Quantum oscillations of electrical resistivity in an insulator

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Science  05 Oct 2018:
Vol. 362, Issue 6410, pp. 65-69
DOI: 10.1126/science.aap9607

Insulator or a metal?

When a metal is cooled to low temperatures and placed in an external magnetic field, its resistivity may oscillate as the magnitude of the field is varied. Seeing these so-called quantum oscillations in an insulating material would be very unusual. Xiang et al. report such findings in the insulator ytterbium dodecaboride (YbB12) (see the Perspective by Ong). In addition to oscillations in resistivity, the authors observed oscillations in the magnetic torque. The results present a challenge to theories that aim to explain the insulating state of YbB12.

Science, this issue p. 65; see also p. 32


In metals, orbital motions of conduction electrons on the Fermi surface are quantized in magnetic fields, which is manifested by quantum oscillations in electrical resistivity. This Landau quantization is generally absent in insulators. Here, we report a notable exception in an insulator—ytterbium dodecaboride (YbB12). The resistivity of YbB12, which is of a much larger magnitude than the resistivity in metals, exhibits distinct quantum oscillations. These unconventional oscillations arise from the insulating bulk, even though the temperature dependence of the oscillation amplitude follows the conventional Fermi liquid theory of metals with a large effective mass. Quantum oscillations in the magnetic torque are also observed, albeit with a lighter effective mass.

In Kondo insulators, the hybridization between itinerant and localized electrons opens an insulating gap, and consequently, their resistivity diverges at low temperature (1, 2). Recently, there has been a lively debate about the nature of the ground state of Kondo insulator samarium hexaboride (SmB6) in intense magnetic fields. Although mixed-valence SmB6 is a good insulator (its resistance increases by five orders of magnitude when cooled down to 300 mK from room temperature), Landau-level (LL) quantization still occurs, and clear quantum oscillations are observed in its magnetization [the de Haas–van Alphen (dHvA) effect] (3, 4). The origin and interpretation of the dHvA oscillations in SmB6 have been highly controversial, owing to a number of peculiar features. First, the oscillations are observed only in magnetization and not in electrical resistivity [the Shubnikov–de Haas (SdH) effect]. Second, unlike the heavy carriers revealed by the thermoelectric studies (5), oscillations appear to arise from quasiparticles with a very light effective mass (m << me, where me is the free electron mass) (3). Third, in floating zone–grown SmB6 samples, the dHvA signal exhibits a striking deviation from the standard Lifshitz-Kosevich (LK) formula in Fermi-liquid theory (4). These observations point to either a topologically protected surface state (3) or the presence of an unconventional Fermi surface in an insulator (4). A number of intriguing physical origins have been proposed, including exciton-based magnetic breakdown (6, 7), Majorana-type charge-neutral Fermi surfaces (8), a failed superconducting ground state (9), and spinon Fermi surfaces (10, 11). A key to solving the most fundamental problem, the existence of “a Fermi surface in an insulator,” lies in clarifying whether quantum oscillations—in particular, in charge transport—are observable in another insulating system.

Here, we present quantum oscillation studies of ytterbium dodecaboride (YbB12), another cubic-structured rare-earth intermetallic compound. YbB12 has long been known as a mixed-valence Kondo insulator (1214). It behaves as a monovalent metal with localized magnetic moments at room temperature, whereas a nonmagnetic insulating ground state develops at low temperatures. The opening of a narrow energy gap of 10 to 15 meV at the Fermi level has been confirmed with many experiments (1517) and is attributed to the hybridization between the itinerant 5d and the localized 4f electrons. The mean valence of Yb ions in YbB12 is +2.9 (18), close to 4f13 (+3) configuration. Therefore, the f electrons are mostly localized, and the crystalline electric field (CEF) ground state is well defined. By contrast, because the mean valence of Sm ions in SmB6 is +2.6 (19), f electrons are more itinerant, and the CEF scheme is not well defined. Therefore, the electronic structure of YbB12 is much simpler than that of SmB6. Furthermore, early calculations in YbB12 predicted the existence of topological surface states owing to mirror-symmetry protection (20).

