## Abstract

The Kondo insulator SmB_{6} is purported to develop into a robust topological insulator at low temperatures. Yet there are several puzzling and unexplained physical properties of the insulating bulk. It has been proposed that bulk spin excitons may be the source of these anomalies and may also adversely affect the topologically protected metallic surface states. Here, we report muon spin rotation measurements of SmB_{6} that show thermally activated behavior for the temperature dependence of the transverse-field relaxation rate below 20 K and a decreasing contact hyperfine field contribution to the positive muon Knight shift below 5–6 K. Our data are consistent with the freezing out of a bulk low-energy (~1 meV) spin exciton concurrent with the appearance of metallic surface conductivity. Furthermore, our results support the idea that spin excitons play some role in the anomalous low-temperature bulk properties of SmB_{6}.

## Introduction

Due to a combination of spin–orbit coupling and time reversal symmetry, a topological insulator (TI) supports protected metallic edge and surface states in two-dimensional (2D) and three-dimensional (3D) systems, respectively.^{1} The ideal 3D TI has a truly insulating bulk gap, as this restricts applications of the transport properties to the topologically protected surface, where the electron spin is uniquely locked to the charge momentum. Yet true bulk insulating behavior is not realized in established TIs due to bulk impurity conduction.^{2} The homogeneous intermediate-valence compound SmB_{6} is a strong candidate for a 3D TI with a robust bulk insulating gap.^{3} In contrast to a conventional band insulator, the insulating gap in SmB_{6} is created via Kondo hybridization of localized Sm-4f and itinerant Sm-5d electrons, with the Fermi level residing in the hybridization gap.

Experimental evidence for SmB_{6} being a TI is provided by transport measurements that have demonstrated predominant surface electrical conduction below ~5–7 K,^{4,5,6} and the detection of in-gap surface states by angle-resolved photoemission spectroscopy (ARPES).^{7,8,9,10,11} However, recent high-resolution ARPES results suggest that the surface conductivity is not associated with topological surface states.^{12} Moreover, at low *T*, there is a sizable metallic-like *T*-linear specific heat of bulk origin,^{13} and significant bulk ac-conduction.^{14} Quantum oscillations are observed in the magnetization of SmB_{6} as expected for 2D metallic surface states,^{15} but subsequent measurements suggest that the origin is a bulk 3D Fermi surface.^{16} These findings have raised the possibility of charge-neutral fermions in the insulating bulk.^{17,18,19}

The Sm ions in SmB_{6} rapidly fluctuate between non-magnetic Sm^{2+} (4f^{6}) and magnetic Sm^{3+} (4f^{5}5d^{1}) electronic configurations, resulting in an average intermediate valence that varies with temperature.^{20,21} Interestingly, SmB_{6} exhibits magnetic fluctuations below 20–25 K where the Kondo gap is fully formed, as observed by muon spin rotation (μSR).^{22,23} This was first presumed^{22} to be due to the bulk magnetic in-gap states detected by nuclear magnetic resonance below 20 K,^{24} and later specifically speculated to be due to bulk spin excitonic excitations.^{25} Spin excitons in SmB_{6} are induced by residual dynamic AFM exchange interactions between the hybridized quasiparticles and are a precursor to an AFM instability.^{26,27} A 14 meV bulk collective mode observed within the hybridization gap by inelastic neutron scattering (INS) has been interpreted as a spin exciton.^{28,29} Bulk spin excitons are expected to adversely affect the protected topological order by causing spin-flip scattering of the surface states.^{30} There is some indirect evidence for this from angle-integrated photoemission^{31} and planar tunneling^{32} spectroscopy studies.

Since an implanted positive muon (μ^{+}) does not create a spin-exciton excitation, μSR is only sensitive to thermally activated spin excitons. Consequently, the 14 meV bulk spin exciton observed by INS should not be detectable by μSR below 20–25 K. An additional lower energy (≤1 meV) spin-exciton branch has recently been predicted and suggested to contribute to bulk quantum oscillations and cause the anomalous upturn in the specific heat at low *T*.^{33} On the other hand, the low-*T* specific heat is enhanced by Gd impurities^{34} and drastically reduced in isotopically enriched SmB_{6},^{35} suggesting that magnetic impurities play some role but spin excitons do not. Nevertheless, decoupling of the surface states from an ~4 meV bulk spin exciton has been argued to explain the rapid increase in the surface conductance below 5–6 K and subsequent saturation below 4 K.^{32} While the reduction in energy from 14 to 4 meV is assumed to be due to a diminished Kondo temperature at the surface,^{30} a distinct low-energy bulk spin exciton is another possibility.

