Introduction

Recently discovered1,2,3,4,5,6,7 three-dimensional (3D) topological insulators (TI) have generated great excitement. Strong spin-orbit coupling in a TI gives rise to a non-trivial and robust conducting surface state4,5, reminiscent of the edge channel8,9 found in quantum Hall (QH)10,11 and quantum spin Hall (QSH)12,13 states. However, such surface transport properties have remained challenging to separate from residual bulk impurity conduction14,15,16,17 despite rapid recent improvements7,18,19,20,21, promoting us to search for TI materials with better bulk insulation. More importantly, most theoretical and experimental efforts to date have been dedicated to materials with the underlying physics of non-interacting electrons1,4,5,12. A big question thus concerns the role of electron interaction and competing orders in new TI materials with strong correlation, from which new types of topological phases are expected to emerge4,5.

Recent seminal theoretical works22,23 have predicted in the strongly correlated material SmB6 the existence of a topological insulator phase: topological Kondo insulator (TKI). SmB6 is a heavy fermion material first studied 40 years ago24. In SmB6, highly renormalized f-electrons hybridize with conduction electrons and form a completely filled band of quasiparticles with a transport excitation gap Δ of about 40 Kelvin. However, its many exotic properties25,26,27 still defy a satisfactory understanding within the framework of conventional insulators. One of these mysteries is the peculiar residual conduction at the lowest temperatures. Behaving electronically like an insulator at high temperatures, at low temperature its resistance mysteriously saturates: a curiosity that is usually attributed to the existence of density of states within the band gap. According to the recent TKI theory22,23,28,29, the hybridization and odd parity wavefunction lead to strong spin-orbit coupling in SmB6 and give rise to a topological surface state, which naturally explains the origin of the in-gap state and pinpoints its location to be on the surface of SmB6. Our recent capacitance measurements on high quality SmB6 crystals revealed intriguing anomalous capacitance effects30 that could be explained by assuming the in-gap-states exist on the surface. In this paper we present evidence of thickness-independent surface Hall effect and nonlocal transport in high quality SmB6 crystals of various geometries and from different growth batches. These results reveal in SmB6 an insulating bulk and a conducting surface.

Results

Hall effect measurements were carried out in wedge-shaped SmB6 crystals. As depicted in the inset in Fig. 1(a), the sample is placed in a perpendicular magnetic field and current I flows between the two ends of the wedge. The Hall resistances Rxy = Vxy/I are measured at different thicknesses d to distinguish between surface and bulk conduction. For bulk conduction, while Rxy/B is d-independent if surface conduction dominates. In both cases, one can define a Hall coefficient RH = Ey/jxB independent of B, where jx is the current density (surface or bulk) and Ey is the transverse electric field. The Hall voltage Vxy is found to be linear with B (Fig. 1(c) (d)) at small fields at all temperatures, but becomes significantly nonlinear for larger fields around 5 K, indicating a temperature regime of multichannel conduction. At high (20 K) or low (2 K) temperatures, the extreme linearity of the Hall effect indicates single channel conduction, either from the bulk or surface. For the simplest case of one surface conduction channel (top and bottom surfaces combined) and one bulk channel with Hall coefficients RHS, RHB and resistivity ρS, ρB respectively, the Hall resistance Rxy at magnetic field B is . Nonlinearity is expected at large B, but at small fields it simplifies to , which indeed gives thickness-independent Rxy/B = RHS if the surface channel dominates (i.e. ρB ≫ ρSd). From B < 1 T data we extract the value Rxy/B at various temperatures T. Representative results in sample S1 are plotted in Fig. 1(a) for d = 120, 270 and 320 μm respectively, showing clearly that while at high temperatures Rxy/B differ at different d, they converge to a same universal value of 0.3 Ω/T below 4 Kelvin, consistent with surface conduction. Since more than one surface channels may exist, as predicted by theory28,29, it is difficult to quantitatively extract the surface carrier density and mobility at this stage. Replotting the Hall resistance ratios Rxy(d1)/Rxy(d2) in Fig. 1(b), we found these ratios to be equal to d2/d1 at high T and become unity at low T, proving the crossover from 3D to 2D Hall effects when T is lowered. The temperature dependence is well described by a two-channel (bulk and surface) conduction model in which the bulk carrier density decreases exponentially with temperature with an activation gap Δ = 38 K. Using this simple model, we could reproduce the curious “peak” in Rxy/B at 4 K (solid lines in Fig. 1(a)), which lacks31 a good explanation until now. Low temperature surface-dominated conduction would also give rise to a longitudinal resistance Rxx that is independent of sample thickness, which we have demonstrated recently32.

Figure 1
figure 1

Surface Hall effect.

(a), Markers, Hall resistances Rxy divided by magnetic field B versus temperature T at three different thicknesses d in a wedge shaped sample S1. Lines are simulations using a two conduction channel model (see text). Left inset, picture of the crystal before wiring. Right inset, measurement schematic. (b), Markers, ratios between Hall resistances Rxy at different d, showing the transition from bulk to surface conduction as temperature is lowered. Lines are calculated from simulations as in (a). (c), Rxy versus B at various T for d = 120 μm, showing nonlinearity at around 5 K. (d), Rxy/B normalized to small field values to demonstrate the nonlinearity.

