Spin excitations in the quantum dipolar magnet Yb(BaBO₃)₃

The temperature dependence of the dc magnetic susceptibility $\chi$ at $\mu_{0}H$ = 1 T down to 2 K is shown in Fig. 2(a). Neither a phase transition nor bifurcations between ZFC and FC curves was observed, indicating the lack of a magnetic ordering or spin freezing down to 2 K. In the inset of Fig. 2(a), we present the inverse of magnetic susceptibility $\chi^{-1}$ at $\mu_{0}H$ = 1 T applied parallel and perpendicular to the $c$ axis of single crystal Yb(BaBO₃)₃. A uniaxial easy axis anisotropy of temperature-dependent susceptibility is observed, $\chi_{H{\parallel}c}>\chi_{H{\perp}c}$. Above 200 K, the magnetic susceptibility data can be well fitted by Curie-Weiss law. $\chi^{-1}$ deviates from a linear dependence on temperature with temperature decreasing below 200 K due to the CEF effect Guo et al. (2019a). But the Curie-Weiss law is again valid at low temperatures (2-20 K). Such a behavior is also seen in NaBaYb(BO₃)₂ Guo et al. (2019a), KBaYb(BO₃)₂ Pan et al. (2021), and RbBaYb(BO₃)₂ Guo et al. (2019b), which share a similar chemical composition with Yb(BaBO₃)₃.

The Curie-Weiss fitting in the 200-300 K temperature range yielded the effective magnetic moments for $H\perp c$ and $H\parallel c$ are 4.38 $\mu_{\rm B}$ and 5.06 $\mu_{\rm B}$ respectively, which are comparable to the free Yb³⁺ ion (4$f^{13}$, ²F_7/2, $J=7/2$, 4.54 $\mu_{\rm B}$). The negative Curie-Weiss temperatures $\theta_{\perp,H}$ = -127.1(2) K and $\theta_{\parallel,H}$ = -182.9(4) K indicate antiferromagnetically coupled Yb³⁺ spins in Yb(BaBO₃)₃. The low-temperature Curie-Weiss fitting yields $\theta_{\perp,L}$ = -1.40(7) K and $\theta_{\parallel,L}$ = -1.16(4) K, suggesting small exchange interactions with effective magnetic moments 2.13 $\mu_{\rm B}$ ($H\perp c$) and 2.54 $\mu_{\rm B}$ ($H\parallel c$). The large difference compared with the moment of a free Yb³⁺ ion, was also reported in other Yb-based triangular lattices Guo et al. (2019b); Baenitz et al. (2018); Ding et al. (2019).

The isothermal magnetization up to 7 T applied parallel and perpendicular to the $c$ axis of the sample at 2 K is shown in Fig. 2(b). $M(H)$ curves start to show a non-linear behavior with magnetic field above 2 T for both orientations, and magnetization tends to saturate above 4 T, indicating small exchange interactions in Yb(BaBO₃)₃. The saturation value of magnetization is 1.60 $\mu_{\rm B}$/Yb and 1.05 $\mu_{\rm B}$/Yb for field along and perpendicular to $c$ axis respectively, which is consistent with the result reported previously Zeng et al. (2020).

In Fig. 3(a), we present the specific heat coefficient $C/T$ of Yb(BaBO₃)₃ at several applied magnetic fields, and $C/T$ of Lu(BaBO₃)₃ for comparison. No sharp anomaly for magnetic transitions is observed, which is consistent with the magnetization measurements. With magnetic fields applied, the magnetic specific heat $C_{\rm mag}$ of Yb(BaBO₃)₃ obtained by subtracting the phonon contribution shows a peak, which retains its amplitude but shifts to higher temperatures with increasing applied magnetic fields (Fig. 3(b)), indicating a typical Schottky anomaly. The peaks can be fitted using the two-level Schottky function:

