Sample dependence of the half-integer quantized thermal Hall effect in the Kitaev candidate α -RuCl 3
SSample dependence of the half-integer quantized thermal Hall e ff ect in a Kitaev candidate α -RuCl M. Yamashita , ∗ N. Kurita , and H. Tanaka The Institute for Solid State Physics, The University of Tokyo, Kashiwa, 277-8581, Japan and Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan (Dated: May 5, 2020)We have investigated the sample dependence of the half-integer thermal Hall e ff ect in α -RuCl under a mag-netic field tilted 45 degree from the c axis to the a axis. We find that the sample with the largest longitudinalthermal conductivity ( κ xx ) shows the half-integer quantized thermal Hall e ff ect expected in the Kitaev model. Onthe other hand, the quantized thermal Hall e ff ect was not observed in the samples with smaller κ xx . We suggestthat suppressing the magnetic scattering e ff ects on the phonon thermal conduction, which broaden the field-induced gap protecting the chiral edge current of the Majorana fermions, is important to observe the quantizedthermal Hall e ff ect. Non-trivial topology in a condensed-matter state realizes aquantization of a physical quantity. One of the most funda-mental examples is the quantized Hall conductivity in a quan-tum Hall system, where the quantized Hall conductivity isgiven by the Chern number determined by the topology of thesystem [1].A new intriguing case of this topological quantization isa Kitaev magnet [2, 3]. In the Kitaev model, localized spin-1 / ff ect of this Kitaev Hamiltonian prevents thespins to order even at the zero temperature, realizing a quan-tum spin liquid state. Remarkably, this ground state of theKitaev Hamiltonian is exactly solvable. The ground state hasbeen shown to be characterized by the two kinds of elemen-tary excitations; itinerant Majorana fermions and localized Z fluxes. In a magnetic field, this itinerant Majorana fermionshave topologically non-trivial gapped bands with the Chernnumber C = ±
1, giving rise to a quantized chiral edge cur-rent. In contrast to a quantized chiral edge current of elec-trons in a quantum Hall system, this chiral edge current is car-ried by the charge neutral Majorana fermions. Therefore, thisquantized chiral edge current has been predicted to appear inthe 2D thermal Hall conductivity as κ Dxy / T = ( C / q t , where q t = ( π/ k B / (cid:126) .Materializing the Kitaev model has been suggested in sev-eral Mott insulators with a strong spin-orbit coupling [4]. Oneof the most studied Kitaev candidates is α -RuCl in whicha 2D honeycomb structure of edge-sharing RuCl octahedrahas been shown to have a dominant Kitaev interaction [5].Various measurements [6–11] have reported Kitaev-like sig-natures above the antiferromagnetic (AFM) ordering temper-ature of T N ∼ ∼ a – b plane [12, 15, 16], enabling one to study the Kitaev QSLdown to lower temperatures. Most remarkably, thermal Hallmeasurements done under an in-plane field have shown thehalf-integer quantized thermal Hall conduction [17, 18], indi-cating the presence of a chiral edge current of the Majoranafermions protected by the field-induced gap. However, de-tails of this field-induced gap are unknown because the KitaevHamiltonian loses its exact solvability in a magnetic field. It has been reported that this quantized thermal Hall ef-fect has a sample dependence associated with the longitudinalthermal conductivity ( κ xx ) [18]. This κ xx dependence may im-ply a scattering e ff ect on the field-induced gap protecting thechiral edge current. A similar scattering e ff ect has been dis-cussed in the intrinsic anomalous Hall e ff ect (AHE) in ferro-magnetic metals [19] where a broadening of the gap by scat-tering e ff ects is suggested to destroy the intrinsic AHE in aless conductive metal. Therefore, further studies of the κ xx dependence of this quantized thermal Hall e ff ect may provideinformation with respect to the unknown field-induced gap. Itis also important to confirm the reproducibility of the quan-tized thermal Hall e ff ect.In this Letter, we report the sample dependence of the longi-tudinal ( κ xx ) and transverse ( κ xy ) thermal conductivity of threesingle crystals of α -RuCl . We confirm the reproducibilityof the half-integer quantized thermal Hall e ff ect in a sam-ple showing the largest κ xx among the three crystals. On theother hand, the other samples with smaller κ xx show κ xy muchsmaller than the value expected for the quantization. We alsofind that a sample with a larger κ xx shows a larger decrease ofthe