Oscillations of the thermal conductivity observed in the spin-liquid state of α-RuCl_3
Peter Czajka, Tong Gao, Max Hirschberger, Paula Lampen-Kelley, Arnab Banerjee, Jiaqiang Yan, David G. Mandrus, Stephen E. Nagler, N. P. Ong
OOscillations of the thermal conductivity observed in the spin-liquid state of α -RuCl Peter Czajka , ∗ , Tong Gao , ∗ , Max Hirschberger , † , Paula Lampen-Kelley , , ArnabBanerjee , ‡ , Jiaqiang Yan , David G. Mandrus , , Stephen E. Nagler , and N. P. Ong , § Department of Physics, Princeton University, Princeton, NJ 08544, USA Department of Materials Science and Engineering,University of Tennessee, Knoxville, Tennessee 37996, USA Materials Science and Technology Division, and Neutron Scattering Division,Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Dated: February 24, 2021)
In the class of materials called spin liquids,a magnetically ordered state cannot be attainedeven at milliKelvin temperatures because of con-flicting constraints on each spin (for e.g. fromgeometric or exchange frustration). The result-ing quantum spin-liquid (QSL) state is currentlyof intense interest because it exhibits novel ex-citations as well as wave-function entanglement.The layered insulator α -RuCl orders as a zigzagantiferromagnet below ∼ H , analogous to quantum oscillations in met-als, even though α -RuCl is an excellent insulatorwith a gap of 1.9 eV. By tilting H out of the plane,we find that the oscillation period is determinedby the in-plane component H a . As the tempera-ture is raised above 0.5 K, the oscillation ampli-tude decreases exponentially. The decrease anti-correlates with the emergence above ∼ (cid:107) a. To exclude extrinsic artifacts,we carried out several tests. The implications ofthe oscillations are discussed. The Quantum Spin Liquid (QSL), first described by An-derson [1], is an exotic state of matter in which the spinwave functions are highly entangled, but long-range mag-netic order is absent [2–4]. The Kitaev honeycomb modelHamiltonian H K for a class of spin liquids has attractedintense interest because the exact solution of its groundstate in zero magnetic field features anyonic excitationsthat are Majorana fermions and Z vortices [5].The magnetic insulator α -RuCl is proximate to theKitaev honeycomb model [6–16]. Interaction betweenthe spins on Ru ions are described by the Kitaev ex-change terms, e.g. K X σ Xi σ Xj where X, Y and Z definethe spin axes [17, 18]. Additional exchange terms Γ andΓ (cid:48) [19] stabilize a zig-zag antiferromagnetic state below7 K when the magnetic field H =0 (Fig. 1a, inset). In a field H (cid:107) a , the zig-zag state is suppressed when H exceeds the critical field H C = 7.3 T. Within the fieldinterval (7.3, 11) T, experiments [8, 16] reveal a mag-netically disordered state, identified as a quantum spinliquid (QSL) in which magnons give way to very broadmodes [10, 16]. Above 11 T, the local moments are par-tially field-polarized. Interest was heightened by a re-port [20] that the thermal Hall conductivity κ xy is quan-tized within a 2-Kelvin window (from 3.3 to 5.5 K).Here we report measurements of both κ xy and the ther-mal conductivity κ xx to temperatures T ∼ κ xx with amplitudes thatare strongly peaked when H lies in the interval (7.3, 11)T. Tilting H reveals that the oscillation period is deter-mined by the in-plane component H a . We note that,despite the similarity to Shubnikov de Haas (SdH) oscil-lations in metals, the free-carrier population at 1 K isexponentially suppressed by the large gap of 1.9 eV [21].Above ∼ λ xx and thermal Hallresistivity λ yx as H was slowly varied at fixed T (Secs.A and B of Methods). As seen in Fig. 1b (for Sample 1with H (cid:107) a ), strong oscillations emerge in κ xx ( H ). Below ∼ κ xx /T in Fig. 1c, show that the oscillation amplitudes continueto grow until they comprise 30-60% of κ xx at 0.43 K. At ∼ κ xx displays a step-increase to a flat