Intrinsic Low-Temperature Magnetism in SmB6

TL;DR

This study uses muon spin relaxation to distinguish intrinsic from extrinsic low-temperature magnetism in SmB6, revealing persistent magnetic fluctuations and their possible impact on surface conductivity in this topological Kondo insulator.

Contribution

It provides new insights into the intrinsic low-energy magnetic fluctuations in SmB6 and their relation to surface conductivity, using advanced muon spin relaxation techniques.

Findings

Detection of quasi-static magnetism from impurities and vacancies below 10 K

Identification of intrinsic low-energy magnetic fluctuations at millikelvin temperatures

Observation of a bulk magnetic excitation at 2.6 meV affecting surface conductivity

Abstract

By means of new muon spin relaxation experiments, we disentangle extrinsic and intrinsic sources of low-temperature bulk magnetism in the candidate topological Kondo insulator (TKI) SmB6. Results on Al-flux grown SmB6 single crystals are compared to those on a large floating-zone grown ^{154}Sm^{11}B6 single crystal in which a 14 meV bulk spin exciton has been detected by inelastic neutron scattering. Below ~ 10 K we detect the gradual development of quasi-static magnetism due to rare-earth impurities and Sm vacancies. Our measurements also reveal two additional forms of intrinsic magnetism: 1) underlying low-energy (~ 100 neV) weak magnetic moment (~ 10^{-2} mu_B) fluctuations similar to those detected in the related candidate TKI YbB12 that persist down to millikelvin temperatures, and 2) magnetic fluctuations consistent with a 2.6 meV bulk magnetic excitation at zero magnetic field that appears to hinder surface conductivity above ~ 4.5 K. We discuss potential origins of the magnetism.

Equations

$$A(t)=a_0{G}_{\mathrm{KT}}(\Delta ,t,H_{\mathrm{LF}})e^{-[\lambda (T)t{]}^{\beta }},$$

$$\lambda (H_{\mathrm{LF}})=\frac{\lambda (H_{\mathrm{LF}}=0)}{1+{\left({\gamma }_{\mu }{H}_{\mathrm{LF}}\tau \right)}^2},$$

Full text

1. underlying low-energy ($\sim\!100$ neV) weak magnetic moment ($\sim 10^{-2}$ $\mu_{\rm B}$) fluctuations similar to those detected in the related candidate TKI YbB12 that persist down to millikelvin temperatures, and 2) magnetic fluctuations consistent with a 2.6 meV bulk magnetic excitation at zero magnetic field that appears to hinder surface conductivity above $\sim\!4.5$ K. We discuss potential origins of the magnetism.

In recent years there has been a concerted effort to determine whether the intermediate-valence compound SmB6 is a strongly correlated three-dimensional (3-D) topological insulator (TI). A 3-D TI possesses an insulating bulk and topologically-protected metallic surface states, where the electron spin is locked perpendicular to the crystal momentum by strong spin-orbit coupling Hasan:10 ; Qi:11 . What makes SmB6 so different from known 3D TIs Hsieh:2008 ; Xia:2009 ; Chen:2009 is that it hosts an unconventional insulating bulk gap that forms due to Kondo hybridization of itinerant Sm $5d$ electrons with localized Sm $4f$ states. As expected for a 3-D TI, experiments on SmB6 have established that metallic surface states dominate the electrical transport below $T\!\sim\!5$ K Kim:13 ; Wolgast:13 ; Syers:15 and a truly insulating bulk exists down to at least 2 K Eo:19 . Yet there is ongoing debate as to whether the surface states are of topological origin. While scanning tunneling microscopy experiments support the existence of heavy in-gap topological Dirac fermion states at the predicted locations in the surface Brillouin zone Pirie:18 , Dirac points have yet to be clearly observed by angle resolved photoemission spectroscopy (ARPES) Xu:14a . Furthermore, spin-polarized ARPES experiments aimed at determining whether the surface states have the topological property of spin-momentum locking have reached very different conclusions Xu:14b ; Hlawenka:18 .

