UserFacilities, Basic Energy Sciences US DOE, P.L.-K. andD.G.M. by the Gordon and Betty Moore Foundation’sEPiQS Initiative through Grant GBMF4416, J.-Q.Y. andC.A.B. by the U.S. Department of Energy, Office of Sci-ence, Office of Basic Energy Sciences, Materials Sciencesand Engineering Division. We would like to thank D.Wulferdung and P. Lemmens for sharing their experimen-tal data. We acknowledge discussions with M. Vojta, A.Chernyshev, M. Zhitomirsky, and A. Kolezhuk. Supplemental Material
JJJJ D B c EA DD DJ
931 GHz887 GHz856 GHz1124 GHz D H [100]h H A Magnetic field (T)
825 GHz 480 GHz T r an s m i ss i on ( a r b . un i t s )
80 GHz186 GHz
A A
270 GHz769 GHz
A D J DDE C C
DC F FCB
740 GHz A
622 GHz B H [100]h H
FCBB C C F
942 GHz708 GHz595 GHz T r an s m i ss i on ( a r b . un i t s ) Magnetic Field (T)
480 GHz
B FC
816 GHz
C F B c FIG. 6. Examples of polarized ESR spectra in α -RuCl for h ω k H (a) and h ω ⊥ H (b) ( h ω is the magnetic componentof the THz radiation). H k [100], T = 1 . ∗ Present Address: Institute of Radiation Physics,Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden,Germany. † Present Address: Department of Physics and Astronomy,Purdue University, West Lafayette, IN 47907, USA. ‡ Corresponding author: [email protected] JJ
586 GHz CD
881 GHz DE H [110]h H
A C AE T r an s m i ss i on ( a r b . un i t s ) Magnetic field (T) D B *c B c C
902 GHz A
186 GHz
A CCCE DDE T r an s m i ss i on ( a r b . un i t s )
587 GHz C H [110]h H
FA B DD CC C
Magnetic Field (T)
710 GHz 500 GHz
B F FDD C FB
790 GHz D
960 GHz
D C B *c B c FIG. 7. Examples of polarized ESR spectra in α -RuCl for h ω k H (a) and h ω ⊥ H (b) ( h ω is the magnetic componentof the THz radiation). H k [110], T = 1 . , 027204 (2010).[2] A. Kitaev, Ann. Phys. (Amsterdam) , 2 (2006).[3] K. W. Plumb, J. P. Clancy, L. J. Sandilands,V. V. Shankar, Y. F. Hu, K. S. Burch, H. Y. Kee,and Y. J. Kim, Phys. Rev. B , 041112(R) (2014).[4] P. A. Maksimov and A. L. Chernyshev,arXiv:2004.10753.[5] J. A. Sears, M. Songvilay, K. W. Plumb, J. P. Clancy,Y. Qiu, Y. Zhao, D. Parshall, and Y.-J. Kim, Phys.Rev. B , 144420 (2015).[6] R. D. Johnson, S. C. Williams, A. A. Haghighirad,J. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li,H. O. Jeschke, R. Valent´ı, and R. Coldea, Phys. Rev. B , 235119 (2015).[7] Because of the presence of the finite Kitaev coupling, afull spin polarization can be achieved only at infinitelylarge magnetic field.[8] A. Banerjee, P. Lampen-Kelley, J. Knolle, Ch. Balz,A. A. Aczel, B. Winn, Y. Liu, D. Pajerowski,J. Yan, C. A. 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Shiba, J. Phys. Soc. Jpn. , 867 (1994).[37] Contrary to our interpretation, the mode M ′ in Ref. [21]is identified as excitation of a two-magnon bound state.[38] The dip J is more pronounced for H k [100], but somesignature of this feature was observed for H k [110] (see,e.g., the spectrum at 881 and 902 GHz in Fig. 2(a), Sup-plemental Material).[39] Y. Kasahara, T. Ohnishi, N. Kurita, H. Tanaka, J.Nasu, Y. Motome, T. Shibauchi, and Y. Matsuda, Na-ture (London)559