Abstract
Molecular based spin1/2 triangular lattice systems such as LiZn_{2}Mo_{3}O_{8} have attracted research interest. Distortions, defects, and intersite disorder are suppressed in such molecularbased magnets, and intrinsic geometrical frustration gives rise to unconventional and unexpected ground states. Li_{2}AMo_{3}O_{8} (A = In or Sc) is such a compound where spin1/2 Mo_{3}O_{13} clusters in place of Mo ions form the uniform triangular lattice. Their ground states are different according to the A site. Li_{2}InMo_{3}O_{8} undergoes conventional 120° longrange magnetic order below T_{N} = 12 K whereas isomorphic Li_{2}ScMo_{3}O_{8} exhibits no longrange magnetic order down to 0.5 K. Here, we report exotic magnetisms in Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8} investigated by muon spin rotation (μSR) and inelastic neutron scattering (INS) spectroscopies using polycrystalline samples. Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8} show completely different behaviors observed in both μSR and INS measurements, representing their different ground states. Li_{2}InMo_{3}O_{8} exhibits spin wave excitation which is quantitatively described by the nearest neighbor anisotropic Heisenberg model based on the 120° spin structure. In contrast, Li_{2}ScMo_{3}O_{8} undergoes shortrange magnetic order below 4 K with quantumspinliquidlike magnetic fluctuations down to the base temperature. Origin of the different ground states is discussed in terms of anisotropies of crystal structures and magnetic interactions.
Introduction
When quantum spins are aligned on geometrically frustrated lattices, unusual ground state eventually emerges among energetically competed states^{1,2,3}. Twodimensional (2D) spin1/2 triangular lattice Heisenberg antiferromagnet (TLHAF) is a prototypical system of geometrically frustrated magnets. Theoretically, the ground states of 2D TLHAF with both quantum and classical spins are known to be socalled 120° longrange order^{4,5,6,7}. When perturbations such as the second nearestneighbor interaction^{8}, ring exchange interaction^{9}, spatially anisotropic interactions^{10}, and randomness of the strength of the nearestneighbor interaction^{11} are set in, the system undergoes a quantum spin liquid (QSL) ground state where the system does not show static longrange magnetic order but shows longrange entanglement and fractional excitations^{1,2}. Extensive experimental studies have also been conducted on spin1/2 TLHAFs; the 120° longrange magnetic order is reported in Ba_{3}CoSb_{2}O_{9}^{12,13,14} whereas QSL state is proposed for the ground states of κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}^{15,16}, EtMe_{3}Sb[Pd(dmit)_{2}]_{2}^{17,18}, YbMgGaO_{4}^{19,20,21,22} and 1TTaS_{2}^{23}. Furthermore, spin1 TLHAF Ba_{3}NiSb_{2}O_{9} also shows QSL behaviors^{24,25,26}. QSL with spinon Fermi surface^{27,28} was proposed and succeeded in understanding the QSL behaviors in such compounds^{20,21,26}. However, experimental realization of the QSL ground state in spin1/2 TLHAF systems is still limited and remains an intriguing pursuit.
Recently, cluster magnet LiZn_{2}Mo_{3}O_{8} has attracted considerable research interest as spin1/2 TLHAF^{29}. Seven 4d electrons in a Mo_{3}O_{13} cluster occupy their orbitals, resulting in one unpaired electron. Unpaired electron with spin S = 1/2 remains in the total symmetry of the Mo_{3}O_{13} cluster (A_{1} irreducible representation) with equal contributions from all three Mo atoms, and network of the magnetic clusters forms a uniform triangular lattice in LiZn_{2}Mo_{3}O_{8}. The dominant magnetic interaction between spin1/2 Mo_{3}O_{13} clusters is antiferromagnetic^{29}, yielding geometrical frustration. LiZn_{2}Mo_{3}O_{8} is therefore an ideal 2D spin1/2 TLHAF system. Magnetic susceptibility and heat capacity measurements suggested that 2/3 of S = 1/2 spins are quenched below 96 K, and condensed valence bond state (VBS) where resonance valencebond states^{30,31} coexist with remnant paramagnetic spins is proposed for the possible ground state^{29,32}. Gapless spin excitations were reported by electron spin resonance^{32}, ^{7}Li nuclear magnetic resonance (NMR)^{32}, muon spin rotation (μSR)^{32}, and inelastic neutron scattering (INS)^{33} measurements. Emergent honeycomb lattice is theoretically proposed for the origin of the condensed VBS^{34}. Recently, a 1/6filled extended Hubbard model in an anisotropic kagome lattice is also proposed to account for the low temperature phase of LiZn_{2}Mo_{3}O_{8}^{35}. However, intersite disorder between Li^{+} and Zn^{2+} ions is reported^{29,32}, which may affect on the intrinsic magnetism in LiZn_{2}Mo_{3}O_{8}.