magnetic susceptibility below T N , in addition to a largerfield-increase e ff ect of κ xx , showing that magnetic scatteringe ff ects are more strongly suppressed by magnetic fields in asample with a better quality. From these results, we suggestthat suppressing this magnetic scattering e ff ect plays an im-portant role to realize the quantized thermal Hall e ff ect.Single crystals used in this work were synthesized by aBridgeman method as described in Ref. [12]. We have mea-sured both κ xx and κ xy of three single crystals (sample A–C)of α -RuCl . A typical sample size was 2.5 mm × × a axis of the sample, and amagnetic field H was applied 45 degree from the c -axis to the a -axis. We denote the in-plane field µ H (cid:107) as µ H (cid:107) = µ H / √ χ ) was checked for all samples prior to the thermal con-ductivity measurements (Fig. 1(a)). As shown in Fig. 1(a), noanomaly is observed at 14 K, showing the absence of the addi- a r X i v : . [ c ond - m a t . s t r- e l ] M a y ! " c &’& ( ) * & ) + , - % &. $ % k &’& & - % & ) - % &. %7%8%$% 678$ /&’&0&.1&2& &/&3&4&5 $! k & : ! ; &<& k & : ; % 678$ m ! == &:/;&3>&8!7&0&4>& !" FIG. 1. (Color online) (a) The temperature dependence of the magnetic susceptibility at 0.1 T applied parallel to the a – b plane. (b) Thetemperature dependence of the longitudinal thermal conductivity κ xx at zero field. (c) The field dependence of κ xx at 4.6 K (sample A) and at5.0 K (sample B and C). The vertical axis is normalized by the zero-field value κ xx (0). The horizontal axis shows the in-plane field µ H (cid:107) = µ H / √ tional magnetic transition caused by stacking faults [12, 14].The AFM transition at T N ∼ χ ( T ) below T N is observed insample A. This decrease is smaller in sample B and the small-est in sample C.Figure 1(b) shows the temperature dependence of κ xx atzero field. As shown in Fig. 1(b), κ xx of all samples showsa very similar temperature dependence with that of previousworks [11, 17, 18, 21, 22]. The magnetic transition to theAFM phase is clearly seen by the onset of the increase of κ xx below T N . On the other hand, the magnitude of κ xx is very dif-ferent for each sample; κ xx of sample A is the largest amongthe samples, which is 4 times larger than that of sample C.This sample dependence of κ xx well correlates to that of thedecrease of χ below T N . A sample with a larger decrease of χ below T N shows a larger κ xx .Figure 1(c) shows the field dependence of κ xx at ∼ κ xx ( H ) /κ xx (0) of all samples shows the minimum of κ xx at the in-plane field of H min = κ xx ( H ) is increased as increasing field. This increase islarger in a sample with a larger κ xx .The field dependence of the thermal Hall conductivity atdi ff erent temperatures is shown in Figs. 2. For a comparison,the value corresponding to the half-integer quantized thermalHall κ Dxy / ( T d ) = ± q t / (2 d ), where d = .
57 nm is the distancebetween the 2D honeycomb layers of RuCl , is shown as thedotted lines.As shown in Figs. 2, the sign of κ xy / T at 10 K is negativein sample A and sample B whereas it is positive in sample C.This sample dependence may be related to the angle betweenthe a axis and the magnetic field, which is discussed to benegative (positive) for 45 (135) degree [18]. In this work, weonly discuss the magnitude of κ xy / T .As shown in Fig. 2(a), sample A shows the largest | κ xy | / T .At 12 K, | κ xy | / T of sample A becomes larger than the half- FIG. 2. (Color online) The field dependence of the thermal Hall con-ductivity divided by the temperature κ xy / T of sample A (a) and sam-ple B and C (b). The dotted lines show the value corresponding tothe half-integer quantized thermal Hall e ff ect (see the main text fordetail). integer quantized value q t / (2 d ) for H > H min . The field de-pendence of κ xy / T of sample A becomes flat for µ H (cid:107) ∼ κ xy / T at the flat region becomes close to q t / (2 d ). On the other hand,as shown in Fig. 2(b), the magnitudes of κ xy / T of sample Band C remain much smaller than q t / (2 d ) for all temperature FIG. 3. (Color online) The field dependence of | κ xy | / T of sample A at 4.6 (a), 4.1 (b), and 3.3 K (c). The data in the previous report [17] is alsoplotted in (a). The dotted lines show the value corresponding to the half-integer quantized thermal Hall e ff ect (see the main text for detail). and field range we measured. Moreover, κ xy / T of sample Bshows a very di ff erent field dependence with sign changes for H > H min .The field dependence of | κ xy | / T of sample A was furtherchecked