Anisotropy in the magnetic interaction and lattice-orbital coupling of single crystal Ni3TeO6

This investigation reports on anisotropy in the magnetic interaction, lattice-orbital coupling and degree of phonon softening in single crystal Ni3TeO6 (NTO) using temperature- and polarization-dependent X-ray absorption spectroscopic techniques. The magnetic field-cooled and zero-field-cooled measurements and temperature-dependent Ni L3,2-edge X-ray magnetic circular dichroism spectra of NTO reveal a weak Ni-Ni ferromagnetic interaction close to ~60 K (TSO: temperature of the onset of spin ordering) with a net alignment of Ni spins (the uncompensated components of the Ni moments) along the crystallographic c-axis, which is absent from the ab-plane. Below the Néel temperature, TN~ 52 K, NTO is stable in the antiferromagnetic state with its spin axis parallel to the c-axis. The Ni L3,2-edge X-ray linear dichroism results indicate that above TSO, the Ni 3d eg electrons preferentially occupy the out-of-plane 3d3z2−r2 orbitals and switch to the in-plane 3dx2−y2 orbitals below TSO. The inherent distortion of the NiO6 octahedra and anisotropic nearest-neighbor Ni-O bond lengths between the c-axis and the ab-plane of NTO, followed by anomalous Debye-Waller factors and orbital-lattice in conjunction with spin-phonon couplings, stabilize the occupied out-of-plane (3d3z2−r2) and in-plane (3dx2−y2) Ni eg orbitals above and below TSO, respectively.

] bond distances and bond angles, as mentioned in ref. 14 . Wu et al. 14 also found that magnetic dipole-dipole interactions, which are stronger than spin-orbit coupling, are responsible for the orientation of the Ni spin axis parallel to the c-axis. Figure 1(c) displays all these spin exchange paths. The unique noncentrosymmetric structure and variation in exchange interaction with various Ni spin sites together give rise to the favorable magnetic field-driven electric polarization properties of NTO [15][16][17] . Yokosuk et al. 15,16 and Kim et al. 17 used extremely high magnetic fields to elucidate the spin-induced electric polarization properties between 9 and 52 T. In a nominal magnetic field, NTO is a collinear antiferromagnet below ~52 K (Néel temperature: T N ) 15 , as revealed by anisotropic magnetization and specific heat capacity measurements 12,13 . Yokosuk et al. 16 and Skiadopoulou et al. 18 performed a combination of infrared, Raman and THz spectroscopic measurements and explained the anomalous temperature-dependent behavior of spin excitation by spin-phonon coupling below T N , whereby local lattice distortion in the ab-plane is induced by magnetic ordering, which subsequently modifies the AFM interaction between Ni ions owing to their displacements from their mean positions in the unit cell 16 . Moreover, numerous reports on related AFM systems have suggested a strong correlation among spin, orbital, charge and lattice degrees of freedom, which are responsible for the intriguing material properties of transition metal oxides [19][20][21][22][23][24][25][26] . Specifically, Ling et al. 19 reported that a structural transition in lanthanum manganate can trigger Mn 3d e g orbital ordering, causing AFM spin ordering. Deshpande et al. 20 also found temperature-and substrate-driven preferential electron occupancy of the Mn 3d e g orbital in La 0.85 Zr 0.15 MnO 3 (LZMO) thin films epitaxially grown on SrTiO 3 (STO) and MgO substrates. As Experimental results further suggested that the strong tensile strain stabilizes the 3d x 2 −y 2 orbital by inducing lattice distortions of the MnO 6 octahedra in LZMO/MgO 20 . Furthermore, in t 2g systems such as rare-earth vanadate, cooperative orbital ordering induces local lattice distortion below a certain transition temperature 21 . Therefore, the origin of all of the aforementioned exotic properties of NTO is believed to be closely associated with the intriguing interplay between the Ni 3d electron's spin, orbital, charge and lattice-related degrees of freedom, which could lead to intriguing magnetic properties, such as the existence of FM and AFM interactions with the spin-axis parallel to the crystallographic c-axis in NTO [27][28][29][30] . Although the spin orientations of Ni 12,14,16 and the spin-phonon coupling in NTO 15,16,18 have recently been studied, the spin-orbit-lattice-charge intercorrelations have not yet been fully explored, to the best of our knowledge. Therefore, this investigation is a detailed study of the temperature-and polarization-dependent electronic and atomic structure of NTO, as well as of the preferential orbital and anisotropic magnetic behaviors, to elucidate correlations among the aforementioned degrees of freedom across the transition temperature of the NTO single crystal.