Here, we report high transverse-field (TF)-μSR measurements of the μ^{+}-Knight shift in Al-flux grown SmB_{6} single crystals that provide evidence for a bulk spin exciton of energy much lower than 14 meV. Specifically, our data demonstrate the occurrence of thermally activated low-energy magnetic excitations that are characteristic of an ~1 meV spin exciton.

## Results

Figure 1 shows the temperature dependence of the bulk magnetic susceptibility *χ*_{mol} at different magnetic fields **H** applied parallel to the *c*-axis. At high temperature, *χ*_{mol} exhibits Curie–Weiss behavior indicative of paramagnetic Sm ions. Opening of the hybridization gap below 110 K gives rise to the broad maximum, followed by a field-dependent upturn below *T* ~17 K that masks the expected low-*T* van-Vleck saturation. The upturn has previously been attributed to impurities.^{36,37}

In the absence of spontaneous magnetic order, the applied field polarizes the conduction electrons and induces spin polarization of the localized Sm-4f magnetic moments. The local field **B**_{μ} sensed by the μ^{+} is the vector sum of the dipolar field **B**_{dip} from the Sm-4f magnetic moments and a contact hyperfine field **B**_{c}. At high *T* where SmB_{6} behaves as a poor metal, the muon’s positive charge is screened by a cloud of conduction electrons. The screening electron cloud acquires a finite spin density due to the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction with the spin-polarized Sm-4f moments, and by direct contact generates a hyperfine field at the μ^{+} site. This is expected to vanish with the development of a bulk insulating gap at lower *T*. In an insulating state, **B**_{c} may instead originate from direct overlap of the μ^{+} with the wavefunction of localized magnetic electrons, or from bonding of the μ^{+} to an ion that is covalently bonded to a local atomic magnetic moment.^{38}

Figure 2 shows Fourier transforms of TF-μSR time spectra recorded on SmB_{6} at *H* = 6 T. Due to the apodization necessary to remove ringing artifacts caused by the short muon time window (~10 μs) and noise caused by fewer counts at later times (due to the short muon lifetime), the Fourier transforms are a broadened visual approximation of the internal magnetic field distribution. Consequently, analysis of the TF-μSR signals was done in the time domain (see Supplementary Information for representative fits). At *T* = 200 K, there are three well-separated peaks in the Fourier transform. The central peak arises from muons stopping in the Ag backing plate. The left and right peaks have an amplitude ratio of 2:1, and are consistent with the μ^{+} stopping at the midpoint of the horizontal or vertical edges of the cubic Sm-ion sublattice. This is in agreement with the identified μ^{+} site in CeB_{6}.^{39} Moreover, we have verified the μ^{+} site assignment by TF-μSR measurements with **H** applied at 45° with respect to the *c*-axis (see Fig. S3).

In a field applied parallel to the *c*-axis, the μ^{+} site is magnetically inequivalent on the horizontal and vertical edges of the cubic Sm sublattice. The dipole field generated by polarization of the Sm-4f moments along the *c*-axis direction is equivalent and antiparallel to **H** at the (⊥) sites \(\left( {\frac{1}{2},0,0} \right)\) and \(\left( {0,\frac{1}{2},0} \right)\), and different in magnitude and parallel to **H** at the (∥) site \(\left( {0,0,\frac{1}{2}} \right)\) (see Fig. 3a inset). Consequently, the TF-μSR time spectra were fit to an asymmetry function with a two-component sample contribution (see Fig. S4).

