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Neutron diffraction study of YVO3, NdVO3, and TbVO3

The structural and magnetic properties of YVO3, NdVO3 and TbVO3 were investigated by single-crystal and powder neutron diffraction. YVO3 shows a structural phase transition at 200 K from an orthorhombic structure with the space group Pbnm to a monoclinic one with the space group P21/b. But supplementary high-resolution synchrotron diffraction experiments showed that the monoclinic distortion is extremely small. A group theoretical analysis shows that this magnetic state in the monoclinic phase is incompatible with the lattice structure, unless terms of higher than bilinear order in the spin operators are incorporated in the spin Hamiltonian. This observation is discussed in the light of recent theories invoking unusual many-body correlations between the vanadium t2g orbitals. A structural phase transition back to the orthorhombic space group Pbnm is observed upon cooling below 77 K. This transition is accompanied by a rearrangement of the magnetic structure into a mode compatible with the lattice structure. The crystal structures of NdVO3 and TbVO3 are closely similar to that of YVO3. However, only a single magnetic phase transition was found in the vanadium sublattice down to 9.5 K. Below 60 K the magnetic moments of the Nd- and Tb-ions are gradually polarized by the ordered vanadium moments. Below 11 K, we found a noncollinear order of the terbium moments.


I. INTRODUCTION
The perovskite structure class with ABX 3  The ideal perovskite structure belongs to the cubic space group Pm−3m, where the Bcations are octahedrally coordinated by six X-anions building a three-dimensional network of corner-sharing BX 6 -octahedra. However, most perovskites have tilted and/or distorted BX 6octahedra leading to a lower symmetry. [1][2][3][4] At room temperature, the principal cause of the deviations from the ideal perovskite structure is the influence of the relative ionic sizes. This "ionic size effect" leads to tilts of the octahedra, as shown in Fig. 1 for YVO 3 . The room temperature crystal structure of the vanadates RVO 3 is thus described by the orthorhombic space group Pbnm. In Fig. 2, the unit cell parameters taken from Ref. 5 are plotted against the atomic number of the R-atoms. From La to Er the cell volume decreases continuously due to the well-known lanthanide contraction. However, only the lattice constants a and c decrease significantly, whereas b increases slightly. This is due to the fact that in the case of the smaller R 3+ -ions the VO 6 -octahedra are more strongly rotated around the band c-axes than along the a-axis of the orthorhombic structure. The materials with the largest R 3+ -ions, LaVO 3 and CeVO 3 , exhibit pseudocubic room-temperature lattice structures with a ~ b ~ c / √2.
The nearly cubic crystal field splits the degenerate manifold of atomic d-electrons on the B-cations into a lower-lying triplet of t 2g symmetry and a high-energy doublet of e g symmetry, with an energy difference that typically exceeds 1 eV. Further distortions away from cubic symmetry split the residual degeneracy of the sub-manifolds. For cations with incompletely filled d-electron shells, electronic energy can thus be gained by a cooperative distortion of the octahedra through the Jahn-Teller (JT) effect, which sets in below an "orbital ordering" temperature. The relative orientation of orbitals on neighboring cations then determines the superexchange interactions between the unpaired d-electrons, which in turn determine the magnetic ordering patterns. For perovskites with partially occupied e g orbitals, such as cuprates and manganites, the JT energy greatly exceeds the superexchange coupling between unpaired electrons, so that orbital and magnetic ordering temperatures are well separated. The two valence electrons of the V 3+ ions in the vanadates, however, occupy the t 2g orbitals (xy, yz, and xz). Owing to the higher degeneracy and weaker JT coupling of the t 2g states, these systems undergo orbital and magnetic ordering transitions in the same temperature range.
Magnetic and orbital degrees of freedom are thus inextricably intertwined.
