Hybridization, Inter-Ion Correlation, and Surface States in the Kondo Insulator ${\mathrm{SmB}}_6$

As an exemplary Kondo insulator, ${\mathrm{SmB}}_6$ has been studied for several decades. However, direct evidence for the development of the Kondo coherent state and for the evolution of the electronic structure in the material has not been obtained due to the compound’s rather complicated electronic and thermal transport behavior. Recently, these open questions have attracted increasing attention as the emergence of a time-reversal-invariant topological surface state in the Kondo insulator has been suggested. Here, we use point-contact spectroscopy to reveal the temperature dependence of the electronic states in ${\mathrm{SmB}}_6$. We demonstrate that ${\mathrm{SmB}}_6$ is a model Kondo insulator: Below 100 K, the conductance spectra reflect the Kondo hybridization of Sm ions, but, below about $30\mathrm{K}$, signatures of inter-ion correlation effects clearly emerge. Moreover, we find evidence that the low-temperature insulating state of this exemplary Kondo-lattice compound harbors conduction states on the surface, in support of predictions of nontrivial topology in Kondo insulators.

Popular Summary

The electrical behavior of a compound discovered in 1969, samarium hexaboride (${\mathrm{SmB}}_6$), has been a 40-year mystery in condensed matter physics. The material was thought to be a Kondo lattice, an insulator in which screening of its localized spins associated with the ion lattice by the itinerant electrons was expected to lead to a temperature-dependent evolution of its electronic structure and “freezing” of the electrons at low temperatures. Yet, a finite electrical conductance was observed at very low temperatures. A theoretical solution to this mystery was proposed two years ago in the form of the novel concept of topological Kondo insulators, materials or physical states where nontrivial surface electrical conduction coexists with the Kondo effect. However, clear evidence for the evolution of the material’s electronic structure with temperature has not been obtained, nor has there been experimental confirmation of the predicted surface conduction. In this paper, we report experimental results that not only show for the first time the systematic Kondo-type evolution with temperature of the electronic structure of ${\mathrm{SmB}}_6$ but also support the prediction of ${\mathrm{SmB}}_6$ as a topological Kondo insulator.

Our experimental results are obtained by the use of point-contact spectroscopy. Soft point-contact junctions between ${\mathrm{SmB}}_6$ and silver are fabricated on a ${\mathrm{SmB}}_6$ single crystal. Conductance through such a junction as the function of the bias voltage across it—a conductance spectrum—probes directly the electronic structure of the bulk crystal. An asymmetric conductance spectrum observed in our junctions is a clear indication of the existence of the expected Kondo screening effect in ${\mathrm{SmB}}_6$. An analysis of the spectra confirms that, by its electronic structure, the ${\mathrm{SmB}}_6$ single crystal is indeed a bulk insulator, a conclusion that strongly suggests that the finite low-temperature conductance observed earlier should be attributed to the conduction of surface states, in other words, that ${\mathrm{SmB}}_6$ is a topological Kondo insulator.

The experimental findings reported here provide a simple but clear example for the existing understanding of the correlation between the electronic structure and the screening effect in Kondo lattices. In addition, the implication of the presence of conducting surface states in ${\mathrm{SmB}}_6$ should also add new incentive to the study of topological insulators and their applications.