New molecular based triangular lattice systems Li_{2}AMo_{3}O_{8} where A = In or Sc are of particular interest in this context^{36,37}. Li_{2}AMo_{3}O_{8} crystallizes in a hexagonal structure P6_{3}mc, and no intersite disorder between Li^{+} and A^{3+} sites exists (see Supplementary Information). As in LiZn_{2}Mo_{3}O_{8}, spin1/2 carrying Mo_{3}O_{13} clusters are arranged on the structurally perfect triangular lattice separated by nonmagnetic Li and A layers in both compounds as shown in Fig. 1(a,b). Susceptibility measurements of both compounds report that the dominant magnetic interactions are antiferromagnetic and the effective moments are 1.61 μ_{B} (In) and 1.65 μ_{B} (Sc), which are close to p_{eff} = 1.73 μ_{B} the ideal value for spin S = 1/2. Spin1/2 TLHAF is therefore realized in Li_{2}AMo_{3}O_{8}, whose ground states are however different from each other. In Li_{2}InMo_{3}O_{8}, longrange magnetic order develops below T_{N} = 12 K with CurieWeiss temperature of Θ_{CW} = −242 K, and ^{7}Li NMR study suggests that the magnetic structure is the 120° structure as described in Fig. 1(c). On the other hand, isostructural Li_{2}ScMo_{3}O_{8} shows no longrange magnetic order down to 0.5 K in spite of large Weiss temperature of Θ_{CW} = −127 K. Instead, both magnetic susceptibility and heat capacity measurements indicate the development of shortrange magnetic order below 10 K. Spin glass state is ruled out as the ground state of Li_{2}ScMo_{3}O_{8} since the magnetic susceptibility shows no splitting between zerofieldcooling and fieldcooling processes^{37}. Lowtemperature heat capacity measurements in Li_{2}ScMo_{3}O_{8} shows sizable Tlinear term γ_{mag} = 35.7 mJ/mol · K^{2}, which is similar to those of QSL candidates κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}^{16}, EtMe_{3}Sb[Pd(dmit)_{2}]_{2}^{18}, and Ba_{3}CuSb_{2}O_{9}^{38}. Furthermore, different magnetic entropies between Li_{2}ScMo_{3}O_{8} and LiZn_{2}Mo_{3}O_{8} suggests that the ground state in Li_{2}ScMo_{3}O_{8} is QSL rather than condensed VBS. Because of easy access to two different ground states of spin1/2 TLHAF, Li_{2}AMo_{3}O_{8} is an intriguing system to investigate 2D spin1/2 TLHAF. However, lack of microscopic measurements prevents us from fully understanding the ground states and dynamics of Li_{2}AMo_{3}O_{8}. In this paper, we investigate quantum magnetisms of polycrystalline Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8} by combination of μSR and timeofflight (TOF) neutron scattering techniques.