plateau.In the high-field partially polarized state, where κ xx isdominated by the phonon conductivity κ ph , oscillationsare rigorously absent (see below and Sec. ?? in Meth-ods). Similar curves are observed in Sample 3 (Fig. ?? ain Methods).The oscillation amplitudes are strongly peaked in theQSL state. To extract the amplitude, we first deter-mined the smooth background curve κ bg ( T, H ) thread-ing the midpoints between adjacent extrema (Fig. ?? din Methods). The oscillatory component, defined as∆ κ = κ xx − κ bg (Fig. ?? c in Methods), allowed accuratedetermination of the amplitude ∆ κ amp , which we plot inFig. 1d for Sample 1 (solid circles). Above 6 T, ∆ κ amp rises steeply to peak at 9.6 T, followed by an abrupt col-lapse to zero above 11 T. Below 6 T, a weak remnant“tail” survives to 4 T in a mixed state in which smallQSL regions coexist with the zig-zag state (we note that4 T is roughly where the averaged zig-zag Bragg intensity a r X i v : . [ c ond - m a t . s t r- e l ] F e b begins to weaken with H [12]). By its profile, ∆ κ amp islargest within the field interval (7.3, 11.5 T) of the QSLstate. The profile in Sample 3 is similar (Fig. ?? a ofMethods). A fourth sample 4 did not exhibit oscillationswith H tilted at 45 ◦ to a (Table in Methods).We next show that the oscillations are periodic in 1 /H .Figure 2a displays plots of the integer increment ∆ n vs. ( µ H n ) − , where H n are fields locating extrema of dκ xx /dB plotted in Fig. 2b. We focus first on data shownas solid symbols. The data from Samples 1 (blue circles)and 3 (red stars), measured with H (cid:107) a , fall on a curvecomprised of straight-line segments separated by a break-in-slope at ∼ S f of the straight segmentsare 41.4 T ( H >
H < H (cid:107) b (Sample 2, green circles), similar behavior is obtained,with the low-field slope S f also at 30.6 T. However, thehigh-field slope is steeper with S f = 64.2 T. As shown inFigs. 1b and c, the periods are T independent from 0.43to 4.5 K.Taken together, the data shown in Figs. 2a to 2d pro-vide strong evidence that the oscillations are intrinsic andreproducible across samples. The five data sets discussedin Panel a were derived from extrema of the derivativecurves dκ xx /dB displayed in Figs. 2b. The profiles showthe close agreement in both period and phase betweenSamples 1 and 3. The matching of the extrema is espe-cially evident in Fig. 2c, which also shows that period-icity vs. H (as opposed to 1 /H ) can be excluded. InSample 2 the period and phase also agree with 1 and 3for H < H tilted in the a - c plane (atan angle θ to a ) provide tests in an independent direc-tion. Figure 2c shows curves of κ xx measured in Sample1 with θ = 0 (blue curve), 39 ◦ (purple) and 55 ◦ (orange)(curves of κ xx at various T are in Figs. ?? a and ?? b).By plotting the curves vs. H a = H cos θ , we find that theperiods match quite well (with a possible phase shift forthe curve at 55 ◦ ). The corresponding derivatives at 39 ◦ and 55 ◦ are plotted in Panel (b). We infer that, in tilted H , the periods depend only on H a . Moreover, the closematching of the blue and purple curves strongly supportsan intrinsic origin.In Sec. ?? in Methods, we discuss the relation of theoscillations to de Haas van Alphen experiments on thecorrelated insulator SmB [22]. For the proposed mech-anism [23, 24] to apply to α -RuCl , we would need H ∼ κ amp vs. H actually imposes a tightconstraint on possible mechanisms. Above 11.5 T in thepolarized state, the oscillations vanish abruptly. Below H C the oscillations survive as a weak tail extending to4 T in the zig-zag state. The amplitude profile suggestsa close connection to the QSL state. The 1 /H period-icity suggests an intriguing analogy with Shubnikov deHaas oscillations, despite the absence of free carriers. We note that Landau-level oscillations have been predictedin the insulating 2D QSL state with H normal to theplane [27, 28]. A spinon Fermi surface in the QSL stateof α -RuCl is widely anticipated [29–31]. Our findingthat S f is determined by H a suggests either a fully 3DQSL state (or possibly a different mechanism). Nonethe-less, quantization of a spin Fermi surface is currently ourleading interpretation.The amplitude ∆ κ amp is much larger in Sample 3 thanin 1 (from Figs. ?? a and 1d, the peak values are 100 and16.5 mW/Km, respectively). We have uncovered a cor-relation with lattice disorder, as estimated from κ ph (seeSec. ?? ). In the QSL state, it is difficult to separatereliably the phonon term κ ph from the spin-excitationconductivity κ sxx because of strong spin-phonon coupling(which causes oscillations in both). However, in the po-larized state above 11.5 T, κ xx is strongly dominatedby κ ph (the profile becomes H independent). Hence theplateau value of κ xx /T measures reliably the lattice dis-order. At 1.0 K, κ xx /T is much higher in Sample 3 (2.2W/mK ) than in 1 (0.7 W/mK ). The lower disorder inSample 3 correlates with a 6-fold increase in the oscilla-tion amplitude.The observed status of H a seems empirically relatedto the planar thermal Hall effect (PTHE), which appearsonly with H (cid:107) a . At a fixed H , the ratio ∆ κ/κ bg decayswith T at a rate consistent with an effective mass m ∗ ∼ . m e where m e is the free electron mass (blue circlesin Fig. 3a). The decay in ∆ κ/κ bg is accompanied bya rapid growth in the PTHE observed with H (cid:107) a (redcircles). Recently, Yokoi et al. [32] reported that κ xy /T measured with H (cid:107) a seems to be quantized, within anarrow interval in T (3.8-6 K) and and in H (10 < µ H < . T = 300 mK to gain a broader perspective.Below 4 K, it is necessary to use the method describedin Eqs. ?? – ?? in Methods [33] to isolate the intrinsicthermal Hall signal δ y (defined in Eq. ?? ) from artifactsarising from hystereses in κ xx as shown for e.g. in Fig.3c. For H (cid:107) b , the intrinsic thermal Hall signal is foundto be zero for 0 < H <
14 T and 0.3 < T < H (cid:107) a , a finite δ y emergesabove ∼ δ y with respect to H identifies it as a truePTHE. This is the thermal-conductivity analog of thetrue planar anomalous Hall effect observed in ZrTe [34].Inverting the matrix λ ij ( H ) to obtain κ ij ( H ), we findthat κ xy displays a dome profile that grows with T in theQSL phase (Fig. 3d). Together, Figs. 3a and 3d providea broad view of how the PTHE varies with T . Whilethe trends of our κ xy are consistent with those in Ref.[32] (e.g. the PTHE exists only with H (cid:107) a ), we notethat the strong T dependence evident in Figs. 2d seemsdifficult to reconcile with a quantized value occuring inthe interval 3.8 – 6 K. Where the two data sets overlap(4-5 K), our magnitudes are much smaller ( κ xy ∼ m at T = 5 K).In summary, we have observed quantum oscillations in κ xx in α -RuCl with H in plane. The prominence of the amplitude in the interval (7.3, 11) T implies that theyare specific to the QSL state. [1] Anderson, P. W. Resonating valence bonds: a new kindof insulator? Mater. Res. Bull. , 153 (1973).[2] Zhou, Y., Kanoda, K. & Ng, T.K. Quantum spin liquidstates, Rev. Mod. Phys. , 025003 (2017).[3] Savary, L. & Balents, L., Quantum spin liquids: a review, Rep. Prog. Phys. , 106502 (2017).[4] Wen, X.G., Quantum Field Theory of Many-Body Sys-tems , Oxford Univ. Press (2004), Ch. 9.[5] Kitaev, A., Anyons in an exactly solved model and be-yond,
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Acknowledgements
We thank Jingjing Lin and Stephan Kim for technicalassistance and T. Senthil and I. Sodemann for valuablediscussions. The measurements of κ xx and P.C. and M.H.were supported by a MRSEC award from the U.S. Na-tional Science Foundation (DMR 1420541). T.G. andthe low- T thermal Hall experiments were supported bythe U.S. Department of Energy (DE-SC0017863). A.B.and S.E.N are supported by the DOE, Office of Science,Scientific User Facilities Division. N.P.O. was supportedby the Gordon and Betty Moore Foundation’s EPiQSinitiative through grant GBMF4539. P.L-K. and D.M.were supported by Moore Foundation’s EPiQS initiativethrough grant GBMF4416. Author contributions