While a true Kondo insulator is non-magnetic, low-temperature magnetism is clearly present in SmB6. There is a field-dependent divergence of the temperature dependence of the bulk magnetic susceptibility $\chi(T)$ below $\sim\!15$ K, originally attributed to bare Sm3+ ($4f^{5}$) magnetic moments, but later ascribed to paramagnetic rare-earth impurities incorporated during sample growth Roman:97 ; Gabani:02 . Magnetic impurities, which can destroy the topological protection of surface states by breaking time-reversal symmetry, are also responsible for a large field-induced enhancement of the thermal conductivity Boulanger:18 .

There is also evidence for intrinsic bulk magnetic excitations in SmB6. A 14 meV bulk spin exciton has been detected by INS Alekseev:93 ; Alekseev:95 ; Fuhrman:15 and there are reports of lower-energy bulk magnetic excitations potentially relevant to the temperature range over which SmB6 exhibits topological behavior. Nuclear magnetic resonance (NMR) measurements indicate the existence of intrinsic bulk magnetic in-gap states separated from the conduction band by a 2.6 meV gap that shrinks with increasing field Caldwell:07 and $\mu$SR experiments detect slowly fluctuating internal magnetic fields that persist down to at least 0.02 K Biswas:14 ; Biswas:17 ; Akintola:17 . Recently, muon Knight shift measurements on SmB6 at $H\!=\!60$ kOe have provided evidence for bulk magnetic excitations governed by an $\sim\!1$ meV thermal activation energy Akintola:18 . While an additional $\lesssim\!1$ meV spin exciton is predicted Knolle:17 , the magnetic excitations at $H\!=\!60$ kOe may derive from the zero-field-extrapolated 2.6 meV magnetic in-gap states detected by NMR. A $\lesssim\!2.6$ meV spin exciton may hinder topological behavior via spin-flip scattering of the metallic surface states Kapilevich:15 ; Arab:16 . Surprisingly, however, no collective magnetic excitation has been detected by INS below 14 meV Fuhrman:17 .

Here we report new zero-field (ZF) and longitudinal-field (LF) $\mu$SR measurements of SmB6 that enable us to disentangle extrinsic and intrinic sources of low-temperature magnetism. Our measurements were performed on a mosaic of hundreds of randomly oriented small aluminum (Al) flux-grown SmB6 single crystals, and on a large doubly-isotope enriched 154Sm11B6 single crystal grown by the floating zone (FZ) method. The latter is the same 154Sm11B6 single crystal in which a 14 meV spin exciton has been detected by INS Alekseev:93 ; Alekseev:95 ; Fuhrman:15 .

Flux-grown SmB6 single crystals are known to contain Al inclusions Phelan:16 . Pure Al does not contain electronic moments, and while muons landing in the Al inclusions may sense static nuclear dipole fields, these are decoupled on the same field scale as the B nuclear moments in our LF experiments. Samarium (Sm) vacancies, which act as “Kondo holes” in the strongly-correlated state of SmB6, are more prevalent in FZ-grown single crystals Phelan:16 . Theoretically, a finite concentration of Sm vacancies introduces an impurity band in the hybridization gap and gives rise to a Curie-Weiss-like susceptibility Schlottmann:92 . They also adversely affect spin excitons, as evidenced by a Sm-vacancy induced suppression of the 16-18 meV exciton feature observed by Raman spectroscopy Valentine:16 .

Figure 1 shows representative $\chi(T)$ data for the two samples studied here and for one of the Al-flux grown single crystals investigated in Ref. Akintola:18 . No magnetic hysteresis was found in any of the samples. The magnetic susceptibility over much of the temperature range is a sum of contributions from the $4f^{6}$ Sm2+ and $4f^{5}5d^{1}$ Sm3+ ion configurations Nickerson:71 . There is a pronounced low-$T$ upturn in $\chi(T)$ for the current samples, and the overall susceptibility is greater in the larger 154Sm11B6 single crystal. Both features are clearly of extrinsic origin.