Results and Discussion
Zero field (ZF) μSR time spectra of Li_{2}InMo_{3}O_{8} at several temperatures are shown in Fig. 2(a). The spectra show a damping at around 12 K, and spectral oscillations appear at lower temperatures. It is a direct evidence of the longrange magnetic order as reported in the earlier studies^{36,37,39,40}. Fourier transform of the spectrum at 3.33 K [see the inset of Fig. 2(a)] suggests that at least three different local fields are found in Li_{2}InMo_{3}O_{8}, which is probably due to crystallographically inequivalent muon stopping sites indicated by our density functional theory (DFT) calculation (see Supplementary Fig. S3 for Li_{2}ScMo_{3}O_{8}). The ZFμSR spectra of Li_{2}InMo_{3}O_{8} are fitted by three cosine functions with transverse and longitudinal relaxations
where A_{n} and A_{BG} are the positron decay asymmetries of each oscillation (n = 1~3) and background (mainly from a silver backing plate) components, f_{n} is the precession frequency, ϕ is the initial phase, and λ_{t} (λ_{l}) is the transverse (longitudinal) relaxation rate. Fitting result at each temperature is shown in Fig. 2(a). Local magnetic fields of 84.9(3), 103.1(2), and 151.5(5) G are extracted at 3.33 K, and these values are comparable in magnitude of local fields that are observed in spin1/2 magnets^{41}. Figure 2(c) shows temperature dependences of f_{1}, f_{2}, and f_{3}, representing that longrange magnetic order evolves in Li_{2}InMo_{3}O_{8} below T_{N} = 12 K with the critical exponents β ~ 0.33.
In the meanwhile, ZFμSR time spectrum of Li_{2}ScMo_{3}O_{8} at 0.07 K shows a highly damped oscillation with a pronounced reduction of the 1/3 tail as described in Fig. 2(b). To see the temperature evolution of the local fields in Li_{2}ScMo_{3}O_{8}, the ZFμSR spectrum are fitted by combination of transverse and longitudinal relaxations
The fitting result at each temperature is plotted in Fig. 2(b), and temperature dependence of the local field f_{Sc} is also plotted in Fig. 2(c). One can clearly see the temperature evolution of f_{Sc} below 4 K with the critical exponent β ~ 0.28 which is similar to those of Li_{2}InMo_{3}O_{8}. Therefore, magnetic nature of these compounds are essentially the same, but it should be noted that the ground state of Li_{2}ScMo_{3}O_{8} is shortrange magnetic order by considering the strong damping of the oscillation below 4 K. The anomaly at 4 K was also found in the temperature derivative of the magnetic susceptibility^{37}. Although the shortrange magnetic order develops in Li_{2}ScMo_{3}O_{8} below 4 K, the spectrum shows a moderate tail over a long period of time, suggesting that spin fluctuation survives even at 0.07 K. To explicitly distinguish the spin fluctuation of the Mo_{3}O_{13} cluster, we performed longitudinal field (LF) μSR measurements on Li_{2}ScMo_{3}O_{8} under longitudinal magnetic field (H_{LF}) of 1 kG. Figure 2(b) and its inset display a LFμSR time spectrum measured at 0.07 K. H_{LF} = 1 kG seems to be sufficient to quench (decouple) muon spin relaxations by both nuclear dipoles and the shortrange ordered state. The characteristic LFμSR spectrum of Li_{2}ScMo_{3}O_{8} at 0.07 K was fitted by the following equation
where A_{f} and A_{s} are asymmetries of fast (λ_{f}) and slow (λ_{s}) relaxation components, respectively (A_{f} + A_{s} = 0.16), and A_{BG} is the background asymmetry (A_{BG} = 0.03). The fitting results are described by the solid lines in Fig. 2(b) and its inset. We also fit LFμSR time spectra under H_{LF} = 1 kG at several temperatures, and obtained temperature dependences of λ_{f} and λ_{s} are plotted in Fig. 2(d). λ_{f} shows a rapid relaxation with relative signal amplitude of ~3%. It mainly corresponds to the remnant signal from the shortrange ordered state since λ_{f} exhibits a steep increase at 4 K as temperature goes down. On the other hand, λ_{s} shows a slow relaxation with two orders of magnitude less than λ_{f}, which is related to the intrinsic spin fluctuation of the Mo_{3}O_{13} cluster. Remarkably, temperature dependence of λ_{s} shows a temperatureindependent plateau below 1 K and converges into the finite value of ~0.002 μs^{−1} which is very close to that of triangular lattice QSL 1TTaS_{2} (λ = 0.0023 μs^{−1} at 0.07 K)^{23}. Indeed, such lowtemperature plateau behaviors of muon relaxation rate is common feature in the TLHAF QSL candidates^{20,25}, which will be discussed again. To obtain complementary information to our μSR results on Li_{2}AMo_{3}O_{8}, TOF neutron scattering measurements were also conducted.
