LETTERS PUBLISHED ONLINE: 23 MARCH 2014 | DOI: 10.1038/NMAT3913
Topological surface state in the Kondo insulator samarium hexaboride D. J. Kim*, J. Xia and Z. Fisk a 100
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Topological invariants of electron wavefunctions in condensed matter reveal many intriguing phenomena1,2 . A notable example is provided by topological insulators, which are characterized by an insulating bulk coexisting with a metallic boundary state3,4 . Although there has been intense interest in Bi-based topological insulators5,6 , their behaviour is complicated by the presence of a considerable residual bulk conductivity7–10 . Theories predict11,12 that the Kondo insulator system SmB6 , which is known to undergo a transition from a Kondo lattice metal to a small-gap insulator state with decreasing temperature, could be a topological insulator. Although the insulating bulk and metallic surface separation has been demonstrated in recent transport measurements13–15 , these have not demonstrated the topologically protected nature of the metallic surface state. Here we report thickness-dependent transport measurements on doped SmB6 , and show that magnetic and non-magnetic doping results in contrasting behaviour that supports the conclusion that SmB6 shows virtually no residual bulk conductivity. The Kondo insulator SmB6 is a dense lattice of Sm magnetic moments, which transforms from a poor metal at room temperature to an insulator with some residual conductance at low temperature16 . This transition is one of the most remarkable phenomena of Kondo lattices, which also exhibit unconventional superconductivity17 , hidden order transition18 and quantum criticality19 as a result of immersion of magnetic moments into a conduction band. The resistance of SmB6 increases exponentially as the temperature decreases (Fig. 1a), with a non-universal ratio of low to high temperature resistance. Usually, the higher quality sample exhibits the higher ratio (Supplementary Information). From Ohm’s law, the electrical resistivity of a rectangular parallelepiped shaped bulk conductor is defined by the product of measured resistance R and geometrical factor A/L, where L and A are the length and cross-sectional area. As ideal topological insulators (TIs) do not have bulk conductance, they cannot satisfy the basic Ohm’s law. Thus, if a three-dimensional (3D) TI transforms from a conventional bulk conductor at high temperature to an insulator with only surface conduction at low temperature, the sample thickness should not affect the measured low temperature limiting resistance but be independent of it. Figure 1b shows this most unusual behaviour of SmB6 , manifesting the thickness independence of sample resistance caused by the bulk and surface separation, which is a necessary condition for an ideal TI. The samples are mirror-polished to follow the previous scanning tunnelling microscopy result20 for metal terminated nonreconstructed surface, and the electrical leads are spot-welded on one side. Then the thickness of the samples is adjusted by polishing the other side with the original leads remaining in place (Supplementary Information). The resistance is measured in a
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Figure 1 | Failure of Ohm’s law. a, Resistance versus temperature of a rectangular parallelopiped shaped SmB6 sample; the inset shows the crystal structure. b, Thickness dependences of SmB6 , BaB6 and CeAuSb2 resistance. SmB6 has very clear thickness independence and its resistance ratios (defined by R_thin/R_thick) for three different thicknesses converge to unity, indicating bulk (insulator) and surface (conductor) separation as the temperature is lowered below 10 K. In contrast, BaB6 and CeAuSb2 show conventional bulk conducting behaviour.
Quantum Design PPMS with an LR700 a.c. bridge and automation software21 . From room temperature down to 10 K the comparative resistance ratio (RR, which is defined by R_thin)/R_thick) of three different thicknesses of SmB6 follows the geometric ratios as with usual bulk metallic systems, but below 10 K RR starts dropping and rapidly converges to unity below 5 K. This convergence indicates
Department of Physics and Astronomy, University of California, Irvine, Irvine, California 92697, USA. *e-mail: [email protected] NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
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NATURE MATERIALS DOI: 10.1038/NMAT3913
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Figure 2 | Topological surface state protected by TRS. a, Thickness resistance ratios for Y (3%), Gd (3%) and Yb (4%) doped SmB6 samples. In contrast to the non-magnetic impurities, which do not destroy the surface state, the magnetic impurities make the sample a conventional insulator. b,c, Resistance versus temperature curve of Y doped and Gd doped SmB6 . The Gd doped sample makes a larger resistance rise and does not show resistance saturation at low temperature down to 500 mK.
