Quantum Oscillations of Electrical Resistivity in an Insulator

TL;DR

This paper reports the discovery of quantum oscillations in the electrical resistivity of the insulator YbB12, indicating the presence of a Fermi surface and strongly correlated electrons despite its insulating behavior.

Contribution

It demonstrates that an insulator can exhibit quantum oscillations typically associated with metals, revealing a bipartite ground state with both insulating and metallic properties.

Findings

Quantum oscillations observed in YbB12 resistivity.

Resistivity oscillations follow Fermi liquid temperature dependence.

Presence of large effective masses indicating strongly correlated electrons.

Abstract

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). Despite much larger than that of metals, the resistivity of YbB12 exhibits profound quantum oscillations. This unconventional oscillation is shown to arise from the insulating bulk, yet the temperature dependence of their amplitude follows the conventional Fermi liquid theory of metals. The large effective masses indicate the presence of Fermi surface consisting of strongly correlated electrons. Our result reveals a mysterious bipartite ground state of YbB12: it is both a charge insulator and a strongly correlated metal.

Full text

Strong electronic correlations bring novel physical phenomena. A remarkable example is the unique ground state of Kondo insulators, in which the hybridization between itinerant and localized electrons leads to the opening of an insulating gap and consequently a divergent low-temperature resistance Kondolattice ; riseborough . Recently the nature of the ground state of a Kondo insulator in intense magnetic fields has become a big mystery in condensed matter physics. In mixed-valence compound samarium hexaboride (SmB6), although the sample is a good insulator and its resistance increases by five orders of magnitude during cooling-downs, Landau level (LL) quantization happens under applied magnetic field, and strong quantum oscillations are observed in magnetization (the de Haas-van Alphen, or dHvA, effect) liSmB6 ; tan2015unconventional . Origin and interpretation of the dHvA oscillations in SmB6 have been highly controversial, as there are a number of peculiar features. Firstly, the oscillations are observed only in magnetization, and not in electrical resistivity (the Shubnikov-de Haas, or SdH, effect). Secondly, despite the strong electron correlations, only the oscillations arising from quasiparticles having very light effective masses ($m\ll m_{e}$, $m_{e}$ is the free electron mass) are observed. Thirdly, in floating-zone grown SmB6 samples the dHvA signal exhibits a striking deviation from the standard Lifshitz-Kosevich (LK) formula of the Fermi-liquid theory tan2015unconventional . These observations point to either the topologically protected surface state liSmB6 , or an unconventional Fermi surface in insulators tan2015unconventional . A number of intriguing physical origins have been proposed, including exciton-based magnetic breakdown knolle2015 ; knolle2017 , Majorana type charge-neutral Fermi surfaces baskaran2015majorana , and a failed superconducting ground state erten2017skyrme . Recent theories suggest that an emergent neutral Fermi sea can exist in a mixed-valence gapped system, which may exhibit both dHvA and SdH oscillations under external magnetic field Senthil2017neutralFS ; Senthil2017QO . A key to solving the most fundamental problem, “the Fermi surface in an insulator”, lies in the clarification whether the quantum oscillations, in particular in the charge transport, are observable in another insulating system.

We chose to study a different mixed-valence Kondo insulator, YbB12, another cubic-structured rare-earth intermetallic compound possessing strong interactions between localized 4f and itinerant 5d orbitals Kasaya1983 ; Kasaya1985 ; Kasuya . Early theories predicted an existence of topological surface states due to mirror-symmetry protection in YbB12 TCPYbB12 , which makes it a topological crystalline Kondo insulator. YbB12 single crystals are grown by floating zone method (section S1 in SM ). Using the experimental setup illustrated in Fig. 1A, magnetic torque and magnetoresistance (MR) are measured up to 45 T simultaneously (section S2 in SM ). The temperature dependence of the resistivity shown in Fig. 1B confirms an increase of five orders of magnitude from room temperature to 50 mK. A weakly-temperature-dependent resistivity appears below 2.2 K, resembling the resistive “plateau” well known in SmB6 at $T<$ 3.5 K SmB6pressure , which is an indication of the existence of extended in-gap states. Fitting with the thermal activation model of resistivity, $\rho(T)$ = \rho_{0}$$\exp(\Delta/2k_{B}T), 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 (inset of Fig. 1B), consistent with former transport result IgaGrowth . Upon applying the magnetic field, the negative slope of $\rho(T)$ curve is preserved up to 45 T with no hints of metallic behaviour (Fig. 1C), indicating that the ground state is still gapped.

