Abstract
Topological insulators, with metallic boundary states protected against timereversalinvariant perturbations^{1}, are a promising avenue for realizing exotic quantum states of matter, including various excitations of collective modes predicted in particle physics, such as Majorana fermions^{2} and axions^{3}. According to theoretical predictions^{4}, a topological insulating state can emerge from not only a weakly interacting system with strong spin–orbit coupling, but also in insulators driven by strong electron correlations. The Kondo insulator compound SmB_{6} is an ideal candidate for realizing this exotic state of matter, with hybridization between itinerant conduction electrons and localized felectrons driving an insulating gap and metallic surface states at low temperatures^{5}. Here we exploit the existence of surface ferromagnetism in SmB_{6} to investigate the topological nature of metallic surface states by studying magnetotransport properties at very low temperatures. We find evidence of onedimensional surface transport with a quantized conductance value of e^{2}/h originating from the chiral edge channels of ferromagnetic domain walls, providing strong evidence that topologically nontrivial surface states exist in SmB_{6}.
Main
First reported over 40 years ago^{6}, the longstanding puzzle of saturating electrical resistivity in SmB_{6} at low temperatures^{6} has recently found a possible solution^{7,8,9,10,11,12,13,14,15,16}. Recent transport experiments^{7,8,15,16} have proved the existence of metallic conduction at the surface of SmB_{6} crystals at temperatures much below the opening of the hybridization gap, where surface conductance dominates that of the insulating bulk of the crystal, as shown by nonlocal transport^{7}, sample thickness dependence and surface gating studies^{17}. However, polaritydriven surface states^{14} and lack of direct evidence of the chiral nature of surface conduction has brought into question the topological nature of these states. In this study, we combine observations of a suppression of weak antilocalization by spinflip scattering, an anomalous Hall effect (AHE), a hysteretic irreversibility in magnetoresistance (MR) and an unusual enhanced domain wall conduction to prove the occurrence of longrange ferromagnetic (FM) order that gaps the Dirac spectrum of the topological surface states and relegates conduction to chiral edge channels.
The overall magnetoresistance in SmB_{6} is negative at low temperatures and varies quadratically with field, which can be attributed to the reduction of the Kondo energy gap by magnetic field and the liberation of bulk charge carriers^{18}. The MR measurements in a perpendicular field orientation (H_{⊥} ≡ H [001], I [100]), obtained while applying an increasing (upsweep) field, or H_{up} (Fig. 1a), are qualitatively similar to those taken with a decreasing (downsweep) field, or H_{dn} (Fig. 1b), but with notable differences. For instance, below 500 mK an oscillatory behaviour in the MR is visible in the upsweep data, reminiscent of Shubnikov–de Haas oscillations, whereas it is nearly absent in the downsweep MR data, as shown in Fig. 1b. Furthermore, on close inspection of the downsweep MR data, abrupt transitionlike features are apparent at low H_{dn} fields that are completely absent in the upsweep data. Below 200 mK, the magnetoresistance abruptly drops through transitionlike steps at low H_{dn} fields, as shown in the inset of Fig. 1b. To compare directly, Fig. 2a presents MR data at 100 mK obtained by systematically sweeping through the full ‘fourquadrant’ range, revealing a stark contrast between up and downsweep MR. This takes the form of a hysteretic loop that does not depend on the sign of the field, but only on the sweep direction. The hysteretic loop seems to close at a field of 8–10 T, with no difference in H_{up} or H_{dn} MR above that range, and vanishes if the turning field is less than 4 T (see Supplementary Methods).