Figure 2 shows typical ZF- and weak LF-$\mu$SR asymmetry spectra for the 154Sm11B6 single crystal, which are reasonably described by

[TABLE]

where $G_{\rm KT}(\Delta,t,H_{\rm LF})$ is the static Gaussian Kubo-Toyabe function Hayano:79 intended to account for the temperature-independent relaxation caused by the nuclear moments. It assumes a Gaussian field distribution of width $\Delta/\gamma_{\mu}$ (where $\gamma_{\mu}/2\pi$ is the muon gyromagnetic ratio) and is dependent on the applied longitudinal field $H_{\rm LF}$. The LF-$\mu$SR spectra in Fig. 1(b) were recorded for $H_{\rm LF}\!=\!100$ Oe, which is sufficient to completely decouple the muon spin from the nuclear dipole fields. The stretched-exponential function in Eq. (1) accounts for additional sources of magnetic field in the sample. Global fits of the ZF and 100 Oe LF spectra assuming $\beta$ is independent of temperature, yield $\beta\!=\!0.562(5)$ and 0.552(9) for the Al-flux grown sample, and $\beta\!=\!0.699(2)$ and 0.658(3) for the FZ-grown single crystal. The ZF fits also yield $\Delta\!=\!0.2336(6)$ $\mu$s*-1* and $\Delta\!=\!0.2589(7)$ $\mu$s*-1* for the Al-flux and FZ grown samples, respectively. The ZF values of $\Delta$ and $\beta$ are somewhat different from those obtained in previous $\mu$SR studies of SmB6 Biswas:14 ; Biswas:17 ; Akintola:17 , highlighting variations in sample quality.

The temperature dependence of the fitted values of $\lambda$ for the two samples is shown in Fig. 3. Below $\sim\!20$ K, the ZF value of $\lambda$ increases more rapidly in the 154Sm11B6 single crystal. This behavior is qualitatively similar to previous findings Biswas:14 , although the difference between the FZ and Al-flux grown single crystals here is more extreme. In the earlier ZF-$\mu$SR studies, a broad peak in $\lambda(T)$ was observed near 4 to 5 K Biswas:14 ; Biswas:17 ; Akintola:17 . The sharpness of this feature, however, is sample dependent. Here both samples display a maximum in $\lambda(T)$ for ZF near 4.5 K. Below $\sim\!10$ K, the 100 Oe LF and ZF values of $\lambda(T)$ diverge in both samples. The significant reduction of $\lambda$ by the 100 Oe field indicates the gradual development of weak local quasi-static magnetic fields as the temperature is lowered toward 2 K. In what follows, we demonstrate via LF-$\mu$SR results up to 4 kOe that this magnetism is dependent on the sample preparation.

Above $\sim\!20$ K we find the stretched-exponential relaxation function in Eq. (1) may be replaced by a pure exponential. Figure 4(a) shows $\lambda(H_{\rm LF})$ at 50 K for both samples, obtained from an analysis with $\beta\!=\!1$. Biswas et al. Biswas:14 previously showed that the relaxation rate $\lambda(H_{\rm LF})$ below $H_{\rm LF}\!\sim\!100$ Oe exhibits a broad peak centered near 40 Oe due to an avoided level crossing resonance (ALCR) — presumably due to a matching of the Zeeman splittings of the muon and B nuclear spins. While the influence of the ALCR is evident below 100 Oe, at higher field $\lambda(H_{\rm LF})$ is independent of field and identical in the two samples. Moreover, the average value of $\lambda(H_{\rm LF})$ between 100 Oe and 4 kOe is in good agreement with the ZF values of $\lambda$ at 50 K in Fig. 3. Thus it is clear that the muons sense fast fluctuating internal fields of a similar rate in both samples at 50 K.

Figures 4(b) and (c) show $\lambda(H_{\rm LF})$ at 4.5 K and 2 K obtained from fits to Eq. (1) with $\Delta$ and $\beta$ fixed to the values determined from the analysis of the ZF-$\mu$SR spectra for each sample. In addition, we show results for the 154Sm11B6 single crystal at 2 K from fits assuming the Al-flux grown value $\beta\!=\!0.562$. Above 100 Oe there is good agreement between $\lambda(H_{\rm LF})$ for the two samples and the data is well described by the Redfield formula Schenck:1985