Field dependence of the magnetic torque in the insulating state of YbB12 is shown in Fig. 1D. Subtle metamagnetic transitions and/or crossover can be seen at 20 T and 27 T, potentially related to the predicted field-induced staggered magnetism in Kondo insulator PatrickLee . Above approximately 37 T, the dHvA oscillations is clearly resolved (Fig. 1D). We note that the dHvA oscillations appear well below the magnetic-field-driven gap closing, i.e., the insulator-metal transition in YbB12 IgaPulsedField (for details see section S3 and Fig . S1 in SM ). Fast Fourier Transform (FFT) on the $\phi$ = 11.3∘ torque curve gives a dHvA frequency of $F$ = 720 T (inset of Fig. 1D).

In Fig. 2A the MR data at $\phi$ = 27.7∘ is plotted between 11.5 T and 45 T. Given the zero-field resistivity $\rho(0)$ = 4.67 $\Omega\cdot$cm of this sample, a significant negative MR ($\{\rho(H)-\rho(0)\}/\rho(0)$) of -95.9$\%$ is achieved at 45 T (A detailed angular dependence of MR is shown in Fig. S2 in SM ). The negative MR is a hallmark in Kondo insulators as a result of the field suppression of the hybridization gap Sugiyama ; CeNiSn ; Boebinger ; Cooley . We note that the negative MR in YbB12 is much larger then that in SmB6 SmB6pulsed ; wolgast2015magnetotransport . This is probably due to the larger effective Landé $g$-factor and the more localized nature of $f$ electrons in YbB12, which lead to the stronger impact of magnetic field Cooley ; IgaPulsedField .

The most striking result in Fig. 2A is the oscillating wiggles appear in the MR under strong magnetic fields. From 40.8 T up to 45 T, two valleys and one peak in total can be clearly observed (upper inset of Fig. 2A), and the FFT on MR in such a field regime yields a profound frequency peak at $F$ = 913 T (lower inset of Fig. 2A). With magnetic field direction close to the crystal axes, up to four oscillation periods can be seen, as shown in Fig. 2B. Also, the overall SdH patterns are almost identical for $\phi$ = 11.3∘ and $\phi$ = 78.4∘, and only have a slight valley position shift between $\phi$ = 18.3∘ and $\phi$ = 70.5∘, suggesting an axis of symmetry along [101] direction for the SdH oscillations. This is consistent with the cubic structure of YbB12 crystal. The fact that the valleys on $d\rho$/$dH$ in Fig. 2B have approximately constant interval as a function of $1/H$ provides strong evidence of the SdH oscillations with a dominating single frequency.

The observation of the SdH oscillations is reinforced by the $T$-dependence of oscillation amplitudes. The evolution of the wiggled features in both MR and torque data show typical behaviour of quantum oscillations, with $T$-independent positions of the dominant peaks/valleys within uncertainty and obviously attenuated amplitude from base temperature up to 1.5 K (Fig. S3 in SM ). The $T$-dependent amplitudes of normalized oscillatory torque (Fig. 3A) and oscillatory MR (Figs. 3B and 3C) are fit using the conventional LK formula shoenberg2009magnetic . The fittings are reasonably good down to 60 mK, indicating that the LK theory, established based on the basic Fermi liquid framework, appears to be valid in Kondo insulator YbB12. The agreement with LK description firmly confirms that the features we resolve are quantum oscillation in nature, instead of successive field-induced Lifshitz transitions.

