A quantum spin liquid (QSL) is an exotic magnetic state, in which spins are highly entangled and remain disordered even in disorder-free lattices at zero temperature due to strong quantum fluctuations [1][2][3][4]. Beyond the traditional Landau symmetry breaking paradigm, QSL is characterized by fractionalized spin excitations instead of an order parameter [5,6]. The triangular-lattice antiferromagnet with isotropic Heisenberg interactions was proposed to host a QSL ground state by Anderson in the resonating valence bond picture [7], but a magnetically ordered ground state was subsequently demonstrated [8] even for a spin-1/2 system where the quantum effects are most significant. The magnetic order is fragile, however, and can be "melted" by other interactions, such as next-nearest-neighbor couplings [9,10], spatially anisotropic exchange interactions [11], or magnetic anisotropy [12,13]. These properties provide alternative ways to realize a QSL.
Rare-earth (RE)-based frustrated lattices represent promising habitats for QSL, since the strong atomic spin-orbit coupling (SOC) gives rise to both spatial and spin anisotropies [14,15]. YbMgGaO 4 and the chalcogenides family NaYbCh 2 (Ch = O, S, Se) have attracted much attention due to potential QSL properties [16][17][18][19][20][21]. SOC together with the crystal electric field (CEF) leads to a Kramers ground-state doublet characterized by an effective spin-1/2 in these magnets [14]. However, the effective spin exhibits easy-plane magnetic anisotropy for the RE-based triangular lattices that have been extensively studied so far.
The Nd-based triangular-lattice tantalate NdTa 7 O 19 has recently been reported to be a potential QSL candidate [22]. In * Contact author: leishu@fudan.edu.cn it the effective spin-1/2 Nd 3+ moments are Ising-like, giving rise to spin excitations down to 40 mK. As a triangular lattice with Ising spins, NdTa 7 O 19 provides a promising system in which to realize QSL and other novel quantum states, which have not been studied before due to limited availability of materials [23]. An early theoretical study of the classical Ising model on a triangular lattice by Wannier [24] led to a classical spin-liquid state. Since then theoretical studies [25,26] have predicted exotic states of Ising spins on the triangular lattice. Synthesis and study of new Ising triangular-lattice materials are needed.
This paper reports comprehensive studies of magnetic susceptibility, specific heat, and muon spin relaxation (μSR) in polycrystalline Ba 6 Nd 2 Ti 4 O 17 , which is an essentially disorder-free Nd 3+ triangular lattice. An Ising-like spin anisotropy is supported by magnetization and specific heat results, which complement a previous study [27]. The absence of magnetic order was confirmed down to 30 mK by zero-field (ZF) μSR measurements, despite a low-temperature Curie-Weiss (CW) temperature CW,L = -1.8 K. A two-level Schottky anomaly was observed in the field dependence of the magnetic specific heat, indicating the spin is effectively 1/2 at low temperatures. The magnetically disordered state with effective spin-1/2 was further studied by longitudinalfield (LF) μSR measurements, which reveal persistent spin dynamics down to 37 mK. Our results provide strong evidence for a QSL in Ba 6 Nd 2 Ti 4 O 17 .
Polycrystalline Ba 6 Nd 2 Ti 4 O 17 was synthesized by the solid state reaction method [28]. The starting materials BaCO 3 , TiO 2 , Nd 2 O 3 were first dried overnight at 700 • C. Stoichiometric amounts of reagents were then mixed, thoroughly ground, and prereacted at 900 • C for 12 h. The samples were finally reground and heated to 1250 • C for 48 h in air, repeating 2-3 times. Powder x-ray diffraction (XRD) data were collected using a Bruker D8 advanced x-ray diffraction spectrometer (λ = 1.5418 Å) at room temperature. Structural refinements were performed using the FULLPROF software package [29].