YbB12 single crystals are grown in a floating-zone furnace [(21), section 1]. Using the experimental setup shown in Fig. 1A, magnetic torque and magnetoresistance (MR) are measured up to 45 T simultaneously [(21), section 2]. The temperature dependence of the resistivity (Fig. 1B) confirms an increase of five orders of magnitude from room temperature to 50 mK. The resistivity has a weak temperature dependence below 2.2 K, resembling the resistive “plateau” well known in SmB6 at temperature (T) < 3.5 K (22). This “plateau” is an indication of the existence of extended in-gap states. Fitting with the thermal activation model of resistivity, ρ(T) = ρ0exp(Δ/2kBT), where kB is the Boltzmann constant, reveals a two-gap feature with the gap width 12.5 meV (20 K < T < 40 K) and 4.7 meV (6 K < T < 12.5 K), respectively (Fig. 1B, inset), which is consistent with previous transport results (23). Upon applying the magnetic field, the negative slope of the ρ(T) curve is preserved up to 45 T, with no hints of metallic behavior (Fig. 1C), indicating that the ground state is still gapped (fig. S5).

Fig. 1 Electrical transport and magnetic torque measurements in YbB12.

(A) Sketch of the experimental setup (21) and the definition of tilt angle ϕ with respect to the magnetic field. (Inset) Photograph of a YbB12 single crystal (sample N2) mounted onto a cantilever beam magnetometer, with four gold wires attached to the crystallographic (001) surface for the transport measurement. (B) Resistivity of two YbB12 single crystals plotted as a function of temperature. (Inset) The Arrhenius plot, lnρ versus 1/T. According to the thermal activation model, the slope of the Arrhenius plot equals Δ/2kB, where Δ is the bandgap width and kB is the Boltzmann constant. Linear fitting in two different temperature ranges, 20 < T < 40 K and 6 < T < 2.5 K, yields two characteristic gap widths, 12.5 and 4.7 meV, respectively. (C) Resistivity of YbB12 sample N2 under different magnetic fields from 0 to 45 T, plotted against temperature. Open and solid symbols are data taken in 3He cryostat at Embedded Image and in dilution fridge at Embedded Image, respectively. Solid lines are guides to the eye. (D) Magnetic torque in YbB12 measured at T = 350 mK and at two different tile angles, Embedded Image and Embedded Image. Both exhibit strong quantum oscillations under high magnetic fields. The amplitude of the oscillatory part of magnetic torque at Embedded Image is ~6 × 10−8 N · m at the highest field, corresponding to an effective transverse magnetization of ~1.4 × 10−9 A · m2 (1.51 × 1014 μB). (Inset) FFT on the magnetic torque signal with Embedded Image reveals a major peak at F = 720 T and its harmonics.

The field dependence of the magnetic torque in the insulating state of YbB12 is shown in Fig. 1D. We observed a step increase at 20 T followed by a decrease at 28 T. These features are weak metamagnetic transitions and/or crossovers that could potentially be related to the predicted field-induced staggered magnetism in Kondo insulators (24). Above ~37.5 T, the dHvA oscillations are clearly resolved (Fig. 1D). The dHvA oscillations appear well below the insulator-metal (I-M) transition field in our YbB12 samples, which was determined by means of pulsed field studies to be 45.3 to 47.0 T [(21), sections 4 and 5]. Fast Fourier transform (FFT) on the Embedded Image torque curve gives a dHvA frequency of F = 720 T (Fig. 1D, inset).

In Fig. 2A, the MR data at Embedded Image is plotted between 11.5 and 45 T. Given the zero-field resistivity ρ(0) = 4.67 ohm · cm of this sample at 350 mK, a significant negative MR {[ρ(H) – ρ(0)]/ρ(0)} of –95.9% is achieved at 45 T (a detailed angular dependence of MR is shown in fig. S3). The negative MR is a hallmark of Kondo insulators that results from field suppression of the hybridization gap (2528). The negative MR in YbB12 is much larger than that in SmB6 (29, 30). This is probably a consequence of the larger effective Landé g-factor in YbB12, which increases the influence of the magnetic field (17, 28). As the magnetic field (H) increases, MR displays wiggle-like features at around 16 and 28 T. These wiggles do not arise from quantum oscillations because their positions are obviously temperature dependent (fig. S7); instead, they are likely to be field-induced transitions or crossovers. The feature at 28 T is probably linked to the kink feature in the magnetic torque [(21), section 6].