Here, *A*_{s} and *A*_{Ag} denote the initial asymmetries of the sample and Ag backing plate contributions, respectively. Also, *ν*_{i} = (*γ*_{μ}/2*π*)*B*_{μ,i} and \({\mathrm{\Delta }}_i^2 = \gamma _\mu ^2\left\langle {{\mathbf{B}}_{\mu ,i}^2} \right\rangle\), where *γ*_{μ}/2*π* = 135.54 MHz/T is the muon gyromagnetic ratio, and *B*_{μ,i} and \(\left\langle {{\mathbf{B}}_{\mu ,i}^2} \right\rangle\) are the local magnetic field and width of the field distribution at the μ^{+} sites (*i* = ⊥, ∥, and Ag), respectively. The initial phase of the muon spin polarization is denoted by ϕ. As the temperature is lowered, *ν*_{⊥} (*ν*_{∥}) increases (decreases), and below *T* ~30 K the ⊥ (∥) peak in the Fourier transform broadens and moves to the far right (left). Even so, the two sample components are observed to maintain a population ratio of 2:1 down to 1.9 K. We note that the 6 T applied magnetic field is far below the field of 80–90 T required to close the insulating gap.^{40}

The relative frequency shift is defined as *K*_{μ,i} = (*ν*_{i} − *ν*_{0})/*ν*_{0}, where *ν*_{0} = (*γ*_{μ}/2*π*)*H*. After correcting for the demagnetization and Lorentz fields, the μ^{+}-Knight shift at the magnetically inequivalent muon sites is:

where \(A_{\mathrm{c}}^i\) and *A*_{dip} are the contact hyperfine and dipolar coupling constants, *χ*_{4f} = *χ*_{mol} − *χ*_{0} is the local 4f magnetic susceptibility, and *χ*_{0} and \(K_0^i\) are the Pauli paramagnetic susceptibility and the corresponding Knight shifts, respectively.

Figure 3 shows the temperature dependence of the Knight shifts *K*_{i} and TF relaxation rates Δ_{i}. In contrast to *χ*_{mol}, the Knight shifts below 17 K do not exhibit an appreciable field dependence. We attribute the large uncertainty and jump or dip in Δ_{⊥} and Δ_{∥} between 20 and 40 K to fit parameters playing off against each other, as *ν*_{⊥} ≈ *ν*_{∥} in this temperature range.

Figure 4a shows the Knight shift data plotted vs. *χ*_{mol} with temperature as an implicit parameter (a so-called Clogston–Jaccarino plot). We find that a plot of *K*_{⊥} − *K*_{∥} vs. *χ*_{mol} above 110 K (not shown) is linear as expected from Eqs. (2a) and (2b), but has a slope and intercept incompatible with *A*_{c} and *K*_{0} being isotropic. Using the calculated value *A*_{dip} = 0.395 T/μ_{B} for the μ^{+} site, linear fits of the Knight shifts in SmB_{6} above 110 K to Eqs. (2a) and (2b) yield \(A_{\mathrm{c}}^ \bot = 0.346\,{\mathrm{T}}/{\mathrm{\mu }}_{\mathrm{B}}\) and \(A_{\mathrm{c}}^\parallel = - 0.378\,{\mathrm{T}}/{\mathrm{\mu }}_{\mathrm{B}}\). This anisotropy can be explained by the influence of the Sm 4f^{5}-shell electric quadrupole moment^{41} on the conduction electron spin polarization at the μ^{+} site, which has been observed in other compounds with non-spherical f-electron distributions.^{42}

Below 110 K, the *K*_{i} vs. *χ*_{mol} plots deviate from linearity (see Fig. 4a). Point-contact spectroscopy^{43} and ARPES^{44} measurements on SmB_{6} show that the hybridization gap develops over a fairly wide temperature range of 30 K ≤ *T* ≤ 110 K. This results in a loss of scaling between *K*_{i} and *χ*_{mol}, due to a gradual reduction of both the Pauli susceptibility \(\left( { \propto K_0^i} \right)\) and the electronic spin density at the μ^{+} sites \(\left( { \propto A_{\mathrm{c}}^i} \right)\). Near 30 K, the simultaneous vanishing of *K*_{⊥} and *K*_{∥} implies \(K_0^i = 0\) and *χ*_{4f} = 0 in Eqs. (2a) and (2b).

Below *T* ~25 K, a μ^{+}-Knight shift reappears, which does not linearly scale with *χ*_{mol} (Fig. 4a) and is accompanied by an increase in the TF relaxation rates with decreasing *T* (Fig. 3b).