where $n$ is the concentration of Schottky centers, $R$ is the molar gas constant, and $\Delta$ is the energy separation between two levels. As shown in Fig. 3(c), $n\approx 1$ for the applied magnetic fields larger than 0.7 T, indicating the two levels are completely opened when $\mu_{0}H>0.7$ T. The energy gap $\Delta$ shows a linear behavior with applied magnetic fields, reflecting a Zeeman splitting effect. It is worth noting that the magnetic specific heat at zero field cannot be fitted well by Eq. (1), which will be discussed later. The entropy calculated by integrating $C_{\rm mag}/T$ reaches $R$ln2 for data with magnetic fields larger than 0.7 T, which is consistent with $n\approx 1$. While for zero field, the entropy at 0.3 K only reaches 2.5$\%$ of $R$ln2, indicative of a considerable remaining entropy below 0.3 K.

In general, hyperfine interactions in lanthanide with $4f$ electrons provide nuclear energy levels, which result in nuclear Schottky anomaly Tari (2003). The nuclear Schottky anomalies at zero field usually occur in the temperature region of 10^-2 K, and an external applied field on the order of 10 T cannot make a significant difference on the nuclear Schottky effect Tari (2003); Grivei et al. (1995). For $C/T$ of Yb(BaBO₃)₃, we observe remarkable shifts of the Schottky peak with the applied magnetic fields of a few Tesla. Hence the nuclear Schottky can be excluded in accounting for the observed anomaly shown in Fig. 3(b). On the other hand, energy levels due to CEF effect can also lead to a Schottky anomaly. In Yb-based triangular lattices, $4f^{13}$ electrons of the Yb³⁺ ions form four Kramers doublets due to crystal electrical fields Ross et al. (2011); Li et al. (2016b); Onoda and Tanaka (2010). The energy gap between the ground-state Kramers doublet and the first excited state is on the order of 10 meV (i.e., 100 K) Li et al. (2017b); Ding et al. (2019); Baenitz et al. (2018); Dai et al. (2021). However, the energy gap of Yb(BaBO₃)₃ at zero field $\Delta(0)$ derived from the field dependence of the energy gap $\Delta(H)$ is nearly 0, which is particularly far from 100 K. Therefore the low temperature Schottky anomaly is not due to the the energy gap between the ground-state Kramers doublet and the first excited state.

We attribute the Schottky anomalies observed in specific heat of Yb(BaBO₃)₃ to the opening of the ground-state Kramers doublet with magnetic fields applied. At zero field, the increase of $C_{\rm mag}/T$ below 1 K is due to the electronic spin excitations, which may exist in some quantum magnets where exchange interaction is weak, and dipole-dipole interaction is dominant to create an anisotropy Quilliam (2010). Low-energy spin excitations with a gap resulting from such anisotropy have been reported in Gd₂Sn₂O₇ Quilliam et al. (2007). The magnetic specific heat, similar to the one for Yb(BaBO₃)₃, is proportional to $(1/{T^{2}})e^{-\frac{\Delta}{T}}$ at zero field. Spin excitations have also been reported in a dipolar magnet Yb₃Ga₅O₁₂ from the specific heat and inelastic neutron-scattering experiments Lhotel et al. (2021).

In Yb(BaBO₃)₃, both the small Curie-Weiss temperatures determined from low temperature susceptibility and the isothermal magnetization data suggest that the exchange interaction is weak. Considering that $J_{\pm}$ = 0.90 K, $J_{zz}$ = 0.98 K for YbMgGaO₄ Li et al. (2015b) and $J_{\pm}$ = 0.18 K, $J_{zz}$ = 0.23 K for NaBaYb(BO₃)₂ Pan et al. (2021), which have a similar structure and comparable distances between Yb³⁺ ions with Yb(BaBO₃)₃, the exchange interaction in Yb(BaBO₃)₃ is also likely to be on the order of 0.1 K. Indeed a thermodynamic property study has revealed that the dominant interaction is the long-range dipole-dipole coupling in Yb(BaBO₃)₃ Bag et al. (2021). The dipole-dipole interaction can induce an anisotropy for the magnetic properties Quilliam (2010). Such anisotropy may affect the degeneration of the ground-state Kramers doublet, creating gapped low-energy spin excitations at zero field and small magnetic fields (less than 0.7 T in the case of Yb(BaBO₃)₃).