[TABLE]

where $\lambda(H_{\rm LF}\!=\!0)\!=\!2\gamma_{\mu}^{2}\langle B_{\rm loc}^{2}\rangle\tau$ and $\langle B_{\rm loc}^{2}\rangle$ is the mean of the square of the transverse components of a local magnetic field fluctuating at a rate $1/\tau$. Equation (2) is strictly valid for fast field fluctuations in a Gaussian distribution with a single fluctuation rate $1/\tau$, whereas a stretched-exponential relaxation often signifies a distribution of fluctuation rates. Nevertheless, the Redfield equation is adequate for achieving an approximate quantitative understanding of the data, provided the applied field does not modify the magnetic fluctuation spectrum. A fit of the $\lambda(H_{\rm LF}\!>\!100$ Oe) data for the Al-flux grown single crystals to Eq. (2) yields $\lambda(H_{\rm LF}=0)\!=\!0.0361(6)$ $\mu$s*-1*, $\tau\!=\!2.15(6)\times 10^{-8}$ s and $B_{\rm loc}\!=\!10.8(5)$ G at 2 K, and $\lambda(H_{\rm LF}=0)\!=\!0.072(4)$ $\mu$s*-1*, $\tau\!=\!0.9(1)\times 10^{-8}$ s and $B_{\rm loc}\!=\!23(5)$ G at 4.5 K. We note that similar values are obtained from fits where $\beta$ is free to vary with $H_{\rm LF}$ Supplemental . The fitted value of $\lambda(H_{\rm LF}\!=\!0)$ at 2 K is nearly 3.5 times smaller than the ZF value of $\lambda$ for the Al-flux grown SmB6 single crystals, and $\sim\!23$ times smaller than the ZF value of $\lambda$ for the FZ-grown 154Sm11B6 single crystal (see Fig. 3). This implies that a weak LF completely decouples the muon spin from a source of bulk magnetism, distinct from that of the nuclear moments. Since the difference between the fitted value of $\lambda(H_{\rm LF}\!=\!0)$ and the ZF value of $\lambda$ is much greater for the 154Sm11B6 single crystal, the magnetism is likely due to a greater concentration of Sm vacancies and perhaps rare-earth impurities. Contrarily, the similarity of $\lambda(H_{\rm LF})$ for the two samples above 100 Oe indicates that there is at least one other source of intrinsic bulk magnetism, which at 2 K gives rise to fluctuating magnetic fields of frequency on the order of $10^{7}$ Hz.

In the absence of a low-energy spin exciton, the existence of the intrinsic magnetism is surprising. The implanted positive muon ($\mu^{+}$) senses the localized Sm-$4f$ moments via the magnetic dipole interaction and through an indirect RKKY interaction that spin polarizes the conduction electrons at the muon site. In zero field, the total field at the $\mu^{+}$ site ${\bf B}_{\mu}$ is the vector sum of the corresponding dipolar (${\bf B}_{\rm dip}$) and hyperfine contact (${\bf B}_{\rm hf}$) fields. The opening of the Kondo gap in SmB6 is complete by $T\!\sim\!30$ K Xu:14a ; Zhang:13 . Consequently, ${\bf B}_{\rm dip}$ and ${\bf B}_{\rm hf}$ are expected to vanish at $T\!<<\!30$ K, due to complete screening of the Sm-$4f$ moments by the conduction electrons and the absence of a screening cloud of conductions electrons about the $\mu^{+}$.