Fig. 2 Resistivity and electrical oscillations in intense magnetic fields in YbB12.

(A) Resistivity of sample N2 as a function of magnetic field measured up to 45 T taken at T = 350 mK at a tilt angle Embedded Image. Quantum oscillations are clearly observed at high magnetic field beyond 40.8 T. The extrema are marked by arrows. (Top inset) First magnetic field derivative of resistance has two prominent valleys and one peak. (Bottom inset) FFT on the magnetoresistance data presented between μ0H = 40.5 T and 45 T. A single peak frequency of F = 913 T is resolved. (B) Field derivative of sample resistivity at four different tilt angles. Dotted lines mark the approximately evenly spaced valleys of SdH oscillation. Three to four periods in total can be observed, depending on the field orientation. The oscillation pattern at ϕ is repeated at Embedded Image, which is consistent with the cubic symmetry of crystal structure.

The most striking result in Fig. 2A is the oscillations that appear in the MR under strong magnetic fields. From 40.8 up to 45 T, two valleys and one peak in total can be clearly observed (Fig. 2A, top inset). The FFT on the MR oscillations in this field regime yields a clear frequency peak at F = 913 T (Fig. 2A, bottom inset). With the magnetic field direction close to the crystal axes, up to four oscillation periods can be seen (Fig. 2B). The overall SdH patterns are almost identical for Embedded Image and Embedded Image, whereas there is a small valley position shift between Embedded Image and Embedded Image. This suggests an axis of symmetry along [101] direction for the SdH oscillations, which is consistent with the cubic structure of YbB12 crystal (fig. S10). The valleys in dρ/dH in Fig. 2B being approximatly uniformly spaced as a function of 1/H provides strong evidence that the SdH oscillations have a single dominating frequency. These overall patterns of the SdH oscillations, as well as those of the dHvA oscillations, are well reproduced between different YbB12 samples [(21), section 3].

The observation of the SdH oscillations is reinforced by the temperature dependence of oscillation amplitudes. The evolution of the high-field wiggle features in both MR and torque data shows the typical behavior of quantum oscillations, with temperature-independent positions of the dominant peaks and valleys within the uncertainty and an attenuated amplitude from base temperature up to 1.5 K (figs. S7 and S8). The temperature-dependent amplitudes of normalized oscillatory torque (Fig. 3A) and oscillatory MR (Fig. 3, B and C) are fitted by using the conventional LK formula (31). The fittings are reasonably good down to 60 mK, indicating that the LK expression, based on the Fermi liquid framework, appears to be valid in the Kondo insulator YbB12. The agreement with the LK description confirms that the features we resolve are quantum oscillations rather than successive field-induced Lifshitz transitions.

Fig. 3 Effective mass fitting on dHvA and SdH oscillations in YbB12.

(A) Normalized dHvA oscillation amplitude Δτ as a function of temperature. Open circles and solid circles are the data taken in 3He cryostat at Embedded Image and in portable dilution fridge at Embedded Image, respectively. The LK model fitting by using parameters m* = 6.6 me, B = 40.4 T is indicated with a solid line based on the 3He cryostat data. In comparison, the LK fitting based on the portable dilution fridge data gives m* = 6.7 me, B = 41.9 T, which is indicated with the dashed line. (B and C) Temperature dependence of normalized SdH amplitude Δρ/ρ0. (B) is taken at the same tilt angle as in (A), and (C) is taken at a tilt angle close to that in Fig. 2A. Open and solid symbols are data measured in portable 3He cryostat and dilution fridge, respectively. Here, ρ0 is the zero-field resistivity at corresponding temperatures. Solid lines are fittings based on the LK formula, with parameters shown in each panel. According to the fittings, SdH effective mass exhibits a weak anisotropy between Embedded Image [(B), m* = 14.6 me] and Embedded Image and [(C), m* = 15.0 me]. In (A) to (C), the quantum oscillation amplitudes (raw data are shown in fig. S8) are taken as the averaged value of an adjacent peak and valley obtained after subtracting a polynomial background from the raw data, and the effective magnetic field for each data set is an inverse average of the peak and valley positions. [Details of the background subtraction and the determination of the B parameters in the LK fittings are presented in (21), section 6.] The error bars on the temperature are estimated based on the MR effect on the ruthenium oxide thermometer above 11.4 T.