The magnetization and dc magnetic susceptibility were measured using a Magnetic Property Measurement System (MPMS, Quantum Design). The specific heat measurements were performed by the adiabatic relaxation method in a Physical Property Measurement System (PPMS) (DynaCool, Quantum Design) equipped with a Helium-3 option. μSR measurements were carried out at TRIUMF, Vancouver, Canada, using the LAMPF spectrometer at the M20 beamline and the dilution refrigerator (DR) spectrometer at the M15 beamline. The samples were mounted on a silver holder in the DR spectrometer and encased in thin silver tape in the LAMPF spectrometer. The μSR data were analyzed using the MUSRFIT software package [30].
Polycrystalline Ba 6 Nd 2 Ti 4 O 17 was synthesized in 2002 [28], and single crystals have been grown recently [27]. The crystal structure of Ba 6 Nd 2 Ti 4 O 17 (space group P6 3 /mmc) is shown in Fig. 1(a) (see Ref. [27] for details of the structure determination). The Nd 3+ ions form triangular layers in the ab plane, as shown in Fig. 1(b). The layers are stacked along the c axis, and are separated by two different interlayer distances d 1 = 7.40 Å and d 2 = 7.56 Å due to different interlayer Ti coordinations. The geometry is such that spin exchange is strongest within Nd 3+ layers, and superexchange between layers is considerably stronger across the shorter distance. The two possible positive-muon (μ + ) stopping sites S1 and S2 discussed in Sec. IV B are shown in Fig. 1(c). Site S1 is located between two Nd 3+ layers, and S2 is in a Nd 3+ layer. Powder x-ray diffraction (XRD) measurements were performed on polycrystalline Ba 6 Nd 2 Ti 4 O 17 to determine the sample quality. The XRD pattern and Rietveld refinements FIG. 2. XRD pattern and Rietveld refinement results for Ba 6 Nd 2 Ti 4 O 17 . Inset: enlarged view of the region 2θ = 24 • -34 • . Arrows: small peaks of unknown origin.
are shown in Fig. 2, and refinement parameters are listed in Table I. The results are essentially the same as previously reported [27], although the refinement exhibits three unexpected small peaks at 2θ = 24 • -34 • , shown in the inset of Fig. 2. None of the residual starting materials are matched by these peaks, and Nd 3+ /Ba 2+ or Nd 3+ /Ti 4+ site-mixing disorder is also not matched. The absence of site-mixing disorder was also reported for single crystal Ba 6 Nd 2 Ti 4 O 17 [27]. The unexpected peaks may originate from dilute impurity phases with superlattice structures.
The temperature dependences of the dc magnetic susceptibility χ and inverse susceptibility χ -1 of Ba 6 Nd 2 Ti 4 O 17 are shown in Fig. 3(a). The magnetic susceptibility was measured at 1 T with zero-field cooling (ZFC) and field cooling (FC). No sharp magnetic transition was observed down to 2 K and no obvious difference between ZFC and FC curves was detected, indicating the absence of spin ordering or freezing. A CW fit at high temperatures (100-200 K) yields an effective magnetic moment μ eff,H = 3.43 μ B , close to the Hund's-rules
TABLE I. Rietveld refinement results for Ba 6 Nd 2 Ti 4 O 17 XRD data measured at room temperature. Cell parameters: α = β = 90 • , γ = 120 • , a = b = 5.9922(4) Å, c = 29.927(2) Å. Overall B = 0.33(6) Å 2 . The space group is P6 3 /mmc. χ 2 = 2.40. Atom Wyckoff position x y z Occ. Nd 4e 0 0 0.1263(2) 1 Ba1 2a 0 0 0 1 Ba2 4f 0.6667 0.3333 0.0861(3) 1 Ba3 4f 0.3333 0.6667 0.1853(3) 1 Ba4 2b 0 0 0.25 1 Ti1 4f 0.3333 0.6667 0.0512(8) 1 Ti2 4f 0.6667 0.3333 0.2079(6) 1 O1 4f 0.3333 0.6667 -0.003(2) 1 O2 12k 0.620(6) 0.810(3) 0.0755(7) 1 O3 12k 0.349(7) 0.175(4) 0.1669(8) 1 O4 6h 0.51(5) 0.019(8) 0.25 1 155148-2 value 3.62 μ B , and a CW temperature CW,H = -28.1(1) K, suggesting antiferromagnetic coupling of Nd 3+ moments. The susceptibility deviates from CW behavior as the temperature drops below 30 K, but becomes CW again at low temperatures. A CW fit over the range 2-20 K yields CW,L = -1.8 K.
