Incommensurate antiferromagnetic order in the manifoldly-frustrated SrTb$_2$O$_4$ with transition temperature up to 4.28 K

The N$\acute{\rm e}$el temperature of the new frustrated family of Sr\emph{RE}$_2$O$_4$ (\emph{RE} = rare earth) compounds is yet limited to $\sim$ 0.9 K, which more or less hampers a complete understanding of the relevant magnetic frustrations and spin interactions. Here we report on a new frustrated member to the family, SrTb$_2$O$_4$ with a record $T_{\rm N}$ = 4.28(2) K, and an experimental study of the magnetic interacting and frustrating mechanisms by polarized and unpolarized neutron scattering. The compound SrTb$_2$O$_4$ displays an incommensurate antiferromagnetic (AFM) order with a transverse wave vector \textbf{Q}$^{\rm 0.5 K}_{\rm AFM}$ = (0.5924(1), 0.0059(1), 0) albeit with partially-ordered moments, 1.92(6) $\mu_{\rm B}$ at 0.5 K, stemming from only one of the two inequivalent Tb sites mainly by virtue of their different octahedral distortions. The localized moments are confined to the \emph{bc} plane, 11.9(66)$^\circ$ away from the \emph{b} axis probably by single-ion anisotropy. We reveal that this AFM order is dominated mainly by dipole-dipole interactions and disclose that the octahedral distortion, nearest-neighbour (NN) ferromagnetic (FM) arrangement, different next NN FM and AFM configurations, and in-plane anisotropic spin correlations are vital to the magnetic structure and associated multiple frustrations. The discovery of the thus far highest AFM transition temperature renders SrTb$_2$O$_4$ a new friendly frustrated platform in the family for exploring the nature of magnetic interactions and frustrations.

The Néel temperature of the new frustrated family of SrRE 2O4 (RE = rare earth) compounds is yet limited to ∼ 0.9 K, which more or less hampers a complete understanding of the relevant magnetic frustrations and spin interactions. Here we report on a new frustrated member to the family, SrTb2O4 with a record TN = 4.28 (2) K, and an experimental study of the magnetic interacting and frustrating mechanisms by polarized and unpolarized neutron scattering. The compound SrTb2O4 displays an incommensurate antiferromagnetic (AFM) order with a transverse wave vector Q 0.5K AFM = (0.5924(1), 0.0059 (1), 0) albeit with partially-ordered moments, 1.92(6) µB at 0.5 K, stemming from only one of the two inequivalent Tb sites mainly by virtue of their different octahedral distortions. The localized moments are confined to the bc plane, 11.9(66) • away from the b axis probably by single-ion anisotropy. We reveal that this AFM order is dominated mainly by dipole-dipole interactions and disclose that the octahedral distortion, nearest-neighbour (NN) ferromagnetic (FM) arrangement, different next NN FM and AFM configurations, and in-plane anisotropic spin correlations are vital to the magnetic structure and associated multiple frustrations. The discovery of the thus far highest AFM transition temperature renders SrTb2O4 a new friendly frustrated platform in the family for exploring the nature of magnetic interactions and frustrations.

I. INTRODUCTION
Revealing the magnetic coupling mechanism is often a critical step towards understanding the role of magnetism in intriguing phenomena such as colossal magnetoresistance (CMR), high T C superconductivity, multiferroicity or frustration in correlated electron materials [1][2][3][4][5]. By way of example, the indirect double-and super-exchange interactions were successfully elaborated in qualitatively explaining the CMR effect and associated magnetic orders based only on the spin and charge degrees of freedom [1]. In 4f -based insulators, the indirect oscillating interaction [4][5][6][7] between pairs of localized 4f moments via the intermediary of valence electrons is blocked. Therefore, possible super-, dipole-dipole and multipolar, and Dzyaloshinsky-Moriya (DM) exchange interactions are primarily responsible for potential magnetic ordering [4]. Without detailed knowledge of the structural and magnetic parameters, it is hard to uniquely determine which interaction acts as the major exchange mechanism [8]. In this case, the origins of the related incommensurable spin structures become elusive [4]. In addition, the competition between spin-orbital coupling and crystal elec-tric field (CEF) at low temperatures largely affects the highly-degenerate Hund's rule ground state, and besides the anisotropic dipolar and DM interactions, determine the magnitude of the magnetic anisotropy [9,10]. This anisotropy strongly influences the degree of magnetic frustration. Sometimes, it may disorder or even quench potential magnetic moments, leading to a virtually nonmagnetic ground state [11].
ting a technically easier study of the two coupling mechanisms.