The SdH oscillations are much more suppressed at higher temperatures as compared with dHvA oscillations at the same angle (Fig. 3, A and B), revealing a heavier effective mass in the electrical transport channel. The effective masses of the quasiparticles estimated from the dHvA and SdH oscillations at the same tilt angle are ~6.6 me and 14.6 me, respectively. Therefore, it is unlikely that both types of oscillations originate from the same band. The SdH and dHvA frequencies have different angle dependences (Fig. 4). The dHvA frequencies F can be tracked with a two-dimensional (2D) Fermi surface model (Embedded Image) with the in-plane cross section area A[001] = 6.67 nm–2 (Fig. 4, solid line). Given the lack of dHvA oscillations observed above Embedded Image (fig. S9), this inverse sinusoidal dependence can be explained by either a 2D Fermi cylinder or a heavily elongated 3D Fermi pocket. On the other hand, the angular dependence of the SdH frequencies displays a clearly nonmonotonic behavior: A frequency maximum appears at θ ~ 15° to 20° from the crystal axes, resulting in an “M”-shape with a local dip at Embedded Image and a fast decrease in the frequency beyond the maximum. The 2D Fermi surface model apparently cannot describe this behavior. Our attempt to model the SdH frequencies (Fig. 4, dashed lines, and figs. S1 and S10) points to hyperbolic “neck” orbits [(21), section 8].

Fig. 4 Angular dependence of quantum oscillation frequencies.

The frequencies of the quantum oscillations appearing at high field in YbB12. Magnetic field H is rotated in a plane perpendicular to the current direction, and the effective tilt angle θ is defined as the angle between H and the equivalent crystal axes [001]/[100] in a cubic structure. Circles are dHvA frequencies obtained from FFT (green) and the slope of a linear fitting of the LL index versus inverse magnetic field (blue). The solid line is a calculation by using 2D Fermi surface model: F = F0/cosθ with F0 = 700 T. Up triangles (red) and down triangles (brown) are SdH frequencies acquired from the FFT and from the linear fitting of the LL index plot, respectively. The dashed line is the fitting by a hyperboloid model representing a Fermi surface neck region with the principal axis along the [001] direction, whereas the dash-dot lines are simulation for the high-angle data points by using an oblate spheroid model with the principal axis along [001] direction. The detailed parameters in these models are provided in (21), section 8. The error bars come from the difference between different sampling windows for the FFT results and from the linear fitting error for the LL analysis.

The amplitude of the quantum oscillations is determined by a combination of temperature, band curvature, Dingle, and spin factors. Assuming Dingle and spin factors do not depend on the band, the oscillation amplitude of the light band with the 2D character should be much larger than that of the heavy band with the 3D character. Because the observed SdH oscillations arise from a relatively heavy band (m* ≈ 15 me), it is highly unusual that the oscillations observed in the dHvA effect, which detects the orbits from a much lighter band (m* ≈ 6.6 me), are not present in the SdH effect. We discuss in section 11 of (21) the possible origins of the dHvA oscillations, based on either surface state (32), charge-neutral Fermi surface (10), or a minority phase. Further, on the basis of the symmetry analysis and other tests in a even stronger magnetic field, the bulk SdH oscillations do not arise from metallic impurity phases [(21), sections 7, 9, and 10] or from a minority portion of sample, which has a lower I-M transition field [(21), sections 4 and 5].

There are several notable features in the SdH oscillations that we observed. The effective masses obtained from LK fittings are large, which is in agreement with the nature of a Kondo insulator in which strong electron correlations make the quasiparticles heavy. Even in the insulating state, a finite electronic specific heat coefficient γ is observed in YbB12 (33). Assuming a spherical Fermi surface with kF ~ 0.156 Å–1 (F = 800 T) and effective mass m* = 15 me obtained from the SdH oscillations, the value of γ is calculated to be 7.6 mJ/mol K2, comparable with the observed value of γ ~ 8 mJ/mol K2 at 39 T (33). The background resistivity ρ still has a magnitude of more than 100 milliohm · cm above 40 T, which is well beyond that of normal metals (34). If we estimate the mean free path l by considering a spherical Fermi surface with m* = 15 me and setting ρ = 0.4 ohm · cm (Fig. 2A), we will obtain an unphysically short mean free path, Embedded Image nm.