The magnetic and orbital ground states of YVO 3 , in particular, have been the subject of much attention. Investigations of the structural and magnetic properties of this compound started in the 1950's. [6][7][8] Antiferromagnetic ordering of the vanadium moments was first observed below about 110 K. 9 In later studies, two magnetic phase transitions were identified at T 1 = 78 K and T 2 = 114 K. [10][11][12] Specific heat measurements showed that the transition at T 1 is of first and that at T 2 of second order. Further, neutron powder diffraction showed a change of the magnetic order of the V 3+ -sublattice from a C-type to a G-type with decreasing temperature at T 1 , whereas another paper 13 claimed a transition from a G-type to a C-type.
Kawano et al. 14 finally confirmed a C-type ordering below T 2 and a transition into a G-type below T 1 . Recently we reinvestigated the magnetic structure of YVO 3 by single-crystal neutron diffraction. 15 Below 77 K a pure G z -type was confirmed, while in the intermediate phase a noncollinear structure was discovered. Here, a weak z-component of the G-type was observed along with the previously reported strong C-modes in the ab-plane. In addition to the change of magnetic ground state, the transition at T 1 is associated with a modification of the lattice structure, and with profound changes in the magnetic excitation spectrum and the optical properties. Based on these observations, it has been identified as an orbital ordering transition. Another structural transition at T 3 = 200 K also leads to a modification of the optical spectrum and has also been attributed to orbital ordering. In the magnetically ordered phases, multiple magnetization reversals and unusual magnetization memory effects are observed.
The nature of the orbitally ordered phases of YVO 3 , the microscopic mechanism of the orbital ordering transitions, and the origin of the magnetization memory effects are still poorly understood. It is generally believed that the degeneracy of the t 2g levels is successively lowered upon cooling below the structural phase transitions at T 3 and T 1 , and that the degeneracy is completely lifted at low temperatures. The orbital ordering pattern below T 1 is also relatively uncontroversial. In the low-temperature phase, the vanadium xy orbital is believed to be occupied by one of the two valence electrons, while the other electron occupies the xz and yz orbitals in an alternating fashion in the ab-plane. Along the c-axis direction, the occupied xz and yz orbitals are uniformly stacked (C-type orbital order). The classical Goodenough-Kanamori rules yield a consistent description of this orbital state as well as the magnetic ground state and excitations. In contrast, the microscopic magnetic properties of the intermediate-temperature phase (T 1 < T < T 3 ) are difficult to describe in this classical picture, and several different proposals have been made for the orbital and magnetic ground state in this temperature range. [16][17][18][19][20][21] An intriguing aspect of some of these proposals is the existence of strong quantum fluctuations in both spin and orbital sectors. While these fluctuations may be responsible for the anomalously small ordered moment and other unusual features of the intermediate phase, direct evidence has thus far proven elusive.
Miyasaka et al. recently studied the evolution of the sequence of phase transitions with the size of the R 3+ -ion. 22 While the lowest-temperature transition at T 1 is only observed for the smallest cations, the magnetic transition at T 2 and structural transition at T 3 evolve smoothly as a function of ionic size. For the largest cations, Ce 3+ and La 3+ , the sequence of magnetic and structural transitions was reported to be reversed, which implies a magnetically ordered "orbital liquid" phase in a narrow temperature range above the Néel transition. The low-temperature phase of the end member LaVO 3 exhibits C-type magnetic order 23,24 , similar to the intermediate-temperature phase of YVO 3 .
The purpose of this paper is to provide a more complete basis for the theoretical description of the electronic properties of the vanadates. To this end, we have used singlecrystal neutron diffraction to carry out a comprehensive investigation of the crystal structures and magnetic ground states of YVO 3 as well as its sister compounds TbVO 3 and NdVO 3 , which undergo different sequences of phase transitions as a function of temperature. In the latter compounds, the exchange interactions between the Tb-and Nd-moments, respectively, and the V-moments were also studied. The paper is organized as follows. Section II contains a description of the experimental setup. In Section III, we present data on the crystal structures (part A) and magnetic structures (part B) of the title compounds and discuss these in the light of prior studies in and the literature. Finally, Section IV provides a brief summary of the major conclusions.