P.C. and T.G. performed the measurements and analyzedthe data together with N.P.O. who proposed the experi-ment. M.H. greatly enhanced the experimental techniqueemployed. A.B., P.L-K. and S.E.N. provided guidance onprior results. Crystals were grown at ORNL by P.L-K.,J.Y. and D.M. at ORNL. The manuscript was written byN.P.O., P.C. and T.G.
Additional Information
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The authors declare no competing financial interests.
Figure CaptionsFigure 1: Quantum oscillations in the quantumspin liquid (QSL) phase in α -RuCl (Sample 1). Panel (a): The phase diagram showing the QSL phasesandwiched between the zig-zag and polarized states with H (cid:107) a (axes a and b shown in inset). The ZZ2 phase ly-ing between critical fields B c and B c is outlined by thedashed curve [16]. Panel (b) shows the emergence of os-cillations in κ xx ( H ) ( H (cid:107) a ) as T falls below 4 K. Datarecorded using the stepped-field technique to correct formagnto-caloric effects (Sec. B in Methods). Panel (c) dis-plays the oscillations over the full field range at selected T (data recorded continuously as well as with stepped-field method). At ∼
11 T, κ xx displays a step-increase toa plateau-like profile in the polarized state in which oscil-lations are strictly absent. The amplitude ∆ κ amp (solidcircles in Panel (d)) is strikingly prominent in the QSLstate. Its profile (shaded orange) distinguishes the QSLfrom adjacent phases. A weak remnant tail extends be-low 7 T to 4 T in the zig-zag state. The derivative curves d ( κ xx /T ) /dB show that the oscillations onset abruptly at4 T. The large derivative peak centered at ∼ κ xx mentioned, and isnot part of the oscillation sequence. Figure 2: Periodicity and intrinsic nature of oscil-lations (panels labelled clockwise).
Panel (a) plotsthe integer increment ∆ n versus 1 /H n (or 1 /H n,a ) where H n are the fields identifying extrema of the derivativecurves dκ xx /dB ( H n,a = H n cos θ for tilted H ). Solidsymbols represent data taken with H strictly in-plane.The blue circles (Sample 1) and red stars (Sample 3)were measured with H (cid:107) a , whereas the green circleswere measured in Sample 2 with H (cid:107) b . Open symbolsare measurements in Sample 1 with H tilted in the a - c plane at angles θ = 39 ◦ (triangles) and 55 ◦ (circles),relative to a . The data sets fall on the same segmentedcurve (comprised of line segments with slope 31 T below7 T and 41 T above 7 T). The exception is the high-fieldslope of 64 T in Sample 2 with H (cid:107) b . For clarity, the3 data sets are shifted vertically by ∆ n = 1. Panel (b):Curves of the derivative dκ xx /dB vs. 1 /H (or 1 /H a ) forSamples 1, 2 and 3 ( H a = H cos θ ). For Sample 1, weshow dκ xx /dB measured with θ = 0 , ◦ and 55 ◦ . Theextrema of dκ xx /dB are plotted in Panel (a). Verticallines mark the values of 1 /H n and 1 /H n,a read off fromthe straight-line fits in Panel (a) for integer increments∆ n . Panel (c): Replot of integer increment ∆ n vs. H n (fields locating the extrema of dκ xx /dB ) in Samples 1(blue circles) and 3 (red stars) to check for periodicityvs. H (as opposed to 1 /H ). In both data sets (mea-sured with H (cid:107) a ), the curve diverges to large negative∆ n as H decreases to 4 T. The narrowing of the spacingbetween adjacent extrema is strongly incompatible withperiodicity vs. H . Panel (d) shows the effect of tilting H out of the plane in Sample 1 by angle θ (relative to a ) at T ∼ θ = 0 ◦ (blue),39 ◦ (purple) and 55 ◦ (orange). When they are plottedvs. H a , the periods of the oscillations in κ xx match wellfor the 3 angles. Figure 3: The planar thermal Hall response.