Fielddependent hysteretic phenomena can have different origins, but are typically associated with the presence of ferromagnetism. As no bulk magnetic order is observed in SmB_{6} according to muon spin resonance experiments performed down to 20 mK (ref. 19), a surfacebased FM order is the likely explanation, as discussed in detail below. Its hallmark signature, the anomalous Hall effect^{20}, is one direct way to confirm its presence. As shown in Fig. 2b, the Hall resistance of SmB_{6} at 1 K is completely linear and negative, consistent with previous reports^{17,21,22}. However, at 100 mK a kink is clearly discernible in the raw data precisely near 8 T, the field at which the hysteretic loop in MR closes. Plotted with a linear background subtracted, the difference ΔR_{yx} = R_{yx} − AH (where A is a linear coefficient obtained from fitting R_{yx} below 5 T) exhibits an abrupt onset, yet lack of hysteretic behaviour (see Supplementary Methods), as seen in Fig. 2c, suggesting that the observed Hall resistance has an AHE term R_{yx}^{A} associated with FM domain alignment. Together with the observation of a fieldhistorydependent dynamics, as indicated by the presence of a strongly asymmetric time relaxation in MR between H_{up} and H_{dn} (see Supplementary Methods), these observations confirm without a doubt the presence of FM domains.
Shown in Fig. 3a, b, the MR hysteresis also depends on the magnetic field orientation with respect to the sample. When the field is oriented parallel to the surface with electrical contacts (H_{} ≡ H [010], I [100]), the difference in up and downsweep MR becomes vanishingly small in magnitude. As it is very unlikely that a bulkorigin anomaly would break the cubic symmetry of the crystal, it is clear that this anomalous MR hysteresis stems from a surfacebased origin.
The observation of weak antilocalization (WAL) confirms this picture. In twodimensional conductors, weak localization appears as a quantum correction to classical magnetoresistance caused by the constructive or destructive interference between timereversed quasiparticle paths. The presence of strong spin–orbit coupling or a π Berry’s phase associated with the helical states of a topological insulator (TI)^{23} changes the sign of the correction and gives rise to the signature WAL enhancement of conductance, which is suppressed by a timereversalsymmetrybreaking perturbation.
As shown in Fig. 3c, d, we observe signatures of WAL at low fields in both field orientations, with a strong anisotropy evident. Because the WAL effect is normally sensitive only to the perpendicular field component, as it is an orbital effect, it is surprising to see any WAL correction in the H_{} field orientation at all, as shown in Fig. 3d. However, one must consider the finite width of the surface conducting state wavefunction that penetrates into the bulk with a characteristic length λ, which combined with the rather small insulating gap of SmB_{6} leads to a situation where orbital motion of electrons can occur even in the H_{} field orientation.
Using the appropriate version of the Hikami–Larkin–Nagaoka (HLN) equation (see Supplementary Methods), we fit the 20 mK lowfield sheet conductance for each field orientation, as shown in Fig. 3c, d, extracting the dephasing length L_{φ} and the conduction channel α parameter for each. Whereas L_{φ} in each field orientation is comparable to previous results^{16}, the value α_{⊥} = 0.17 is much smaller than the α = 2 × 1/2 = 1 value expected in the presence of top and bottom surface conduction channels (per Dirac cone)^{16}. Surprisingly, the parallel field orientation fit yields α_{} = 0.29, which is small but still much larger than α_{⊥}. We attribute the overall strong suppression (α < 1) and the unusual anisotropy (α_{} > α_{⊥}) of these values to the presence of spinflip scattering, which is known to reduce the WAL effect as a result of destructive interference, leading to α = 0 in the extreme limit where spinflip scattering is much stronger than spin–orbit scattering^{24,25}.
Together with MR hysteresis, the presence of spinflip scattering suggests the existence of magnetic moments on the surface of SmB_{6}. These are probably associated with samarium ions, either via the presence of unscreened felectron Sm moments (socalled ‘Kondo holes’^{26}) proposed to explain logarithmic corrections to surface conductance at low temperatures^{16}, or possibly Sm^{3+} moments in a native surface oxide layer observed by Xray photoelectron spectroscopy^{27}. With ample charge carriers present on the surface of SmB_{6}, these moments can play a similar role to that of magnetic impurities on the surface of a TI system in stabilizing ferromagnetism^{28}: in the presence of conducting Dirac electron surface states, magnetic order can be stabilized via Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions, and is guaranteed to be of the FM type if the chemical potential is close to the Dirac point owing to a small Fermi wavenumber^{28}.
