Bulk Rotational Symmetry Breaking in Kondo Insulator ${\mathrm{SmB}}_6$

The Kondo insulator samarium hexaboride (${\mathrm{SmB}}_6$) has been intensely studied in recent years as a potential candidate of a strongly correlated topological insulator. One of the most exciting phenomena observed in ${\mathrm{SmB}}_6$ is the clear quantum oscillations appearing in magnetic torque at a low temperature despite the insulating behavior in resistance. These quantum oscillations show multiple frequencies and varied effective masses. The origin of quantum oscillation is, however, still under debate with evidence of both two-dimensional Fermi surfaces and three-dimensional Fermi surfaces. Here, we carry out angle-resolved torque magnetometry measurements in a magnetic field up to 45 T and a temperature range down to 40 mK. With the magnetic field rotated in the (010) plane, the quantum oscillation frequency of the strongest oscillation branch shows a fourfold rotational symmetry. However, in the angular dependence of the amplitude of the same branch, this fourfold symmetry is broken and, instead, a twofold symmetry shows up, which is consistent with the prediction of a two-dimensional Lifshitz-Kosevich model. No deviation of Lifshitz-Kosevich behavior is observed down to 40 mK. Our results suggest the existence of multiple light-mass surface states in ${\mathrm{SmB}}_6$, with their mobility significantly depending on the surface disorder level.

Popular Summary

Strong interactions among electrons have led to novel phenomena such as high-temperature superconductivity, while topological materials constitute a new type of substance that protects exotic electronic states. Kondo insulators—a type of insulator based on rare-earth elements—bridge these two exciting frontiers. They are the only experimentally demonstrated materials in which strong electronic interactions coexist with topological protections; the interior is a perfect insulator and yet the surface is conductive. This is not only fundamentally new, but it also opens doors for electronics that use the conductive surfaces for circuits. Researchers have debated the origin of magnetic oscillations, known as the de Hass-van Alphen (dHvA) effect, observed in Kondo insulators, a curious behavior since it is a key feature of metals. We provide crucial evidence—an angular dependence of quantum oscillation amplitude—that suggests these oscillations originate in the surface of samarium hexaboride (${\mathrm{SmB}}_6$), a typical Kondo insulator.

We measured the dHvA signals using torque magnetometry in single crystals of ${\mathrm{SmB}}_6$ at many different tilt angles between magnetic fields and crystal axes. Besides the 2D-like angular dependence of dHvA oscillation frequencies, the angle-dependent amplitude breaks the fourfold rotational symmetry of the cubic crystal structure. The angular and temperature dependence of dHvA amplitudes are both consistent with the behavior of a 2D Fermi-liquid system down to 40 mK in a strong magnetic field of 45 T.

Our results show that quantum oscillations in a Kondo insulator come from the surface, which is consistent with the interpretation of topologically protected surface states and should shed light on future work pursuing topological protection in strongly correlated systems.

Figure 1

Torque measurements on ${\mathrm{SmB}}_6$. (a) Photograph showing the beryllium-copper cantilever and the ${\mathrm{SmB}}_6$ single crystal S5 we use for the torque magnetometry measurement. In this setup, the magnetic field is rotated in the (010) plane of the sample. Arrows sketch the definition of the tilt angle $\varphi$. (b) Magnetic torque of sample S5 measured up to 45 T at different tilt angles. The torque curves are shifted vertically for clarity. The absolute value of torque is calibrated by the sample weight. (c) Angular dependence of all resolved dHvA oscillation frequencies on the fast Fourier transform (FFT) of $M_{\mathrm{eff}}$ with field rotated in (010) plane. FFT peaks are indexed with the same labels as in Ref. [12]. Solid symbols denote the data points at $\varphi <45°$, while hollow symbols are data taken at $\varphi >45°$. Harmonics of branches $\alpha$, $\beta$, and $\gamma$ are presented by diamonds, circles, and triangles, respectively. Dashed lines are fittings based on 2D FS model: $F=F_0/\cos (\varphi -{\varphi }_0)$. For branch $\beta$, ${\varphi }_0=\pm 45°$ and $F_0=285\mathrm{T}$. The two “split” branches ${\gamma }_1$ (blue) and ${\gamma }_2$ (navy) both have symmetric axis along [100], i.e., ${\varphi }_0=0°$ and 90°, while $F_0$ is 374 T for ${\gamma }_1$ and 414 T for ${\gamma }_2$.

Figure 2

Angular dependence of oscillation amplitudes. (a) The amplitude of peak $\beta$ in the FFT spectra of $M_{\mathrm{eff}}$, plotted against the tilt angle $\theta$ between $H$ and [101] direction. Inset shows how $\theta$ is defined. The solid (fitted by the solid line) and hollow (fitted by the dashed line) symbols denote the amplitude of $F^{\beta }$ assumed to come from surfaces $(101)(\overline{1}0\overline{1})$ and $(10\overline{1})(\overline{1}01)$, respectively. The fittings (dark cyan) are made using the 2D LK model in Eq. (1) and yield $\xi =0$. Results with $\xi =0.1$ (purple) and 0.2 (blue) are also shown for comparison. (b) Amplitude analysis using the same model applied on the data of an old ${\mathrm{SmB}}_6$ sample, S1, measured up to 18 T. Data are extracted from Fig. S3 in Ref. [12].