Elastic neutron scattering spectra of Li_{2}InMo_{3}O_{8} below and above T_{N} are shown in Fig. 3(e). A magnetic Bragg peak appears at momentum transfer Q = 0.719(1) Å^{−1} below T_{N}. The Q position corresponds to (1/3, 1/3, 0), indicating the 120° magnetic structure consistent with the previous ^{7}LiNMR measurements^{37}. By comparing the intensity of the magnetic peak with those of nuclear Bragg peaks, the ordered moment at 4.6 K is estimated to be 0.51(3) μ_{B}. Theoretically, the magnetic moment is reduced by about 59% for the spin1/2 TLHAF^{6}, which is close to the observed ordered moment (reduced by 49% assuming g = 2). The reduced moment originates in a combination of geometrical frustration and quantum fluctuation. Neutron scattering intensity (I) map from Li_{2}InMo_{3}O_{8} as a function of Q and energy transfer (\(\hslash \omega \)) at 4.5 K (<T_{N}) is shown in Fig. 3(a). Dispersive excitation centered at the magnetic zone center (1/3, 1/3, 0) was observed. Because of the Q position, the excitation is assigned to be the spin wave excitation in the longrange magnetic ordered state. Energy spectrum at the magnetic zone center exhibits a substantial peak at \(\hslash \omega =2.08(3)\) meV as shown in Fig. 3(f). This result claims that one branch (or some branches) of the spin wave excitation has spin gap at the magnetic zone center due to the magnetic anisotropy. On the other hand, magnetic signals at the magnetic zone center become quasielastic above T_{N} as shown in Fig. 3(b,f). Therefore, the gaplike excitation is a characteristic feature of the longrange magnetic ordered state. To observe the whole structure of the spin wave excitation at 4.5 K, \(I(Q,\hslash \omega )\) map using higher E_{i} is presented in Fig. 3(c). The spin wave excitation survives up to ~9 meV. Q dependences of the spin wave intensities at various \(\hslash \omega \)s are plotted in Fig. 3(g). The spectra are asymmetric at \(\hslash \omega > 3.0\) meV, and the peak shifts to lower Q at higher \(\hslash \omega \). This result suggests that the squared magnetic form factor (F(Q)^{2}) of the Mo_{3}O_{13} cluster decreases quickly and is negligible at high Q, representing the unpaired electron with equal contributions from all three Mo atoms in Li_{2}InMo_{3}O_{8}.
For quantitative analysis on the spin wave excitation in Li_{2}InMo_{3}O_{8}, semiclassical linear spin wave (LSW) analysis was performed considering the 120° spin structure on the spin1/2 2D Mo_{3}O_{13}based triangular lattice [Fig. 1(c)]. The gaplike excitation at the magnetic zone center in the longrange magnetic ordered state is also observed in the other spin1/2 triangular lattice system Ba_{3}CoSb_{2}O_{9}^{13}, and the peak energy (E_{0}) roughly scales with T_{N} in these compounds: E_{0} = 0.65 meV and T_{N} = 3.8 K in Ba_{3}CoSb_{2}O_{9}^{13} whereas E_{0} = 2.08 meV and T_{N} = 12 K in Li_{2}InMo_{3}O_{8}. This suggests that the origin of the gaplike excitation in Li_{2}InMo_{3}O_{8} is the same as that in Ba_{3}CoSb_{2}O_{9}^{13}. Therefore, as in Ba_{3}CoSb_{2}O_{9}^{12,13,14}, the nearestneighbor anisotropic exchange interaction was considered as the model Hamiltonian for Li_{2}InMo_{3}O_{8}
where α, J, and δ represent the renormalization factor, the nearest neighbor exchange coupling constant, and the anisotropic factor. J was fixed to 112 K determined by the magnetic susceptibility measurement^{37}. By fitting the calculated powderaveraged Q dependences to the experimental results at different \(\hslash \omega \)s (2~7.5 meV) simultaneously, optimum parameters were yielded
Fitting results together with the experimental results are shown in Fig. 3(g), and calculated LSW \(I(Q,\hslash \omega )\) map is also shown in Fig. 3(d). Satisfactory agreements with calculation and experiment were confirmed. Obtained α is smaller than 1, indicating a negative quantum renormalization effect theoretically proposed for 2D spin1/2 TLHAF^{42,43,44}. Similar negative quantum renormalization effect (α ~ 0.65) was also reported in Ba_{3}CoSb_{2}O_{9}^{14}. Therefore, observed magnetic excitations of Li_{2}InMo_{3}O_{8} in the accessible \((Q,\hslash \omega )\) region are well understood by the semiclassical LSW theory assuming the 120° magnetic structure on the spin1/2 Mo_{3}O_{13} triangular lattice.