We define a Curie temperature of T_{C} ≃ 600 mK by the onset of magnetotransport hysteresis, as shown in Fig. 4a. The associated hysteretic MR loop is, however, quite different from the conventional butterfly shape commonly observed in FM materials. First, it is not centred around zero magnetic field, but rather closes abruptly before zero field is reached on downsweep. The type of magnetization leading to such hysteresis is not consistent with the usual overshoot that is necessary to overcome a coercive field, but does indeed occur in certain situations (see Supplementary Methods). Second, the increased scattering observed in SmB_{6} on decreasing field (that is, R(H_{dn}) > R(H_{up})) is opposite to that usually observed in a ferromagnet, where scattering associated with domain walls is typically enhanced on magnetization reversal. Rather, there is an enhanced conductance in SmB_{6} on upsweep that is diminished on reaching the closing field and returning to low fields. Interestingly, equivalent behaviour was observed in Mndoped Bi_{2}(Te, Se)_{3} thin films tuned by ionic liquid gating techniques^{29}. In this ferromagnetic TI system, a reversal of the usual butterfly shape occurs on gating the system into the bulk gap regime, where the TI chiral conducting modes trapped by domain walls result in an anomalous Hall conductance associated with a quantum Hall droplet^{1}. In this picture, the domain wall conductance is enhanced during reversal of the magnetization because the number of domain walls increases; at the coercive field, where the number of the domain walls is a maximum, the conductance exhibits a maximum. In SmB_{6}, similar behaviour is readily shown by plotting the difference in conductance, ΔG = G_{up}–G_{dn} (see Fig. 4b), where G_{up} (G_{dn}) is the magnetoconductance for H_{up} (H_{dn}). With decreasing temperature, ΔG is gradually enhanced and the peak position shifts to higher field. The temperature dependence of this characteristic field H^{∗} follows a meanfieldlike order parameter dependence that terminates at the Curie temperature T_{C} = 600 mK, as shown in the inset of Fig. 4b. We therefore interpret H^{∗} as a coercive field, in terms of the enhancement of the conductance due to a maximum number of minority domains.
Below 100 mK, the peak conductance at H^{∗} reaches a value of ∼e^{2}/h, a value observed in several samples with a wide range of residual sheet resistance values and sample dimensions (see Supplementary Methods), indicating that the observed conductance of e^{2}/h is not coincidental and possibly quantized. We suggest that the quantized conductance signature originates from transport of massive Dirac carriers along onedimensional channels that lie between FM domains. In this scenario, the domain wall conductance is equal to the anomalous quantum Hall conductance, quantized as (n + 1/2)e^{2}/h, as expected in a massive Dirac spectrum induced by FM ordered moments pointing out of the surface plane^{1,30}.
In an ideal system with no dissipation, ballistic transport should occur and exactly quantized conductance should be observed, as is the case for refs 31,32. However, in nonideal systems the chiral surface states suffer from abundant inelastic scattering due to electron correlations^{33} or the ‘puddling’ effect of spatial variations in the electronic structure^{34}, both of which easily suppress ballistic quantum transport and lead to deviations from an exactly quantized conductance. The peculiar temperature and fielddependent nature of the domain wall conductance, as well as the strong sample variation (see Supplementary Methods), provide a picture quite consistent with this situation.
Furthermore, the presence of a gridlike FM domain structure introduces another degree of complexity in the presence of dissipation. Provided the characteristic FM domain size is sufficiently smaller than the approximately mmscale sample dimension, the domain walls will effectively form an infinite resistor (conductor) network with R_{1} = h/e^{2}(G_{1} = e^{2}/h) elements, as shown in Fig. 4c. The dissipation or puddling is equivalent to virtual removal of resistors at random from the network, yielding the formation of a random network which consists of a conductance G_{1} = e^{2}/h distributed randomly with a probability of p and a conductance G_{2} = 0 distributed randomly with a probability of 1 − p. In such a random network, the percolating conduction gives the equivalent conductance between any two nonadjacent nodes of (2p − 1)G_{1} (ref. 35), namely, of the order of, but not exactly G_{1}. This naturally explains not only the nonexact e^{2}/h quantization at the coercive field, but also the further deviation from this value on further diminishment from maximum minority domain count with field.