Figure 3

Temperature dependence of oscillation amplitudes. (a) Capacitance signals in Sample S5 measured up to 45 T at different temperatures ranged from 41 mK to 656 mK. The tilt angle for this data set is $\varphi =32.6°$. Inset: The oscillatory part of magnetic torque extracted from the capacitance curves, as a function of inverse magnetic field. A polynomial background is subtracted. (b) The FFT amplitude curves of magnetic torque shown in the inset of (a) in a field range between 16.7 T and 45 T. (c) The FFT amplitudes of $F^{\beta }$ and $F^{\gamma }$ plotted as a function of temperature. Dashed lines are fittings based on Lifshitz-Kosevich formula with effective mass $m^{*}=0.13$ $m_e$ and 0.19 $m_e$ for oscillation branches $\beta$ and $\gamma$, respectively. (d) Temperature dependence of the FFT amplitude of $F^{\beta }$ tracked up to 30 K at $\varphi =41°$. Fitting by LK formula yield an effective mass of $0.138{\mathrm{m}}_e$.

Figure 4

(a) The FFT spectra of $M_{\mathrm{eff}}$ for various tilt angles. Two peaks on the high-frequency side of the major peak $\beta$ are labeled as ${\gamma }_1$ and ${\gamma }_2$, respectively. One unknown small feature on the $\varphi =48.4°$ at 435 T is marked by a star. Inset: The FFT spectrum at $\varphi =28.6°$, between 11.4 T and a varied cutoff field. The split of peak $\gamma$ is more evident as the cutoff field becomes larger. (b) The angle dependence of lowest frequency oscillation branch $\alpha$ and its second harmonic, plotted together with the frequency interval between split ${\gamma }_1$ and ${\gamma }_2$ and data of frequency ${\rho }^{\prime }$ extracted from Ref. [13]. The red curves are fittings using a 2D Fermi surface (FS) model, whereas the black curves are 3D fittings based on an ellipsoid FS. For the 3D model, we follow the previous report in which the geometry of the FSs contributing to ${\rho }^{\prime }$ are figured out to be small ellipsoidal pockets with their long axes along the cubic [101] directions [13]. The fitting (black solid line) gives a ratio of long axis (along [$10\overline{1}$]) to short axis (along [101]) of 2.58. However, the other set of equivalent FSs with long axis along [101] (black dashed line) are absent from our data.

Figure 5

Absence of high-frequency oscillations. The oscillatory effective magnetization $M_{\mathrm{eff}}$ as a function of inverse magnetic field at (a) $\varphi =40.5°$ and (c) $\varphi =84.9°$. Polynomial backgrounds are subtracted from the raw data. Insets: Low-frequency component of $M_{\mathrm{eff}}$ obtained by a low-pass FFT filter with threshold frequency 2 kT (red) and the residual component after subtracting the low-frequency oscillation from $M_{\mathrm{eff}}$ (blue), both shown in a magnetic field range between 40 and 45 T. The FFT results of the oscillatory part of $M_{\mathrm{eff}}$ at (b) $\varphi =40.5°$ and (d) $\varphi =84.9°$ between 25 and 45 T. No oscillation frequency higher than 2 kT can be resolved from the background at both tilt angles. Insets: Low-frequency peaks with $F<1200\mathrm{T}$ in a full range FFT from 11.4 to 45 T.

Figure 6

A comparison of the FFT peaks resolved from the magnetic torque data in the floating-zone-grown ${\mathrm{SmB}}_6$ single crystals, extracted from Ref. [13], and the Al-flux-grown samples studied in Ref. [12]. In (a) and (b) the FFT peaks $\rho$ and ${\rho }^{\prime }$ in Ref. [13] are plotted, respectively, together with the dHvA oscillation features in the flux-grown samples that are within the same frequency range. In both data sets, the magnetic field is rotated from crystal [100] axis towards [011] axis. Dashed lines are fittings by the 2D cylinder FS model [12], and dotted lines are fittings of the harmonics.

Figure 7

The dHvA oscillation amplitudes of (a) 2D surface state on the bisectrix plane and (b) 3D ellipsoidal bulk FS in ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$, fitted by Eq. (f7). Data points are extracted from Ref. [27]. Tilt angle $\theta$ is the angle between the magnetic field, which is rotated in the binary plane, and the crystal axis $C_3$. The carrier mobility in each panel is calculated from the parameters obtained in the same work. (c) Fitting of the angle-dependent amplitude of FFT peak $\beta$ in ${\mathrm{SmB}}_6$ sample S2 by Eq. (f6), with parameter $\xi =0$. Definitions of ${\mu }_0$ and $\xi$ are the same as in Fig. 2. Data are extracted from Fig. S4 in Ref. [12].

Figure 8

Dingle plots. (a) Dingle plot $\ln (\mathrm{\Delta }{M}_{\perp }/R_T)$ vs $1/B$ for sample S5 at $\theta =31°$. Solid line is the linear fit which has a slope of $-(2{\pi }^2{k}_B{m}^{*}{T}_D)/e\hslash$. The carrier mobility $\mu$ is calculated as $\mu =e{\tau }_Q/m^{*}=e\hslash /2\pi k_B{m}^{*}{T}_D$. Inset: Raw data of $M_{\perp }$ after subtracting the nonoscillatory background and the bandpass-filtered oscillation pattern used in the Dingle plot. (b) Dingle plots for sample S1 and S2 at $\theta =15°$.

Figure 9

(a) Magnetic torque below ${}^3\mathrm{He}$ temperature in sample S6 measured by the cantilever capacitor at $\varphi =88.2°$. Inset: Oscillatory part of magnetic torque plotted against inverse magnetic field. (b) FFT curves of magnetic torque in a field range between 16.7 and 45 T. Inset: Normalized FFT amplitudes of $F^{\beta }$ as a function of temperature. Dashed line is fitting based on LK formula with effective mass $m^{*}=0.14m_e$.