at lower temperatures (Figs. 3). As shown in Figs. 3,the flat field dependence of | κ xy | / T observed for 8–9 T per-sists down to 3.3 K at q t / (2 d ) within our experimental errorof ± | κ xy | / T with respect to both mag-netic field and temperature. On the other hand, compared tothe data in the previous report [17], the quantization of | κ xy | / T is observed at higher fields despite the similar H min . Quan-tization of | κ xy | / T at higher fields has also been reported inRef. [18].Here we discuss the sample dependence of κ xx and κ xy .From the previous κ xx measurements for both in-plane andout-of-plane transport [21], the dominant heat carrier in α -RuCl has been shown as phonons. The di ff erence of thephonon thermal conductivity of the same compound is givenby the di ff erent length of the phonon mean free path whichis limited by scattering e ff ects on phonons [23]. Therefore,the di ff erent magnitudes of κ xx of di ff erent samples are de-termined by the di ff erent scattering strength on the phonons.As shown in Fig. 1(b), all samples show a very similar fielddependence with a large reduction of κ xx at H min . This fielddependence of κ xx indicates that a magnetic-field dependentscattering mechanism on phonons is dominant in all samples.In fact, the analysis of the temperature dependence of κ xx bythe Callaway model done in Ref. [21] has suggested that aresonant magnetic scattering is the most dominant. There-fore, the di ff erent magnitudes of κ xx in di ff erent samples areattributed to di ff erent strengths of the magnetic scatterings onphonons.As shown in Fig. 1(b), the increase of κ xx above H min islargest in sample A, and is smaller in sample B and C in or-der of the magnitude of κ xx . This sample dependent increaseabove H min shows that the magnetic-field dependent scatter-ing is more strongly suppressed in a sample with a larger κ xx . In addition to this relation between the magnitude and the field dependence of κ xx , a sample with a larger κ xx showsa larger decrease of χ ( T ) below T N as shown in Fig. 1(a).This decrease of χ ( T ) below T N reflects the magnitude of theAFM order, showing that a larger decrease of χ ( T ) is observedin a sample with a better quality. Therefore, a larger field-suppression on the magnetic-field dependent scattering is ob-served in a better-quality sample. Given that the quantized κ xy is observed only in sample A showing the largest suppressionof the magnetic-field dependent scattering, we conclude thatthe suppression of the magnetic-field dependent scattering in ahigh-quality sample is necessary to realize the quantized ther-mal edge current. The di ff erent field region of the quantizedthermal Hall in this work from that of the previous work [17]may imply that a larger magnetic field is required to stabilizethe chiral edge current in our sample.The intrinsic AHE in ferromagnetic metals has been sug-gested to be dissipated when the energy broadening by scat-tering e ff ects, which is estimated by magnitude of the longi-tudinal conductivity, exceeds the energy gap formed by thespin-orbit interaction [19]. In contrast to the electric AHEwhere both longitudinal and transverse conductions are givenby electrons, the thermal Hall conductivity in α -RuCl is car-ried by the itinerant Majorana fermions whereas the longitu-dinal thermal conducitivity is by phonons. Thus, the scatter-ing e ff ects on Majorana fermions cannot be estimated fromthe magnitude of κ xx . Meanwhile, it has been pointed outthat a large coupling between the Majorana fermions and thephonons is necessary to observe the quantized thermal Hall ef-fect [24, 25]. We therefore speculate that the scattering e ff ectson Majorana fermions are correlated to those on phonons.The e ff ects of disorders, such as bond randomness or va-cancies, on the Kitaev model have been extensively studiedin theory [26–32]. Recently, it has been pointed out that thequantized thermal Hall e ff ect is very sensitive to these disor-ders [31, 32]. Clarifying further details of the disorder e ff ectson the quantized thermal Hall e ff ect by investigating the struc-ture of the candidate materials or by artificially introducingdisorders will be an important future issue.In summary, we have investigated the sample dependenceof the thermal Hall conductivity of the Kitaev candidate ma-terial α -RuCl . We confirm the reproducibility of the half-integer quantized thermal Hall e ff ect in the sample with thelargest longitudinal thermal conductivity. 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