Kondo Hybridization and the Origin of Metallic States at the (001) Surface of ${\mathrm{SmB}}_{6}$

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Kondo Hybridization and the Origin of Metallic States at the (001) Surface of ${SmB}_{6}$

E. Frantzeskakis, N. de Jong, B. Zwartsenberg, Y. K. Huang, Y. Pan, X. Zhang, J. X. Zhang, F. X. Zhang, L. H. Bao, O. Tegus, A. Varykhalov, A. de Visser, and M. S. Golden

Phys. Rev. X 3, 041024 – Published 9 December 2013

Abstract

${SmB}_{6}$ , a well-known Kondo insulator, has been proposed to be an ideal topological insulator with states of topological character located in a clean, bulk electronic gap, namely, the Kondo-hybridization gap. Since the Kondo gap arises from many-body electronic correlations, ${SmB}_{6}$ would be placed at the head of a new material class: topological Kondo insulators. Here, for the first time, we show that the $k$ -space characteristics of the Kondo-hybridization process is the key to unraveling the origin of the two types of metallic states experimentally observed by angle-resolved photoelectron spectroscopy (ARPES) in the electronic band structure of ${SmB}_{6} (001)$ . One group of these states is essentially of bulk origin and cuts the Fermi level due to the position of the chemical potential 20 meV above the lowest-lying $5 d − 4 f$ hybridization zone. The other metallic state is more enigmatic, being weak in intensity, but represents a good candidate for a topological surface state. However, before this claim can be substantiated by an unequivocal measurement of its massless dispersion relation, our data raise the bar in terms of the ARPES resolution required, as we show there to be a strong renormalization of the hybridization gaps by a factor 2–3 compared to theory, following from the knowledge of the true position of the chemical potential and a careful comparison with the predictions from recent local-density-approximation $(LDA) + Gutzwiller$ calculations. All in all, these key pieces of evidence act as triangulation markers, providing a detailed description of the electronic landscape in ${SmB}_{6}$ and pointing the way for future, ultrahigh-resolution ARPES experiments to achieve a direct measurement of the Dirac cones in the first topological Kondo insulator.

Received 12 August 2013

DOI:https://doi.org/10.1103/PhysRevX.3.041024

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

E. Frantzeskakis^1,*, N. de Jong¹, B. Zwartsenberg¹, Y. K. Huang¹, Y. Pan¹, X. Zhang², J. X. Zhang², F. X. Zhang², L. H. Bao³, O. Tegus³, A. Varykhalov⁴, A. de Visser¹, and M. S. Golden^1,†

¹Van der Waals—Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, Netherlands
²Key Laboratory of Advanced Functional Materials, Ministry of Education, College of Materials Science and Engineering, Beijing University of Technology, Beijing 100124, China
³Inner Mongolia Key Laboratory for Physics and Chemistry of Functional Materials, Inner Mongolia Normal University, Hohhot 010022, China
⁴Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein-Strasse 15, 12489 Berlin, Germany

^*e.frantzeskakis@uva.nl
^†m.s.golden@uva.nl

Popular Summary

For the past 40 years, $Sm B_{6}$ has presented the condensed-matter-physics community with a scientific puzzle. This material is thought to be an archetypal Kondo insulator, whose electrical resistivity should diverge as its temperature approaches zero. $Sm B_{6}$ , however, defies this expectation. While its electrical resistivity indeed rises sharply as it is cooled below 50 K, the rise ends at a finite value of resistivity at lower temperatures. Recently, theoreticians have proposed a new solution for this old mystery: that the Kondo behavior coexists with the existence of additional conducting states hosted by the surfaces of the $Sm B_{6}$ crystal that possess special topological properties. If these theoretical predictions can be verified experimentally, $Sm B_{6}$ will become the first in a new material class: topological Kondo insulators. Experiments are still scarce, however. In this experimental paper, we add significant weight to the experimental side of the research effort by providing detailed measurements on the electronic structure of $Sm B_{6}$ and identifying the origins of different electronic states relevant to the puzzling behavior of $Sm B_{6}$ .