As shown in Figs. 3A and 3B, the SdH oscillation is much easier to be suppressed at higher $T$ compared to dHvA oscillations at the same angle, revealing a remarkably 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 approximately 6.6 $m_{e}$ and 14.6 $m_{e}$, respectively. Therefore it is unlikely that both oscillations originate from the same band. Indeed the SdH and dHvA frequencies have distinct angle dependence (Fig. 4). The dHvA frequencies $F$ can be tracked by a two-dimensional (2D) Fermi surface model ($F\propto\frac{1}{\cos\theta}$) with the in-plane cross section area $A_{[001]}$ = 6.67 nm*-2* (solid line in Fig. 4). Given the lack of dHvA oscillations observed above $\simeq$ 20∘ (Fig. S4A in SM ), this inverse sinusoidal dependence can be explained by either 2D Fermi cylinder, or a heavily elongated 3D Fermi pocket. On the other hand, the angle dependence of SdH frequencies displays an evident nonmonotonic behaviour: frequency maxima appears at $\theta\sim$ 15-20∘ from crystal axes, resulting in an “M”-shape with a local dip at $H\parallel$ [100] and potentially a second minimum at higher tilt angle. The 2D Fermi surface model apparently can not describe this behaviour. Attempt to model the SdH frequencies (dash lines in Fig. 4, see also Fig. S5 in SM ) points to hyperbolic “neck” orbits (section S5 in SM ).

We can rule out a possibility that the bulk SdH oscillations arise from metallic impurity phases. Firstly, as the crystals are grown by floating zone method, additional elements are not included. The synchrotron X-ray diffraction measurements reveal the absence of metallic Yb-B alloys, including YbB2 and YbB4, within the resolution of 10 ppm in volume (section S6 and Fig. S6 in SM ). The only indexed impurity YbB6 shows a much lower dHvA frequency that we observed in YbB12 (Fig. S7 in SM ). Secondly, the angle dependence of both the amplitude and frequency of the observed oscillations coincide with the cubic symmetry of YbB12, inconsistent with hexagonal and tetragonal structures of YbB2 and YbB4, respectively (section S4 and Fig. S5 in SM ). Thirdly, according to our simulation assuming the presence of tiny but highly conducting impurities, SdH signal should be detectable at much lower fields (section S7 and Fig. S8 in SM ), which contradicts with our observations.

The present result is, to our knowledge, the first report of the observation of SdH oscillations in the insulating state. There are several remarkable features in the SdH oscillations. The effective masses attained by LK fittings are considerably large, in agreement with the nature of Kondo insulator in which the strong electron correlations make the quasiparticles heavy. Even in the insulating state, finite specific heat coefficient $\gamma$ is observed in YbB12. Interestingly, $\gamma$ calculated from the SdH oscillations assuming a spherical Fermi surface with $k_{F}\sim 0.17$ Å*-1* ($F=952$ T) and effective mass $m^{\ast}$ = 15 $m_{e}$, is comparable to the observed value of $\gamma\sim 8$ mJ/mol K2 at 40 T TerashimaThesis . The background resistivity $\rho$ still has a magnitude of 10-100 m$\Omega\cdot$cm above 40 T, which is far beyond that of normal metals Ioffe-Regel . Indeed if we estimate the mean free path $\ell$ by considering a spherical Fermi surface with $m^{*}$ = 15 $m_{e}$, then $\rho=0.4\,\Omega\cdot$cm (Fig. 2A) will give a unphysically short mean free path, $\ell\sim 0.01$ nm, indicating highly unusual situation. A proper theory is yet to be established to describe the quantum oscillations observed in YbB12 under high magnetic field (section S8 in SM ).

The observed unconventional existence of metallic quantum oscillations in insulating ground state reveals that the Kondo insulator is both a charge insulator and a correlated metal, providing a new mysterious and fertile feature of a strongly correlated insulator.