that the surface conductance dominates, with a much smaller conductance being left from the finite temperature effect arising from the tiny band gap in the bulk, which indicates complete insulating bulk and metallic surface separation at low temperature. These temperatures are consistent with our previous reports13,14 . In contrast to SmB6 , semiconducting BaB6 and quantum critical CeAuSb2 (ref. 22) show only the bulk metallic behaviour over the entire temperature range. What the thickness reduction data clearly show is that Ohm’s law fails, and the cleanest interpretation of this is that a conducting surface state forms at low temperature coincident with the full formation of the Kondo insulating energy gap, which shorts out the much less conducting bulk conduction. It is perhaps remarkable that the clear evidence of the conducting surface is present for low temperature resistivity which differs by orders of magnitude. However, this separation of bulk and surface is not sufficient to claim that SmB6 is indeed a topological Kondo insulator (TKI). The most convincing way to prove SmB6 to be a TKI is with angle-resolved photoemission spectroscopy to observe the Dirac cones. However, if SmB6 is an ideal TKI with virtually zero bulk conductance at low temperature, it should be possible to use transport measurements to investigate the underlying physics as well as that of hybrid systems with other superconducting and magnetic materials. Even though the thickness independence is exotic, it could arise from an accidental layer23 on the surface. Thus, it is necessary to find firmer evidence to support the existence of surface Dirac fermions in addition to the thickness independent resistance at low temperature. In conventional TIs (refs 4–6), the surface Dirac fermion has usually been probed through quantum oscillations from the Berry phase and Landau quantization using scanning tunnelling microscopy or magneto-optics experiments. However, the Kondo insulator is a strongly correlated system with a small band gap arising from Kondo screening, and magnetic field may not only affect the non-trivial band topology but also quench the Kondo effect. Previous magnetoresistance measurement up to 9 T did not show evidence of quantum oscillations13 , and at even higher field quenching of the Kondo insulating property leading 2
to more metallic behaviour was reported24 . Therefore, conventional transport measurements in high magnetic field may not be the way to detect Dirac fermions through quantum oscillation. TIs have three aspects of topological protection of the surface state4–6,9,10 . First, their fundamental Z2 topology preserves a gapless surface state unless time reversal symmetry (TRS) is broken. Second, helical spin polarization prevents momentum backscattering from k to −k by non-magnetic impurities. Finally, the Berry phase protects the surface state from weak localization through time reversed paths. These collectively provide a robust surface state with TRS conservation. The simplest manifestation of this TRS protected surface state would probably be positive magneto-resistance in an external magnetic field, and a transition from positive to negative would be expected for TKI from the above reasoning. This does indeed happen for pure SmB6 below 1 K, but the observation of this transition does not uniquely support a topological surface state (Supplementary Information). Another trivial surface state may exist intrinsically on 001 surfaces of all metal hexaborides25 . However, this polarity driven metallic surface cannot explain the surface transport for various surface orientations including the nonpolar 110 surface15 , nor can it disappear through broken TRS. To examine the effect of broken TRS, we use magnetic and nonmagnetic impurity doping in SmB6 and measure resistance versus temperature with the thickness reduction method to check if the low temperature resistance value converges to unity as for the pure samples. Figure 2a shows a clear contrast between magnetic Gd and non-magnetic Y and Yb doped on Sm sites in SmB6 samples. The overall RRs for two different thicknesses in Yb and Y doped SmB6 exhibit a bulk conducting property at high temperature which converges to unity at low temperature. In contrast, the RR of the Gd doped sample follows the geometrical ratio over the entire temperature range. Usually both magnetic and non-magnetic doping can weaken the Kondo insulating phase and make doped samples quite metallic at concentration higher than 30%, eventually leading to antiferromagnetic (GdB6 ), superconducting transition (YB6 ) and semiconducting (YbB6 ) states. At intermediate doping concentrations, the resistance rise at low temperature tends to
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NATURE MATERIALS DOI: 10.1038/NMAT3913 3% Gd
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Figure 3 | Behaviour of magnetic impurities in SmB6 . a, Inverse magnetic susceptibility of subtracted Gd contribution with temperature obeying Curie’s law. Insets, inverse susceptibility of 3% Gd (upper) and 3% Y (lower). b, Normalized magnetic susceptibility of 40% Gd doped sample shows saturation below 3 K; magnetization curve versus field for the sample has a straight line portion at 2 K (inset), indicating possible magnetic interaction between Gd sites at lower temperature. In contrast, the susceptibility of the low concentration (3%) sample exhibits divergence at low temperature. c–e, Magnetization versus field for subtracted 3% Gd portion at 1.8 K, 5 K and 10 K. The red lines are Brillouin fits to each curve.