In this study, we report on a new frustrated member to the family of SrRE 2 O 4 , namely SrTb 2 O 4 , which has not been studied yet by neutron scattering. The singlecrystal SrTb 2 O 4 displays a long-range magnetic order relative to the underlying lattice. The noncollinear incommensurate AFM structure forms at T N = 4.28(2) K upon cooling. The synthesis of SrTb 2 O 4 with the highest Néel temperature in the family opens up an easier route to elucidate the magnetic coupling and frustrating mechanisms. By polarized and unpolarized neutron scattering we uniquely determine the detailed structural and magnetic parameters to understand the magnetism in SrTb 2 O 4 .

II. EXPERIMENTAL
Polycrystalline samples of SrTb 2 O 4 were synthesized from stoichiometric mixtures of SrCO 3 (99.99%) and Tb 4 O 7 (99.99%) compounds by standard solid-state reaction [43]. Both raw materials were preheated at 800 • for 12 h and weighted at ∼ 200 • . The mixed and milled raw materials were calcined twice at 1473 and 1573 K for 48 h each in air in order to perform decarbonization and prereaction. The resulting powder was pressed into cylindrical rods with an isostatical pressure of ∼ 78 MPa. The rods were sintered two times at 1573 and 1673 K for 48 h at each temperature in air. After each round of the isostatic pressing and subsequent firing, the product was reground and ball-remilled, which results in a dense and homogenous sample and ensures a complete chemical reaction. The single crystal of SrTb 2 O 4 was grown by optical floating-zone method with an atmosphere of ∼ 98% Ar and ∼ 2% O 2 . The growing speed is ∼ 4 mm/h with rotations of the feed and seed rods at +32 and -28 rpm, respectively. The phase purity of the polycrystalline and single-crystalline samples was checked by in-house X-ray powder diffraction. The electrical resistivity of a bar-shaped single crystal by standard dc fourprobe technique was measured on a commercial physical property measurement system.
High-resolution neutron powder diffraction (NPD) patterns were collected with a pulverized SrTb 2 O 4 single crystal (∼ 5 g) mounted in a 3 He insert on the structure powder diffractometer (SPODI) [44] with constant wavelength λ = 2.54008(2)Å at the FRM-II research reactor in Garching, Germany.
The SrTb 2 O 4 single crystal (∼ 2.2 g) for the neutronscattering studies was oriented in the (H, K, 0) scattering plane with the neutron Laue diffractometer OrientExpress [45] and the IN3 thermal triple-axis spectrometer at the Institut Laue-Langevin (ILL), Grenoble, France. The mosaic of this single crystal is 0.494(5) o full width at half maximum (FWHM) for the nuclear (2, 0, 0) Bragg reflection at 1.5 K. Longitudinal XYZ neutron polarization analysis was carried out on the D7 (ILL) diffractometer [46] with a dilution fridge and λ = 4.8Å. Unpolarized elastic neutron-scattering studies were performed at the two-axis D23 diffractometer (ILL) with incident wavelength 1. Here the wave vector cell.