II. EXPERIMENTAL DETAILS
For our experiments cylindrical crystals of YVO 3 , NdVO 3 and TbVO 3 with dimensions d = 4 mm and h = 5 mm were used. The crystals were grown by a floating-zone method as described elsewhere. 25 All crystals were untwinned. Single-crystal neutron diffraction data were collected on the four-circle diffractometer E5 at the BER II reactor of the Hahn-Meitner-Institut in Berlin. The crystal structures were determined from data sets using the neutron wavelength λ = 0.89 Å (Cu-monochromator). The refinements of the crystal structures were  27 For the investigation of the magnetic structure, we collected data sets using a pyrolytic-graphite monochromator selecting the longer neutron wavelength λ = 2.4 Å. At 150 K, well above the Néel temperature, we collected data sets of YVO 3 , NdVO 3 and TbVO 3 in order to determine the overall scale factor from the crystal structure refinements. With the absorption and extinction corrected magnetic structure factors, we were able to obtain the magnetic moments of the metal ions in the magnetically ordered range. The moments of the V-, Nd-and Tbatoms were refined with the program FullProf. 28 The magnetic form factors of the V 3+ -, Nd 3+ , and Tb 3+ -ions were taken from Ref. 29. K, and λ = 0.49950 Å at 300 K. Rietveld refinements of the powder diffraction data were carried out with the program FullProf using the atomic scattering factors provided by the program for the x-ray data and the neutron scattering lengths given above.

III. RESULTS AND DISCUSSION
A. Crystal structure of YVO 3 and TbVO 3 We have carried out a comprehensive investigation of the crystal structure of YVO 3 at 5, 85, 230, and 295 K. For comparison, the crystal structure of TbVO3 was also investigated at 295 K. We first describe the results of single-crystal neutron diffraction experiments on both compounds at room temperature. A total of 565 and 429 averaged independent structure factors were collected for the yttrium and terbium vanadate, respectively. Refinements of the diffraction patterns were performed in the orthorhombic space group Pbnm (standard setting Pnma), using the atomic parameters of GdFeO 3 as starting values. 30  In Table I the parameters of YVO 3 are compared with those of TbVO 3 . Here it can be seen that the positional parameters of the O1-and O2-atoms in the yttrium compound deviate more strongly from the ideal sites (0, ½, ¼) and (½, ½, 0) of the undistorted cubic structure.
The observation that a smaller R 3+ -ion favours stronger tiltings of the VO 6 -octahedra indicates the influence of the ionic-size effect. Table II lists   As reported earlier, a monoclinic structure of YVO 3 is stable between T 1 = 77 K and T 3 = 200 K. 31 The temperature dependence of the intensities of some prominent reflections shown in Fig. 3 indeed indicates structural changes with an onset at T 3 . For instance, the reflections 022 and 002 show a strong change of the intensity at 200 K, while no change can be observed for 020 and 202. This indicates a spontaneous shift of the atoms in particular directions. In agreement with Blake et al. 31 , we also observed additional reflections for T 1 < T < T 3 which are forbidden in the orthorhombic space group Pbnm. These measurements confirm the presence of a structural phase transition from an orthorhombic to a monoclinic structure. 31,32 Assuming a the same monoclinic structure with the space group P2 1 /b (P2 1 /b 1 1) as found in Ref. 33 for LaVO 3 , the Rietveld refinements resulted at 85 K in a monoclinic angle α = 89.980(3) °. This is in very good agreement with the value α = 89.977(3) ° (at 80 K) given in Ref. 31. In order to check the presence of monoclinic splittings, we recorded high-resolution powder diagrams using synchrotron radiation, but we could not find any direct evidence of splitting of reflections down to 25 K.
