Two-dimensional Fermi surfaces in Kondo insulator SmB6

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Science  05 Dec 2014:
Vol. 346, Issue 6214, pp. 1208-1212
DOI: 10.1126/science.1250366


In the Kondo insulator samarium hexaboride (SmB6), strong correlation and band hybridization lead to an insulating gap and a diverging resistance at low temperature. The resistance divergence ends at about 3 kelvin, a behavior that may arise from surface conductance. We used torque magnetometry to resolve the Fermi surface topology in this material. The observed oscillation patterns reveal two Fermi surfaces on the (100) surface plane and one Fermi surface on the (101) surface plane. The measured Fermi surface cross sections scale as the inverse cosine function of the magnetic field tilt angles, which demonstrates the two-dimensional nature of the conducting electronic states of SmB6.

Teasing out the topological character

When theoretical physicists proposed the existence of an exciting class of materials called topological insulators (TIs), they had in mind a material that is electrically insulating in the bulk but conducts electricity on its surface. Experimentally discovered TIs, however, still have considerable bulk conductivity. Theoreticians then noticed that the material SmB6, which has long been known as an insulator with peculiar conduction properties, may be a TI. However, confirming that SmB6 is a TI has been an arduous process. Li et al. traced the electronic structure of SmB6 in high magnetic fields and found that it does indeed have two-dimensional surface states.

Science, this issue p. 1208

The recent development of topological insulators is a triumph of single electron band theory (18). Kondo insulators can be used to explore whether similar exotic states of matter can arise in the presence of strong electronic interactions. In these strongly correlated heavy-fermion systems (9), the hybridization between itinerant electrons and localized orbitals opens a gap and makes the material insulating. Once the sample temperature is cold enough, the electronic structure can be mapped to a state that resembles a normal topological insulator (TI) (10), with a bulk insulating state and a conductive surface state. In samarium hexaboride (SmB6), the existence of the surface state has been suggested by recent experimental observations of the surface conductance as well as a mapping of the hybridization gap (1113). We report the observation of quantum oscillations in Kondo insulator SmB6 using torque magnetometry via the de Haas–van Alphen (dHvA) effect. The observed Fermi surfaces are shown to be two-dimensional (2D) and arise from the crystalline (101) and (100) surfaces.

One major difference between SmB6 and the conventional topological insulators is the crystal structure, which for SmB6 is simple cubic (Fig. 1A). SmB6 single crystals were grown by conventional flux methods. Each sample was etched with acid to remove the leftover flux. Figure 1B shows a photo of a piece of SmB6 single crystal. Beside a flat (001) surface, there are four (101) planes.

Fig. 1 Experimental setup.

(A) The crystal structure of Kondo insulator SmB6. The B6 cluster and rare earth element Sm form a simple cubic structure. (B) Photograph of an SmB6 single crystal. Guide lines are drawn to label the crystalline plane of each surface. (C) The field dependence of the magnetic torque τ in SmB6. The data were taken at temperature T = 300 mK. Inset: sketch of the measurement setup. The sample stage is rotated to tilt magnetic field Embedded Image in the crystalline Embedded Image - Embedded Image plane. The magnetic field was applied to the sample with a tilt angle Embedded Image ~44° relative to the crystalline Embedded Image axis. The spring constant of the cantilever was determined using the sample weight. (D) The field dependence of the magnetic moment M, taken at 0.3 K and under a magnetic field as high as 45 T. The low-field linear MH dependence reflects the paramagnetism in SmB6. The vertical gray scale bar shows the M value corresponding to 0.5 μB per surface unit cell.