The strength of the exchange interaction J ex between nearest-neighbor Nd 3+ spins can be roughly estimated from the mean-field approximation J ex = 3k B CW,L /(zJ eff (J eff + 1)) ≈ 1.2 K, where z = 6 is the number of nearest-neighbor Nd ions and J eff = 1/2 is the effective spin [31]. The effective moment from the low-temperature CW fit is μ eff,L = 2.54 μ B , considerably less than the Hund's-rules value.
The field dependence of the isothermal magnetization M(H ) up to 7 T at several temperatures is shown in Fig. 3(b). At 2 K M(H ) exhibits a nonlinear field dependence above 1 T, and tends to saturate above 4 T. The energy scale μ eff,L B of a 1 T magnetic field for Nd 3+ spins at low temperatures is 1.7 K, which is of the same order as J ex estimated above. A rough value of the low-temperature saturation moment is 1.3 μ B , about half the value of μ eff,L , which is expected from powder-averaged Ising spins in a polycrystalline sample [32]. The field response of the magnetization becomes linear as the temperature increases up to 100 K as a consequence of the decrease in spin-spin correlation. These results are consistent with previously reported data for polycrystalline Ba 6 Nd 2 Ti 4 O 17 [27].
The absence of spin ordering or spin freezing is indicated by magnetization data down to 2 K (Sec. III B). To confirm this, and to detect potential magnetic excitations, we performed specific heat measurements on polycrystalline Ba 6 Nd 2 Ti 4 O 17 down to 0.4 K.
Figure 4(a) shows the measured specific heat at low temperatures and fields up to 8 T. The lack of any sharp anomaly is consistent with the absence of spin ordering. The field dependence is characteristic of Schottky anomalies (but see below).
The temperature dependences of the specific heat at ZF and 8 T up to 300 K are shown in Fig. 4(b). The data for ZF and 8 T basically overlap for 20 K < T < 200 K, indicating the absence of excited-state CEF energy levels in this range. The phonon contribution, shown in Fig. 4(c), was obtained by fitting ZF specific heat (2 -200 K) with a combination of two Debye and two Einstein functions
(
The weight factors were fixed at the ratios f D1 : f D2 : f E1 : f E2 = 6:15:2:6, with a sum of 29 atoms per formula unit [33]. The fitting yields Debye temperatures θ D1 = 172 K, θ D2 = 726 K, and Einstein temperatures θ E1 = 82 K, θ E2 = 227 K. The temperature dependence of the magnetic specific heat C m , obtained by subtracting C ph [Eq. ( 1)] from the measured total specific heat, is shown in Fig. 5(a). A peak is observed in C m above 1 T that moves to higher temperatures with increasing field. This is characteristic Schottky-anomaly behavior, and was previously reported for single crystal Ba 6 Nd 2 Ti 4 O 17 [27]. However, the two-level Schottky function
where n is the moment concentration, R is the molar gas constant, and is the energy gap between the two levels [34], does not provide good fits to the data (not shown). This is attributed to the spatial distribution of spin directions in the polycrystalline sample. The Zeeman energy depends on this direction, resulting in a distribution of energy gaps . A uniform angular distribution of spin directions yields a two-level Schottky specific heat given by
Fits of Eq. ( 3) to the C m data, shown by dashed curves in Fig. 5(a), are good for fields 0.1 T, suggesting a ground-state Kramers doublet that is dominant at low temperatures and leads to an effective spin J eff = 1/2. The field dependences of and n are shown in Fig. 5(b). A linear fit of (H ) is shown for the n ∼ 1 data, as expected for Zeeman splitting by the field. The fit yields μ = 2.56 μ B , essentially the same as 2.54 μ B from magnetization (Sec. III B).
The degeneracy of the ground-state doublet can be lifted by magnetic fields, but at zero and low fields the system cannot be regarded as an assembly of noninteracting two-level ions; a weak field cannot completely lift the degeneracy, resulting in a concentration dependence. Similar behavior has been reported for a number of RE-based oxide insulators [35][36][37].