In contrast to Li_{2}InMo_{3}O_{8}, no magnetic Bragg peak evolves in the elastic channel down to 0.3 K in Li_{2}ScMo_{3}O_{8} as plotted in Fig. 4(c), in agreement with our μSR results. On the other hand, diffuse scattering expected for the shortrange order is not observed in our neutron measurements. Strong incoherent scattering may smear out such magnetic diffuse scattering in Li_{2}ScMo_{3}O_{8}. Figure 4(a) depicts \(I(Q,\hslash \omega )\) map at 0.3 K. Clear diffuse scattering was observed in the inelastic channel. Although both magnetic excitations in Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8} are centered at Q ~ 0.7 Å^{−1} [Figs 3(a) and 4(a)], the overall structures are different, representing their different ground states. In Li_{2}ScMo_{3}O_{8}, steep continuum excitation was observed. The Q dependences of the magnetic excitations are invariant in the different energy windows as shown in Fig. 4(d). Steep continuum excitation, or spinon continuum, is the common feature of the magnetic excitations in the QSL candidates^{3,21,22,26,45}. \(I(Q,\hslash \omega )\) map at high temperature (22 K) is also shown in Fig. 4(b). Although overall magnetic fluctuation at 22 K is similar to that at 0.3 K, there are some differences. Scattering intensity decreases at 22 K. In addition, as shown in Fig. 4(e), spectrum weight of the Q dependence at 2 meV slightly shifts to Q = 0 at high temperature, which is also observed in other QSL candidates^{22,26,45}.
To investigate in more detail the characteristic energy (or time) scale of the steep continuum in Li_{2}ScMo_{3}O_{8}, the dynamical spin susceptibilities \(\chi ^{\prime\prime} (\hslash \omega )=[1\exp (\,\,\hslash \omega /{k}_{B}T)]/F(Q){}^{2}I(\hslash \omega )\) at Q = [0.6, 0.8] Å^{−1} where the magnetic signal is maximal are plotted for different temperatures in Fig. 4(f). The spectra are well fitted by the quasielastic Lorentzian \(\chi ^{\prime\prime} (\hslash \omega )=\chi ^{\prime} \hslash \omega {\rm{\Gamma }}/[{(\hslash \omega )}^{2}+{{\rm{\Gamma }}}^{2}]\) where χ′ is the static susceptibility and Γ the spin relaxation rate [or peak position of \(\chi ^{\prime\prime} (\hslash \omega )\)]. The temperature dependences of the resulting parameters are shown in Fig. 4(g). Upon decreasing temperature, Γ decreases while χ′ increases. Contrary to the conventional longrange ordered magnets, no divergent behavior was observed in the temperature dependences of χ′ and Γ. It should be noted that χ′ scales with bulk magnetic susceptibility χ_{bulk} over the temperature range of 3 ≤ T ≤ 40 K [see solid line in Fig. 4(g)] and Γ is also scaled by the muon relaxation rate λ_{s} as discussed below. These fittings also extract two important features of the steep continuum in Li_{2}ScMo_{3}O_{8}: (1) the magnetic excitation is gapless consistent with the heat capacity measurement^{37} and (2) the dynamical spin susceptibility extends from the elastic channel up to at least 9.5 meV which is about 1.6 J where J (=67 K) is determined by the magnetic susceptibility measurement^{37}.