As muon is extremely sensitive to small local magnetic fields, we performed ZF-$\mu$SR measurements on Yb(BaBO₃)₃ down to 270 mK. ZF-$\mu$SR spectra at several representative temperatures are shown in Fig. 4(a). Neither oscillations nor an initial asymmetry loss was observed throughout the whole temperature range, confirming the absence of long-range magnetic order or spin-freezing Zheng et al. (2005). Furthermore, the lack of polarization recovery to 1/3 of the initial asymmetry value rules out a static random field distribution Uemura et al. (1985), demonstrating the dynamic nature of magnetism in Yb(BaBO₃)₃.

The normalized ZF-$\mu$SR spectra after subtracting the background signal can be well fitted by the function:

where $\lambda_{1}$ and $\lambda_{2}$ are muon spin relaxation rates, $f_{1}$ is the fraction of the first exponential component. The value of $f_{1}$ was found to be temperature-independent, therefore it was fixed at its average value 0.3. We attribute the two exponential components with their temperature-independent fractions to two muon sites in Yb(BaBO₃)₃, which will be discussed later.

The temperature dependence of relaxation rates $\lambda_{1}$ and $\lambda_{2}$ is shown in Fig. 4(b). $\lambda_{1}$ is significantly larger than $\lambda_{2}$, while they show similar temperature dependence and both have a plateau-like behavior in the intermediate temperature range. $\lambda_{2}$ remains almost temperature-independent for $1\leq T\leq 40$ K. Above 40 K, $\lambda_{2}$ decreases gradually as the temperature increases. $\lambda_{1}$ remains independent of temperature over a wider temperature range, at least up to 100 K, then a drop of $\lambda_{1}$ was found at 140 K. Muon diffusion can lead to such a behavior, however, both $\lambda_{1}$ and $\lambda_{2}$ should drop at the same temperature in this case. The temperature dependence of $\lambda_{1,2}$ above 1 K can be described by an Orbach process Dalmas de Réotier et al. (2003). Orbach process is a two-phonon scattering process with an excited CEF level, which was generally observed in some insulating oxides Orbach and Bleaney (1961); Arh et al. (2022). A global fit ($1\leq T\leq 140$ K) was performed for $\lambda_{1}$ and $\lambda_{2}$ with the form:

where $\lambda$ denotes the relaxation rate $\lambda_{1}$ or $\lambda_{2}$, $\lambda_{0}$ is the saturation value of the relaxation rate, $B_{\rm me}$ models the magnetoelastic coupling of the Yb³⁺ electronic spin with the phonon bath. $\Delta_{\rm CEF}$ is the energy difference between CEF levels involved in Orbach process. We get $\Delta_{\rm CEF,2}$ = 17.2(8) meV from the fit of temperature dependence of $\lambda_{2}$. $\Delta_{\rm CEF,2}\approx$ 170 K (in $k_{B}T$) is consistent with the temperature where $\chi^{-1}$ deviates from a Curie-Weiss dependence due to CEF, and comparable with the CEF energy gap between the ground state doublet and the first excited state of other Yb-based triangular lattices Li et al. (2017b); Ding et al. (2019); Baenitz et al. (2018); Dai et al. (2021). Because there are too few data in the decline range of $\lambda_{1}$, we can only get qualitative estimation. The fit using Eq. (3) yields $\Delta_{\rm CEF,1}$ = 107(17) meV, which may not only be the energy difference of the ground-state doublet and the first excited state, higher excited states may also be associated.