Figure 5(c) shows the magnetic specific heat coefficient C m /T and the corresponding entropy at ZF and 8 T. In 8 T the magnetic entropy at 20 K is nearly R ln 2, the entropy of a spin-1/2 system. The entropy is slightly less than R ln 2 due to the lack of data below ∼0.6 K. The ZF magnetic entropy only reaches 3% of R ln 2, suggesting considerable residual entropy below 0.4 K.
μSR is an ideal technique to probe magnetic order, and is particularly sensitive to slow spin fluctuations [38]. ZF-and LF-μSR experiments were carried out to further investigate the low-temperature magnetic state of Ba 6 Nd 2 Ti 4 O 17 .
ZF-μSR asymmetry spectra at several representative temperatures are shown in Fig. 6(a). Neither oscillations nor a drastic loss of initial asymmetry is observed down to 30 mK, ruling out long-or short-range magnetic order (magnetic freezing) [39]. The lack of polarization recovery to 1/3 indicates μ + spin relaxation is dynamic rather than due to static random fields [40,41].
The normalized ZF-μSR spectra P(t ) with background signal subtracted were fit by the function
where λ 1 and λ 2 are μ + spin relaxation rates and f 1 is the fraction of the first exponential term. f 1 was found to be temperature-independent and was fixed at the average value 0.63. The temperature dependences of the two ZF rates λ 1 and λ 2 are shown in Fig. 6(b). They exhibit very similar behavior: both increase drastically with decreasing temperature below T 2 ∼ 30 K, and saturate at T 1 ∼ 4 K, exhibiting a temperatureindependent plateau down to 30 mK. A low-temperature plateau of μ + spin relaxation rate is generally regarded as the evidence for persistent spin dynamics and a correlated disordered state [42].
To confirm the dynamical nature of the μ + spin relaxation and further investigate the Nd 3+ spin dynamics, LF-μSR experiments were carried out. Figure 7(a) shows LF-μSR spectra measured at 37 mK. The relaxation persists at high applied fields; a longitudinal field of 1 T does not completely "decouple" the μ + spin depolarization, i.e., reduce the rate to zero.
If the observed ZF exponential depolarization function originated from a Lorentzian distribution of static fields [43], then the distribution widths H Li = λ i /γ μ are ∼10 mT and ∼2 mT for i = 1 and 2, respectively, where γ μ /2π = 135.54 MHz/T is the gyromagnetic ratio of the muon. An applied field an order of magnitude larger than the static distribution width fully decouples the μ + spin depolarization [44], but the maximum field [Fig. 7(a)] of 1 T is two (three) orders of magnitude larger than H L,1 ( H L,2 ) and is still insufficient to decouple the μ + spin depolarization. This is strong evidence that dynamic μ + spin relaxation persists down to 37 mK in Ba 6 Nd 2 Ti 4 O 17 .
Representative LF-μSR spectra measured at 10 K are shown in Fig. 7(b). The relaxation is hardly suppressed by an external longitudinal field of 400 mT, exhibiting a different behavior from that at lower temperatures. The LF-μSR spectra are also well fitted by Eq. ( 4) with f 1 fixed at the ZF value 0.63.
The field dependences of λ 1 and λ 2 at four temperatures are shown in Figs. 7(c decreases monotonically with increasing external longitudinal field. However, the field dependences of λ 1 and λ 2 both exhibit unusual maxima, which shift to higher fields with increasing temperature up to 10 K. Both rates are quenched rapidly by external longitudinal field above 100 mT below 6 K, but at 10 K they remain comparable to their ZF values even at 400 mT. This unusual field dependence will be discussed in detail below in Sec. IV D.
The electron configuration of Nd 3+ is 4 f 3 . According to Hund's rules, the spin angular momentum s = 3/2, the orbital angular momentum L = 6, and SOC leads to a total angular momentum J = 9/2. The ground state of free Nd 3+ is tenfold degenerate 4 I 9/2 . A point-charge calculation for Ba 6 Nd 2 Ti 4 O 17 (Appendix) reveals that the ten J = 9/2 states are CEF-split into five Kramers doublets. The specific heat results show that only the ground-state Kramers doublet is occupied below 200 K. Thus the system can be treated as effective spin-1/2 at low temperatures.