Complementary analysis of μSR and INS results enables us to exclusively clarify the quantum fluctuations in Li_{2}ScMo_{3}O_{8}. Muon spin relaxation rate λ_{s} in Fig. 2(d) is related to the spin relaxation rate of the magnetic fluctuation Γ in Fig. 4(g) on the basis of following Redfield’s formula^{46}
where γ_{μ} and δ_{μ} are the gyromagnetic ratio of muon (=2π × 135.54 MHz/T) and average distribution of local magnetic fields at muon sites. We performed electrostatic potential calculations using a pointcharge model^{47} and estimated δ_{μ} = 204.8 G for Li_{2}ScMo_{3}O_{8} (see Supplementary Information). Since H_{LF} (=1 kG = 8.5 × 10^{8} Hz) is much smaller than Γ (=2.7 meV = 6.5 × 10^{11} Hz at 0.3 K) in Li_{2}ScMo_{3}O_{8}, Eq. (6) is reformulated as
We plotted calculated temperature dependence of λ using Γ obtained by our INS measurements and compared with λ_{s} obtained by our LFμSR measurements [see solid line for calculation and circles for μSR results in Fig. 2(d)]. Quantitative agreement can be seen; the anomaly around 4 K is artificial feature owing to λ_{f}. Therefore, both μSR and INS measurements exhibit that quantum fluctuations persist at the lowest measured temperature. As mentioned above, such lowtemperature plateaus of the relaxation rates were widely observed in the triangularlattice^{20,25} and kagomelattice^{45,48,49,50,51} QSL candidates.
To account for the QSLlike excitations in Li_{2}ScMo_{3}O_{8}, we now consider the spinon Fermi surface QSL model. In Li_{2}ScMo_{3}O_{8}, no static longrange order was detected even down to 0.07 K [Figs 2(b) and 4(c)]. Alternatively, gapless continuum in Li_{2}ScMo_{3}O_{8} was observed at Q = 0.726(4) Å^{−1} corresponding to the (1/3, 1/3, 0) position [Fig. 4(a,d,f)]. Moreover, both λ_{s} and Γ exhibit temperatureindependent plateaus at low temperature [Figs 2(d) and 4(g)]. These features are well explained by QSL with spinon Fermi surface^{27,28}. As discussed in earlier works^{20,21,26}, the spinon Fermi surface QSL model on the spin1/2 TLHAF expects that (1) absence of static longrange magnetic order, (2) muon spin relaxation rate approach a finite value as temperature approaches zero, (3) magnetic excitation is gapless continuum, and (4) \(\chi ^{\prime\prime} (Q,\hslash \omega )\) shows the maximum intensity at the corner of the 2D Brillouin zone [e.g. (1/3, 1/3, 0)]. All observed features of the magnetic fluctuation in Li_{2}ScMo_{3}O_{8} can be well described by the spinon Fermi surface QSL model. Although the second peak of the spinon continuum in Ba_{3}NiSb_{2}O_{9} was also observed at (2/3, 2/3, 0)^{26}, the second peak in Li_{2}ScMo_{3}O_{8} was not detected at (2/3, 2/3, 0) corresponding to Q = 1.45 Å^{−1} as shown in Fig. 4(d) because of the quick decay of the squared magnetic form factor of the Mo_{3}O_{13} cluster^{33}. By performing complementary analysis on μSR and INS results, we conclude that Li_{2}ScMo_{3}O_{8} undergoes the shortrange magnetic order below 4 K with the QSLlike fluctuations which persist down to the lowest temperature.