## Discussion

The μ^{+}-Knight shift below 25 K is a property of the insulating bulk. In insulators and semiconductors, the μ^{+} sometimes forms a bound state with an electron, known as a *muonium* atom (Mu).^{38} The signature of Mu in high TF is a pair of frequencies separated by the Mu hyperfine splitting and centered on the precession frequency of the free μ^{+} in the applied field—the latter being close to the μ^{+} precession frequency in the Ag backing plate. This is clearly not observed in Fig. 2. Consequently, the μ^{+}-Knight shift must still be induced by the Sm-4f moments.

The lack of scaling of *K*_{i} with the bulk magnetic susceptibility *χ*_{mol} below 25 K could potentially arise from the charged muon significantly altering the Sm^{3+} crystal electric field (CEF) level scheme and hence *χ*_{4f}. A significant influence of the μ^{+} on the local magnetic susceptibility has been identified in a few Pr^{3+}-ion systems.^{45,46} The CEF level scheme of Sm^{3+} (4f^{5}) in SmB_{6} is similar to Ce^{3+} (4f^{1}) in CeB_{6}. In both cases, the spin–orbit interaction splits the 4f states into *J* = 5/2 and *J* = 7/2 multiplets. The *J* = 5/2 multiplet is further split in the cubic crystalline field into a Γ_{7} doublet, and a ground-state Γ_{8} quartet that has magnetic and quadrupolar moments. The energy difference between the Γ_{8} quartet and excited Γ_{7} doublet is about 15 meV in SmB_{6},^{47} and 46 meV in CeB_{6},^{48} which in both compounds exceeds the Kondo energy scale (temperature). Thus, only modifications of the Zeeman split Γ_{8} quartet are relevant in the low *T* regime. In CeB_{6}, which does not develop a Kondo insulating gap, *K*_{i} linearly scales with *χ*_{mol} above 10 K. Hence, it is unlikely that the μ^{+} induces the Knight shift observed in SmB_{6} below 20 K. We note that the loss of scaling between *K*_{i} and *χ*_{mol} in CeB_{6} below 10 K is due to the development of antiferroquadrupolar ordering,^{49} which does not occur in SmB_{6}.

According to Eqs. (2a) and (2b), there must be a new contact hyperfine field **B**_{c} to cause the sign change in the values of *K*_{⊥} and *K*_{∥} below 25 K. As mentioned earlier, in an insulating state, this may result from the μ^{+} bonding to an ion that is covalently bonded to a localized magnetic electron. A super-transferred hyperfine field at the μ^{+} site through a Sm-B-μ^{+} connection could arise from field-induced moments at the B sites. In CeB_{6}, field-induced magnetic moments inside or around the B_{6} octahedron have been ruled out by polarized neutron diffraction,^{50} which is presumably also the case in SmB_{6}. Moreover, the formation of a B-μ^{+} bond is incompatible with the μ^{+} site, which is ~2 Å from the nearest B atom.

The alternative possibility in an insulating state is that **B**_{c} originates from direct overlap of the μ^{+} with the wavefunction of the localized magnetic electrons. While the Sm-4f orbitals are highly localized, the 5d orbitals of the nearest-neighbor Sm ions partially overlap the μ^{+} site. In a spin exciton, the spin polarization of the bound 5d electron is coherently coupled to the localized 4f electrons, and a contact hyperfine field may result from an exchange interaction between the μ^{+} and the extended magnetic 5d electrons.

The temperature dependence of the TF-μSR relaxation rate (Fig. 3b) provides evidence for a low-energy spin exciton. The marked increase of Δ_{⊥} and Δ_{∥} below 20 K corresponds to an increase in the width of the local field distribution, indicative of a gradual slowing down of magnetic fluctuations. As shown in the inset of Fig. 3b, the *H* = 6 T data for Δ_{⊥} above 5 K can be fit with a thermally activated law: Δ_{⊥} = *A* + *B*exp(*E*_{A}/*k*_{B}*T*), yielding *A* = 0.35 ± 0.07 μs^{−1}, *B* = 0.12 ± 0.04 μs^{−1}, and *E*_{A} = 0.99 ± 0.12 meV. The temperature-independent contribution comes from the nuclear dipole moments and the spatial inhomogeneity of the applied magnetic field. The thermally activated decrease of Δ_{⊥} is consistent with a rising fluctuation rate 1/*τ*, where 1/*τ*∝exp(−*E*_{A}/*k*_{B}*T*) and *τ* is the average correlation time. Spin excitons create fluctuating regions of AFM correlations extending over a few unit cells, which modify the local field sensed by the μ^{+}. The increase in Δ_{i} is explained by AFM amplitude fluctuations perpendicular to **H**, producing a small temporary canted moment. This is presumably of order *H*/*J*_{RKKY}, where *J*_{RKKY} is the virtual RKKY-like magnetic exchange interaction between the 4f moments in the theory of Riseborough.^{26,27} The nearly constant difference between Δ_{i} at 6 T and 4 T above 6 K is primarily caused by a difference in the inhomogeneity of the applied field. Below 5–6 K, however, Δ_{i} saturates and exhibits an intrinsic increase with *H*. The saturation indicates that the average fluctuation period of the spin excitons (*τ*) has become large with respect to the muon time window, such that Δ_{i} is no longer significantly affected by fluctuations. The corresponding field dependence of Δ_{i} below 5–6 K reflects an inhomogeneous broadening of the local field distribution.