The singular domain wall conductance element G_{1} of the chiral modes is determined by the chemical potential of the Dirac electrons because multiple quantized channels with (n + 1/2)e^{2}/h could contribute to the conduction in SmB_{6}. A halfquantized conductance of (1/2)e^{2}/h would be expected if the surface state Fermi energy is in the gap and in the lowest Landau level of n = 0. Assuming both top and bottom surfaces contribute equally to the total conductance, this gives the observed value of e^{2}/h = 2 × (1/2)e^{2}/h. Note that three Dirac bands are calculated to reside at the Γ and X/Y points in SmB_{6}(refs 10,11,12,13). The quantized conductance of e^{2}/h suggest that the chemical potential sits in the gap only at the Γ points, whereas it is above the gaps at the X/Y points, as shown in Fig. 4d. In fact, the saturation of sample resistance indicates that there remains surface conduction channels at very low temperatures, as shown in the inset of Fig. 1a. In this case, a quantized anomalous Hall effect in the Hall resistance, such as observed in gatetuned Crdoped Bi_{2}Se_{3} (ref. 31) and BiSbTeSe_{2} (ref. 32), is probably masked by the conduction of Dirac electrons at the X/Y points. Finally, we note that the abrupt, sharp transitions observed in MR on downsweep (Fig. 1b inset) often exhibit a jump in conductance very close to (1/2)e^{2}/h; whether this indicates a true quantization, or a unique signature of a chiral domain wall reconfiguration, remains a provocative observation to be explained. With a truly insulating bulk band structure, future gating experiments^{17} utilizing singlecrystal surfaces of SmB_{6} should readily facilitate the observation of these and other quantized properties in this system.
References
 1.
Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
 2.
Wilczek, F. Majorana returns. Nature Phys. 5, 614–618 (2009).
 3.
Wilczek, F. Two applications of axion electrodynamics. Phys. Rev. Lett. 58, 1799–1802 (1987).
 4.
Dzero, M., Sun, K., Galitski, V. & Coleman, P. Topological Kondo insulators. Phys. Rev. Lett. 104, 106408 (2010).
 5.
Takimoto, T. SmB_{6}: A promising candidate for a topological insulator. J. Phys. Soc. Jpn 80, 123710 (2011).
 6.
Menth, A., Buehler, E. & Geballe, T. H. Magnetic and semiconducting properties of SmB_{6}. Phys. Rev. Lett. 22, 295–297 (1969).
 7.
Wolgast, S. et al. Lowtemperature surface conduction in the Kondo insulator SmB_{6}. Phys. Rev. B 88, 180405 (2013).
 8.
Kim, D. J. et al. Surface Hall effect and nonlocal transport in SmB_{6}: Evidence for surface conduction. Sci. Rep. 3, 3150 (2013).
 9.
Zhang, X. et al. Hybridization, interion correlation, and surface states in the Kondo insulator SmB_{6}. Phys. Rev. X 3, 011011 (2013).
 10.
Xu, N. et al. Surface and bulk electronic structure of the strongly correlated system SmB_{6} and implications for a topological Kondo insulator. Phys. Rev. B 88, 121102 (2013).
 11.
Neupane, M. et al. Surface electronic structure of the topological Kondoinsulator candidate correlated electron system SmB_{6}. Nature Commun. 4, 2991 (2013).
 12.
Jiang, J. et al. Observation of possible topological ingap surface states in the Kondo insulator SmB_{6} by photoemission. Nature Commun. 4, 3010 (2013).
 13.
Frantzeskakis, E. et al. Kondo hybridization and the origin of metallic states at the (001) surface of SmB_{6}. Phys. Rev. X 3, 041024 (2013).
 14.
Zhu, Z.H. et al. Polaritydriven surface metallicity in SmB_{6}. Phys. Rev. Lett. 111, 216402 (2013).
 15.