Results and Discussion
Figure 2(a) displays the X-ray diffraction (XRD) pattern of the NTO single crystal at room temperature, showing the [003] Bragg reflection. The diffraction line shape is symmetrical, and Gaussian fitting yields a small full-width-at-half-maximum (FWHM = 0.11 0 ), as observed in the θ scan of the (006) Bragg reflection in the inset of Fig. 2(a). This very small FWHM indicates good crystallinity and chemical homogeneity of the NTO single crystal. Figure 2(b) also shows the temperature-dependent XRD pattern (all indexed peaks 7 are tabulated in Table 1 of the Supplementary Information) of crushed and finely ground NTO powder obtained from its single crystal. All data are similar at all temperatures of interest with no appearance or disappearance of peaks, which shows that NTO exhibits no structural phase transition in the investigated temperature range of 11-300 K. However, the intensities of some peaks [such as (1 0 1), (0 1 2) and (1 1 0)] relative to the most intense (1 0 4) peak vary with temperature; the implications of this finding will be discussed later in this manuscript. Sankar et al. 13 conducted XRD analyses of NTO using the general structure analysis system code 31 and identified a trigonal crystal structure in a rhombohedral lattice with space group R3 and cell parameters a = b = 5.11 Å and c = 13.74 Å at 300 K. Figure 3(a) plots the magnetic susceptibility (M/H) as a function of temperature [in the field-cooling (FC) and zero-field-cooling (ZFC) modes] using a nominal external magnetic field H = 100 Oe aligned parallel and perpendicular to the c-axis, thus revealing the anisotropic magnetic properties of NTO. The FM interaction [mostly a result of a J 2 exchange interaction, as depicted in Fig. 1(c)] is rather weak and appears as a hump close to ~60 K (T SO : spin-ordering temperature) when the magnetic field is applied parallel to the c-axis [inset in Fig. 3(a)] and it is absent when the magnetic field is applied perpendicular to the c-axis (H⊥ c). The weaker FM interaction suggests that the Ni spins may not be collinear. Instead, the uncompensated component of the Ni spin moment is aligned parallel to the c-axis (hereafter referred to simply as FM Ni spins) and the interaction between these components of the moments will be referred to as FM interaction in this manuscript. Below ~52 K (T N ), the M/H curves (for H// c and H⊥ c) turn downward, revealing AFM ordering (caused by J 3 -J 5 exchange interactions).  The different magnetization features revealed by the H// c and H⊥ c curves below T N (~52 K) suggest that the AFM spin axis is primarily parallel to the c-axis 12 . This alignment of the magnetic spin axis parallel to the c-axis in the FM and AFM phases is consistent with the earlier calculations of Wu et al. 14 . Sankar et al. 13 claimed that in a high magnetic field, most of the Ni ions in NTO are in the Ni 3+ state in either the high-spin (S = 3/2; t 2g 5 e g 2 ) or low-spin (S = 1/2; t 2g 6 e g 1 ) configuration, with the remaining minority in the Ni 2+ state with the S = 1 (t 2g 6 e g 2 ) spin configuration. Their analysis reflects that at low magnetic fields, however, the Ni 2+ spin state in NTO is responsible for magnetization. Temperature-and polarization-dependent Ni K-edge X-ray absorption near-edge structure (XANES) analyses of the NTO single crystal, as well as Ni K-and L 3,2 -edge XANES analyses of powdered NiO, Ni 2 O 3 and NTO at room temperature, have also been performed, as described in Fig. S1 Fig. S1(a,b)] in Ni 2 O 3 are higher than those in NiO (20.08 ± 0.05 and 0.05 ± 0.02 at the Ni L 3,2 -and K-edge, respectively). Clearly, for NTO, the areas under the Ni L 3,2 -edge (21.05 ± 0.05) and K pre-edge (0.05 ± 0.01 and 0.06 ± 0.01 for E// c and E⊥ c, respectively) features are close to those for NiO, suggesting that most of the Ni ions in NTO are in the 2 + valence state. Additionally, the Ni L 3,2 -edge absorption line shapes of NTO are consistent with those of NiO in the results of Hu et al. 35 and Abbate et al. 36 , who compared the Ni L 3,2 -edge absorption line shapes of NiO with those of other Ni compounds to estimate the valence states.