We use torque magnetometry to resolve the Landau Level quantization and the resulting quantum oscillations in magnetization. Electronic transport measurements have been used to detect quantum oscillations in conventional topological materials (1422). In contrast, magnetization is simply the derivative of the magnetic free energy G with respect to magnetic field H. Therefore, torque magnetometry probes the oscillations in the free energy and is sensitive to Fermi surface (FS) topology of both 3D and 2D electronic systems (2327). Torque magnetometry measures directly the anisotropy of the magnetic susceptibility of a sample (25). With the tilted magnetic field Embedded Image confined to the Embedded Image - Embedded Image plane, the torque τ of a paramagnet is shown as follows:Embedded Image (1)where µ0 is the vacuum permeability, Embedded Image is the tilt angle of Embedded Image away from the crystalline Embedded Image - Embedded Image axis, and Δχ = χa − χc is the magnetic susceptibility anisotropy. Therefore, any change of the FS topology caused by the Landau Level quantization is revealed by torque magnetometry.

In our experimental setup, an SmB6 single crystal is glued to the tip of a thin brass cantilever (Fig. 1C, inset). The magnetic torque τ is measured by tracking the capacitance change between the cantilever and a gold film underneath. An example torque τ versus magnetic field μ0H curve (Fig. 1C), taken at temperature T = 0.3 K and Embedded Image ~ 44°, is quadratic overall, reflecting the linear H dependence of the sample magnetization and the paramagnetic nature of SmB6. Large oscillations and small wiggles start to appear as the magnetic field goes beyond 5 T; for μ0H > 10 T, the fast oscillation patterns dominate.

The first question that arises is whether it is possible for the thin atomic layer of the surface state to be responsible for the observed magnitude of the magnetic torque and magnetization. The magnitude of the observed magnetic torque was recorded by the relative capacitance change, then converted to absolute values using the calibrated spring constant of the cantilevers (see supplementary materials). Figure 1D shows the field dependence of the effective magnetic moment Embedded Image in a magnetic field H as high as 45 T. The maximum oscillatory M is around 3 × 10−12 A·m2. Using the total area of all the surfaces of the SmB6 sample, including the dominating top and bottom (100) surfaces and other small edge surfaces, we can estimate that there are 7.5 × 1011 unit cells on the sample surfaces. By normalizing the magnetic moment M to the number of surface unit cells, we find the maximum change of oscillatory M is around ΔM ~ 0.4 μB per surface unit cell, where μB is the Bohr magneton. The gray scale bar in Fig. 1D marks 0.5 μB per surface unit cell for comparison.

Our measured oscillatory M is consistent with the surface carrier density measured by the Fermi pocket size. As we will demonstrate later, the measured pocket size (300 T ~ 400 T) leads to a total carrier density of ~1 × 1014/cm2, which is ~0.17 electrons per unit cell. The theoretical value of the oscillatory M is 2 μB per electron in 2D electronic systems, confirmed experimentally in (26). The carrier density implies a magnetization oscillation of 0.34 μB per surface unit cell. Our observed maximum ΔM ~ 0.4 μB per surface unit cell is consistent with this simple estimate.

Figure 2 presents the oscillation pattern of the magnetic torque. The oscillatory torque τOSC is defined by subtracting a quadratic background from the torque τ. τOSC is periodic in 1/μ0H (Fig. 2A), reflecting the quantization of the Landau Levels. The magnetic torque is measured in H up to 45 T. For metals, the oscillation frequency F is determined by the cross section area A of the Fermi surface (23): Embedded Image (2)where ħ is the reduced Planck constant, and e is the electrical charge. The fast Fourier transformation (FFT) of the oscillatory τOSC, measured at 0.3 K and Embedded Image ~ 32° (Fig. 2B) shows a number of peaks, including a small pocket α at Fα ~ 35 T, a larger pocket β at Fβ ~ 300 T, and the largest pocket γ at Fγ ~ 450 T. In the higher-frequency range, we observed a series of new features: the second harmonic of the β pocket at 2β ~ 600 T and the third harmonic of the β pocket at 3β ~ 900 T. Most notably, the other peak (marked as β′) appears at ~520 T and arises from the oscillation due to the β pocket in a neighboring (101) surface plane. This determination is based on the angular dependence of the oscillation frequencies, as discussed in detail below and in Fig. 3.

Fig. 2 Quantum oscillation pattern observed by torque magnetometry.