If the intrinsic magnetism is associated with populating a low-energy spin exciton state, the ZF relaxation rate should obey an Arrhenius law $\lambda\!=\!\lambda_{0}\exp(E_{\rm a}/k_{\rm B}T)\!\propto\!\tau$. Unfortunately, any such behavior for the 154Sm11B6 single crystal is masked by the large Sm vacancy/impurity contribution. The ZF relaxation rate for the Al-flux grown sample does exhibit an Arrhenius behavior for $T\!\geq\!10$ K, characterized by an activation energy $E_{\rm a}\!=\!2.25(8)$ meV (Fig. 3, inset). Combined with the lower value $E_{\rm a}\!\sim\!1$ meV determined from 60 kOe transverse-field $\mu$SR measurements of the relaxation rate in similar Al-flux grown single crystals Akintola:18 , these results are compatible with the field-dependent contribution to the 11B NMR spin-lattice relaxation rate — which has been explained by in-gap magnetic states separated from the conduction band by a 2.6 meV gap that shrinks with increasing field and closes by 140 kOe Caldwell:07 . An $\sim\!2.6$ meV zero-field gap has also been observed by magnetotransport measurements Shahrokhvand:17 , and in the low-energy electrodynamic response spectra of SmB6 in the far-infrared range Gorshunov:99 . A 2.6 meV magnetic exciton is predicted to arise from a competition between the magnetic $4f^{5}$ and non-magnetic $4f^{6}$ multiplets Shick:15 . Based on solutions of the single-site Anderson impurity model, 1.74 $\mu_{B}$ local magnetic moments are generated via excitation of a triplet state situated 2.6 meV in energy above an intermediate valence non-magnetic singlet ground state. However, one then expects a sharp neutron energy transfer peak at 2.6 meV or a dispersive peak associated with a collective mode of the localized moment system due to intersite interactions. The $\sim\!2.6$ meV magnetic excitation may instead be connected to recent theoretical work that shows donor impurities in SmB6 produce a mid-gap impurity band with a corresponding ionization energy of 1 to 5 meV Skinner:19 .

Below 4.5 K, the LF results clearly show that spin freezing in SmB6 has an extrinsic origin and there exist weak intrinsic fields ($B_{\rm loc}\!\sim\!10.8$ G) that fluctuate too slow ($1/\tau\!\sim\!47$ MHz) to originate from a magnetic excitation gap on the order of 1 meV. The latter is similar to $\mu$SR findings in the Kondo insulator YbB12, which indicate the presence of slowly fluctuating ($1/\tau\!\sim\!60$ MHz) weak internal fields ($B_{\rm loc}\!=\!5.4$ G) below $\sim\!5$ K Yaouanc:99 . The values of $B_{\rm loc}$ are consistent with very small localized Sm and Yb magnetic moments ($\sim\!10^{-2}\mu_{\rm B}$). Since there is no further change in $\lambda$ down to millikelvin temperatures Biswas:14 ; Akintola:17 ; Yaouanc:99 , the small magnetic moment may indicate that the carrier density in these compounds is insufficient to completely Kondo screen the localized $4f$ magnetic moments. On the other hand, $\mu$SR experiments on CaB6 and BaB6 show the emergence of small random magnetic moments ($\sim\!10^{-2}\mu_{\rm B}$/B) below $\sim\!130$ K Kuroiwa:06 , perhaps associated with intrinsic defects detected in transport measurements of alkaline-earth-metal hexaborides Stankiewicz:16 . Here the similarity of the data for the FZ and Al-flux grown samples in Fig. 4(c) seems to rule out Sm vacancies, but the weakly dynamic small-moment magnetism may originate from other kinds of intrinsic defects.

In summary, we have shown there are multiple sources of low-$T$ magnetism in SmB6. In addition to underlying persistent slowly fluctuating weak moments, our experiments provide additional support for a bulk $\sim\!2.6$ meV magnetic excitation. The results on the neutron 154Sm11B6 sample demonstrate that enhanced Sm vacancies do not explain why a magnetic excitation of this energy is not observed by INS. Since optical conductivity experiments link the $\sim\!2.6$ meV excitation to charge carrier localization Gorshunov:99 , the origin might be connected to intrinsic defects and the creation of very small magnetic moments — such that the neutron cross section is below the noise floor even in the large 154Sm11B6 sample. A very weak broad “hump” seen in the specific heat of a FZ-grown isotopically-enriched SmB6 sample between 4 K to 10 K Orendac:17 may be a manifestation of this magnetic excitation, as this feature is also observed in the specific heat of our samples. However, in contrast to the strong magnetic field dependence of the 11B NMR $1/T_{1}$ maximum and $\mu$SR relaxation rate in this temperature range, the field dependence of this very small specific heat hump is almost negligible up to 140 kOe. Lastly, we note that there is a similar 2.7 meV electronic excitation Gorshunov:06 and an analogous field-dependent 11B NMR $1/T_{1}$ maximum Shishiuchi:02 in YbB12, suggesting the sources of the intrinsic low-$T$ magnetism in these two candidate TKIs are the same.