To determine the valence state of the Ni ions in NTO at various temperatures, as shown in the bottom panels of Fig. S1(a,b) based on the position of the threshold feature, the derivative of the threshold feature of the Ni K-edge absorption feature is used. Apparently, the Ni K-edge threshold energy and the line shapes of the NTO single crystal, as well as the area under the pre-edge peak, do not vary with temperature for two electric polarizations (E// c and E⊥ c), confirming that the valence state of the Ni ions is insensitive to temperature and orientation and is mostly 2+ in NTO. Furthermore, we have also carried out temperature-dependent Te K-edge XANES analyses [ Fig. S2(a,b) for E// c and E⊥ c, respectively] as complementary evidence to support the Ni 2+ state. The threshold/ peak position of the Te K-edge XANES is 31825.0 ± 0.5 eV for both the E// c and E⊥ c polarizations, which is consistent with the XANES spectra for the Te 6+ state as reported by Grundler et al. 37 for their Te(OH) 6 sample. Assuming oxygen in the O 2− state, the 6+ valence state of Te evidently suggests that Ni will be in the 2+ valence state to satisfy charge compensation in NTO. Moreover, similar to the Ni K-edge XANES, the Te K-edge XANES is also insensitive to changes in temperature within the measured 40-300 K range. This result further indicates that Te and Ni are stable in their respective 6+ and 2+ states and do not vary with temperature within this range. Evidently, the temperature-and polarization-dependent Ni K-edge XANES studies do not support any substantial effect of the disproportionation of Ni 3+ and Ni 2+ on the anisotropic magnetic properties of NTO below the transition temperature T N because the Ni ions in NTO are primarily not in the Ni 3+ state and therefore do not exhibit high-spin or low-spin configurations at various temperatures.
In the FC and ZFC curves of NTO for H// c in Fig. 3(a), the FM region is barely observable relative to the dominant AFM feature in NTO. Therefore, for better understanding of the FM feature in NTO, magnetization measurements were performed in various magnetic fields, and the first derivatives (dM/dT) for H// c and H⊥ c are plotted in Fig. 3(b,c), respectively. dM/dT is useful in the sense that it can be used to identify minor features or fluctuations of magnetization with temperature, which are difficult to observe from raw data 38,39 . Although the measured absolute intensity of the FM feature for H// c gradually increases with the magnetic field, as shown of Fig. 3(b), the corresponding normalized dM/dT plots [normalized dM/dT = (dM/dT)/(dM/dT at T N )] in NTO gradually decrease, as shown in the inset in Fig. 3(b). The normalized dM/dT plots enable the identification of the variation in the FM feature with respect to the AFM feature in NTO under various magnetic fields. The plots in the inset of Fig. 3(b) indicate that in a stronger magnetic field, the AFM feature dominates the FM feature and since they are close to each other on the temperature scale, the FM feature is generally invisible. This fact may explain why the results of Sankar et al. 13 indicate several different magnetic behaviors of NTO in high and low magnetic fields. In contrast, the field-dependent normalized dM/dT plots for H⊥ c do not show any features close to ~60 K, as presented in Fig. 3(c), suggesting a lack of alignment among the FM Ni spins of NTO in the ab-plane. To further understand the correlation of the anisotropic FM and AFM phases with the preferential orbital and lattice degrees of freedom in the NTO single crystal, temperature-dependent X-ray magnetic circular dichroism (XMCD), X-ray linear dichroism (XLD) and extended X-ray absorption fine structure (EXAFS) analyses were performed and are described below. Figure 4(a) displays the temperature-dependent Ni L 3,2 -edge XANES spectra of the NTO single crystal, with the photohelicity of incident X-rays parallel (μ + ) and antiparallel (μ − ) to the direction of