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Oscillations of the thermal conductivity in the spin-liquid state of α-RuCl3

Nature Physics volume 17, pages 915–919 (2021)Cite this article

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Abstract

In the class of materials called spin liquids1,2,3, a magnetically ordered state cannot be attained even at millikelvin temperatures because of conflicting constraints on each spin; for example, from geometric or exchange frustration. The resulting quantum spin-liquid state is currently of intense interest because it exhibits unusual excitations as well as wave-function entanglement. The layered insulator α-RuCl3 orders as a zigzag antiferromagnet at low temperature in zero magnetic field4. The zigzag order is destroyed when a magnetic field is applied parallel to the zigzag axis. At moderate magnetic field strength, there is growing evidence that a quantum spin-liquid state exists. Here we report the observation of oscillations in its thermal conductivity in that field range. The oscillations, whose amplitude is very large within this field range and strongly suppressed on either side, are periodic. This is analogous to quantum oscillations in metals, even though α-RuCl3 is an excellent insulator with a large gap. As the temperature is raised above 0.5 K, the oscillation amplitude decreases exponentially, anticorrelating with the emergence of an anomalous planar thermal Hall conductivity above approximately 2 K.

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Fig. 1: Quantum oscillations in the QSL phase in α-RuCl3 (Sample 1).
Fig. 2: Periodicity and intrinsic nature of oscillations.
Fig. 3: The planar thermal Hall response.

Data availability

The data in the plots in this paper are available via the Harvard DataVerse at https://doi.org/10.7910/DVN/CWLZCI.

References

  1. Zhou, Y., Kanoda, K. & Ng, T. K. Quantum spin liquid states. Rev. Mod. Phys. 89, 025003 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  2. Savary, L. & Balents, L. Quantum spin liquids: a review. Rep. Prog. Phys. 80, 106502 (2017).

    Article  ADS  Google Scholar 

  3. Wen, X. G. Quantum Field Theory of Many-Body Systems Ch. 9 (Oxford Univ. Press, 2004).

  4. Takagi, H., Takayama, T., Jackeli, G., Khaliullin, G. & Nagler, S. E. Concept and realization of Kitaev quantum spin liquids. Nat. Rev. Phys. 1, 265–280 (2019).

    Article  Google Scholar 

  5. Anderson, P. W. Resonating valence bonds: a new kind of insulator? Mater. Res. Bull. 8, 153–160 (1973).

    Article  Google Scholar 

  6. Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  7. Plumb, K. W. et al. α-RuCl3: a spin-orbit assisted Mott insulator on a honeycomb lattice. Phys. Rev. B 90, 041112 (2014).

    Article  ADS  Google Scholar 

  8. Sears, J. A. et al. Magnetic order in α-RuCl3: a honeycomb-lattice quantum magnet with strong spin–orbit coupling. Phys. Rev. B 91, 144420 (2015).

    Article  ADS  Google Scholar 

  9. Banerjee, A. et al. Proximate Kitaev quantum spin liquid behavior in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016).

    Article  ADS  Google Scholar 

  10. Sears, J. A. et al. Phase diagram of α-RuCl3 in an in-plane magnetic field. Phys. Rev. B 95, 180411 (2017).

    Article  ADS  Google Scholar 

  11. Wang, Z. et al. Magnetic excitations and continuum of a possibly field-induced quantum spin liquid in α-RuCl3. Phys. Rev. Lett. 119, 227202 (2017).

    Article  ADS  Google Scholar 

  12. Leahy, I. A. et al. Anomalous thermal conductivity and magnetic torque response in the honeycomb magnet α-RuCl3. Phys. Rev. Lett. 118, 187203 (2017).

    Article  ADS  Google Scholar 

  13. Banerjee, A. et al. Excitations in the field-induced quantum spin liquid state of α-RuCl3. npj Quant. Mater. 3, 8 (2018).

    Article  ADS  Google Scholar 

  14. Hentrich, R. et al. Unusual phonon heat transport in α-RuCl3: strong spin–phonon scattering and field-induced spin gap. Phys. Rev. Lett. 120, 117204 (2018).

    Article  ADS  Google Scholar 

  15. Lampen-Kelley, P. et al. Anisotropic susceptibilities in the honeycomb Kitaev system α-RuCl3. Phys. Rev. B 98, 100403 (2018).

    Article  ADS  Google Scholar 

  16. Lampen-Kelley, P. et al. Field-induced intermediate phase in α-RuCl3: non-coplanar order, phase diagram, and proximate spin liquid. Preprint at https://arxiv.org/abs/1807.06192 (2018).

  17. Balz, C. et al. Finite field regime for a quantum spin liquid in α-RuCl3. Phys. Rev. B 100, 060405(R) (2019).

    Article  ADS  Google Scholar 

  18. Jackeli, G. & Khaliullin, G. Mott insulators in the strong spin–orbit coupling limit: from Heisenberg to a quantum compass and Kitaev models. Phys. Rev. Lett. 102, 017205 (2009).