Due to twinning problems below the structural phase transition at 200 K, we refined the atomic positions in YVO 3 by neutron powder diffraction. Powder patterns were collected at 5 K ( Fig. 4) in the low-temperature orthorhombic phase and at 85 K in the monoclinic phase. A third one was collected at 230 K in the high-temperature orthorhombic phase. The additional reflections characteristic of the monoclinic structure were found to be rather weak and could be observed only in the single-crystal diffraction experiments. Thus we first assumed the crystal structure at 85 K to be pseudo-orthorhombic, and we refined it in the space group Pbnm. From the refinements we obtained almost the same residuals and similar V-O distances as observed in the high-temperature orthorhombic phase. The V-O22 (and V-O24) distances in the orthorhombic phase at 230 K and 295 K (Table II) are found to be the largest, followed by V-O21 (and V-O23), both almost lying in the ab-plane. The apical bond lengths of V-O11 and V-O12 are clearly the smallest.
We then repeated the refinements in the monoclinic space group P2 1 /b and found V-Obond lengths in good agreement with those of Blake et al. 31 . Here the length of V-O22 (and V-O24) is strongly elongated, while it is strongly shortened for V-O21 (and V-O23) reaching almost the same bond lengths of the apical O1-atoms. However, the determination of the positional parameters of the O2 and O3 atoms, which show the strongest shifts at the orthorhombic-monoclinic transition, is problematic within P2 1 /b, as these parameters are highly correlated, with correlation coefficients between 84 and 97%. (The correlation coefficient between two parameters p i and p j is obtained from the covariance, i.e. the second moment with respect to p i and p j by normalization corr where σ denotes the standard deviation.) As a consequence, the standard deviations of the positional parameters of the O-atoms given in Table I as well as the V-O-bond lengths in Table II  In the monoclinic phase, the vanadium atoms are located in the Wyckoff positions 2c (½, 0, 0 and 0, ½, 0) and 2d (½, 0, ½ and 0, ½, ½) of the space group P2 1 /b. Despite the lower symmetry, the coordinates of these atoms are still the same as in the orthorhombic space group Pbnm. Recently, an even lower symmetric structure with the non-centrosymmetric space group Pb (P b 1 1) was deduced from ellipsometric measurements. 32 The vanadium atoms as well as the yttrium and oxygen atoms are located in the only possible Wyckoff position 2a (x, y, z and −x, ½ + y, z). For both atoms V1 and V2 the parameters x and y are expected to be close to the values ¼ and ¾, respectively; the parameter z of V1 is close to zero, while it is close to ½ for V2. In this symmetry, the vanadium atoms form periodic chains along the xand ydirections with constant interatomic distances within the chains. This is not the case for the vanadium chains along z, where one expects alternating slightly shortened and elongated V1-V2-distances. In fact, the crystal structure in Pb allows the possibility of dimerization along z, in agreement with the results of neutron spectroscopy measurements that can be consistently interpreted in terms of an "orbital Peierls state". 15 In our experiment, however, the positional parameters of the vanadium atoms could not be refined with sufficient accuracy to resolve this issue, because the neutron scattering power of vanadium is rather weak and the expected distortion is extremely small. The unusual spin correlations observed in the intermediate-temperature phase of YVO 3 are therefore at most weakly reflected in the lattice structure. This may be due to the weak lattice coupling of the t 2g valence orbitals, in conjunction with strong orbital fluctuations.
The thermal variation of the intensities of the reflections in Fig. 3 clearly indicates a strong change in the crystal structure of YVO 3 at about 77 K. At this temperature, Kawano et al. also observed a spontaneous change of the lattice constants. 14 They found that the values a and c spontaneously increase by about 0.15 -0.16 %, while b decreases strongly by about 0.64 %. Below this transition temperature the crystal structure again can be described in the orthorhombic space group Pbnm. 31,32 The results of the Rietveld refinements of the data set collected at 5 K are summarized in Tables I and II. Here it can be seen that the V-O1-bond distances at 5 K do not strongly differ from the values of the monoclinic structure determined at 85 K. But strong shifts could be observed for the O2-atoms. The length of V-O22 (and V-O24), the largest at higher temperature, strongly decreases reaching almost the same value of that of V-O11 (and V-O12). On the other hand the value of V-O21 (and V-O23) is now clearly the largest. This is in contrast to the result of Blake et al. 31 In contrast to the intermediate-temperature phase, the neutron data in the low-temperature phase therefore exhibit an unequivocal signature of the Jahn-Teller effect, in line with theoretical expectations.