magnetization under a magnetic field of H = 100 Oe applied parallel and antiparallel to the c-axis, respectively. All temperature-dependent Ni L 3,2 -edge XANES spectra of NTO exhibit two broad features in the ranges of 852-857 eV and 868-873 eV, which are attributed to Ni 2p 3/2 → 3d 5/2 and 2p 1/2 → 3d 3/2 dipole transitions, respectively. Figure 4(b) also shows the corresponding Ni L 3,2 -edge XMCD spectra [(μ − − μ + )/(μ − + μ + )] obtained at various temperatures. From the field-dependent normalized dM/dT in the inset of Fig. 3(b), the FM feature gradually becomes obscured by the AFM feature at higher magnetic fields. Therefore, a considerable XMCD signal from NTO can be obtained at an applied magnetic field of 100 Oe without being buried under the dominating AFM feature. XMCD reveals the expectation value of the magnetic moment, <M> 40,41 , and thereby provides information about individual magnetic ion spins and magnetic orbital moments in the overall FM interactions 42,43 . Clearly, the temperature-dependent Ni L 3,2 -edge XMCD spectra obtained by switching the magnetic field from parallel to antiparallel to the c-axis reveal a weak but clear XMCD signal at 59 K [close to T SO (60 K) , marked by a green arrow in Fig. 4(b)] that diminishes as temperature varies upward or downward. The directionality of the FM spins of the Ni ions along the c-axis in NTO is consistent with the magnetization measurements herein (Fig. 3) and in other studies 12,13 . Importantly, in contrast, the temperature-dependent Ni L 3,2 -edge XMCD measurements made when a magnetic field of 100 Oe is applied perpendicular to the c-axis (two opposite magnetic field directions) do not reveal an XMCD signal (see Fig. S3 in the Supplementary Information). In magnetic materials, the FM moment is typically correlated with the degree of the long-range magnetic order of the magnetic ions, which is determined by competitive exchange coupling between electron spins and thermal fluctuation. However, the XMCD measurements in Fig. 4(b) clearly demonstrate weak FM interaction between the Ni 3d electron spins in NTO at ~59 K with the magnetic field along the c-axis.
Typically, XMCD is sensitive only to the expectation value of the local magnetic moment, <M>, and therefore disappears in the AFM regime 40,41 at or below T N ~ 52 K, as shown in Fig. 4(b). The simultaneous determination of the spin orientation in an AFM system, unlike in an FM system, and local orbital structure is a challenging experimental task because of their compensated magnetic dichroism nature. However, with the development of synchrotron sources with high photon fluxes, soft XLD spectroscopy has emerged as a powerful tool for detecting the spin axis and orbital symmetry of all uniaxial magnetic systems because linearly polarized photons have only The sign of the XLD spectra is negative in the paramagnetic regime (above 60 K), indicating that the Ni e g holes preferentially occupy 3d x 2 −y 2 (that is, preferential occupancy of the Ni e g electrons in 3d 3z 2 −r 2 ) orbitals. However, upon cooling below T SO , the sign of the XLD spectra is reversed to positive, suggesting that the Ni e g holes preferentially occupy the out-of-plane 3d 3z 2 −r 2 (that is, preferential occupancy of the Ni e g electrons in 3d x 2 −y 2 ) orbitals. This result is reproducible and consistent with similar temperature-dependent Ni L 3,2 -edge XANES and corresponding XLD measurements with electrically polarized X-rays on a different beamline [BL-20A at the NSRRC (not presented here)]. To elucidate this phenomenon, the integrated intensity of the area under the Ni L 3 -edge of XLD spectra (A L3 , in the region from 848-858 eV) is plotted in Fig. 5(b), which clearly indicates the temperature dependency of the preferential electron occupancy in the Ni 3d e g orbitals. Negative and positive values of A L3 at various temperatures demonstrate the preferential Ni e g electron occupancy in the 3d 3z 2 −r 2 and 3d x 2 −y 2 