The specific heat of single-crystal Ba 6 Nd 2 Ti 4 O 17 exhibits perfect two-level Schottky behavior, as expected for spin-1/2 system [27], whereas in the present study the specific heat of a polycrystalline sample can be fit by a modified Schottky model [Eq. ( 3)]. If the spin directions are assumed to be randomly oriented in the polycrystalline sample, the effective fields experienced by nonparallel spins are reduced and the energy gap of the ground-state doublet split by the magnetic field is distributed.
Random orientation and Ising-like spin anisotropy are both required for a distribution of energy gaps. Ising-like spins basically only experience the z component of the magnetic field. Consequently, the Zeeman energy is highly dependent on the angle between the spin and the magnetic field, which results in a significant distribution of the energy gap. In contrast, there are a number of polycrystalline materials with effective spin 1/2 [37,[45][46][47] for which the standard two-level Schottky function provides a good fit. For a quantum spin with isotropic exchange the spin coupling is also isotropic.
Ising anisotropy of Nd 3+ spins in Ba 6 Nd 2 Ti 4 O 17 is also supported by the magnetization results. The saturation value of the magnetization at 2 K is μ sat = 1.3 μ B , about half the susceptibility value of μ eff,L = 2.54 μ B , which is a sign of powder-averaged Ising spins in a polycrystalline sample [32]. Ising anisotropy is also consistent with the evidence for Ising-like spins from electron spin resonance measurements on single crystals [27].
The ground-state Kramers doublet created by CEF and SOC is a dipole-octupole (DO) doublet for Nd 3+ in a triangular lattice with space group of P6 3 /mmc or R 3m [48]. The wave functions of ground-state doublet for Ba 6 Nd 2 Ti 4 O 17 from point-charge calculations are given in the Appendix, Table III. They are linear superpositions of the states with J z = 3n/2, where n is an odd integer, indicating the DO doublet nature. Dipolar order together with emergent octupolar order is predicted for DO doublets on triangular lattices with negligible interlayer interactions [48]. However, no dipolar order is observed down to 30 mK in Ba 6 Nd 2 Ti 4 O 17 . A possible reason is that the exchange interaction between Nd 3+ spins with interlayer distance d 1 cannot be ignored.
The fitting function for μSR asymmetry spectra [Eq. ( 4)] includes two exponential terms. The fractions of these terms are nearly temperature-independent, which is usually a sign for signals from distinct μ + stopping sites [49], the fitting function for LF-μSR and ZF-μSR is the same, and the relaxation rates show similar temperature and field dependences. These properties are consistent with their attribution to two inequivalent μ + stopping sites.
μ + sites generally maximize the number of near-neighbor O 2-ions. In the Ba 6 Nd 2 Ti 4 O 17 structure the two sites S1 and S2 shown in Fig. 1(c) satisfy this criteria. S1 is located between two Nd 3+ layers and close to TiO 4 tetrahedra or Ti 2 O 9 dimers, and S2 is in the Nd layer and close to NdO 6 octahedra. The ratio of the two kinds of O-O bonds is 1.38, comparable to the ratio of the two fractions of exponential terms in the asymmetry spectra (≈1.7). λ 1 is several times larger than λ 2 , which may be related to the strength of the local magnetic field at each μ + site.