(A) The oscillatory magnetic torque τosc is plotted as a function of 1/μ0H. τosc is calculated by subtracting a polynomial background from the full torque signal. (B) The Fast Fourier Transform (FFT) of the oscillatory torque τosc shows three major peaks, labeled α, β, and γ. The higher harmonics of the β peak are also observed, together with the β′ peak from the neighboring surface plane. (C) The temperature dependence of the FFT amplitude of the three major peaks, normalized by their 0 K limit. The magnetic field tilt angle Embedded Image ~ 32°. Fitting the oscillatory amplitude to the LK formula (solid lines) yields the effective mass m = 0.119me for the low-frequency α pocket, m = 0.129me for the high-frequency β pocket, and m = 0.192me for the higher-frequency γ pocket. (D) At 0.3 K and Embedded Image ~ 30°, the oscillation amplitude is tracked as a function of field H, generally known as the Dingle plot. Solid lines are fits to the Dingle factor.

Fig. 3 Angular dependence of the oscillating frequencies in SmB6..

(A) Oscillation frequency Fα of the observed Fermi surfaces α is shown as a function of Embedded Image. The error bar of Fα is defined as the half-width at half-height of the peak in the FFT plot. To improve the statistics, two SmB6 samples were cooled down twice. All the oscillation frequencies are inferred from the FFT spectra at many tilt angles Embedded Image from –30° to 180° (figs. S3 to S5) (37). The data are in agreement with a fit to the solid lines (described by Embedded Image, where Embedded Image = 0°, 90°, and 180°). The lines correspond to 2D Fermi surfaces on the surface planes (100), (001), and (–1 0 0). (B) Oscillation frequency Fβ of the observed Fermi surfaces β measured in two SmB6 samples. Fβ is consistent with the expected 2D Fermi surface sitting at the four surface planes of the (101) surface family. The solid line is the fitting curve Embedded Image, where Embedded Image = –45°, 45°, 135°, and 225°. (C) The angular dependence of the oscillation frequency Fγ of the pocket γ. Solid lines are fits to Embedded Image with Embedded Image = 0°, 90°, and 180°. (D) Sketch of the Brillouin zone of SmB6. Band inversion occurs at the X points. The projection of these X points to the (101) plane shows a possible location of the Fermi surfaces in the (101) surface state.

The electronic properties of these three Fermi surfaces are revealed by tracking the temperature and field dependence of the oscillatory torque τosc (Fig. 2C). Even though an insulating gap is observed in the temperature dependence of the conductivity of SmB6 (1113), the temperature dependence of τosc is very much like that of a normal metal. In metals, the oscillating magnetic torque is well described by the Lifshitz-Kosevich (LK) formula (23). The temperature and magnetic field dependences of the oscillation amplitude are determined by the product of the thermal damping factor RT and the Dingle damping factor RD, defined as Embedded Image(3)where the effective mass m = m*me and the Dingle temperature Embedded Image . τS is the scattering rate, kB is the Boltzmann Constant, me is the bare electron mass, B = μ0H is the magnetic flux density, and Embedded Image ~14.69 T/K (23).

Fitting the temperature dependence of the normalized oscillation amplitudes to the thermal damping factor RT (Fig. 2C) yields m = 0.119me for Fermi surface α, m = 0.129me for Fermi surface β, and m = 0.192me for Fermi surface γ. Figure 2D displays the field dependence of the oscillation amplitude, normalized by the thermal damping factor RT. Fitting the curves to RD yields TD ~15.9 K for Fermi surface α, TD ~18.6 K for Fermi surface β, and TD ~ 29.5 K for Fermi surface γ. Based on the extracted oscillation frequencies, effective masses, and Dingle temperatures, we are able to characterize the observed Fermi surfaces in SmB6 (Table 1).

Table 1 Parameters in the three Fermi pockets in SmB6.

The oscillation frequency F and Fermi wavevector kF are obtained from the dHvA period. The effective mass m, Fermi velocity vF, mean free path l, and the mobility μ are inferred from the T and H dependences of the oscillation amplitude. The Landau level index plot yields the infinite field limit δ.