orbitals, respectively. Above T SO (60 K), the negative A L3 s are fairly independent of temperature, as shown in Fig. 5(b). The evolution of this anisotropic behavior can also be attributed to an additional local CF effect and ligand-metal p-d hybridization, which stabilize electron occupancy in the 3d 3z 2 −r 2 orbitals in the NTO. Although the origin of this CF effect and ligand-metal p-d hybridization is currently not considered, the distortion of the crystal structure is primarily responsible for the lowering of the energy of either the in-plane or out-of-plane e g orbitals of transition metal oxides 45 . We believe that the inherent distortion of NiO 6 octahedra in an environment of trigonal symmetry also plays an important role, just as VO 6 octahedral distortion contributes importantly to orbit-lattice coupling in rare-earth vanadates 24,25 . Previous studies 6,14 have revealed that the NiO 6 octahedra in NTO are distorted and that the Ni-O-Ni bond distances and angles vary substantially among the three Ni sites (Fig. 1). This phenomenon may be responsible for additional CF and ligand-metal p-d hybridization effects and thereby stabilize Ni e g electron occupancy in the 3d 3z   Fig. 2(b)], the switching of the preferred orbital is not caused by a structural transition, unlike in lanthanum manganite 19 . Several reports have verified strong orbital-lattice coupling in rare-earth perovskites of the families RVO 3 24,25 and RTiO 3 (R = La, Pr, Sm, Yb and Lu) 26 , in which VO 6 octahedral distortion is most likely responsible for the orbital ordering. Therefore, the spin-orbit-lattice couplings that accompany local lattice distortion, owing to static disorder and the breakdown of lattice symmetry, certainly seem to favor (or strongly correlate with) preferential orbital occupancy and orbital ordering in various transition metal oxide systems 20,[22][23][24]26,49 .
To elucidate the correlation between the Ni 3d electron spin and the lattice, as well as the factors that drive the switching of the preferential orbital occupancy from out-of-plane to in-plane in the NTO single crystal below T SO , temperature-dependent Ni K-edge EXAFS measurements were performed with linearly polarized X-ray beams oriented along E// c and E⊥ c. Figure 6(a,b) display the temperature-dependent Fourier transform (FT) plots (solid lines) of the Ni K-edge EXAFS spectra of the NTO single crystal at θ = 70° (E// c) and θ = 0° (E⊥ c), respectively, within a k range of 3.3-11.6 Å −1 . The insets present corresponding EXAFS k 3 χ data. The first main feature of the FT plots of the Ni K-edge EXAFS spectra corresponds to the nearest-neighbor (NN) bond length of Ni-O in the NTO single crystal. We have analyzed the EXAFS region of the spectra by using the ATHENA and ARTHEMIS program packages 50 to extract quantitative local information, such as the mean NN Ni-O bond length (R), its mean-squared fluctuation, called the Debye-Waller factor (DWF), and the coordination number (N) around the Ni sites in NTO for E// c and E⊥ c. Figure 6(a,b) also show fitted plots (circles) of the temperature-dependent Ni K-edge FT for two polarization orientations, E// c and E⊥ c, respectively. This work primarily focuses on the NN oxygen coordination number around the Ni sites, the variation in the NN Ni-O bond length, and the corresponding DWFs for different temperatures in NTO, for both the E// c and E⊥ c polarizations. Therefore, we fit the first main FT feature within an R range of 1.26-2.30 Å, which resembles the first shell around the Ni site. The goodness of fit, i.e., the R-factor, lies between 0.009 and 0.014 (see SI Table 2), which indicates an excellent match between the experimental data and the model system (constructed with a calculated NN oxygen coordination number and known lattice parameters 13 ) used in fitting. Table 2 of the Supplementary Information and Fig. 7 present the fitted results for the first main FT feature, shown in Fig. 6(a,b). Notably  Fig. 