Figure 4b shows a strong temperature dependence of *K*_{⊥}/*K*_{∥} at 6 ≤ *T* ≤ 20 K, which with \(K_0^i = 0\) reflects the behavior of \(\left( {A_{\mathrm{c}}^ \bot - A_{{\mathrm{dip}}}/2} \right)/\left( {A_{\mathrm{c}}^\parallel + A_{{\mathrm{dip}}}} \right)\). Changes in the lattice parameter^{20,51,52} below 20 K are too small to cause an appreciable change in *A*_{dip}. The remaining possibility is that \(A_{\mathrm{c}}^ \bot\) and \(A_{\mathrm{c}}^\parallel\) change with decreasing temperature. The anisotropy of *A*_{c} above 110 K indicates coupling of the non-spherical 4f-electron distribution to the field-induced Sm magnetic moments. Hence, the strong temperature dependence of \(A_{\mathrm{c}}^ \bot\) and \(A_{\mathrm{c}}^\parallel\) below 20 K is likely due to a rotation of the 4f-electron distribution with the canted moment induced by AFM fluctuations. We note that the values of \(A_{\mathrm{c}}^i\) are dependent on the overlap integral of the non-spherical 4f-electron distribution and the 5d electrons with wavefunctions that overlap the μ^{+} site. Below 6 K, there is an abrupt increase in *K*_{⊥}/*K*_{∥} to a value comparable to that at 25 K. This suggests that the 4f-electron distribution returns to being more closely aligned with the applied field.

As shown in Fig. 3a, there is an abrupt decrease (increase) in *K*_{⊥} (*K*_{∥}) below 5–6 K. According to Eqs. (2a) and (2b), these simultaneous behaviors cannot be explained by a rapid change in *χ*_{4f}(*T*). Instead, these behaviors appear to reflect the temperature dependence of *K*_{⊥}/*K*_{∥} (and hence \(A_{\mathrm{c}}^ \bot\) and \(A_{\mathrm{c}}^\parallel\)) that is apparent at *T* ≤ 5 K in Fig. 4b. As shown in Fig. 4, the temperature dependence of *K*_{⊥} for *T* ≤ 5 K is well fit with a thermally activated Arrhenius equation: *K*_{⊥}(*T*) = *K*_{⊥}(0) + *C*exp(−*E*_{A}/*k*_{B}*T*) that assumes the same value *E*_{A} = 0.99 meV obtained from the fit of the temperature dependence of Δ_{⊥}. This suggests that unlike the TF-μSR relaxation rate, the μ^{+}-Knight shift is sensitive to thermal activation of the spin exciton below 6 K, and in particular the shift in the time-average and field sensed by the muon ensemble associated with the hyperfine coupling between the μ^{+} and the bound 5d electron of the spin exciton. Nevertheless, there are insuffient μ^{+}-Knight shift data points below 6 K to establish true exponential-activated behavior.