The evidence for these stopping sites is of course very qualitative. Density functional theory calculations were unsuccessful because of the large number of atoms in the unit cell. Precise knowledge of the sites is, however, not essential to the present work.
a. Frustration vs Nd 3+ distances. Frustration and long distances between magnetic ions can both be involved in suppression of magnetic freezing. In Table II we list ion spacing and frustration parameters for some Nd-based
TABLE II. Structural and magnetic properties of some Nd-based triangular-lattice antiferromagnets. Compound d inter (Å) d intra (Å) θ CW (K) f Ba 6 Nd 2 Ti 4 O 17 7.40 5.99 -1.8 > 60 NdTa 7 O 19 a 9.97 6.62 -0.46 >11.5 CsNdSe 2 b -4.35 -0.66 (H c) >16.5 NdZnAl 11 O 19 c 11.43 5.59 -1.23 >0.6 KNdSe 2 d 7.60 4.31 -1.9 (H c) >4.7 a Reference [22]. b Reference [50]. c Reference [51]. d Reference [52]. triangular lattices for comparison. None of them exhibit magnetic ordering at low temperatures. Although the Nd-Nd distance within the triangular layer is about 6 Å for Ba 6 Nd 2 Ti 4 O 17 , the frustration parameter f > 60, indicating a significant suppression of magnetic ordering. Therefore, the magnetic disorder is not due to the isolation of magnetic ions. b. Temperature dependence. Two characteristic temperatures T 1 ∼ 4 K and T 2 ∼ 30 K are observed in ZF-μSR measurements [Fig. 6(b)]. The μ + spin relaxation rates rise rapidly below T 2 ∼ 30 K, and exhibit a plateau below T 1 ∼ 4 K. k B T 1 is of the order of the exchange interaction J ex ∼ 1.2 K, suggesting the onset of spin correlation. The lowtemperature plateau of λ 1,2 indicates the spin fluctuation no longer slows down as temperature drops, and signals a disordered magnetic ground state with persistent dynamics. This is further confirmed by LF-μSR measurements.
It is informative to compare the behavior of λ 1,2 at T 2 with the estimated μ + spin relaxation rate λ ∞ for a local-moment paramagnet in the high-temperature limit T → ∞. λ ∞ is temperature independent and approximately 2(γ μ H rms ) 2 /ν [53], where H rms is the rms field at a μ + site and the exchange fluctuation rate ν ≈ √ zJ ex S/ h; z and S are the local-moment coordination number and spin, and J ex is the exchange energy.
With z = 6, and using the low-temperature J ex = 1.2 K (Sec. III B) and S = J eff = 1/2, we find ν ≈ 1.9 × 10 11 s -1 . Using the high-temperature Nd moment 3.43 μ B , μ 0 H rms = 103 mT assuming a dipolar field and uncorrelated fluctuations. This yields λ ∞ ≈ 0.13 µs -1 , much less than the observed rates at 10 K. It would be an order of magnitude smaller if CW,H were used to evaluate the exchange instead of CW,L .
The values and strong temperature dependences of the observed rates λ 1,2 below T 2 [Fig. 6(b)] do not characterize the high-temperature paramagnetic state. The onset of Nd 3+ spin correlations slows their fluctuations with decreasing temperature, thus increasing λ(T ) so that it becomes observable at lower temperatures.
The local magnetic field in Ba 6 Nd 2 Ti 4 O 17 at low temperatures has been shown to be dynamical by LF-μSR measurements, but the field dependence of μ + spin relaxation rates shows an unusual maximum. Level-crossing resonance (LCR) can lead to a resonance peak in the field dependence of μ + spin relaxation rates at B res = ω Q /γ μ , where ω Q is the quadrupolar frequency [54]. LCR occurs when the μ + Zeeman frequency in an external field is equal to the quadrupolar frequency of nuclei with spins larger than 1/2 [55]. A resonance peak due to LCR centered at ∼8 mT was previously reported for Cu metal [54], due to a μ + -induced electric field gradient (EFG) at 63,65 Cu sites.
A LCR peak might be expected in Ba 6 Nd 2 Ti 4 O 17 . The quadrupole moments of Ba, Nd, Ti, and Cu nuclei are of the same order (0.1 barn) as Cu, and ω Q is proportional to the quadrupole moment [44]. The extrema of the LF μ + relaxation rates Ba 6 Nd 2 Ti 4 O 17 are observed around 100 mT, but the higher field may reflect the intrinsic EFG at the noncubic lattice sites. Thus LCR cannot be excluded. NMR/NQR measurements in Ba 6 Nd 2 Ti 4 O 17 are needed to clarify this issue.