As shown in Fig. 1(b), Rietveld refinement of XRD on Yb(BaBO₃)₃ has revealed that Yb³⁺ ions are located at two different sites Gao et al. (2018). The surrounding charge distributions of the two distinct Yb³⁺ ions are different, thus resulting in two different CEFs. Different CEF energy gaps have also been observed in a Yb-based pyrochlore Yb₂Ti₂O₇ with two Yb³⁺ sites Gaudet et al. (2015), while the difference between the CEF energy gaps is less significant. For Yb(BaBO₃)₃, a sketch of the triangular lattice with two inequivalent Yb³⁺ ions and two different muon sites is shown in Fig. 1(b). Since $\mu$SR is a local probe on an atomic scale, $\mu 1$ and $\mu 2$, which are close to Yb1 and Yb2, respectively, experience different CEFs. The ratio of the occupation number of two Yb³⁺ sites is $1:2$, which is almost consistent with the ratio of two muon sites $3:7$ in Eq. (2), indicating that different CEFs do exist in Yb(BaBO₃)₃.

Below $T=1$ K, both $\lambda_{1}$ and $\lambda_{2}$ exhibit a slightly upward trend (Fig. 4(b)). Although the detail behavior of $\lambda_{1,2}(T)$ at low temperatures can not be determined due to the limited number of points, the deviation of plateau-like behavior below 1 K can be seen clearly. We note that the specific heat $C/T$ at zero field also starts to increase below 1 K with temperature decreasing. The anomaly of $\lambda_{1,2}$ and $C/T$ share the same temperature range, indicating they are induced by the same physics. Spin excitation has been suggested to account for the anomaly in $C/T$ as discussed in Sec. III.2, and it may also lead to an increase of muon-spin relaxation rates Yaouanc and De Reotier (2011).

To further investigate the spin dynamics, LF-$\mu$SR measurements were performed at $T=270$ mK and 2 K. As seen in Fig. 5(a)-(b), the relaxation persists also in an applied magnetic field. Even a field of 400 mT is insufficient to decouple the muon spins, suggesting that the magnetic field at muon stopping site is dynamic. The LF-$\mu$SR spectra are also well fitted by function Eq. (2), whereas $\lambda_{1}$ and $\lambda_{2}$ here are muon spin relaxation rates with longitudinal fields applied. $f_{1}$ is 0.3, which is consistent with the ZF value.

In Fig. 5(c)-(d), we present the field dependence of $\lambda_{1}$ and $\lambda_{2}$ at 270 mK and 2 K, respectively. The field dependences of the two relaxation rates are similar but quite unusual. Generally, the magnetic field suppresses the dynamical relaxation rate Yaouanc and De Reotier (2011); Li et al. (2016a). However, here both relaxation rates show an unexpected initial increase instead of being quenched monotonically with magnetic field. A maximum was observed at 150 mT, after which two relaxation rates start to decrease with increasing the magnetic filed. The field-induced hump in the muon spin relaxation rates can be due to the level-cross resonance (LCR) between the muon Zeeman splitting and the induced quadrupole coupling of the muon’s nearest-neighbor nuclei Kreitzman et al. (1986). The LCR occurs at a field of $B\approx\omega_{Q}/\gamma_{\mu}$, where $\omega_{Q}$ is the transition frequencies between the nuclear quadrupole energy levels and $\gamma_{\mu}$ is the muon’s gyromagnetic ratio. However, The transition frequency $\omega_{Q}$ is on the order of 1$\sim$10 MHz, which means the LCR occurs at a magnetic field of 1$\sim$10 mT Kreitzman et al. (1986); Lindon et al. (2016); Kundu et al. (2020); Kuroiwa et al. (2006). The field maximum of muon spin relaxation rates in Yb(BaBO₃)₃ is observed only around 150 mT, an order of magnitude larger than the field where LCR occurs. Thus, the LCR effect can be excluded.

We attribute the initial increase of $\lambda_{1}$ and $\lambda_{2}$ to a field-induced increase of the density of spin excitations, which competes with the quenching effect of the applied magnetic field. A maximum of the relaxation rates at 150 mT is eventually formed due to such competition. A similar behavior of field dependence of the muon spin relaxation rate was reported in frustrated pyrochlore Tb₂Sn₂O₇, where the relaxation rate reached a maximum at about 50 mT at low temperatures Dalmas de Réotier et al. (2006). Since we find a maximum at higher magnetic field, its effect on spin excitations is stronger than that of Tb₂Sn₂O₇.