1(a)], the projections of the O atoms onto the c-axis and the ab-plane are calculated based on the angles they make with the c-axis. This results in fractional coordination numbers of NN O atoms for the E// c and E⊥ c polarizations. Figure 7(a,b) plot the NN Ni-O bond lengths (weighted according to the angle between the electric polarization and bond direction) and DWFs, respectively, as functions of temperature. Clear anisotropy is observed The out-of-plane mean NN Ni-O bond length in the 60-300 K temperature range is fairly independent of temperature, but compression is observed upon cooling below T SO (60 K). However, the in-plane mean NN Ni-O bond length is rather temperature-independent throughout the entire measured temperature range (40-300 K). Consistent with our earlier discussion of the preferential orbital occupancy of the Ni 3d e g electrons in NTO, this anisotropy of the NN Ni-O bond lengths above T SO may lead to additional CF and ligand-metal p-d hybridization effects, causing preferential Ni e g electron occupancy in the 3d 3z 2 −r 2 orbitals in the paramagnetic (PM) region, as mentioned above. Below T SO , a tensile-like strain (compression of the Ni-O bond lengths projected out-of-plane) in the NiO 6 octahedra can favor the preferential Ni e g electron occupancy in the 3d x 2 −y 2 orbitals 51 . Additionally, as shown in Fig. 7(b), the DWFs at Ni sites in NTO (for E ⊥ c) deviate from their expected decreasing trend at and below T SO . The variation in the DWF can be understood with reference to the intensity of the first main FT feature (better views of the variation in intensity of the first Ni K-edge FT features for both polarizations are shown in Fig. S4 in the Supplementary Information), which is determined by the oxygen coordination number N and the DWF [σ 2 (T)] at Ni sites 51,52 . Due to the absence of structural phase transitions in the measured temperature range [11-300 K, see Fig. 2(b)], it is fair to consider the oxygen coordination number, N, as independent of temperature. Therefore, a change in σ 2 (T) is associated with a variation in FT intensity with temperature. The DWF has two components, σ 2 s and σ 2 v (T), which are associated with static disorder in atomic structure and thermal lattice vibration, respectively 52 . Since σ 2 s is independent of temperature, thermal variation in the DWF is commonly associated with variation in σ 2 v (T), which, according to the Einstein and Debye models 51,52 , decreases as temperature decreases. As expected, the DWFs under both polarizations decrease (the intensity of the first main FT feature increases, as shown in Fig. S4) as the temperature decreases from 300 to 80 K because of the σ 2 v (T) component. Interestingly, below T SO , the intensity of the FT feature decreases as temperature decreases, causing the DWF to increase (for E⊥ c). This anomalous result clearly indicates that the static disorder component, σ 2 s , dominates the decrease in the σ 2 v (T) component caused by the disorder produced by thermal vibration of the lattice. Therefore, the overall DWF, σ 2 (T), increases meaningfully as temperature decreases below T SO (60 K). The large static DWF, particularly in the ab-plane (E⊥ c), can be understood to arise from the large static distortion of the octahedral oxygen network around the Ni sites in NTO below T SO , which strongly influences the intensity of the FT feature associated with NN Ni-O bonds. The evolution of this static disorder can be understood with respect to phonon-assisted behavior 53,54 , which is related to a decrease in or breaking of the crystal lattice symmetry, especially in the ab-plane of NTO. Infrared, Raman and time-domain THz spectroscopic studies have previously revealed phonon-softening modes in NTO in the ab-plane below T N 16,18 , which were explained by spin-phonon coupling. Spin-phonon coupling induces local lattice distortion in the ab-plane, and the corresponding local displacements modulate the magnetic