decrease markedly without universality. Thus, the contrasting RRs or bulk conducting behaviour for the Gd doped sample could rather be misinterpreted as a more effective reduction of the resistance rise at low temperature. However, as shown in Fig. 2b,c, the Gd doped sample makes an almost three orders of magnitude jump whereas the Y doped sample has a less than two orders of magnitude increase. Remarkably, below 4 K the resistance of the Y doped sample saturates but the Gd doped sample shows further marked divergence even to the base temperature. In the temperature regime from 20 K to 500 mK the resistance increases (R(T )/R(20 K)) are 237 and 23 for Gd doping and Y doping respectively. The low temperature resistance increase in SmB6 is due to the opening of a hybridization gap arising from f and conduction electron hybridization through Kondo screening, and introduction of dopants on Sm sites disturbs this hybridization and eventually reduces the low temperature resistance increase in turn. Substituting Sm with Y or Gd decreases the resistance significantly but a band gap still survives from the slightly disturbed hybridization at ∼3% doping concentration. Thus, the Y doped sample’s RR convergence and low temperature saturation is consistent with the prediction for a much more conductive surface state, but when TRS is broken along with weakened Kondo quenching by the magnetic impurities, the bulk resembles a tiny gap insulator with thermally induced conduction at finite temperature whereas the resistance continues increasing with lowering temperature owing to the absence of a metallic surface state. The chemistry and size of dopants are quite similar, which suggests that segregation is unlikely. The thickness control results indicate that any extrinsic carriers that might be introduced by the dopants do not dominate the robust surface conduction. This result
is in agreement with the required property of a topological surface state protected by TRS invariance. An important question concerning impurities on Sm sites is whether or not they are dynamically coupled to the conduction electrons and involved in the Kondo quenching in the bulk. If they participate in a more complex way or generate another competing order, as for example in CeB6 and EuB6 , the interpretation of Fig. 2 might be more subtle. All possible magnetic rare earth hexaborides have their own unique magnetic characteristics. Gd is the most simple and a controllable case, with ideal non-interacting Gd ions at low concentration but antiferromagnetic interactions at higher concentration. The surface state arises from the topological property of the bulk. Thus, the ideal doping condition is to keep the Kondo insulating property intact and introduce just enough impurities to demonstrate the effect on the surface state. Figure 3a shows the inverse of magnetic susceptibility obtained by subtracting the magnetic susceptibility χ of the 3% Y doped sample from that of the 3% Gd doped sample to elucidate the pure Gd behaviour at 1,000 Oe. The inverse susceptibility for the Gd doped sample exhibits a straight line passing through the origin, which indicates that the magnetic impurities obey a simple paramagnetic Curie law. However, 40% Gd takes the sample beyond the percolation limit and the Curie law behaviour disappears at low temperature (Fig. 3b) with magnetization saturation, in contrast to the 3% Gd doping. The magnetization curve (Fig. 3b inset) suggests antiferromagnetic interactions between Gd moments for the 40% doped sample. To confirm the non-interacting impurity picture in the low concentration doped samples, we measured the magnetization for both Gd and Y doped samples and subtracted again to get Brillouin function fits up to 2 T perfectly, as shown in
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NATURE MATERIALS DOI: 10.1038/NMAT3913
LETTERS Resistance ratio
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Methods Single crystal growth. We used Al flux growth in a continuous Ar purged vertical high temperature tube furnace with high purity elements to grow all single crystals. The samples are leached out in sodium hydroxide solution.
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Thickness reduction. The sample thickness is reduced by polishing. After the first resistance versus temperature measurement, the same sample is flipped and mounted on a polishing fixture to reduce the thickness from the backside to keep the original surface and electrical leads made by spot welding intact.
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Thus, as the non-magnetic impurity concentration increases, the scattering in the dense bound states can form an extra impurity band of the overlapping bound states, and this makes it possible to induce backscattering to destroy the surface state. Thus, the possible insulating state in Fig. 4c is most naturally interpreted as a quantum percolation limit from a topological insulator to a conventional insulator.
2.5
Measurement. Quantum Design PPMS and MPMS were used for transport and magnetic property measurements. All resistance measurements were made with an LR700 a.c. bridge with automatic current excitation to ensure minimum heating and 5½ digit resistance resolution at all acquisition temperatures.
18% Yb 120 μm 1
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Figure 4 | Quantum percolation of high concentration Yb doped SmB6 . a, Thickness resistance ratios for 4% Yb and 18% Yb doped SmB6 samples. b,c, Resistance versus temperature curves for each concentration. At low temperature 18% doping resistance diverges when compared with 4% doping. This indicates a transition to a conventional insulator by forming an impurity band above the percolation limit.