In a Kondo insulator, the interaction between localized and itinerant electrons, known as the Kondo-hybridization interaction, is responsible for the insulating behavior. This interaction is reflected in the material’s electronic band structure. Our experiments employ angle-resolved photoelectron spectroscopy (ARPES), the most direct tool to access the electronic band structure of complex materials and to shed light on the topological character of electronic states. The technique is based on the photoelectric effect: a photon-in–electron-out experiment where measurements of the kinetic energy and the emission angle of the outgoing photoelectrons are translated via conservation laws into their energy vs momentum relation within the material, namely, the electronic band structure. Using high-quality, floating-zone-grown $Sm B_{6}$ crystals, we have shown for the first time that the Kondo-hybridization interaction is the key to unraveling the origins of two observed types of metallic states in the material’s electronic structure. One type of these states is related to the bulk electronic structure, while the other indeed represents a good candidate for topological surface states.

Our findings provide an important part of the puzzle that has been missing until now: a state-of-the-art, direct measurement of the electronic landscape in this material. Such a detailed map of the relevant electronic states points the way forward for future work on this first candidate topological Kondo insulator.

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Vol. 3, Iss. 4 — October - December 2013

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Condensed Matter Physics

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Figure 1
${SmB}_{6}$ transport, surface quality, and ARPES data overview. (a) Temperature dependence of the resistivity of a floating-zone, optical-furnace-grown ${SmB}_{6}$ crystal. There is a sharp increase below 50 K that results in a very high resistivity ratio of $10^{5}$ between the saturation regime and room temperature. The inset shows the Hall resistivity, which is linear, yielding an $n$ -type carrier concentration as indicated (low $T$ regime). (b) Very sharp LEED spots reflect the simple ( $1 \times 1$ ) surface structure—free of reconstructions—and attest to the long-range order of the cleavage surface. The LEED primary beam energy is 97 eV. (c) Sketch of the 3D Brillouin zone of ${SmB}_{6}$ and its projection onto the (001) plane. The colored planes denote the relevant cuts through the zone that would be made for photon energies in ARPES corresponding to a $k_{z}$ value of an even number of $π / a$ (green plane), an odd number of $π / a$ (purple plane), and the surface projection (red plane). (d) Three-dimensional ARPES data block displaying $I (E_{B}, k_{x}, k_{y})$ . Two types of bands are seen with the strongly dispersing (flat) bands originating from Sm $5 d$ ( $4 f$ ) states, as indicated. The Sm $5 d$ -related states give rise to elliptical contours at higher binding energies (the cutaway in the lower-right corner of the data block showing contours for a binding energy of 0.8 eV), which then evolve to give elliptical Fermi-surface contours around the $\bar{X}$ points. These states are discussed in the text as the $X$ states and are argued to be essentially bulk-derived bands, well described by bulk, ab initio theoretical approaches. Where the flat $4 f$ bands intersect the $5 d$ -dominated $X$ states, complex band structures result because of the hybridization effects between $d$ and $f$ states of the equal symmetry, as will be discussed in detail in the context of Fig. 2. The photon energy for the data block is $h ν = 70 eV$ .Reuse & Permissions
Figure 2
$k$ -space fingerprint of $5 d − 4 f$ hybridization. (a)–(e) The elliptical contours from Sm $5 d$ -dominated states centered at $\bar{X}$ increase in size as the binding energy decreases. The dashed yellow square in (e) shows the surface Brillouin zone (BZ). (f)–(h) When the $d$ and $f$ states interact strongly—in hybridization zones—strong spectral weight redistribution occurs, giving significant intensity at $\bar{Γ}$ . In (i), the constant-energy contours within the lowest-lying hybridization zone are indicated with yellow lines. (j) These same lines (now in blue) are also an excellent fit to the energy contours of the deeper-lying hybridization zone. (k) Exemplary hybridized (solid blue) and nonhybridized (dashed red) contours are overlaid. The broad, green-shaded lines highlight the resulting fourfold-symmetric $k$ -space fingerprint of the $5 d − 4 f$ hybridization, characterized by a gapping out of the ARPES intensity underlying parts of the dashed red ellipses. (l)–(o) Zooms of data covering the $k$ -space area indicated with a dashed green box in (a). In (n), dashed red ellipses are drawn to match the data at 430-meV binding energy and the same ellipses are copied onto the other topmost panels, reiterating the trend that is also evident in (a)–(e) and in Fig. 1d that the $X$ -state ellipses grow in size as the binding energy is increased. (l) Just after a hybridzation zone, the $X$ -state ellipses shrink again, which leads to a $k$ discontinuity for these states. This second characteristic sign of strong $5 d − 4 f$ interaction is also shown in Fig. 3 and discussed in detail in the context of Fig. 4. The ARPES data are acquired with $h ν = 70 eV$ .Reuse & Permissions