interactions among the Ni ions in NTO, causing them to align along the c-axis, thus resulting in low-temperature polarization 16 . As stated earlier, the theoretical analysis of Wu et al. 14 suggested that the magnetic dipole-dipole interaction is stronger than the spin-orbit coupling and is therefore responsible for the orientation of the Ni spins parallel to the c-axis in NTO. Their calculations also revealed that the J 2 interaction between the Ni spins (Ni II -Ni III ) along the c-axis is more strongly FM than the J 1 interaction (Ni I -Ni II in the ab-plane). Consistent with these calculations, weak FM (~60 K) interactions are observed herein with the uncompensated component of the Ni spins aligned only along the c-axis, as shown in the magnetization plots (Fig. 3) and the XMCD spectra [for H// c, as shown in Fig. 4(b), at 59 K but not for H⊥ c, as shown in Fig. S3 of the Supplementary Information]. These interactions are followed by strong AFM (T N ~ 52 K) interactions with the Ni spin axis aligned parallel to the c-axis, as presented in the magnetization plots in Fig. 3(a). Notably, in NTO, the NiO 6 octahedra are inherently distorted with off-centered Ni atoms above T SO , so the evolution of phonon-softening modes below T SO in the lattice is expected to correlate with the (phonon-mediated) interaction between additionally off-centered Ni atoms with respect to O atoms in the NiO 6 octahedra in the Ni I -Ni II and Ni III -Te layers. Although EXAFS analysis does not reveal the exact Ni sites responsible for the phonon-softening modes, earlier studies 16,18 have established that the phonon-softening modes in the ab-plane of NTO are associated with the displacement of Ni I atoms with respect to Ni II and Ni III atoms and the octahedral stretching mode, which involves the shifting of O atoms with respect to Ni atoms. Nevertheless, Ni displacement and octahedral stretching modes in the ab-plane are evidently associated with the variation in Ni-O bond lengths. Figure 7(a) indicates that the NN Ni-O bond lengths are, on average, compressed along the c-axis (E// c) upon cooling below T SO . These variations in Ni-O bond lengths below T SO may explain the additional off-centering/displacement of the Ni ions, which causes phonon softening below T SO . Typically, the process of off-centering of metal ions preserves the overall crystal symmetry, but it may change the structure factor, resulting in a modification of the intensity of the powder XRD feature 53 . Variation in the intensities of the powder XRD feature of NTO is observed in Fig. 2(b), as mentioned earlier, while the crystal structure is maintained throughout the range of measurement temperatures. The unusually high DWFs in the ab-plane relative to those along the c-axis throughout the specified temperature range further suggest anisotropic local structural ordering of NTO. In addition, the higher slope of the DWF as a function of temperature between 80 and 300 K suggests that the in-plane (E⊥ c) NN Ni-O bond length is more sensitive to temperature than the out-of-plane (E// c) Ni-O bond length. This anomalous variation in DWFs at in-plane Ni sites and the compression of the apical projection (E// c) of Ni-O bonds, which are correlated with lattice-orbit coupling and occur in conjunction with spin-phonon coupling, stabilize the preferential occupancy of the Ni e g electrons in the 3d x 2 −y 2 orbitals in NTO below T SO . In contrast, the inherent distortion of NiO 6 octahedra stabilizes the Ni 3d 3z 2 −r 2 orbital above T SO . Phonon-softening behavior and, consequently, anomalous DWF behavior in the ab-plane below a transition temperature were recently observed in our study of a SrFeO 3−δ single crystal 45 . A detailed theoretical investigation must be conducted in the future to develop our understanding of the correlation between local lattice distortion and preferential orbital occupation and its effect on the spin-spin correlation function in the NTO single crystal.