Fig. 3c–e. Higher field alters the Kondo effect in the SmB6 host and changes the free Gd response from its ideal paramagnetic behaviour. Independent of dopant type (magnetic or non-magnetic), a high concentration of impurities in a Kondo insulator can affect its properties. Substituting for Sm with impurities decreases the resistance significantly, and high doping (above 30%) destroys the insulating properties and leads to a metallic state26 . When the doping concentration is modest and lies between the strong Kondo insulating state and bulk metallic state, the surface state can be altered even with non-magnetic impurities. Divalent Yb does not have magnetic moments in SmB6 , Interestingly, YbB6 was recently claimed to have a non-trivial topological surface state27 . However, in thickness control transport magneto-resistance and bulk magnetometer measurements we find that YbB6 does not show the bulk–surface separation or mixed valence state (Supplementary Information). The resistance of a 4% Yb doped sample has an almost four orders of magnitude rise with lowering temperature and very clear bulk and surface separation with concentration (Fig. 4a,b). However, the RR for an 18% doped sample does not have the bulk and surface separation characteristic (Fig. 4a). As with the 3% Gd and Y doped samples in Fig. 2, Fig. 4b,c shows a clear difference at low temperature. The 4% Yb doped sample shows the usual Kondo insulating surface state with saturating resistance, but 18% Yb instead has a diverging feature in the low temperature range. This increasing resistance indicates a bulk insulating property and this is probably connected with the fact that pure YbB6 is apparently a typical band semiconductor. A single impurity or defect behaves as a boundary in the system. As the Z2 index varies at the boundary, localized in-gap bound states with opposite spins with gapless Dirac dispersion can be formed in topological insulators with the wavefunction decaying exponentially away from the centre of the impurity on a characteristic length scale. When the length scale is smaller than the distance between impurities, the energy overlap of these bound states leads to possible quantum scattering between them at each impurity site28 . However, when the impurity concentration is low enough, this quantum scattering is prohibited to ensure the robustness of the surface state. 4
Received 08 October 2013; accepted 12 February 2014; published online 23 March 2014
References 1. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982). 2. Xiao, D., Chang, M. C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010). 3. Moore, J. E. The birth of topological insulators. Nature 464, 194–198 (2010). 4. Ando, Yoichi Topological insulator materials. J. Phys. Soc. Jpn 82, 102001 (2013). 5. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010). 6. Qi, X-L. & Zhang, S-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011). 7. Hsieh, D. et al. A topological Dirac insulator in a quantum spin hall phase. Nature 452, 970–974 (2008). 8. Zhang, H. et al. Topological insulators in Bi2 Se3 , Bi2 Te3 and Sb2 Te3 with a single Dirac cone on the surface. Nature Phys. 5, 438–442 (2009). 9. Zhang, T. et al. Experimental demonstration of topological surface states protected by time-reversal symmetry. Phys. Rev. Lett. 103, 266803 (2009). 10. Okada, Y. et al. Direct observation of broken time reversal symmetry on the surface of a magnetically doped topological insulator. Phys. Rev. Lett. 106, 206805 (2011). 11. Dzero, M., Sun, K., Galitski, V. & Coleman, P. Topological Kondo insulators. Phys. Rev. Lett. 104, 106408 (2010). 12. Lu, F., Zhao, J., Weng, H., Fang, Z. & Dai, X. Correlated topological insulators with mixed valence. Phys. Rev. Lett. 110, 096401 (2013). 13. Kim, D. J., Grant, T. & Fisk, Z. Limit cycle and anomalous capacitance in the Kondo insulator SmB6 . Phys. Rev. Lett. 109, 096601 (2012). 14. Wolgast, S. et al. Low temperature surface conduction in the Kondo insulator SmB6 . Phys. Rev. B 88, 180405(R) (2013). 15. Kim, D. J. et al. Surface hall effect and nonlocal transport in SmB6 . Sci. Rep. 3, 3150 (2013). 16. Aeppli, G. & Fisk, Z. Kondo insulators. Comments Condens. Matter Phys. 16, 155–165 (1992). 17. Fisk, Z. & Ott, H. R. Superconductivity in New Materials Vol. 4 (Elsevier, 2010). 18. Mydosh, J. A. & Oppeneer, P. M. Superconductivity and magnetism-unsolved case of URu2 Si2 . Rev. Mod. Phys. 83, 1301–1322 (2011). 19. Coleman, P. & Schofield, A. J. Quantum criticality. Nature 433, 226–229 (2005). 20. Ozcomert, J. S. & Trenary, M. Atomically resolved surface structure of LaB6(100). Surf. Sci. 265, L227 (1992). 21. Kim, D. J. & Fisk, Z. A labview based template for user created experiment automation. Rev. Sci. Instrum. 83, 123705 (2012). 22. Balicas, L. et al. Magnetic field-tuned quantum critical point in CeAuSb2 . Phys. Rev. B 72, 064422 (2005). 23. von Klitzing, K. & Landwehr, G. Surface quantum states in tellurium. Solid State Commun. 9, 2201–2205 (1971). 24. Cooley, J. C. et al. High field gap closure in the Kondo insulator SmB6 . J. Supercond. 12, 171–173 (1999). 25. Zhu, Z-H. et al. Polarity-driven surface metallicity in SmB6 . Phys. Rev. Lett. 111, 216402 (2013).
NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
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NATURE MATERIALS DOI: 10.1038/NMAT3913 26. Yeo, S., Song, K., Hur, N., Fisk, Z. & Schlottmann, P. Effects of Eu doping on SmB6 single crystals. Phys. Rev. B 85., 115125 (2012). 27. Hongming, W., Jianzhou, Z., Wang, Z., Fang, Z. & Dai, X. Topological crystalline Kondo insulator in mixed valence ytterbium borides. Phys. Rev. Lett. 112, 016403 (2014). 28. Chu, R-L., Lu, J. & Shen, S-Q. Quantum percolation in quantum spin Hall antidote systems. Europhys. Lett. 100, 17013 (2012).
Acknowledgements We thank M. Dzero, I. Krivorotov and S. Thomas for discussions. This research was supported by NSF-DMR-0801253, UC Irvine CORCL grant MIIG-2011-12-8 and Sloan Research Fellowship BR2013-116, J.X.
LETTERS Author contributions All authors conceived the idea of the experiment together, D.J.K. and Z.F. grew crystals and D.J.K. made the measurements. All authors discussed the results, participated in data analysis and wrote the manuscript.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to D.J.K.
Competing financial interests The authors declare no competing financial interests.
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5
Topological surface state in the Kondo insulator samarium hexaboride D. J. Kim*, J. Xia and Z. Fisk a 100
SmB6
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Topological invariants of electron wavefunctions in condensed matter reveal many intriguing phenomena1,2 . A notable example is provided by topological insulators, which are characterized by an insulating bulk coexisting with a metallic boundary state3,4 . Although there has been intense interest in Bi-based topological insulators5,6 , their behaviour is complicated by the presence of a considerable residual bulk conductivity7–10 . Theories predict11,12 that the Kondo insulator system SmB6 , which is known to undergo a transition from a Kondo lattice metal to a small-gap insulator state with decreasing temperature, could be a topological insulator. Although the insulating bulk and metallic surface separation has been demonstrated in recent transport measurements13–15 , these have not demonstrated the topologically protected nature of the metallic surface state. Here we report thickness-dependent transport measurements on doped SmB6 , and show that magnetic and non-magnetic doping results in contrasting behaviour that supports the conclusion that SmB6 shows virtually no residual bulk conductivity. The Kondo insulator SmB6 is a dense lattice of Sm magnetic moments, which transforms from a poor metal at room temperature to an insulator with some residual conductance at low temperature16 . This transition is one of the most remarkable phenomena of Kondo lattices, which also exhibit unconventional superconductivity17 , hidden order transition18 and quantum criticality19 as a result of immersion of magnetic moments into a conduction band. The resistance of SmB6 increases exponentially as the temperature decreases (Fig. 1a), with a non-universal ratio of low to high temperature resistance. Usually, the higher quality sample exhibits the higher ratio (Supplementary Information). From Ohm’s law, the electrical resistivity of a rectangular parallelepiped shaped bulk conductor is defined by the product of measured resistance R and geometrical factor A/L, where L and A are the length and cross-sectional area. As ideal topological insulators (TIs) do not have bulk conductance, they cannot satisfy the basic Ohm’s law. Thus, if a three-dimensional (3D) TI transforms from a conventional bulk conductor at high temperature to an insulator with only surface conduction at low temperature, the sample thickness should not affect the measured low temperature limiting resistance but be independent of it. Figure 1b shows this most unusual behaviour of SmB6 , manifesting the thickness independence of sample resistance caused by the bulk and surface separation, which is a necessary condition for an ideal TI. The samples are mirror-polished to follow the previous scanning tunnelling microscopy result20 for metal terminated nonreconstructed surface, and the electrical leads are spot-welded on one side. Then the thickness of the samples is adjusted by polishing the other side with the original leads remaining in place (Supplementary Information). The resistance is measured in a
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Figure 1 | Failure of Ohm’s law. a, Resistance versus temperature of a rectangular parallelopiped shaped SmB6 sample; the inset shows the crystal structure. b, Thickness dependences of SmB6 , BaB6 and CeAuSb2 resistance. SmB6 has very clear thickness independence and its resistance ratios (defined by R_thin/R_thick) for three different thicknesses converge to unity, indicating bulk (insulator) and surface (conductor) separation as the temperature is lowered below 10 K. In contrast, BaB6 and CeAuSb2 show conventional bulk conducting behaviour.