Figure 3
Energy dispersion and dichroic signature of the $X$ state and identification of an additional $Γ$ -state feature. (a),(b) Data recorded for $k_{z} = 6 π / a$ ( $h ν = 70 eV$ ) for the $k$ -space cuts along the $\bar{M X M}$ and $\bar{X Γ X}$ high-symmetry directions as indicated by the color-coded arrows underlying the Fermi-surface map in between the panels (integrated over $\pm 5 meV$ ). In each case, a zoom of the dashed region is included, as is a momentum-distribution line cut in green for $E = E_{F}$ . In (a), the zoom panel clearly shows the $X$ states crossing $E_{F}$ , and the same states seen as the outer two features in (b) are joined by an enigmatic, third feature centered on the zone center dubbed the $Γ$ state. (c) Data along the same direction as that in (b) but now recorded for $k_{z} = 4.5 π / a$ ( $h ν = 34 eV$ ). It is evident that these Sm $5 d$ -dominated states now have an inverted dispersion with respect to (b). Plus, both the Fermi surface and the 350-meV energy contour (both integrated over $\pm 10 meV$ ) possess significant intensity around the $\bar{Γ}$ point, rather than centered on the $\bar{X}$ point. A schematic of the surface Brillouin zone is added between (c) and (d), showing the color-coded $k$ -space coverage of the constant-energy surfaces and the $k$ -space trajectories, giving the $E (k)$ images as dashed lines. Finally, (d) illustrates the dichroic signature of the Sm $5 d$ -related bands, shown for different binding-energy regimes and $k_{z}$ values. Plotted is the difference in ARPES intensity between data recorded using left or right circular light. It is clear that this dichroic signal has the same sign at all energies below 1 eV. In particular, the fact that the same dichroic response is seen for high binding energy (left panel, $h ν = 34 eV$ , $k_{z} = 4.5 π / a$ ) and for the 20 meV closest to the Fermi level (right panel, $h ν = 21 eV$ , $k_{z} = 2 π / a$ ) shows that the states that give rise to the elliptical Fermi surfaces around the $\bar{X}$ points are no different than those that make up the large-scale $5 d$ -related dispersive features pointed out in Fig. 1d.Reuse & Permissions
Figure 4
$k$ discontinuities, position of the chemical potential, and hybridization renormalization from comparison with bulk, ab initio theory. (a),(b) Solid lines in the left (right) parts illustrate hybridized (nonhybridized) $d$ and $f$ bands along two high-symmetry directions in $k$ space. The lines are adaptations of the bulk $LDA + Gutzwiller$ theory data of Ref. 18. The hybridized cases are overlaid as thin dashed lines in the right parts to aid identification of the hybridization gapping. (c1),(d) The same theory curves are now overlaid on the experimental band structure along $\bar{X Γ X}$ and $\bar{M X M}$ ; $k_{z} = 6 π / a$ ( $h ν = 70 eV$ ). A very good match is obtained when the total bandwidth of the theoretical $4 f$ multiplet is renormalized by a factor of one-third compared to the original theory data [18], with a simultaneous rigid shift of the theoretical data 20 meV to higher binding energy, carried out to account for a difference in chemical potential between theory and experiment. This renormalization means the $5 d − 4 f$ hybridization gaps are only 5–10 meV. From (c1) and (d), it is clear that the $X$ states, including the $k$ discontinuity that is evident on comparing below and above the hybridization zone, are described very well by the energy-scaled bulk theory, removing them from the suspect lineup as far as topological surface states go. In contrast, the $Γ$ states, which are highlighted in the zoomed portion of the $\bar{X Γ X}$ data in (c2), are missing in the bulk theory. The absence of $Γ$ states in bulk calculations is a signal of their promise in terms of identifying potential topological surface states in future ultrahigh-resolution ARPES experiments. (e) A zoom of the theory around $\bar{Γ}$ , together with a sketch of a massless Dirac cone centered at the $\bar{Γ}$ point as a guide for future experiments.Reuse & Permissions

Physical Review X

Kondo Hybridization and the Origin of Metallic States at the (001) Surface of SmB6

E. Frantzeskakis, N. de Jong, B. Zwartsenberg, Y. K. Huang, Y. Pan, X. Zhang, J. X. Zhang, F. X. Zhang, L. H. Bao, O. Tegus, A. Varykhalov, A. de Visser, and M. S. Golden

Phys. Rev. X 3, 041024 – Published 9 December 2013

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Kondo Hybridization and the Origin of Metallic States at the (001) Surface of ${SmB}_{6}$