    Article  ADS  Google Scholar 

  19. Chun, S. W. et al. Direct evidence for dominant bond-directional interactions in a honeycomb lattice iridate Na2IrO3. Nat. Phys. 11, 462–467 (2015).

    Article  Google Scholar 

  20. Kim, H. S. & Kee, H. Y. Crystal structure and magnetism in α-RuCl3 an ab initio study. Phys. Rev. B 93, 155143 (2016).

    Article  ADS  Google Scholar 

  21. Kasahara, Y. et al. Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid. Nature 559, 227–231 (2018).

    Article  ADS  Google Scholar 

  22. Sinn, S. et al. Electronic structure of the Kitaev material α-RuCl3 probed by photoemission and inverse photoemission spectroscopies. Sci. Rep. 6, 39544 (2016).

    Article  ADS  Google Scholar 

  23. Tan, B. S. et al. Unconventional Fermi surface in an insulating state. Science 349, 287–290 (2015).

    Article  ADS  Google Scholar 

  24. Johannes, K. & Cooper, N. R. Quantum oscillations without a Fermi surface and the anomalous de Haas–van Alphen effect. Phys. Rev. Lett. 115, 146401 (2015).

    Article  ADS  Google Scholar 

  25. Zhang, L., Song, X. -Y. & Wang, F. Quantum oscillation in narrow-gap topological insulators. Phys. Rev. Lett. 116, 046404 (2016).

    Article  ADS  Google Scholar 

  26. Kubota, Y., Tanaka, H., Ono, T., Narumi, Y. & Kindo, K. Successive magnetic phase transitions in α-RuCl3: XY-like frustrated magnet on the honeycomb lattice. Phys. Rev. B 91, 094422 (2015).

    Article  ADS  Google Scholar 

  27. Cao, H. B. et al. Low-temperature crystal and magnetic structure of α-RuCl3. Phys. Rev. B 93, 134423 (2016).

    Article  ADS  Google Scholar 

  28. Motrunich, O. I. Orbital magnetic field effects in spin liquid with spinon Fermi sea: possible application to κ-(ET)2Cu2(CN)3. Phys. Rev. B 73, 155115 (2006).

    Article  ADS  Google Scholar 

  29. Sodemann, I., Chowdhury, D. & Senthil, T. Quantum oscillations in insulators with neutral Fermi surfaces. Phys. Rev. B 97, 045152 (2018).

    Article  ADS  Google Scholar 

  30. Liu, Z. X. & Normand, B. Dirac and chiral quantum spin liquids on the honeycomb lattice in a magnetic field. Phys. Rev. Lett. 120, 187201 (2018).

    Article  ADS  Google Scholar 

  31. Patel, N. D. & Trivedi, N. Magnetic field-induced intermediate quantum spiin liquid with a spinon Fermi surface. Proc. Natl Acad. Sci. USA 116, 12199–12203 (2019).

    Article  ADS  Google Scholar 

  32. Takikawa, D. & Fujimoto, S. Impact of off-diagonal exchange interactions on the Kitaev spin-liquid state of α-RuCl3. Phys. Rev. B 99, 224409 (2019).

    Article  ADS  Google Scholar 

  33. Yokoi, T. et al. Half-integer quantized thermal Hall effect in the Kitaev material α-RuCl3. Preprint at https://arxiv.org/abs/2001.01899v1 (2020).

  34. Hirschberger, M. Quasiparticle Excitations with Berry Curvature in Insulating Magnets and Weyl Semimetals. PhD thesis, Princeton Univ. (2017).

  35. Liang, T. et al. Anomalous Hall effect in ZrTe5. Nat. Phys. 14, 451–455 (2018).

    Article  Google Scholar 

Download references

Acknowledgements

We thank J. Lin and S. Kim for technical assistance and T. Senthil and I. Sodemann for valuable discussions. P.C. and M.H., and the measurements of κxx, were supported by a MRSEC award from the US National Science Foundation (DMR 1420541 and DMR 2011750). T.G. and the low-T thermal Hall experiments were supported by the US 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 supported by the Gordon and Betty Moore Foundation’s EPiQS initiative through grant GBMF9466. P.L.-K. and D.G.M. were supported by Moore Foundation’s EPiQS initiative through grant GBMF4416.

Author information

Author notes

  1. Max Hirschberger

    Present address: Department of Applied Physics and Quantum-Phase Electronics Center, The University of Tokyo, Tokyo, Japan

  2. These authors contributed equally: Peter Czajka, Tong Gao.

Authors and Affiliations

  1. Department of Physics, Princeton University, Princeton, NJ, USA

    Peter Czajka, Tong Gao, Max Hirschberger & N. P. Ong

  2. Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA

    Paula Lampen-Kelley & David G. Mandrus

  3. Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA

    Paula Lampen-Kelley, Jiaqiang Yan & David G. Mandrus

  4. Department of Physics, Purdue University, West Lafayette, IN, USA

    Arnab Banerjee

  5. Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA

    Arnab Banerjee & Stephen E. Nagler

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Contributions

P.C. and T.G. performed the measurements and analysed the data together with N.P.O., who proposed the experiment. M.H. greatly enhanced the experimental technique employed. A.B., P.L.-K. and S.E.N. provided guidance on prior results. Crystals were grown at Oak Ridge National Laboratory by P.L.-K., J.Y. and D.G.M. The manuscript was written by N.P.O., P.C. and T.G.