Quantum Design PPMS with an LR700 a.c. bridge and automation software21 . From room temperature down to 10 K the comparative resistance ratio (RR, which is defined by R_thin)/R_thick) of three different thicknesses of SmB6 follows the geometric ratios as with usual bulk metallic systems, but below 10 K RR starts dropping and rapidly converges to unity below 5 K. This convergence indicates
Department of Physics and Astronomy, University of California, Irvine, Irvine, California 92697, USA. *e-mail: [email protected] NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
© 2014 Macmillan Publishers Limited. All rights reserved.
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NATURE MATERIALS DOI: 10.1038/NMAT3913
LETTERS
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Figure 2 | Topological surface state protected by TRS. a, Thickness resistance ratios for Y (3%), Gd (3%) and Yb (4%) doped SmB6 samples. In contrast to the non-magnetic impurities, which do not destroy the surface state, the magnetic impurities make the sample a conventional insulator. b,c, Resistance versus temperature curve of Y doped and Gd doped SmB6 . The Gd doped sample makes a larger resistance rise and does not show resistance saturation at low temperature down to 500 mK.
that the surface conductance dominates, with a much smaller conductance being left from the finite temperature effect arising from the tiny band gap in the bulk, which indicates complete insulating bulk and metallic surface separation at low temperature. These temperatures are consistent with our previous reports13,14 . In contrast to SmB6 , semiconducting BaB6 and quantum critical CeAuSb2 (ref. 22) show only the bulk metallic behaviour over the entire temperature range. What the thickness reduction data clearly show is that Ohm’s law fails, and the cleanest interpretation of this is that a conducting surface state forms at low temperature coincident with the full formation of the Kondo insulating energy gap, which shorts out the much less conducting bulk conduction. It is perhaps remarkable that the clear evidence of the conducting surface is present for low temperature resistivity which differs by orders of magnitude. However, this separation of bulk and surface is not sufficient to claim that SmB6 is indeed a topological Kondo insulator (TKI). The most convincing way to prove SmB6 to be a TKI is with angle-resolved photoemission spectroscopy to observe the Dirac cones. However, if SmB6 is an ideal TKI with virtually zero bulk conductance at low temperature, it should be possible to use transport measurements to investigate the underlying physics as well as that of hybrid systems with other superconducting and magnetic materials. Even though the thickness independence is exotic, it could arise from an accidental layer23 on the surface. Thus, it is necessary to find firmer evidence to support the existence of surface Dirac fermions in addition to the thickness independent resistance at low temperature. In conventional TIs (refs 4–6), the surface Dirac fermion has usually been probed through quantum oscillations from the Berry phase and Landau quantization using scanning tunnelling microscopy or magneto-optics experiments. However, the Kondo insulator is a strongly correlated system with a small band gap arising from Kondo screening, and magnetic field may not only affect the non-trivial band topology but also quench the Kondo effect. Previous magnetoresistance measurement up to 9 T did not show evidence of quantum oscillations13 , and at even higher field quenching of the Kondo insulating property leading 2
to more metallic behaviour was reported24 . Therefore, conventional transport measurements in high magnetic field may not be the way to detect Dirac fermions through quantum oscillation. TIs have three aspects of topological protection of the surface state4–6,9,10 . First, their fundamental Z2 topology preserves a gapless surface state unless time reversal symmetry (TRS) is broken. Second, helical spin polarization prevents momentum backscattering from k to −k by non-magnetic impurities. Finally, the Berry phase protects the surface state from weak localization through time reversed paths. These collectively provide a robust surface state with TRS conservation. The simplest manifestation of this TRS protected surface state would probably be positive magneto-resistance in an external magnetic field, and a transition from positive to negative would be expected for TKI from the above reasoning. This does indeed happen for pure SmB6 below 1 K, but the observation of this transition does not uniquely support a topological surface state (Supplementary Information). Another trivial surface state may exist intrinsically on 001 surfaces of all metal hexaborides25 . However, this polarity driven metallic surface cannot explain the surface transport for various surface orientations including the nonpolar 110 surface15 , nor can it disappear through broken TRS. To examine the effect of broken TRS, we use magnetic and nonmagnetic impurity doping in SmB6 and measure resistance versus temperature with the thickness reduction method to check if the low temperature resistance value converges to unity as for the pure samples. Figure 2a shows a clear contrast between magnetic Gd and non-magnetic Y and Yb doped on Sm sites in SmB6 samples. The overall RRs for two different thicknesses in Yb and Y doped SmB6 exhibit a bulk conducting property at high temperature which converges to unity at low temperature. In contrast, the RR of the Gd doped sample follows the geometrical ratio over the entire temperature range. Usually both magnetic and non-magnetic doping can weaken the Kondo insulating phase and make doped samples quite metallic at concentration higher than 30%, eventually leading to antiferromagnetic (GdB6 ), superconducting transition (YB6 ) and semiconducting (YbB6 ) states. At intermediate doping concentrations, the resistance rise at low temperature tends to