Unconventional quantum oscillations have been proposed in insulators with hybridization gaps (611, 35, 36). However, our discovery of quantum oscillations in charge transport cannot be effectively interpreted by the theories that either invoke charge-neutral quasiparticles (79) or associate the oscillations with the grand canonical potential (6). All these theories of exotic quantum oscillations predict that in a gapped system, the temperature dependence of the oscillation amplitude deviates from the conventional LK-formula at certain elevated temperatures. By contrast, our observations show that in YbB12, the LK formula works from T = 1.5 K (0.13 meV) down to T = 60 mK (0.0052 meV), a range of energy lower than both the hybridization gap width and the cyclotron energy of charge carriers at a field of Embedded Image T [(21), section 11]. Recent theories suggest that an emergent neutral Fermi sea can exist in a mixed-valence gapped system and exhibit both dHvA and SdH oscillations (10, 11), which may shed light on the exotic SdH oscillations we resolved, although a deviation from LK formula is still required in this scenario. A proper theory is yet to be established to describe the quantum oscillations observed in YbB12 under a high magnetic field.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S13

Table S1

References (3750)

Data File S1

References and Notes

  1. Materials, methods, and additional data are available as supplementary materials.
Acknowledgments: We thank R. Peters, H. Shishido, Kai Sun, T. Senthil, Patrick Lee, and Y. Tokiwa for valuable discussions. We are grateful for the assistance of T. Murphy, H. Baek, G. Jones, and J.-H. Park of the National High Magnetic Field Laboratory (NHMFL). Funding: This work is mainly supported by the National Science Foundation under award DMR-1707620 (high field magnetization and resistivity measurements), by the Office of Naval Research through the Young Investigator Prize under award N00014-15-1-2382 (sample structure and low field electrical transport characterizations), by Grants-in-Aid for Scientific Research (KAKENHI) (25220710, 15H02106, 15H03688, 16K05460, 16K13837 18H01180, and 18H05227) and on Innovative Areas Topological Material Science (15H05852) from the Japan Society for the Promotion of Science (JSPS). The synchrotron radiation x-ray powder diffraction measurements were carried out at BL02B2/SPring-8 in Japan (proposal 2017A1856). Some results are obtained with equipment supported by the National Science Foundation Major Research Instrumentation award under DMR-1428226 (the equipment of the thermodynamic and electrical transport characterizations). The development of the torque magnetometry technique in intense magnetic fields was supported by the U.S. Department of Energy (DOE) under award DE-SC0008110. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement DMR-1644779 and DMR-1157490, the DOE, and the state of Florida. J.S. acknowledges support from the DOE BES Program “Science in 100 T.” The experiment in NHMFL is funded in part by a QuantEmX grant from the Institute for Complex Adaptive Matter and the Gordon and Betty Moore Foundation through grant GBMF5305 to Z.X., T.A., L.C., C.T., and L.L. T.A. thanks the Nakajima Foundation for support. B.L. acknowledges support by the National Science Foundation Graduate Research Fellowship under grant No. F031543 and the National Science Foundation East Asia and Pacific Summer Institute Fellowship award 1614138. Author contributions: L.L. and Y.M. designed the experiments. Z.X. performed the experiments on the quantum oscillation measurements and analyzed the data on YbB12 with assistance by T.A., B.L., C.T., L.C., J.S., and L.L. G.L. performed the supporting oscillation measurements in YbB6. Y.K. performed the supporting x-ray studies and SdH simulations with assistance by K.S., S.K., Y.S., and Y.M. S.Y., Y.L.C., and F.I. provided the samples. Z.X., Y.M., and L.L. wrote the manuscript with assistance from all the authors. Competing interests: The authors declare no competing interests. Data and materials availability: All the data in the manuscript main text are available as .csv ASCII files in the supplementary materials.
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