ReportHeavy Fermions

Unconventional Fermi surface in an insulating state

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Science  02 Jul 2015:
DOI: 10.1126/science.aaa7974


Insulators occur in more than one guise; a recent finding was the class of topological insulators, which host a conducting surface juxtaposed with an insulating bulk. Here we report the observation of an unusual insulating state with an electrically insulating bulk that simultaneously yields bulk quantum oscillations with characteristics of an unconventional Fermi liquid. We present quantum oscillation measurements of magnetic torque in high purity single crystals of the Kondo insulator SmB6, which reveal quantum oscillation frequencies characteristic of a large three-dimensional conduction electron Fermi surface similar to the metallic rare earth hexaborides such as PrB6 and LaB6. The quantum oscillation amplitude strongly increases at low temperatures, appearing strikingly at variance with conventional metallic behavior.

Kondo insulators, a class of materials positioned close to the border between insulating and metallic behavior, provide fertile ground for novel physics (114). This class of strongly correlated materials is thought to be characterised by a ground state with a small energy gap at the Fermi energy owing to the collective hybridisation of conduction and f-electrons. The observation of quantum oscillations has traditionally been associated with a Fermi liquid state; here we present the surprising measurement of quantum oscillations in the Kondo insulator SmB6 (15) that originate from a large three-dimensional Fermi surface occupying half the Brillouin zone and strongly resembling the conduction electron Fermi surface in the metallic rare earth hexaborides (16, 17). Our measurements in SmB6 reveal a dramatic departure from conventional metallic Lifshitz Kosevich behavior (18); instead of the expected saturation at low temperatures, a striking increase is observed in the quantum oscillation amplitude at low temperatures.

Single crystals of SmB6 used in the present study were grown by the image furnace technique (19) in order to achieve high purities as characterised by the high inverse residual resistivity ratio. Single crystals with inverse resistance ratios [IRR = R(T = 1.8 K)/R(T = 300 K), where R is resistance and T is temperature] of the order of 105were selected for this study; the IRR has been shown to characterize crystal quality, with the introduction of point defects by radiation damage (20), or by off-stoichiometry (21) resulting in a decrease in low temperature resistance and an increase in high temperature resistance. Figure 1B shows the resistance of a SmB6 single crystal measured as a function of temperature at zero magnetic field and in an applied DC magnetic field of 45 T, demonstrating that activated electrical conductivity characteristic of an energy gap ≈40 K at the Fermi energy persists up to high magnetic fields. The non-magnetic ground state of SmB6 is evidenced by the linear magnetisation up to 60 T (Fig. 1B, lower inset).

Fig. 1 Quantum oscillations in the magnetic torque in SmB6.

(A) Quantum oscillations in magnetic torque are visible against a quadratic background. Inset: schematic of the magnetic torque measurement setup using a capacitive cantilever and the notation for angular rotation by angle θ. (B) Resistance as a function of temperature in zero magnetic field (blue line), and at 45 T (green line) using an unchanged measurement configuration on a SmB6 sample of dimensions 1.1 mm × 0.3 mm × 0.1 mm. Upper inset: measured resistance from 80 mK up to high temperatures, from which the high IRR can be ascertained (see (23) for a fit to activated electrical conductivity). Lower inset: magnetisation of SmB6 at 2.1 K remains linear up to 60 T. (C) Dominant low frequency quantum oscillations can be discerned after background subtraction of a sixth order polynomial. (D) Magnetic torque at the highest measured fields after the subtraction of the low frequency background torque. Quantum oscillations are visible in an intermediate frequency range (between 2,000 and 4,000 T) as well as a high frequency range up to 15,000 T.

We observed quantum oscillations in SmB6 by measuring the magnetic torque. The measurements were done in magnetic fields up to 40 T down to T = 0.4 K, and in magnetic fields up to 35 T down to T = 0.03 K. Quantum oscillations periodic in inverse magnetic field are observed against a quadratic background, with frequencies ranging from 50 T to 15,000 T (Figs. 1A,C,D). Figure 2A shows a Fourier transform of the quantum oscillations as a function of inverse magnetic field, revealing well defined peaks corresponding to multiple frequencies. The periodicity of the quantum oscillations in inverse magnetic field is revealed by the linear Landau index plot in Fig. 2B.