Kim, D. J., Xia, J. & Fisk, Z. Topological surface state in the Kondo insulator samarium hexaboride. Nature Mater. 13, 466–470 (2014).
 16.
Thomas, S. et al. Weak antilocalization and linear magnetoresistance in the surface state of SmB_{6}. Preprint at http://arXiv.org/abs/1307.4133 (2013).
 17.
Syers, P., Kim, D., Fuhrer, M. S. & Paglione, J. Tuning bulk and surface conduction in the proposed topological Kondo insulator SmB_{6}. Phys. Rev. Lett. 114, 096601 (2015).
 18.
Cooley, J. C. et al. High magnetic fields and the correlation gap in SmB_{6}. Phys. Rev. B 52, 7322–7327 (1995).
 19.
Biswas, P. K. et al. Lowtemperature magnetic fluctuations in the Kondo insulator SmB_{6}. Phys. Rev. B 89, 161107 (2014).
 20.
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).
 21.
Allen, J. W., Batlogg, B. & Wachter, P. Large lowtemperature Hall effect and resistivity in mixedvalent SmB_{6}. Phys. Rev. B 20, 4807–4813 (1979).
 22.
Cooley, J. C., Aronson, M. C., Fisk, Z. & Canfield, P. C. SmB_{6}: Kondo insulator or exotic metal? Phys. Rev. Lett. 74, 1629–1632 (1995).
 23.
Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).
 24.
Hikami, S., Larkin, A. I. & Nagaoka, Y. Spin–orbit interaction and magnetoresistance in the two dimensional random system. Prog. Theor. Phys. 63, 707–710 (1980).
 25.
Lu, H.Z. & Shen, S.Q. Weak localization of bulk channels in topological insulator thin films. Phys. Rev. B 84, 125138 (2011).
 26.
Hamidian, M. H. et al. How Kondoholes create intense nanoscale heavyfermion hybridization disorder. Proc. Natl Acad. Sci. USA 108, 18233–18237 (2011).
 27.
Phelan, W. A. et al. Correlation between bulk thermodynamic measurements and the lowtemperatureresistance plateau in SmB_{6}. Phys. Rev. X 4, 031012 (2014).
 28.
Liu, Q., Liu, C.X., Xu, C., Qi, X.L. & Zhang, S.C. Magnetic impurities on the surface of a topological insulator. Phys. Rev. Lett. 102, 156603 (2009).
 29.
Checkelsky, J. G., Ye, J., Onose, Y., Iwasa, Y. & Tokura, Y. Diracfermionmediated ferromagnetism in a topological insulator. Nature Phys. 8, 729–733 (2012).
 30.
Qi, X.L., Hughes, T. L. & Zhang, S.C. Topological field theory of timereversal invariant insulators. Phys. Rev. B 78, 195424 (2008).
 31.
Chang, C.Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).
 32.
Xu, Y. et al. Observation of topological surface state quantum Hall effect in an intrinsic threedimensional topological insulator. Nature Phys. 10, 956–963 (2014).
 33.
Konig, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).
 34.
Martin, J. et al. Observation of electronhole puddles in graphene using a scanning singleelectron transistor. Nature Phys. 4, 144–148 (2008).
 35.
Kirkpatrick, S. Percolation and conduction. Rev. Mod. Phys. 45, 574–588 (1973).
Acknowledgements
The authors would like to acknowledge I. Appelbaum, J. Cummings, L. Fu, L. Li, T. PeregBarnea, J. Sau and I. Takeuchi for extremely valuable discussions. This research was supported by AFOSR (FA95501410332) and NSF (DMR0952716).
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Center for Nanophysics and Advanced Materials, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
 Yasuyuki Nakajima
 , Paul Syers
 , Xiangfeng Wang
 , Renxiong Wang
 & Johnpierre Paglione
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Contributions
Y.N. and R.W. performed the transport measurements and analysed the data. X.W. and P.S. grew and characterized single crystals of SmB_{6}. J.P. and Y.N. conceived and designed the experiments, and all authors contributed to the editing of the manuscript.
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The authors declare no competing financial interests.
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Correspondence to Johnpierre Paglione.
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