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NATURE MATERIALS DOI: 10.1038/NMAT3913 3% Gd
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Figure 3 | Behaviour of magnetic impurities in SmB6 . a, Inverse magnetic susceptibility of subtracted Gd contribution with temperature obeying Curie’s law. Insets, inverse susceptibility of 3% Gd (upper) and 3% Y (lower). b, Normalized magnetic susceptibility of 40% Gd doped sample shows saturation below 3 K; magnetization curve versus field for the sample has a straight line portion at 2 K (inset), indicating possible magnetic interaction between Gd sites at lower temperature. In contrast, the susceptibility of the low concentration (3%) sample exhibits divergence at low temperature. c–e, Magnetization versus field for subtracted 3% Gd portion at 1.8 K, 5 K and 10 K. The red lines are Brillouin fits to each curve.
decrease markedly without universality. Thus, the contrasting RRs or bulk conducting behaviour for the Gd doped sample could rather be misinterpreted as a more effective reduction of the resistance rise at low temperature. However, as shown in Fig. 2b,c, the Gd doped sample makes an almost three orders of magnitude jump whereas the Y doped sample has a less than two orders of magnitude increase. Remarkably, below 4 K the resistance of the Y doped sample saturates but the Gd doped sample shows further marked divergence even to the base temperature. In the temperature regime from 20 K to 500 mK the resistance increases (R(T )/R(20 K)) are 237 and 23 for Gd doping and Y doping respectively. The low temperature resistance increase in SmB6 is due to the opening of a hybridization gap arising from f and conduction electron hybridization through Kondo screening, and introduction of dopants on Sm sites disturbs this hybridization and eventually reduces the low temperature resistance increase in turn. Substituting Sm with Y or Gd decreases the resistance significantly but a band gap still survives from the slightly disturbed hybridization at ∼3% doping concentration. Thus, the Y doped sample’s RR convergence and low temperature saturation is consistent with the prediction for a much more conductive surface state, but when TRS is broken along with weakened Kondo quenching by the magnetic impurities, the bulk resembles a tiny gap insulator with thermally induced conduction at finite temperature whereas the resistance continues increasing with lowering temperature owing to the absence of a metallic surface state. The chemistry and size of dopants are quite similar, which suggests that segregation is unlikely. The thickness control results indicate that any extrinsic carriers that might be introduced by the dopants do not dominate the robust surface conduction. This result
is in agreement with the required property of a topological surface state protected by TRS invariance. An important question concerning impurities on Sm sites is whether or not they are dynamically coupled to the conduction electrons and involved in the Kondo quenching in the bulk. If they participate in a more complex way or generate another competing order, as for example in CeB6 and EuB6 , the interpretation of Fig. 2 might be more subtle. All possible magnetic rare earth hexaborides have their own unique magnetic characteristics. Gd is the most simple and a controllable case, with ideal non-interacting Gd ions at low concentration but antiferromagnetic interactions at higher concentration. The surface state arises from the topological property of the bulk. Thus, the ideal doping condition is to keep the Kondo insulating property intact and introduce just enough impurities to demonstrate the effect on the surface state. Figure 3a shows the inverse of magnetic susceptibility obtained by subtracting the magnetic susceptibility χ of the 3% Y doped sample from that of the 3% Gd doped sample to elucidate the pure Gd behaviour at 1,000 Oe. The inverse susceptibility for the Gd doped sample exhibits a straight line passing through the origin, which indicates that the magnetic impurities obey a simple paramagnetic Curie law. However, 40% Gd takes the sample beyond the percolation limit and the Curie law behaviour disappears at low temperature (Fig. 3b) with magnetization saturation, in contrast to the 3% Gd doping. The magnetization curve (Fig. 3b inset) suggests antiferromagnetic interactions between Gd moments for the 40% doped sample. To confirm the non-interacting impurity picture in the low concentration doped samples, we measured the magnetization for both Gd and Y doped samples and subtracted again to get Brillouin function fits up to 2 T perfectly, as shown in
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NATURE MATERIALS DOI: 10.1038/NMAT3913
LETTERS Resistance ratio
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Thus, as the non-magnetic impurity concentration increases, the scattering in the dense bound states can form an extra impurity band of the overlapping bound states, and this makes it possible to induce backscattering to destroy the surface state. Thus, the possible insulating state in Fig. 4c is most naturally interpreted as a quantum percolation limit from a topological insulator to a conventional insulator.