Fig. 2 Landau quantisation in SmB6.

(A) Fourier transforms of magnetic torque as a function of inverse magnetic field, from which a polynomial background has been subtracted, revealing multiple quantum oscillatory frequencies ranging from 50 T to 15,000 T. Field ranges for analysis have been chosen that best capture the observed oscillations, with the highest frequencies only appearing in the higher field ranges. (B) The maxima and minima in the derivative of magnetic torque with respect to the magnetic field, corresponding to the dominant low frequency oscillation, are plotted as a function of inverse magnetic field; the linear dependence signals Landau quantisation.

The observation, especially of rapid quantum oscillations with frequencies higher than 10 kT (corresponding to approximately half the volume of the cubic Brillouin zone) in SmB6, is striking. This observation is in contrast to previous reports of very low frequency quantum oscillations corresponding to a few % of the Brillouin zone in SmB6, attributed to a two-dimensional surface contribution (22). Our observation of very high quantum oscillation frequencies requiring mean free paths of order a few microns would be challenging to explain from a surface layer of thickness a few atomic lengths, which would typically yield such rapid frequencies only at a special angle of inclination where the cyclotron orbit lies completely within the surface layer. Key to identifying the Fermi surface from which the observed quantum oscillation frequencies originate is a comparison with previous quantum oscillation measurements on metallic hexaborides such as nonmagnetic LaB6, antiferromagnetic CeB6, and antiferromagnetic PrB6 (16, 17). These materials exhibit a metallic ground state involving predominantly conduction electrons, with a low residual resistivity of the order of one (≈106 times lower than in Kondo insulating SmB6), and are characterised by a multiply connected Fermi surface of prolate spheroids (Figs. 3, D and E). Strikingly, the angular dependence of the various quantum oscillation frequencies in SmB6 reveals characteristic signatures of the three-dimensional Fermi surface identified in the metallic rare-earth hexaborides (Figs. 3, A-C). In particular, the high observed α frequencies (Fig. 3A) reveal the characteristic symmetry of large prolate spheroids centered at X-points of the Brillouin zone (Fig. 3D,E), whereas the lower observed frequencies (Fig. 3A) reveal the characteristic symmetry of small ellipsoids located at the neck positions. Both these types of ellipsoids are universal Fermi surface features identified from experiment and band structure calculations in the metallic rare-earth hexaborides (16, 17). Similar features are also revealed in density functional calculations of SmB6 when the Fermi energy is shifted from its calculated position in the insulating gap either up into the conduction or down into the valence bands (Figs. 3, D and E) (23).

Fig. 3 Angular dependence of the quantum oscillation frequencies in SmB6.

(A) Data from two of the SmB6 samples where oscillations were observed are shown, by solid and open circle symbols. Importantly, one of the samples (solid circles) was prepared as a thin plate with the dominant face perpendicular to the [100] axis (sample 1 (23)), and the second sample (open circles) was prepared as a thin plate with the dominant face perpendicular to the [110] axis (sample 2, (23)). The angular dependence strongly resembles that of the three-dimensional Fermi surface in antiferromagnetic PrB6 shown in (B), and nonmagnetic LaB6 shown in (C) (17). The α orbit in red in all the rare earth hexaborides is fit to large multiply connected prolate spheroids centered at the X points of the Brillouin zone, shown in (D); a cross-section in the XM plane is shown in (E). The ρ and ρ′ orbits in each of the rare earth hexaborides are fit to small ellipsoids located at the neck positions (not shown in D and E). For more details of the fits, see (23). The remaining intermediate orbits are shown with lines as a guide to the eye. All orbit identifications have been made after measured frequencies and band structure calculations in PrB6 and LaB6 (17). (D) and (E) show Fermi surfaces calculated for SmB6 using density functional theory (23), with a downward shift of the Fermi energy from its calculated position within the gap to expose the unhybridised bands, and yield pocket sizes similar to experiment.