View this table:

The long mean free path l measured for all three pockets (Table 1) supports the idea of intrinsic metallic surface states. The mean free path is two orders of magnitude larger than the crystal lattice constant of ~0.4 nm. Such a long mean free path is incompatible with the surface/bulk conductance arising from hopping between impurities.

The low effective mass that we measured (Table 1) is quite surprising, because most of the theoretical work on the topological Kondo insulator predicts a heavy mass of the surface states (28). The observed mass in SmB6 is also much smaller than that of the divalent hexaboride compounds such as La-doped CaB6 and EuB6 (29, 30). Pockets with heavier masses may still exist but could not be observed in our experimental conditions because of the thermal and Dingle dampings. Measurements in higher magnetic field and in colder temperatures would help resolve such extra Fermi pockets.

The most important feature of the observed quantum oscillations is the two-dimensionality (2D). For a 2D planar metallic system, the magnetic field projected to the normal axis of the plane determines the Landau Level quantizations. Thus, the oscillation frequency versus tilt angle curve generally follows the inverse of a sinusoidal function (1517). We rotate the whole cantilever setup to track how the oscillation frequency F changes as a function of the tilt angle Embedded Image in a broad range of more than 180°. Using the complete FFT plots (figs. S3 to S5), we obtain the angular dependences of the oscillation frequencies of all the Fermi pockets on the (100) and (101) family surfaces, as displayed in Fig. 3, A to C.

For the dominating oscillatory pattern from pocket β, the oscillation frequency Embedded Image not only displays a large angular dispersion but also closely tracks the 2D angular dependence from the (101) surface families. There are four branches of the FβEmbedded Image patterns, as a result of the fourfold crystalline symmetry of the SmB6 cubic structure. At each Embedded Image, there are two appearances of the β family. For example, at Embedded Image ~ 30°, there is one Fβ from the (101) plane and another one from the neighboring Embedded Image plane (the latter contribution is marked as β′ in Fig. 2B). Most notably, all the Fβ points track the solid lines of the function Embedded Image, Embedded Image, Embedded Image, and Embedded Image. We observe Fβ at frequencies as high as 900 T—a more than 200% increase from the minimum value of 286 T. Such large divergence and the close tracking of the inverse cosine dependence strongly support the 2D nature of the observed β pocket on the (101) surface plane families.

In contrast, the angular dependence of the oscillation frequency of the α pocket Fα follows a different pattern. In Fig. 3A, the Fα values are plotted against the tilt angle Embedded Image of two SmB6 single crystals. The uncertainty of Fα is determined by the half width at half-height of the Fα peak in the FFT plot. Similar to Fβ, the Fα pattern has a fourfold symmetry, but the minima are located at Embedded Image = 0°, 90°, and 180°, that is, along the (100) crystalline axes. Fitting the data to the 2D form for this family, Embedded Image, Embedded Image, and Embedded Image, with Embedded Image = 30.5 T (solid lines in Fig. 3A) results in reasonable agreement. This suggests that the observed α pocket arises from a surface state on the (100) plane families. Similarly, Fig. 3C shows that the angular dependence of the γ pocket Fγ follows the functional form of Embedded Image, Embedded Image, and Embedded Image with Embedded Image ~385 T, suggesting the two-dimensionality of the γ pocket on the (100) surface plane.

Further, given the small value and large uncertainly of Fα, there is still a chance that an extremely elongated 3D ellipsoidal Fermi surface may fit the Fα versus Embedded Image dependence. Experiments with cleaner SmB6 crystals may resolve the issue.