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Measurement. Quantum Design PPMS and MPMS were used for transport and magnetic property measurements. All resistance measurements were made with an LR700 a.c. bridge with automatic current excitation to ensure minimum heating and 5½ digit resistance resolution at all acquisition temperatures.
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Figure 4 | Quantum percolation of high concentration Yb doped SmB6 . a, Thickness resistance ratios for 4% Yb and 18% Yb doped SmB6 samples. b,c, Resistance versus temperature curves for each concentration. At low temperature 18% doping resistance diverges when compared with 4% doping. This indicates a transition to a conventional insulator by forming an impurity band above the percolation limit.
Fig. 3c–e. Higher field alters the Kondo effect in the SmB6 host and changes the free Gd response from its ideal paramagnetic behaviour. Independent of dopant type (magnetic or non-magnetic), a high concentration of impurities in a Kondo insulator can affect its properties. Substituting for Sm with impurities decreases the resistance significantly, and high doping (above 30%) destroys the insulating properties and leads to a metallic state26 . When the doping concentration is modest and lies between the strong Kondo insulating state and bulk metallic state, the surface state can be altered even with non-magnetic impurities. Divalent Yb does not have magnetic moments in SmB6 , Interestingly, YbB6 was recently claimed to have a non-trivial topological surface state27 . However, in thickness control transport magneto-resistance and bulk magnetometer measurements we find that YbB6 does not show the bulk–surface separation or mixed valence state (Supplementary Information). The resistance of a 4% Yb doped sample has an almost four orders of magnitude rise with lowering temperature and very clear bulk and surface separation with concentration (Fig. 4a,b). However, the RR for an 18% doped sample does not have the bulk and surface separation characteristic (Fig. 4a). As with the 3% Gd and Y doped samples in Fig. 2, Fig. 4b,c shows a clear difference at low temperature. The 4% Yb doped sample shows the usual Kondo insulating surface state with saturating resistance, but 18% Yb instead has a diverging feature in the low temperature range. This increasing resistance indicates a bulk insulating property and this is probably connected with the fact that pure YbB6 is apparently a typical band semiconductor. A single impurity or defect behaves as a boundary in the system. As the Z2 index varies at the boundary, localized in-gap bound states with opposite spins with gapless Dirac dispersion can be formed in topological insulators with the wavefunction decaying exponentially away from the centre of the impurity on a characteristic length scale. When the length scale is smaller than the distance between impurities, the energy overlap of these bound states leads to possible quantum scattering between them at each impurity site28 . However, when the impurity concentration is low enough, this quantum scattering is prohibited to ensure the robustness of the surface state. 4
Received 08 October 2013; accepted 12 February 2014; published online 23 March 2014
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NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials
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NATURE MATERIALS DOI: 10.1038/NMAT3913 26. Yeo, S., Song, K., Hur, N., Fisk, Z. & Schlottmann, P. Effects of Eu doping on SmB6 single crystals. Phys. Rev. B 85., 115125 (2012). 27. Hongming, W., Jianzhou, Z., Wang, Z., Fang, Z. & Dai, X. Topological crystalline Kondo insulator in mixed valence ytterbium borides. Phys. Rev. Lett. 112, 016403 (2014). 28. Chu, R-L., Lu, J. & Shen, S-Q. Quantum percolation in quantum spin Hall antidote systems. Europhys. Lett. 100, 17013 (2012).
Acknowledgements We thank M. Dzero, I. Krivorotov and S. Thomas for discussions. This research was supported by NSF-DMR-0801253, UC Irvine CORCL grant MIIG-2011-12-8 and Sloan Research Fellowship BR2013-116, J.X.
LETTERS Author contributions All authors conceived the idea of the experiment together, D.J.K. and Z.F. grew crystals and D.J.K. made the measurements. All authors discussed the results, participated in data analysis and wrote the manuscript.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to D.J.K.
Competing financial interests The authors declare no competing financial interests.
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