The observed angular dependence of quantum oscillations in SmB6 remains the same irrespective of whether the sample is prepared as a thin plate with a large plane face perpendicular to the [110] direction, or to the [100] direction (Fig. S1), and exhibits the same characteristic signatures with respect to the orientation of the magnetic field to the crystallographic symmetry axes of the bulk crystal (Fig. 3A). We note that the bulk quantum oscillations we measure in SmB6 corresponding to the three-dimensional Fermi surface mapped out in the metallic rare-earth hexaborides may not be directly related to the potential topological character of SmB6, which would have as its signature a conducting surface (24). In addition to the magnetic torque signal from the atomically thin surface region being several orders of magnitude smaller than the signal from the bulk, the observation of surface quantum oscillations would be rendered more challenging by the reported Samarium depletion and resulting reconstruction of Samarium ions at the surface layer of SmB6 (25).

The unconventional character of the state we measure in SmB6 becomes apparent on investigating the temperature dependence of the quantum oscillation amplitude in SmB6. We find that between T = 25 K and 2 K, the quantum oscillation amplitude exhibits a Lifshitz-Kosevich like temperature dependence (Fig. 4), characteristic of a low effective mass similar to metallic LaB6, which has only conduction electrons (16). The comparable size of low temperature electronic heat capacity measured for our SmB6 single crystals to that of metallic LaB6 (23), also seems to suggest a large Fermi surface with low effective mass in SmB6. However, instead of saturating at lower temperatures as would be expected for the Lifshitz Kosevich distribution characteristic of quasiparticles with Fermi Dirac statistics (18), the quantum oscillation amplitude increases dramatically as low temperatures down to 30 mK are approached (Fig. 4). Such non-Lifshitz Kosevich temperature dependence is remarkable, given the robust adherence to Lifshitz Kosevich temperature dependence in most examples of strongly correlated electron systems, from the underdoped cuprate superconductors (26), to heavy fermion systems (27, 28), to systems displaying signatures of quantum criticality (29), a notable exception being fractional quantum Hall systems (30, 31). The possibility of a subtle departure from Lifshitz Kosevich temperature dependence has been reported in a few materials (32, 33).

Fig. 4 Temperature dependence of quantum oscillation amplitude.

The dominant 330 T frequency over the magnetic field range 25 to 35 T is shown, revealing a steep increase in amplitude at low temperatures. The measurements in the temperature range 25 K down to 0.35 K were performed in a 3He fridge in the hybrid magnet (sample 1 (23): blue diamonds), whereas the measurements at temperatures in the range from 1 K down to 30 mK were done in a dilution fridge in the resistive magnet on two different samples (sample 1: purple diamonds and sample 3 (23): blue squares). At low temperatures, a strong deviation from the conventional Lifshitz Kosevich form can be seen in the inset by comparison with a simulated Lifshitz Kosevich form for effective mass m* = 0.18me. A logarithmic temperature scale is used in the inset for clarity.

We note that the ground state of SmB6 is fairly insensitive to applied magnetic fields, with activated electrical conductivity behavior across a gap remaining largely unchanged up to at least 45 T (Fig. 1B). Such a weak coupling to the magnetic field is in contrast to unconventional states in other materials that are tuned by an applied magnetic field (611, 13). Furthermore, this rules out the possibility of quantum oscillations in SmB6 arising from a high magnetic field state in which the energy gap is closed. The possibility whereby quantum oscillations arise from static, spatially disconnected metallic patches of at least 1 μm length scale which do not contribute to the electrical transport also appears unlikely. Similar quantum oscillations are observed in all (more than ten) measured high quality samples in multiple high magnetic field experiments, with the best samples yielding magnetic quantum oscillations of amplitude corresponding to a substantial fraction of the expected size from bulk SmB6. The presence of rare-earths other than Sm has been ruled out to within 0.01% by chemical analysis and scanning electron microscopy (23). Off-stoichiometric metallic regions of SmB6 appear an unlikely explanation for our results, given reports that up to 30% Samarium depletion does not close the energy gap (20), whereas scanning electron microscopy of our samples reveals a homogeneity of within 1% of Samarium concentration over the sample area (23). The possibility of spatially disconnected strained regions of SmB6, which is known to become metallic under applied pressures of the order of 10 GPa, or spatially disconnected islands of hybridised and unhybridised Sm f-electrons also seems unlikely. An improvement in the IRR by the removal of strain by electropolishing strengthens the quantum oscillation signal whereas straining the sample by thermal cycling weakens the quantum oscillation signal (23). Further, the interplay between hybridised and unhybridised Sm f-electrons which may be an important ingredient in the physics of SmB6, has been revealed by Mössbauer and muon-spin relaxation experiments to be homogenous and dynamically fluctuating, rather than being manifested as static spatially inhomogeneous regions (34, 35).