The angular dependence of the oscillation frequency suggests that the Fermi surface β is two-dimensional and likely arises from the crystalline (101) plane. In contrast, most of the theoretical modeling focuses on the surface states in the (100) planes (31), in which the mapping of the band inversion X-point gives two Fermi pocket cross sections on the (100) surface plane. Our observed α and γ pockets may be the two predicted pockets. Recent angle resolved photoemission spectroscopy (ARPES) measurements on SmB6 revealed two Fermi pocket areas on the (100) surface Brillouin zone (3236). Our measured Fermi surface area and the mass of pocket γ are comparable to those of a small Fermi surface centered at the Γ point and measured by ARPES. For a detailed comparison of our dHvA result and the ARPES results, see (37).

Questions remain as to the origin of the observed Fermi surface in the (101) plane. In the (101) plane, there are four high symmetry points [(0, 0), (0, π/a), (Embedded Image, 0), and (Embedded Image, π/a)]. For example (Fig. 3D), projecting the bulk band X points to the (101) plane leads to a pocket at (0, π/a) and a pair of pockets at (Embedded Image,0) and (−Embedded Image,0). Based on the time-reversal symmetry, only (0, π/a) has a Dirac point, which is consistent with the experimental observation. However, the topological theory does not prohibit pairs of Dirac points at low symmetry points (37), which offers another possible origin for the observed pockets.

The other important question is whether the 2D electronic state on the surface follows the Dirac dispersion. A general test is to track the Landau Level index plots to find out the infinite field limit, i.e., the geometric Berry phase factor. Using 45 T, the quantum limit is reached for pocket α, which, in the infinite magnetic field limit, points to –0.45 ± 0.07, very close to –1/2, the geometric Berry phase contribution, similar to other 2D Dirac electronic systems such as graphene (18, 19). As the oscillation frequencies of Fermi pocket β and γ are quite close, filtering is needed to isolate the H dependence of the oscillation patterns for each Fermi pocket. This filtering may cause additional uncertainty of the oscillation phase, and the Zeeman effect and correlations may lead to some nonlinear effect of the Landau level index plots (37).

Because Al flux is used in the sample growth, the observed effective masses, oscillation frequencies, and angular dependence bear some similarity to those of pure aluminum (Al) (38). However, the oscillation periods of all the observed Fermi surfaces are symmetric about crystalline symmetry axes in the rotation plane. The observed fourfold symmetry and the behavior Fα ~ Embedded Image, Embedded Image~Embedded Image and Fγ ~ Embedded Image cannot be explained by a residual Al impurity [for more details, see (37)]. Therefore, the observed quantum oscillation pattern is an intrinsic property of single-crystalline SmB6.

We note also that our observed quantum oscillation feature is quite robust against oxidization, as the samples were always in atmosphere for storage. The ordinary surface states known to occur for vacuum clean surfaces of metal hexaborides such as LaB6 disappear under even modest oxygen exposure (37, 39).

Supplementary Materials

Materials and Methods

Supplementary Text

Figures S1 to S14

Table S1

Equation S1

References (4050)

References and Notes

  1. Additional information may be found in the supplementary materials on Science Online.
  2. See, for example, B. Nishitani, et al., Surface Science 95, 341 (1980).
  3. Acknowledgments: This study is based on work supported by the U. S. Department of Energy (DOE) under Award no. DE-SC0008110 (high-field torque magnetometry), by the startup fund and the Mcubed project at the University of Michigan (low-field magnetization characterization), and by the NSF DMR-0801253 and Univ. of California–Irvine CORCL grant MIIG-2011- 12-8 (Z. F. group, sample growth). Part of the work performed at the University of Michigan was supported by NSF grant DMR-1006500 (C.K. group, device fabrication) and by NSF grant ECCS-1307744 (L.L. group, device transport characterization). Z.X. and X.H.C. thank the China Scholarship Council for support and the National Basic Research Program of China (973 Program, grant no. 2012CB922002). B.L. thanks the NSF graduate research fellowship (F031543) for support. T.A. thanks the Nakajima Foundation Scholarship for support. The Corbino samples were fabricated at the Lurie Nanofabrication Facility (LNF), a member of the National Nanotechnology Infrastructure Network, which is supported by the NSF. The high-field experiments were performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement no. DMR-084173, by the State of Florida, and by the DOE.
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