We compare the Mo_{3}O_{13}clusterbased triangular lattice antiferromagnets, Li_{2}AMo_{3}O_{8} and LiZn_{2}Mo_{3}O_{8}, in line with the recent theory by Chen et al.^{35}. They proposed a 1/6filled Hubbard model on an anisotropic kagome lattice with the nearestneighbor electron hopping and repulsions^{35} to account for the magnetism in LiZn_{2}Mo_{3}O_{8}^{29}. Electron is fractionalized into charged boson and spincarring spinons; plaquette charge order emerges as the charge ground state and the spin degree of freedom can be then described by U(1) QSL with spinon Fermi surface, which can explain the unusual magnetic susceptibility in LiZn_{2}Mo_{3}O_{8}^{29}. For comparison with different compounds, they introduce a phenomenological parameter ξ to characterize the anisotropy of the Mo kagome lattice: ξ = d_{inter}/d_{intra} where d_{intra} (d_{inter}) is the intracluster (intercluster) MoMo bond length. Large anisotropy ξ tends to suppress charge fluctuations between clusters leading to the 120° longrange magnetic order whereas small anisotropy ξ corresponds to large charge fluctuation generating the U(1) QSL with spinon Fermi surface. Using the structural parameters summarized in Supplementary Information, we estimated ξ as 1.271, 1.269, and 1.258 for Li_{2}InMo_{3}O_{8}, Li_{2}ScMo_{3}O_{8}, and LiZn_{2}Mo_{3}O_{8}^{29}, respectively. The phenomenological parameter ξ explains the different ground states between the 120° longrange magnetic order in Li_{2}InMo_{3}O_{8} and the condensed VBS in LiZn_{2}Mo_{3}O_{8}. However, the ξ values of Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8} are very close to each other in spite of their different ground states. Nevertheless, ^{115}In and ^{45}Sc NMR measurements on Li_{2}AMo_{3}O_{8} reported that charge fluctuation in Li_{2}ScMo_{3}O_{8} is 2.6 times larger than that in Li_{2}InMo_{3}O_{8}^{52}, and the difference between charge fluctuations can qualitatively explain the different ground states of Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8}. Therefore, the anisotropic parameter for the Mo kagome lattice, ξ, is too simplified to explain the different ground states in Li_{2}AMo_{3}O_{8}, and more detailed parameter is required for Li_{2}AMo_{3}O_{8}.
We also compare the magnetic excitations in Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8} to discuss the origin of the different ground states. Although both magnetic excitations in Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8} center at Q ~ 0.72 Å^{−1}, lowenergy magnetic excitations show opposite behaviors. The magnetic excitation at the magnetic zone center in Li_{2}InMo_{3}O_{8} clearly exhibits the peak at 2.08(3) meV [Fig. 3(a,f)]. Our LSW analysis suggests that the anisotropic exchange interaction is necessary to reproduce the peak. Meanwhile, the gapless magnetic excitation in Li_{2}ScMo_{3}O_{8} indicates that magnetic anisotropy is negligibly small in Li_{2}ScMo_{3}O_{8} [Fig. 4(a,f)]. Thus, the difference in the magnetic anisotropy is another possibility of the origin of the different ground states in Li_{2}AMo_{3}O_{8}. In fact, the gaplike excitation was observed in the longrange ordered state of Ba_{3}CoSb_{2}O_{9}^{13,14}, whereas the gapless magnetic excitations in the QSL systems YbMgGaO_{4}^{21,22} and Ba_{3}NiSb_{2}O_{9}^{26}. INS measurements on magnetic excitations in the substitution system Li_{2}(In_{1−x}Sc_{x})Mo_{3}O_{8}^{40} are effective to further elucidate the origin of different magnetic ground states, which is left for future work.
Conclusion
We performed a comprehensive study on the quantum magnetisms in the Mo_{3}O_{13}clusterbased spin1/2 triangular lattice antiferromagnets, Li_{2}InMo_{3}O_{8} and Li_{2}ScMo_{3}O_{8} by means of μSR and TOF neutron scattering techniques. Spin wave excitation in Li_{2}InMo_{3}O_{8} was well described by the nearest neighbor anisotropic Heisenberg model based on the 120° spin structure. Li_{2}ScMo_{3}O_{8} exhibits the shortrange magnetic order below 4 K with the QSLlike fluctuations which persist down to the lowest temperature. The origin of the different magnetic ground states in Li_{2}AMo_{3}O_{8} is discussed in terms of anisotropies of crystal structures and magnetic interactions.