III. RESULTS Figure 1 shows the neutron polarization analysis in the spin-flip (SF, i.e., Z-flipper on) and non-spin-flip (NSF, i.e., Z-flipper off) channels. Compared with the maps at 300 K ( Figure 1A), it is clear that extra fourfold Bragg peaks around (±1.6, ±1, 0) appear symmetrically in both SF and NSF reciprocal space maps at 50 mK ( Figure 1B) due to a long-range magnetic transition. Polarized neutron magnetic scattering depends on the direction of the neutron polarizationP with respect to the scattering vectorQ, and also the direction of the ordered-momentsμ. In our case,P (Z-component) c-axis [46], and the magnetic Bragg reflections are observed in the (H, K, 0) plane, i.e.,P ⊥Q. In this case, the neutron-scattering cross sections of the NSF and SF channels are respectively. The first and the second terms in each equation refer to the magnetic and spin-incoherent scatterings, respectively. The third term in Eq. (1) denotes nuclear and isotope incoherent contributions [46]. The presence of the incommensurable AFM Bragg peaks in the NSF channel ( Figure 1B) indicates that one component ofμ is parallel to the c axis, while their appearances in the SF channel imply aμ component lying in the ab plane. We observe the magnetic Bragg peak only at 0.5 K in our NPD study (Figure 2). We thereby refine the AFM wave vector exactly as Q AFM = (0.5924(1), 0.0059(1), 0) by the profile-matching mode [47] and a total moment |μ| = 1.92(6) µ B at the maximum amplitude for the Tb1 ions only with the b-and c-components equalling to +1.88 (8) and +0.40(23) µ B (Table 1), respectively. The moment size of the Tb2 site is negligible. Figure 3 schematically shows the resulting crystal and magnetic structures as well as the structural parameters for the bent Tb 6 honeycombs. The temperature dependence of the AFM (1.6, 1, 0) Bragg peak is shown in Figure 4. The extracted integrated intensity (I ) was fit to a power law I = I 0 (1− T TN ) β , which produces a Néel temperature T N = 4.28(2) K, and a critical exponent β = 0.55(2) probably indicative of a second-order type phase transition and possible threedimensional Heisenberg-like spin interactions [48].
We record a reciprocal space map ( Figure 5A) around the AFM (1.6, 1, 0) Bragg peak at 1.7 K using D23, and the central scans along the q H and q K directions ( Figure 5B) were measured at IN12. In both Figures, the FWHM of the magnetic Bragg peak along the q H and q K directions is sharply different. Both magnetic Bragg peaks are broader than the nuclear Bragg (2, 0, 0) reflection in the reciprocal space as shown in Figure 5B, which indicates that the observed magnetic Bragg peaks are beyond the instrument resolution. Therefore, Figure 5B shows a real in-plane magnetic anisotropy.

IV. DISCUSSION
To quantitatively estimate the in-plane anisotropy, we take the FWHM of the nuclear Bragg (2, 0, 0) peak as the detecting accuracy which is convoluted in fitting the magnetic peaks by a Gaussian function shown as the solid lines in Figure 5B. This results in FWHM = 0.0183(1) and 0.0492(2)Å −1 along the q H and q K TABLE 1. Refined structural parameters (lattice constants, atomic positions, Debye-Waller factor B, bond angles, and bond lengths), magnetic momentμ, and the corresponding goodness of refinement by the Fullprof Suite [47] from the NPD data measured at 0.5, 10 and 60 K using SPODI (FRM-II). The calculated average bond-lengths Tb1-O1,2,3 and Tb2-O1,3,4 and the extracted octahedral distortion parameter ∆ are also listed. All atoms reside in the Wyckoff site 4c, i.e., (x, y, 0.25). Number in parenthesis is the estimated standard deviation of the last significant digit.  directions, respectively, implying highly anisotropic inplane spin correlations consistent with the observation that strong magnetic frustration exists in SrTb 2 O 4 . We roughly estimate the spin-correlation length (ξ) by ξ = 2π FWHM , i.e., ξ H = 343.7(22)Å and ξ K = 127.6(4)Å. Therefore, ξ H ξ K = 2.69 (2). Similar in-plane anisotropic magnetic correlations were also observed in the iron-based superconductors [49][50][51][52][53] that are highly frustrated, too, where its microscopic origin, from the ellipticity of the electron pockets or the competing exchange interactions associated with the local-moment magnetism, is still being strongly argued [54][55][56][57]. It is undoubted that the observed in-plane magnetic anisotropy in SrTb 2 O 4 indicates an appearance of the competing spin exchanges and is certainly associated with a description of the purelylocalized magnetism of ionic Tb 3+ ions. A deeper understanding of the insulating state necessitates theoretical band structure calculations. We tentatively estimate the compatibility between ordered magnetic and nuclear crystalline domains based on the non-deconvoluted FWHM (κ) of the Bragg (1.6, 1, 0) (κ m = 0.0300(7)Å −1 ) and (2, 0, 0) (κ n = 0.0238(2)Å −1 ) peaks, i.e., κ n /κ m = 79(2)%, which implies that the incommensurate AFM structure orders with a long-range fashion relative to the underlying lattice of the single crystal.