Figure 1

(a) A schematic view for a $\mathrm{Ag}\mathrm{\text{−}}{\mathrm{Sm}}_6$ junction. A Fano-like conductance spectrum is expected due to the interference of the two electron paths, as indicated by the two green arrows: One leads directly into the itinerant electron states (right side), and thus should reflect the DOS. The other leads indirectly into the electron sea through a spin-singlet Kondo state (left side). Inset: The crystal structure of the ${\mathrm{SmB}}_6$ single crystal. The Sm ions form a simple cubic structure. (b) A schematic view of the DOS for the ground state of ${\mathrm{SmB}}_6$. In point-contact-spectroscopy measurement, quasiparticle broadening should be taken into account, as indicated by the green dashed line. The blue solid curve is the expected spectrum with Fano resonance. $\mu$ is the chemical potential while $\lambda$ is the renormalized $f$ level. (c) The contour map of the normalized conductance in the bias voltage-temperature ($V\mathrm{\text{−}}T$) plane. All conductance data are normalized by the spectrum conductance obtained at 150 K. As indicated by the color legend, red and blue colors represent values below 0.7 and above 1.1, respectively. At temperatures above about $90\mathrm{K}$, the conductance spectroscopy of the junction is highly symmetric, with negligible temperature dependence due to the weak interaction between electrons and local moments. From 90 K to 30 K, the zero-bias conductance is gradually suppressed, while an asymmetric conductance line shape develops. In this temperature range, the conductance spectrum can be described by the single-ion resonance model. Below about $30\mathrm{K}$, the conductance spectra are clearly asymmetric with a dip at zero bias and a peak at 20 mV. As discussed in the main text, this strongly asymmetric conductance line shape provides direct evidence for the inter-ion correlations in the Kondo lattice system.

Figure 2

(a) The observed conductance spectra with the best fits to the classical Fano-resonance line shape [Eq. (1) and dashed lines] and to the tunneling model for Kondo lattices [Eq. (2) and solid lines], respectively. For clarity, only the spectrum and the simulations for 2 K are plotted with the actual value, while other curves are vertically shifted. (b), (c), and (d) show the temperature dependence of $\lambda$ (renormalized $f$ level), $\Delta$ (Kondo resonance width), and $\Phi$ (half of the direct gap size), respectively, used in the Kondo-lattice-tunneling simulations. The dashed line in (b) and (c) suggests that both the renormalized $f$ level and the Kondo gap width are nearly temperature independent in the entire temperature range below 100 K. (e) The temperature dependence of the hybridization gap extracted from the hybridization amplitude and the bandwidth $D$, based on the Kondo-lattice-tunneling model. The dashed line in (e) represents a smooth trend of the size of the hybridization gap as the temperature changes. The extrapolation of the curve suggests that the hybridization gap vanishes at a temperature of around 100 K.

Figure 3

(a) Temperature dependence of electrical resistivity of a ${\mathrm{SmB}}_6$ single crystal (blue data points) and the resistance of a $\mathrm{Ag}\mathrm{\text{−}}{\mathrm{SmB}}_6$ junction (pink). Compared to the 4 order-of-magnitude increase in the resistivity and a significant suppression below 4 K for single ${\mathrm{SmB}}_6$ crystals, the temperature dependence of the $\mathrm{Ag}\mathrm{\text{−}}{\mathrm{SmB}}_6$ junction resistance exhibits a rather different behavior with an overall increase of about 2 in the measured temperature range and a low-temperature saturation below about $10\mathrm{K}$. The dashed lines indicate that the junction resistance holds a constant value at temperatures above about $10\mathrm{K}$, while the resistivity of bulk single crystals shows an activation behavior in the same temperature range. (b) Schematic view of the DOS in ${\mathrm{SmB}}_6$ without the presence of a surface state. It is expected that the coherent transport of the bulk in-gap bound states (green area) leads to the suppression in resistivity at temperatures below 4 K. (c) The DOS in ${\mathrm{SmB}}_6$ with the emergence of a surface state at low temperatures. The finite residue resistivity of the material is due to the presence of a conducting surface state. Both topological surface states (orange) and conventional two-dimensional electron gas (blue) could lead to such a metallic 2D conduction. (d) Simulation of the measured resistivity in the scenario that a metallic-like surface state dominates the conductance at low temperatures. The overall conductivity is the sum of two conduction channels: the surface conduction (green) and the intrinsic bulk conduction (blue). The measured resistivity approaches the saturation point at a temperature of around 4 K.