B. Magnetic order in RVO 3 with R = Y, Nd, Tb
We begin this section by presenting magnetic neutron scattering data on YVO 3 , following our recent brief account of the magnetic properties of this material 15  is simply a scalar multiplying the bilinear term in the spin Hamiltonian. If the crystal field gap is small or absent, and substantial fluctuations are present in the orbital sector, the exchange coupling itself is a quantum variable. This naturally leads to higher-order terms in a freeenergy expansion.
As already mentioned above, thermodynamic experiments suggested that the lowtemperature structural and magnetic phase transition, which is observed for the yttrium compound at T 1 = 77 K, is absent for vanadates with an ionic size larger than that of Dy 3+ . 22 Our neutron diffraction experiments on NdVO 3  In order to obtain the moment values, we determined the overall scale factor from a data set collected at room temperature, where both compounds crystallize in the orthorhombic space group Pbnm. The relatively large wavelength only allowed us to collect a small number of Bragg reflections. Therefore the structure refinement of TbVO 3 resulted in atomic parameters with significantly larger standard deviations than that of the data set collected at the wavelength λ = 0.89 Å, but the refined values of both data sets were in a good agreement. The structure refinements of NdVO 3 resulted in positional parameters similar to those obtained for the terbium compound, but with the tendency that the tiltings of the VO 6 -octahedra in the neodymium compound are less pronounced than in TbVO 3 . For lower temperatures, the magnetic moments of the Nd and Tb sublattices are polarized in magnetic modes compatible with the magnetic order of the V moments. This was confirmed for NdVO 3 from the analysis of our single-crystal neutron data collected at 9.5 K.
Here it could be seen that both the V 3+ -and Nd 3+ -sublattices show a pure C-type ordering, where the moments are ordered antiferromagnetically in the ab-plane and ferromagnetically along the c-axis. The magnetic moments of the metal atoms in NdVO 3 obtained by a refinement of these data are given in Table III. The exchange coupling between Nd and V spins is antiferromagnetic along the xand y-directions, resulting in a noncollinear magnetic structure (Fig. 7).
In contrast to the neodymium compound, the magnetic intensity of the reflections 100 and 010 of TbVO 3 decreases at lower temperature (Fig. 6). Our data analysis shows that this is because the magnetic moments of the Tb 3+ -and V 3+ -ions are coupled ferromagnetically along the xand y-directions, whereas those of the Nd 3+ -and V 3+ -ions in NdVO 3 are coupled antiferromagnetically. Below 11 K the magnetic intensity of the reflection 010 sharply increases in magnitude. This can be ascribed to the spontaneous magnetic order of the terbium sublattice. From the absence of additional magnetic intensity on the 100 position, we infer a C-type ordering of the terbium moments in the x-direction. A strong increase of the magnetic intensity could also be detected at 200, whereas the change of the intensity for the 020 is negligible. This clearly suggests a ferromagnetic order of the terbium moments along the ydirection. The absence of 001 finally excluded the presence of the modes A x and/or A y . We also found additional but weaker intensities for the reflections 111, 201 and 021 indicating A modes, as well as for 101 and 011 indicating G-modes. However, these additional reflections are present only because the terbium atoms occupy the general position x, y, ¼. They carry no magnetic intensity when x(Tb) and y(Tb) are equal to zero. Thus we finally could use 30 magnetic reflections for the data analysis. The refinement of the moments along z, assuming the A-mode, resulted in a value close to zero. From refinements of the modes C x and F y , the magnetic moments of the terbium moments in the ab-plane were calculated to be µ x = 6.08(11) µ B and µ y = 5.03(14) µ B with a residual of R F = 0.09. The total moment µ exp = 7.89(13) µ B is somewhat smaller than the theoretical value of the free Tb 3+ -ion, 9.0 µ B , presumably because the saturation moment is not reached at 9.5 K. The low-temperature magnetic structure of TbVO 3 is shown in Fig. 7. Our results are not in agreement with the magnetic structure given in Ref. 13, where the modes A x and G y were reported.