We further analyze the spin-correlation length with our NPD data ( Figure 2C). Firstly, it is pointed out that the positive and negative momenta cannot technically be differentiated in a NPD study. As shown in Figure 6A, taking into account the corresponding SPODI instrument resolution (dashed line) [44], a Gaussian fit (solid line) to the AFM Bragg (1.5924, 1.0059, 0) peak (squares) results in an average ξ AFM = 864(36)Å in real space. This indicates that the AFM ordering observed in SrTb 2 O 4 is indeed of long range in character in comparison with the reported extremely-broad magnetic diffuse scattering which was attributed to the presence of short-ranged magnetic ordering in polycrystalline SrRE 2 O 4 (RE = Ho, Er, Dy) samples in the study of reference [34]. With the same method utilized in the analysis of the data as shown in Figure 6A, we also analyze the NPD peak of the nuclear Bragg (2, 0, 0) reflection as shown in Figure 6B and extract that ξ (200) = 1304(34)Å. This indicates that ξ AFM /ξ (200) = 66(3)% basically in accord with the compatibility between ordered magnetic and nuclear crystalline domains extracted with our singlecrystal neutron-scattering data. Since our NPD data were collected from a pulverized SrTb 2 O 4 single crystal, that ξ AFM is ∼ 2.5 times larger than ξ H may indicate that there have strong magnetic and crystalline domain effects in single-crystal SrTb 2 O 4 , or a large part of spins are blocked probably due to a pining effect by strains accumulated during single crystal growth. In any case, this difference between single-crystalline and polycrystalline samples in turn supports the fact that there is a strong magnetic frustration in single-crystal SrTb 2 O 4 . Further studies with high pressures would be of great interest. In most cases, the strength of the indirect magnetic interactions such as conventional double-or superexchange [1] can be influenced more or less by the value of the revelent bond angle [58][59][60][61], e.g. the ∠Tb-O-Tb bond angles in SrTb 2 O 4 as listed in Table 1 (see also Figure 7). However, the respective values of ∠Tb-O-Tb display no appreciable difference within accuracy between 0.5 and 10 K ( Table 1), below and above the T N , respectively, which may indicate an invalidity of the two conventional magnetic coupling mechanisms (doubleor super-exchange) in SrTb 2 O 4 . This is consistent with the study of SrTm 2 O 4 [11] and in excellent agreement with our transport study, where any attempts to measure possible resistivity in SrTb 2 O 4 from 2 to 300 K were fruitless. We estimate that the resistance of the single crystal measured is beyond at least 10 6 ohm. We thus conclude that SrTb 2 O 4 is a robust insulator, and the electrons responsible for the incommensurable antiferromagnetism are mainly from the localized 4f 8 shell of the ionic Tb 3+ ions. In this localized picture, the interionic exchange interactions dominate for the formation of the magnetic structure [4,6]. The nearest Tb neighbours are stacked linearly along the c axis ( Figure 3B). The shortest Tb1-Tb1 and Tb2-Tb2 have the same bond length. However, the NN Tb1 ions have a ferromagnetic (FM) arrangement. By contrast, the interaction between the NN Tb2 ions is blocked unexpectedly ( Figure 3B). There is no appreciable difference in the NN Tb-Tb bond length, i.e., the c lattice constant, between 0.5 and 10 K ( Table 1), which probably rules out the potential direct exchange interaction consistent with the fact that unpaired 4f electrons are deeply embedded under the 5s 2 p 6 shells and also indicates that the prevalent dipole-dipole interaction is subjected to some condition, i.e., the octahedral distortion as discussed below, in agreement with the study of SrTm 2 O 4 [11].