In summary, the electronic/atomic structure, preferential Ni 3d-orbital occupation and magnetic properties of the NTO single crystal were elucidated through magnetization measurements and temperature-dependent Ni L 3,2 -edge XANES, XLD and XMCD and K-edge EXAFS spectroscopic techniques. The magnetization measurements reveal a transition from the PM phase at high temperature to the FM phase close to T SO (~60 K), followed by an AFM transition close to T N (~52 K). Consistent with theory, the FM interactions associated with Ni II -Ni III spins (exchange interaction of J 2 ) are responsible for most of the evolution of the XMCD spectra along the c-axis close to T SO , whereas the corresponding signal is absent when a magnetic field is applied perpendicular to the c-axis. The Ni L 3,2 -edge XLD spectra above T SO reveal that Ni 3d e g electrons preferentially occupy the out-of-plane 3d 3z 2 −r 2 orbital and switch to the in-plane 3d x 2 −y 2 orbital at and below T SO . The inherent distortion of NiO 6 octahedra and anisotropic NN Ni-O bond length stabilize the preferential Ni e g electron occupation in the out-of-plane (3d 3z 2 −r 2 ) orbital above T SO . However, at and below T SO , a large static distortion of the NiO 6 octahedra (tensile-like strain due to the compression of the Ni-O bond length projected out-of-plane) network around the Ni sites, associated with phonon-softening behavior, stabilizes the preferential Ni e g electron occupation of the in-plane orbital (3d x 2 −y 2 ). These strong anisotropic lattice-orbital and spin-phonon couplings are responsible for the evolution of anisotropic magnetic properties and orbital switching in the NTO single crystal.

Methods
Sample preparation and characterization. Single crystal NTO with a (001) plane was synthesized using the flux slow cooling method; details on the sample dimensions (~3 × 2 mm surface dimension) and preliminary characterizations can be found elsewhere 13 . A detailed structural study of NTO was carried out using temperature-dependent powder and room temperature single-crystal XRD. Magnetic and electronic properties were obtained from temperature-dependent magnetization, Ni L 3,2 -edge XANES, XMCD and XLD and Ni K-edge XANES and EXAFS measurements.
Single-crystal XRD measurements were made in 2θ and θ scan modes at room temperature on a four-circle X-ray diffractometer with the X-ray beam aligned with the (003) Bragg reflection of NTO. Cu K α X-ray radiation with a spot size of ~2 mm in diameter was focused onto the sample surface. Temperature-dependent powder XRD was also performed at beamline BL-07A of the NSRRC in Taiwan. The energy of the X-ray beam was 14 keV (wavelength ~0.8856 Å), which was later transformed to the wavelength (1.5406 Å) corresponding to the energy of Cu K α to maintain consistency with the single-crystal XRD data. Temperature-dependent FC and ZFC magnetization measurements were made using a Quantum Design superconducting quantum interference magnetometer in the temperature range of 2-300 K with external magnetic fields applied parallel and perpendicular to the crystallographic c-axis of the NTO single crystal.
XANES measurements with a circularly polarized X-ray beam at the Ni L 3,2 -edge were carried out in beamline Dragon-BL-11A at the NSRRC (the X-ray beam spot size was ~0.3 × 0.1 mm on the sample surface), and the spectra were obtained in the total fluorescence yield (TFY) and total electron yield (TEY) modes after a magnetic field of H = 100 Oe was applied to NTO. By switching the direction of the magnetic field, NTO's magnetization direction was made parallel (μ + ) and antiparallel (μ − ) to the helicity of the incident X-rays. The difference between the above two measurements [(μ − − μ + )/(μ − + μ + )] is referred to as XMCD. For the linearly polarized X-ray beam, Ni L 3,2 -edge measurements (in Dragon-BL-11A and BL-20A in the TEY mode at the NSRRC) and K-edge measurements (in BL-17C in the TFY mode at the NSRRC) were made at two angles of X-ray photon incidence (θ) with respect to the normal to the NTO surface, θ = 0 0 (normal incidence, with the electric field E of the linearly polarized photons perpendicular to the c-axis, E⊥ c) and θ = 70° (grazing incidence, with E almost parallel to the c-axis, E// c). L 3,2 -edge XLD denotes the difference between the above two measurements for different θ (E// c − E⊥ c). The Ni K-edge EXAFS spectra were analyzed using the ATHENA and ARTHEMIS program packages 50 to extract quantitative local information, such as the mean NN Ni-O bond length (R), its mean-squared fluctuation, called the DWF, and the coordination number (N) for E// c and E⊥ c.