The dynamic relaxation rate *λ*_{ZF} observed by zero-field (ZF) μSR develops below 20–25 K,^{22} exhibits a short anomalous peak near 4 K, and subsequently saturates. Our findings here suggest that the saturation is due to the freezing out of a bulk spin exciton of much lower energy than that observed by INS, giving way to AFM quantum spin fluctuations. We note that the peak in *λ*_{ZF}(*T*) near 4 K vanishes with the addition of a 0.5% Fe impurity.^{23} This is likely due to the predicted adverse effect of impurities on a fully developed spin exciton.^{27} We have carried out similar TF-μSR measurements on the 0.5% Fe-doped sample. Figure 5 shows a comparison of the low-temperature results with those for the pure compound. While the bulk magnetic susceptibility is significantly modified by the impurities, the changes to the temperature dependences of Δ_{⊥} and *K*_{⊥} are more subtle. A fit of the Δ_{⊥} vs. *T* data for the Fe-doped sample above 5 K to the thermally activated law described earlier yields *E*_{A} = 0.75 ± 0.16 meV. The smaller activation energy compared to pure SmB_{6} is consistent with the expected impurity-induced broadening and decrease of the binding energy of the spin exciton.^{27} Furthermore, the substantial effect of the dilute Fe impurity is consistent with the finding that impurities on the level of 1% are enough to destroy the spin exciton feature observed by Raman spectroscopy.^{53}

Our experiments provide evidence for thermally activated low-energy magnetic excitations in SmB_{6}. The results support the theoretical prediction of a low-energy spin exciton (≤1 meV) in SmB_{6}, which has been argued to account for certain low-temperature thermodynamic and transport anomalies.^{33} As shown in Fig. 4c, the saturation of the electrical sheet resistance *R*_{s} occurs below 4 K, where *K*_{⊥} is greatly reduced. This observation suggests that the resistivity plateau develops when spin exciton scattering of the metallic surface states becomes insignificant.

## Methods

### Samples and experimental technique

The growth and characterization of the crystals were previously reported.^{23} The high TF μ^{+}-Knight shift measurements were performed on a 5 × 5-mm^{2} mosaic of single crystals aligned with the *c*-axis parallel to the applied field and mounted on a pure Ag backing plate. The measurements utilized a He-gas flow cryostat and the so-called “NuTime” spectrometer at TRIUMF in Vancouver, Canada. All of the high TF-μSR measurements were recorded with the initial muon spin polarization **P**(0) perpendicular to the external magnetic field **H**, which was applied parallel to the muon beam momentum. Figure S1 shows a schematic of a custom sample holder designed for the μ^{+}-Knight shift measurements.

The value of the muon spin precession frequency in the applied magnetic field *H* alone, *ν*_{0} = (*γ*_{μ}/2*π*)*H*, was accurately determined by first simultaneously recording the TF-μSR signal in a 99.998% pure Ag mask located upstream of the sample. To correct for the difference in the external field at the Ag mask and sample locations, TF-μSR measurements were also simultaneously performed on the Ag mask with Ag in place of SmB_{6} at the sample location.

### Data availability

All relevant data are available from the authors upon request.

## Additional information

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## Acknowledgements

This work was performed at TRIUMF, the University of Maryland (UofM), and Simon Fraser University (SFU). The authors wish to thank E. Mun, J.S. Dodge, and P.S. Riseborough for informative discussions. J.E.S. acknowledges support from NSERC of Canada. J.P. acknowledges support from AFOSR through Grant No. FA9550-14-1-0332 and the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant No. GBMF4419.

## Author information

### Affiliations

#### Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

- Kolawole Akintola
- , Anand Pal
- , Sarah R. Dunsiger
- , Alex C. Y. Fang
- , Matt Potma
- & Jeff E. Sonier

#### Kwantlen Polytechnic University, Richmond, BC, V6X 3X7, Canada

- Matt Potma

#### Department of Physics, Center for Nanophysics and Advanced Materials, University of Maryland, College Park, MD, 20742, USA

- Shanta R. Saha
- , Xiangfeng Wang
- & Johnpierre Paglione

#### The Canadian Institute for Advanced Research, Toronto, ON, M5G 1Z8, Canada

- Johnpierre Paglione
- & Jeff E. Sonier

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### Contributions

S.R.S., X.F.W., and J.P. grew and characterized the sample. K.A. performed the bulk magnetic susceptibility measurements. K.A., A.P., S.R.D., A.C.Y.F., M.P., S.R.S., and J.E.S. performed the high TF-μSR measurements at TRIUMF. K.A. and J.E.S. carried out the data analysis. K.A. and J.E.S. wrote the manuscript with input from all co-authors.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Jeff E. Sonier.

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