Corresponding author

Correspondence to N. P. Ong.

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Extended data

Extended Data Fig. 1 Experimental details.

Panel (a) shows a schematic of the mounted crystal and the applied in-plane field b. The temperatures TA, TB and TC are read off RX102A thermometers as shown. Panel (b) shows a photo of Sample 2 contacted with thick Au wires (100 μm in diameter) to thermometers and bonded to the bath by stycast. The heater and Delrin post are visible at the top right and left, respectively. Panel (c) shows the Cl octahedra enclosing Ru ions (adapted from 20). The pink and blue planes are normal to the spin axes SZ and SX, respectively. The lower sketch highlights the plane containing the ‘Z’ Ru-Ru bond and the spin axis SZ. Panel (d) displays time-traces of the temperatures TA, ⋯ , Tbath in a λyx measurement with the bath temperature fixed at 280 mK. The field is gradually increased from -13.5 to 13.5 T over 14 hours by a step-wise change of (for example) 125 mT at each step. After each step-increase, transients caused by heating (or cooling) of the spins via the magneto-caloric effect combined with eddy-current heating of the brass bath are seen in all channels. The total transverse signal ΔyT is the difference between the red and black curves (as expressed in Eq. (2), ΔyT is the sum of δy and the ‘pick-up’ of the longitudinal ΔxT caused by contact misalignment). Panel (e) shows an expanded view of 5 transient pulses bracketing H = 0 (vertical dashed line). For t < 26,000 s, the 2 effects partially cancel whereas for t > 26,000 s, they add to give large transients. Readings are recorded within the blue-shaded interval after all transients have decayed. The average over the readings gives ΔyT to a resolution of ± 3 μK. Because of systematic errors, however, the total uncertainty in measuring δy is ± 200 μK.

Extended Data Fig. 2 Data analysis details.

Panel (a): Contributions of hysteretic effects to the signal detected at the Hall contacts (ΔyT) and at the longitudinal contacts (ΔxT). Below 4 K, it is critically important to identify and separate these contributions from the intrinsic PTHE signal δy using the procedure described in Methods. Panel (b) shows traces of the derivative curves dκxx/dB above 2 K in Sample 1. The physical realilty of the oscillations is apparent in the raw data. Panel (c): The oscillatory component Δκ (divided by the background κbg) measured in Sample 1 at selected T. Panel (d) illustrates the procedure for determining κbg (red curve) from the mid-points between derivative extrema of the measured curve of κxx (black curve). The difference of the 2 curves gives Δκ.

Extended Data Fig. 3 Oscillations in Sample 3.

Panel (a): Oscillations in κxx observed in Sample 3 at selected T from 0.51 to 1.57 K. The derivative curves dκxx/dB are shown in Panel (b). Panel (c) compares the AC susceptibility χac reported in ref. 13 with the oscillations in Sample 1 (adapted from Fig. 1c of main text). As shown by the two red dashed lines, sharp peaks in χac occur close (but not exactly at) the minima in κxx near 6 T and 7.2 T. However, no peaks are observed in χac away from these H values where multiple oscillations occur in κxx.

Extended Data Fig. 4 Oscillations in tilted field.

Curves of κxx/T vs. H measured in Sample 1 at tilt angle θ = 39∘ (Panel a) and θ = 55∘ (Panel b) with T fixed at the 6 values indicated. Panel (c) shows the effect of tilting H out of the plane in Sample 1 at angle θ (relative to a). Curves of dκxx/dB vs. 1/H are displayed for θ = 0 (top panel), 39∘ (middle) and 55∘ (bottom panel). Panel (d) displays the magnetization M in Sample 1 (expressed in emu) vs. T measured with H = 0.1 T in the direction ∥a (blue circles) and ∥b (red). The smooth increase in M (for H∥a) as T decreases to 7 K is direct evidence for absence of stacking faults. The presence of stacking faults leads to a distinctive flat-plateau feature extending from 7 to 14 K. The inset plots the curve of κ (in zero H) vs. T in Sample 1. The ratio of the peak value (at 5 K) to the minimum at 7.5 K is 2.04.

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Czajka, P., Gao, T., Hirschberger, M. et al. Oscillations of the thermal conductivity in the spin-liquid state of α-RuCl3. Nat. Phys. 17, 915–919 (2021). https://doi.org/10.1038/s41567-021-01243-x

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