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Surface dominated conduction could also be demonstrated at zero magnetic field with so-called “non-local” transport in the spirit of nonlocal transport experiments performed in QH10,11 and QSH12,13 states that have served as evidence8,9 for the existence of the topological edge states that are one-dimensional analogues to the surface state in a TI. The highly metallic surface conduction in a TI would necessarily invalidate Ohm's law and introduce large nonlocal voltages, which we have indeed found in SmB6 samples. Fig. 2(a) shows a schematic of the nonlocal measurement in sample S4. Current I16 flows between current leads 1 and 6 at the center on opposite faces of the crystal. Contacts 2 and 3 are located close to contact 1 for the detection of “local” voltage V23. Contacts 4 and 5 are put near the sample edge far away from the current leads to detect “nonlocal” voltage V45. As shown in the inset in Fig. 2(b), in the case of bulk conduction, current will concentrate in the bulk near the current leads 1 and 6, resulting in negligibly small nonlocal voltage (V45 ≪ V23). If surface conduction dominates, however, current will be forced to flow between contacts 1 and 6 via the surface, making V45 larger. Fig. 2(a) shows as a function of temperature the measured V45 and V23 divided by I16, both agreeing qualitatively with our finite element simulations (Supplementary Information) incorporating the aforementioned simple model. The ratio V45/V23 is replotted in Fig. 2(b). At low temperature, when surface conduction dominates, the nonlocal voltage V45 becomes large and even surpasses the local voltage V23. Above T = 5 K, when bulk conduction dominates, the magnitude of V45/V23 is very small. The negative sign of V45/V23 is due to the small misalignment of contacts (Supplementary Information). The change of both magnitude and sign of V45/V23 is reproduced by our simulation and highlights the distinction between high T bulk conduction and low T surface conduction.

Figure 2
figure 2

Nonlocal transport due to surface conduction.

(a), Markers, nonlocal resistance V45/I16 and local resistance V23/I16 versus temperature T. Dashed lines are finite element simulations. Inset is a schematic of the measurement configuration on sample S4. (b), Ratio between nonlocal and local voltages V45/V23 versus T. Dashed lines are finite element simulations. Inset, cartoons for current distribution in sample cross-section for surface and bulk dominated conductions.

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An important aspect of TI is the topological protection of the surface state against time-reversal invariant perturbations. The robustness against perturbation distinguishes a topological surface state from “accidental” surface conduction4,5. We found that the above surface Hall effect and nonlocal transport are recurrent phenomena in various samples and persist after chemical and mechanical sample treatments. We have mechanically cut and scratched the surface of sample S10 (Fig. 3 inset) and performed Hall measurements before and after surface abrasion. We found that the low temperature Hall resistance Rxy/B remained unchanged (Fig. 3(c)), indicating that the surface carrier density nS was not affected by abrasion. The abrasion does reduce the surface mobility μS though, as reflected by the height of Rxy/B “peak” at T = 4 K, presumably due to greatly enhanced geometric roughness. As shown in Fig. 3 (a) (b), the surface dominated conduction persists after abrasion, as demonstrated by the thickness-independent Hall effect below 4 K.

Figure 3
figure 3

Surface conduction after surface treatments.

(a), Markers, Hall resistances Rxy divided by magnetic field B versus temperature T at three different thicknesses d in a wedge shaped sample S10, after intentional chemical etching and cutting. (b), Markers, ratios between Hall resistances Rxy at different d, showing the transition from bulk to surface conduction as temperature is lowered. (c), Solid lines, Hall resistance Rxy divided by magnetic field B versus temperature T, before and after surface abrasion. Dashed lines are simulations assuming abrasion only reduces the effective surface mobility μS. Inset, picture of sample S10 during abrasion.

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Discussion

The experimental identification of thickness-independent surface Hall effect and nonlocal transport serve as strong evidence that SmB6 has a metallic surface state surrounding an insulating bulk at low temperatures. The characterization of energetics of the surface state and hence direct tests of the topological nature, however, awaits future investigations using energy and spin resolved techniques like ARPES2,3 and STM33,34. Recent ARPES35,36,37, quantum oscillation38 and STM measurements39 have provided further evidence of surface conduction, as well as insights of the surface electronic structure. Recent transport measurements with doped SmB6 samples32 have demonstrated that the surface conduction persists after non-magnetic doping but is destroyed by magnetic doping, as expected from a topological surface state protected by time-reversal-symmetry. Transport experiments40 at lower temperatures have revealed the weak-anti-localization effect, which is expected from a spin-momentum-locked topological surface state. Unlike weakly interacting TI materials, the strong electron correlation in SmB6 could give rise to exotic emergent phases4,5 with exciting new physics.

Note added: During initial submission of the manuscript, we became aware of a related work41 in which evidence for surface conduction was provided in a SmB6 sample with contacts arranged in a unique configuration and a point contact measurement42 that excludes the possibility that the in-gap state is located in the bulk of SmB6.

Methods

High quality SmB6 crystals were grown using the aluminium flux method. The surfaces of these crystals were carefully etched using hydrochloric acid and then cleaned using solvents to remove possible oxide layer or aluminium residues. These crystals are then inspected using X-ray analysis to make sure SmB6 is the only content. Samples used in the experiments were made from these crystals either by mechanical cleaving or polishing using polishing films containing diamond particles. The exposed surfaces are (100) planes. Gold and platinum wires are attached to the samples using micro spot welding and/or silver epoxy, with no discernable differences in measurements. Low frequency transport measurements were carried out in dilution fridges and helium cryostats using either standard low frequency (37 Hz) lock-in techniques with 50 nA excitation currents or with a resistance bridge.