To summarize the rare-earth magnetism in NdVO 3 and TbVO 3 , we note that the magnetically ordered vanadium sublattice induces, via Nd-V and Tb-V exchange, a polarization of the lanthanide moments resulting in a magnetic structure with the same symmetry, so that one finds a pure C-type order for both metal sublattices in the ab-plane over a wide temperature range below the Néel transitions. A comparable polarization process was observed earlier for other perovskites like NdCrO 3 , NdFeO 3 and NdNiO 3 , [34][35][36] where it proved to be amenable to a mean-field description 35 . We point out two additional aspects. First, the signs of the V-Nd and V-Tb exchange interactions are opposite. The same trend was found earlier for the intermetallic compounds Nd 2 Co 12 P 7 and Ho 2 Co 12 P 7 , where the induced moments of the (light) neodymium and (heavy) holmium ions are aligned antiparallel and parallel to well ordered cobalt moments, respectively. 37 Conversely, the induced alignment between the Fe and Nd spins in Nd 2 Fe 14 B is parallel, and that of the Fe and Ho spins in Nd 2 Fe 14 B is antiparallel. 38 This effect can be attributed to the opposite sign of the spin-orbit coupling parameter for lanthanide ions with less-than-half-filled and more-than-half-filled felectron shells. A second notable aspect is the large polarization of the Nd moments, which significantly exceeds that of the much larger Tb moments at comparable temperatures (Figs. 5 and 6). This indicates that the V-Nd coupling is relatively large.
At 11 K, Tb-Tb interactions drive a spontaneous re-ordering of the terbium moments from a C y to an F y mode. Analogous ordering phenomena have been observed in the intermetallic compounds Pr 2 Co 12 P 7 37 and PrCo 2 P 2 39 as well as in layered cuprates such as  While the G-type magnetic structure of YVO 3 in the low-temperature phase is collinear and fully compatible with the orthorhombic crystal structure, the magnetic structure in the intermediate-temperature phase is highly noncollinear. This may be a manifestation of the unusual orbital correlations in this phase. The group theoretical analysis presented in the Appendix has revealed that this magnetic structure is incompatible with the crystal structure, unless the spin Hamiltonian is expanded beyond the usual bilinear terms. The presence of substantial higher-order spin-spin interactions has not been predicted. Theoretical work is required to assess whether they could be another consequence of the unusual orbital correlations in the intermediate-temperature phase of YVO 3 .
The properties of YVO 3 were contrasted to those of TbVO 3 and NdVO 3 , whose room temperature lattice structure is closely similar. These materials differ from YVO 3 in two respects. First, whereas the Y 3+ ion is nonmagnetic, the Tb 3+ and Nd 3+ ions carry substantial magnetic moments. The exchange interactions between rare-earth and vanadium moments are weaker than the vanadium-vanadium interactions and lead to a polarization of the rare-earth moments. At low temperatures, direct interactions between the rare-earth moments lead to a spontaneous ordering of the Tb-sublattice in TbVO 3 . A similar transition is expected in NdVO 3 , but was not observed due to the limited temperature range covered by our experiments. While the magnetic structures we have found are at variance with some previous reports, they are in general accord with other materials containing both dand f-electron elements.
The second major difference between YVO 3 on the one hand, and TbVO 3 and NdVO 3 on the other hand, is the absence of the low-temperature orthorhombic phase with G-type magnetic structure in the latter two materials. In both of these compounds, the vanadium moments are arranged in a C-type pattern at low temperature, similar to the magnetic structure  Table IV. From our experiment we found for the vanadium moments in YVO 3 the modes C x , C y and G z . It is interesting to see, that only the magnetic ordering in the bc-plane is compatible with Pbnm as deduced from Bertaut's representation analysis 48     Nd x, y, ¼ Pbnm