Panel(a) The T dependence of ∆ κ/κ bg at 8.4 T (blue cir-cles) and the planar thermal Hall conductivity κ xy at9 T (red). The decrease of ∆ κ/κ bg with T (consistentwith an effective mass m ∗ /m e = 0.64) is anti-correlatedwith the increase in κ xy . Panel (b) shows the emergenceof the PTHE signal δ y with H (cid:107) a (upper panel). At T = 4.03 K, δ y in Sample 1 (left axis) displays sharppeaks that are antisymmetric in H for H (cid:107) a (black cir- cles). Corresponding values of λ yx are on the right axis.The lower panel shows the null thermal Hall resistivity(expressed as the thermal Hall signal δ y ) measured with H (cid:107) b at 0.3, 2.6 and 5 K in Sample 2. The total uncer-tainty in δ y is 0.3 mK. Panel (c) shows the hysteresis in κ xx that can contaminate κ xy if not properly subtracted.The right-going (purple) and left-going (red) scans havebeen antisymmetrized with respect to H . In Panel (d), κ xy ( H ) derived from the measured tensor λ ij are plottedfor several T from 3.4 to 5.5 K. The dome-shaped profilesare the planar thermal Hall effect reported in Ref. [32]but in our experiment the values are not quantized. bc d zig-zag AF QuantumSpinLiquid P a r a m agne tt i c s t a t e b a B (T) T ( K ) a H || a κ xx / T ( W / K m ) µ H (T) κ xx ( W / K m ) µ H (T) ∆κ amp (0.43 K) ZZ2 Sample 1Sample 1 d ( κ xx / T ) / d B ( W / K m T ) µ H (T) ∆ κ xx ( m W / K m ) ∆ κ a m p Sample 1 c1 B c2 B FIG. 1. a bcd θ = 55 o , T = 0.60 K θ = 39 o , T = 0.61 K θ = 0 o , T = 0.64 K κ xx ( H ) / κ xx ( ) Field component along a , µ H a (T)Sample 1 -101234567 H || a H || b θ = 55 o θ = 39 o H || a ∆ n ( µ H) -1 or ( µ H a ) -1 (T -1 ) H || b θ = 55 o θ = 39 o H || a H || a d κ xx / d B ( a r b . un i t s ) ( µ H) -1 or ( µ H a ) -1 (T -1 ) I n t ege r n µ H (T) Sample 1Sample 3
FIG. 2. a bc d κ xy ( m W / K m ) µ H (T) -12 -8 -4 0 4 8 12-40-2002040 κ xx AS / T ( m W / K m ) µ H (T)
T = 1.31 K
Sample 1 Sample 1 -12 -8 -4 0 4 8 12-1.5-1.0-0.50.00.51.01.5 -1.5-1.0-0.50.00.51.01.5 -12 -8 -4 0 4 8 12-0.50.00.5 H || a µ H (T) T he r m a l H a ll r e s i s t i v i t y λ yx ( m K . m / W ) δ y ( m K ) aver.T = 4.03 K µ H (T) δ y ( m K ) H || b ∆ κ / κ bg ( . Τ ) T (K)
Sample 1 κ xy / T ( T )( W / m K ))