Methods
The preparation of polycrystalline Li_{2}InMo_{3}0_{8} (Li_{2}ScMo_{3}O_{8}) was carried out by two steps^{37}. First, to synthesize a precursor Li_{2}MoO_{4}, a mixture with a ratio of MoO_{3}:Li_{2}CO_{3} = 1:1 was ground, placed in an alumina crucible, and heated at 873 K for 24 hours in air; we repeated this step for three times. Then, a mixture having a ratio of In_{2}O_{3} (Sc_{2}O_{3}):Li_{2}MoO_{4}:MoO_{3}:Mo = 0.5:1:0.84:1.16 was ground, pressed into a pellet, sealed in an evacuated quartz tube, heated at 923 K for 12 hours, and heated at 1198 K (1173 K) for 24 hours; we repeated this step for two times. Magnetization measurements were performed using a commercial superconducting quantum interference device (SQUID) magnetometer (Quantum Design Magnetic Property Measurement System, MPMS). ZF and LFμSR experiments were performed using the Advanced Research Targeted Experimental Muon Instrument at the S1 line spectrometer (ARTEMIS)^{53} with a conventional ^{4}He flow cryostat and the D1 spectrometer^{53} with a ^{3}He^{4}He dilution refrigerator installed at Materials and Life Science Experimental Facility (MLF), Japan Proton Accelerator Research Complex (JPARC). We used the VASP software^{54} for DFT calculation and the DipElec program^{47} to calculate the local magnetic fields in Li_{2}ScMo_{3}O_{8}. TOF neutron scattering measurements were performed using the Fermi chopper spectrometer 4SEASONS at MLF, JPARC^{55}. Frequencies of the Fermi chopper were 350 and 250 Hz for the In and Sc systems, resulting in the combinations of incident neutron energies of 11.9, 15.8, 22.0, and 32.7 meV, and 7.5, 10.3, 15.0, and 23.9 meV^{56}, respectively. A standard toploading cryostat at 4SEASONS was used for the measurements on Li_{2}InMo_{3}O_{8}, whereas a ^{4}He refrigerator and a ^{3}He cryostat were used for Li_{2}ScMo_{3}O_{8}. Empty can was measured at corresponding temperatures, and then subtracted from raw data of Li_{2}ScMo_{3}O_{8}. TOF data were visualized by software suite Utsusemi^{57}. Neutron scattering intensities are converted to the absolute unit using the incoherent scattering of each sample^{58} after correction of the neutron absorption effect. Squared magnetic form factor of the Mo_{3}O_{13} cluster^{33} and \(\hslash \omega \)dependent energy resolution at 4SEASONS^{59} were included in the LSW calculations for Li_{2}InMo_{3}O_{8}.
Data Availability
The datasets generated and analyzed during the current study are available from the corresponding author.
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Acknowledgements
We thank Seiko OhiraKawamura and Kim Sandvik for helpful discussion and Hua Li for DFT calculations. We also acknowledge technical supports from the MLF sample environment team. Magnetic susceptibility measurements were performed at the CROSS user laboratories. The synchrotron Xray diffraction experiments were conducted at BL5S2 of Aichi Synchrotron Radiation Center, Aichi Science and Technology Foundation, Aichi, Japan (Proposal Nos. 201803046 and 201803047). The μSR measurements at the S1 and D1 beamlines were conducted under the user program with proposal number 2017B0033. The proposal numbers for the TOF neutron scattering experiments at 4SEASONS were 2016I0001, 2017A0004, 2017B0030, and 2018I0001. The DFT calculations were supported by the Condensed Matter Research Center, Institute of Materials Structure Science, KEK, and the KEK Large Scale Simulation Program (Nos 15/1607 and 16/1718). This work was partially supported by JSPS KAKENHI Grant Numbers JP17K14349 and JP18K03529, and the Cooperative Research Program of “Network Joint Research Center for Materials and Devices” (20181072).
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K.I. designed the project. H.Y. and Y. Ishii synthesized the polycrystalline samples. H.Y., Y. Ishii, M.I. and K.I. characterized the samples. N.K. conducted synchrotron Xray diffraction measurements and Rietveld analysis. H.O., A.K., R. Kadono, K.I. and R. Kajimoto conducted muon experiments while K.I., R. Kajimoto, H.Y., Y. Inamura, N.M. and M.I. performed neutron scattering measurements. K.I., H.O. and N.K. analyzed the data and wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Iida, K., Yoshida, H., Okabe, H. et al. Quantum magnetisms in uniform triangular lattices Li_{2}AMo_{3}O_{8} (A = In, Sc). Sci Rep 9, 1826 (2019). https://doi.org/10.1038/s41598018361237
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DOI: https://doi.org/10.1038/s41598018361237
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