As a non-Kramers ion, Tb 3+ (S = 3, L = 3, J = 6, g J = 1.5) in principle keeps the time reversal symmetry and doesn't show any energy degeneracy in the presence of the purely-localized electric field. However, we refine two kinds of octahedra as shown in Figure 7: Tb1O 6 and Tb2O 6 , corresponding to the partially-ordered and totally-frozen Tb1 and Tb2 ions, respectively. The average octahedral distortion [58,59] can be quantitatively measured by the parameter ∆ defined as: ∆ =   [60]. This sharp contrast implies that the Tb1 ions are strongly distorted, while the Tb2 ions behave normally within the non-Kramers scheme. Therefore, the ∆ magnitude that reflects the ion local symmetry and thus the strength of the surrounding CEF directly determines the existence of the magnetic ordering, which is supported by the observation that below T N the respective ∆ values of the Tb1 and Tb2 ions change oppositely with temperature ( Figure 8). We therefore infer that one possible reason for the formation of the incommen-surable magnetic structure is the modulated distribution of the 4f 1 6s 2 valence electrons which modify the surrounding environment experienced by the localized unpaired 4f electrons. The corresponding modulation of the local symmetry may plausibly be attributed to the spatial zigzag-type Tb arrangements along the a and b axes in the process of forming the crystallographic domains. This is supported by the fact that the honeycomb columns run straightly along the c axis, and there is no spin modulation at all in that direction.
Based on the refined Tb-O bond lengths, we deduce two distortion modes for the Tb1O 6 and Tb2O 6 octahedra (Figure 7), respectively. The possible product of the Tb1 subjected stress-vectors (small arrows) should point qualitatively to the direction of the Tb1 moment, implying a strong single-ion anisotropy. This JT-like distortion mode leads to the large ∆ value of the Tb1 ions, and possibly lifts further the degenerate multiplets. By contrast, the Tb2 ions are subjected to opposing stresses in all the three pair-directions. In this case, the octahedral distortion strongly depends on their competing strengths. This mode makes the small ∆ value of the Tb2 ions and their potential total magnetic moments quenched vitally.
The maximum Tb1 moment size is mere 1.92(6) µ B , 21.3(7)% of the theoretical saturation value (g J J = 9 µ B ). It is of particular interest to explore the frustrating mechanism. The virtual non-Kramers state of the Tb2 site reduces the total moment size per molar formula by 50%. The Tb1 moment fluctuates like a wave defined asμ = |μ max |cos(Q AFM · R x + φ), where R x is a spin coordinate along the a axis and φ is a phase parameter. The existence of the strong single-ion anisotropy indicates a large CEF effect which should be comparable to the energy scale of the magnetic interactions. We have shown the clear evidence for a large magnetic exchange anisotropy ( Figure 5), which is ascribed to the anisotropic dipole-dipole interaction. The NN magnetic arrangement is FM ( Figure 3B), implying no possibility for a magnetic frustration. The NNN magnetic configurations display a dual character, i.e., FM and AFM for the equivalent Tb11 and Tb12 sites, respectively. This sharp difference may frustrate the Heisenberg-exchange coupled NNN spins.

V. CONCLUSIONS AND OUTLOOK
To summarize, we have synthesized large enough SrTb 2 O 4 single crystals suitable for neutron scattering studies and revealed a modulated spin structure in SrTb 2 O 4 with the highest AFM transition temperature at T N = 4.28(2) K in the SrRE 2 O 4 family, which provides a technically friendly platform to explore the related magnetic coupling and frustrating mechanisms. Our studies show that the localized Tb1 moments lie in the bc plane with the FM chains along both the b and c directions and the AFM modulation mainly along the a axis. We have found two distinct octahedra for the non-Kramers Tb 3+ ions: Tb1O 6 being strongly distorted, corresponding to the partially-ordered moments; Tb2O 6 being frustrated entirely in the non-Kramers state. Therefore, the octahedral distortion has a decisive influence on the Hund's rule magnetic ground state ( 7 F 6 ) and the related frustrations. The magnetocrystalline anisotropy is crucial in determining the direction of the ordered moments. The direct NN interaction results in a FM arrangement for the Tb1 ions along the c axis, and the different NNN Tb configurations (FM and AFM) further lift the magnetic frustration. The present results make SrTb 2 O 4 a particularly significant compound in the family for theoretical and further experimental studies. Inelastic neutron-scattering studies to determine the detailed crystal-field and magnetic-interaction parameters would be of great interest. The factors that influence the value of the AFM transition temperature would be further explored in combination with theoretical calculations.