The insulating state in SmB6 in which low energy excitations lack long range charge transport as shown by the activated DC electrical conductivity, but display extended character as shown by quantum oscillations, poses a mystery. A clue might be provided by slow fluctuations between a collectively hybridized insulating state, and an unhybridized state where the conduction electrons form a solely conduction electron Fermi surface, similar to that we observe (2, 3539). A fluctuation timescale in the range between 10−8 – 10−11 seconds is suggested by previous x-ray absorption spectroscopy and Mössbauer measurements (40). This timescale is longer or comparable to the inverse cyclotron frequency (1/ωc) which is of the order of 10−11 seconds for the measured cyclotron orbits. Intriguingly, similar slow fluctuations have been invoked to explain quantum critical signatures in the metallic f-electron system β-YbAlB4 (41). SmB6 may be viewed as being on the border of quantum criticality in the sense that it transforms from a non-magnetic insulating phase to a magnetic metallic phase under applied pressures of the order of 10 GPa (4245), in contrast to other metallic rare earth hexaborides where the f-electrons order magnetically in the ambient ground state. Our observation of a large three-dimensional conduction electron Fermi surface revealed by quantum oscillations may be related to reports of a residual density of states at the Fermi energy in SmB6 by measurements of heat capacity (refs. (23, 46)), optical conductivity (47), Raman scattering (48), and neutron scattering (49). Another possibility is that quantum oscillations could arise even in a system with a gap in the excitation spectrum at the Fermi energy, provided that the size of the gap is not much larger than the cyclotron energy (50). Within this scenario, the residual density of states observed at the Fermi energy by complementary measurements, and the steep upturn in quantum oscillation amplitude we observe at low temperatures appear challenging to explain.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S5

Table S1

References (5157)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: BST, YTH, MH, MK, AS, and SES acknowledge support from the Royal Society, the Winton Programme for the Physics of Sustainability, and the European Research Council under the European Unions Seventh Framework Programme (grant number FP/2007-2013)/ERC Grant Agreement number 337425. BZ and LB acknowledge support from DOE-BES through award DE-SC0002613. MCH and GB acknowledge support from EPSRC Grant EP/L014963/1. NH and ZZ acknowledge support from the US Department of Energy, Office of Science, BES-MSE ‘Science of 100 Tesla’ program. MDJ acknowledges support for this project by the Office of Naval Research (ONR) through the Naval Research Laboratory’s Basic Research Program. GGL acknowledges support from EPSRC grant EP/K012894/1. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement No. DMR-1157490 and the State of Florida. We acknowledge valuable inputs from G. Baskaran, D. Benkert, A. K. Cheetham, D. Chowdhury, P. Coleman, N. R. Cooper, M. P. M. Dean, O. Ertem, J. Flouquet, R. H. Friend, R. Golombok, C. Harris, S. A. Hartnoll, T. Kasuya, G. Khaliullin, E. -A. Kim, J. Knolle, P. A. Lee, P. B. Littlewood, C. Liu, K. Miyake, J. E. Moore, O. Petrenko, S. Sachdev, A. Shekhter, N. Shitsevalova, Q. Si, A. Thomson, S. Todadri, C. M. Varma, J. Zaanen. We thank magnet lab personnel including J. Billings, R. Carrier, E. S. Choi, B. L. Dalton, D. Freeman, L. J. Gordon, M. Hicks, J. M. Petty, and J. N. Piotrowski for their assistance. Data will be made available at the institutional data repository
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