For the case of random fluctuations originating from localized and uncorrelated spins, the spin dynamic autocorrelation function generally takes an exponential form, S(t ) ∼ e -νt , where ν is the field fluctuation rate. Then the μ + spin relaxation rate can be expressed by the Redfield formula [55] λ
in the fast fluctuation limit (γ μ H loc ν), where H loc is the Gaussian distribution width of local magnetic fields, ν is the field fluctuation rate, and H 0 is the applied longitudinal magnetic field. λ(H ) described by the Redfield formula monotonically decreases as the external field is increased when H loc and ν are nearly unchanged, which is inconsistent with the observed field dependence of λ 1,2 here.
We recall that the energy scale of exchange interactions is about 1 K for Ba 6 Nd 2 Ti 4 O 17 . At T = 10 K, thermal fluctuations are relatively strong, and the exchange interactions are negligible. The extremum of λ 1,2 persists at temperature up to 10 K, suggesting it does not result from exchange interactions or spin correlation. The extremum of λ 1,2 may be due to the change in the distribution width of local fields H loc . Since the field coupling of Ising spins are highly anisotropic and the sample is polycrystalline, a field dependence of H loc may be significant to drive λ(H ) no longer monotonic.
Both λ 1 and λ 2 remain a sizable value at 10 K under 400 mT. The field independence of λ 1,2 is reminiscent of a similar result in YbZnGaO 4 [56], which was attributed to rapidly fluctuating local spins. As noted above, however, in the high-temperature limit the Nd 3+ spin fluctuation rate ν is essentially the Nd spin precession frequency in the exchange field, so that λ is temperature-independent at a low value.
Polycrystalline antiferromagnetic Ba 6 Nd 2 Ti 4 O 17 with an essentially disorder-free triangular lattice has been synthesized and studied. The absence of long-range magnetic order and spin freezing is confirmed down to 30 mK by ZF-μSR measurements. The negative CW temperature CW = -1.8 K from magnetic susceptibility measurements indicates an antiferromagnetic exchange interaction of Nd 3+ spins. The low-temperature magnetic specific heat is well fitted by a modified two-level Schottky model, suggesting the system has an effective Ising spin J eff = 1/2 at low temperatures due
TABLE III. Calculated eigenvalues and eigenvectors of CEF energy levels for Ba 6 Nd 2 Ti 4 O 17 . E (meV) -9 2 -7 2 -5 2 -3 2 -1 2 1 2 3 2 5 2 7 2 9 2 0 0.753 0 0 -0.655 0 0 0.025 0 0 -0.061 0 0.061 0 0 0.025 0 0 0.655 0 0 0.753 0.401 0 -0.91 0.04 0 0.291 0.046 0 -0.253 0.142 0 0.401 0 -0.142 -0.253 0 0.046 -0.291 0 -0.04 -0.91 0 12.366 0 0.095 -0.822 0 -0.122 0.209 0 -0.481 0.162 0 12.366 0 -0.162 -0.481 0 0.209 0.122 0 0.822 0.095 0 72.528 0.656 0 0 0.749 0 0 -0.098 0 0 0.007 72.528 -0.007 0 0 -0.098 0 0 -0.749 0 0 0.656 76.633 0 -0.059 0.165 0 -0.16 0.911 0 0.029 -0.337 0 76.633 0 -0.337 -0.029 0 -0.911 -0.16 0 0.165 0.059 0 to a ground-state Kramers doublet. Ising spin anisotropy is also supported by the effective moment from magnetization measurements.
The low-temperature plateau of μ + spin relaxation rates in ZF-μSR indicates a quantum disordered ground state, and LF-μSR measurements give further evidence for persistent spin dynamics for temperatures down to 37 mK. These results are consistent with a QSL picture, where the spins remain disordered and fluctuate strongly even at zero temperature [22,42,57,58].
Intriguing behavior has been observed in a triangularlattice antiferromagnet with effective Ising spin 1/2. It provides an excellent example of a system that remains magnetically disordered down to ∼| CW |/60 with persistent spin dynamics. However, more experiments are needed, especially on single crystals, to characterize the spin excitations at low temperatures and to confirm the ground state.