C. Ulrich, G. Khaliullin, H. He, P. Horsch, A.M. Ole´s, and B. Keimer;
M. Reehuis (Hahn-Meitner-Institut, Berlin); M. Ohl (Institut Laue-Langevin, Grenoble);
S. Miyasaka and Y. Tokura (University of Tokyo) The interplay between spin and orbital degrees
of freedom in transition metal oxides has been a subject of investigation since the 1950’s. As one of the salient outcomes of this effort, the
‘Goodenough-Kanamori rules’ provide a de-scription of the exchange interactions between magnetic atoms, and hence the magnetic or-dering pattern, in terms of the relative orienta-tion of valence orbitals on neighboring lattice sites. This field has recently moved back into the center of attention in solid state physics, as advances in materials preparation have made it possible to investigate not only the static spin and orbital arrangements, but also the spin and orbital dynamics in a variety of d-electron sys-tems. In cubic manganites, the focus of much of the recent attention, the spin-wave excitations have been studied extensively, and excitations associated with the d-orbital degrees of freedom have recently also been detected.
In transition metal oxides with perovskite struc-ture, such as the manganites, the cubic crys-tal field splits the d-orbicrys-tal manifold of a free transition metal atom into a lower-lying triplet
of t2gsymmetry and a higher-lying eg doublet.
In the manganites, where the egdoublet is par-tially occupied, coupling to the lattice through the Jahn-Teller effect lifts the orbital degen-eracy and generally pushes the orbital excita-tions to energies much larger than the magnon band width. The spin and orbital dynamics are thus largely decoupled in these materials. A dynamical coupling between both sectors only occurs in special situations, in particular near phase boundaries separating states with dif-ferent orbital ordering patterns where soft or-bital excitations can induce anomalies in the spin excitation spectra. For t2g orbitals, on the other hand, the higher degeneracy and the more isotropic, less bond-directional charge distribu-tion reduces the lattice coupling, and one may expect a more dramatic interplay between the orbital and the spin dynamics. Apart from some experimental and theoretical work on cubic ti-tanates, however, the spin and orbital excita-tions in transition metal oxides with t2gvalence electrons remain largely unexplored to-date.
Here we report elastic and inelastic magnetic neutron scattering experiments on the spin dy-namics of the Mott insulators LaVO3 and YVO3, each of which has two d-electrons par-tially occupying the t2g orbitals of the V3ion.
Prior experiments on these materials provide ample, albeit indirect, evidence of a soft or-bital sector. In particular, YVO3undergoes a se-quence of transitions between states with differ-ent orbital and spin ordering patterns as a func-tion of temperature[Y. Ren et al., Nature 396, 441 (1998)]. Elastic neutron scattering data, mea-sured on the four circle neutron diffractometer E5 at the Hahn-Meitner Institute in Berlin, on a large single crystal grown by the floating zone technique, are shown in Fig. 34.
Figure 34: Integrated intensities of two magnetic Bragg reflections ofYVO3as a function of tempera-ture. The Bragg peaks (0.5 0.5 0) and (0.5 0.5 0.5) correspond to the C-type and the G-type spin ar-rangements, respectively. The sketches (inset) show the spin ordering patterns in the low temperature and intermediate temperature phase, respectively.
At low temperatures, the spin structure is G-type, that is, antiferromagnetic in all three spatial directions. At 77 K, a transition into a
C-type spin structure takes place, that is, the exchange coupling along the c-axis becomes ferromagnetic while it remains antiferromag-netic in the ab-plane. The magantiferromag-netic transition is accompanied by a structural transition that has been interpreted in prior work[Y. Ren et al., Nature 396, 441 (1998)]as evidence of orbital re-ordering from a C-type to a G-type arrangement of the xz and yz valence orbitals occupied by one of the two d-electrons; the other d-electron is assumed to remain in the xy-orbital. Upon fur-ther heating, the C-type antiferromagnetic order disappears at 114 K, and the G-type orbital or-der at 200 K.
Figure 35: Spin wave dispersion relations ofYVO3 in the G-type low temperature phase (T = 50 K). The lines are results of a fit to a standard magnon disper-sion as described in the text.
The elastic neutron scattering data in the in-termediate phase already provide some indica-tion of the unusual nature of this phase. First, the ordered magnetic moment, 1.05 µB, must be regarded as anomalously small, as one would expect only a slight reduction of the full 2 µB moment of the two electrons by zero-point and thermal fluctuations at these temperatures. Sec-ond, the full magnetic structure in the inter-mediate phase, though predominantly C-type, turns out to be highly collinear. The non-vanishing intensity of the121212reflection for 77 KT114 K (Fig. 34) is a manifestation of this non-collinearity. (For simplicity we ne-glect tilting distortions of the VO6 octahedra
and use a pseudocubic unit cell to index the re-flections.) A detailed crystallographic analysis shows that the G-type component of the non-collinear structure is oriented along the c-axis (not shown in Fig. 34). Its amplitude is about 30% of the total sublattice magnetization.
The unconventional nature of the intermediate phase is further highlighted by an inelastic neu-tron scattering study of the spin dynamics per-formed at the triple axis neutron spectrometer IN22 at the Institut Laue-Langevin in Greno-ble, France. The dispersion curve in the low temperature phase (T77 K) extracted from the inelastic scattering data is shown in Fig. 35.
The spectra are well described by a Heisen-berg Hamiltonian with antiferromagnetic ex-change parameter JcJab5.5 meV and mod-erate single-ion and exchange anisotropy terms.
These parameters are within the normal range expected for a S = 1 system. In contrast, the spin wave dispersions in the intermediate phase are very unusual, in particular along the c-axis where the exchange coupling is ferromagnetic.
The measured magnetic intensity and spin wave dispersions are plotted in Fig. 36. Several anomalies are apparent. First, the exchange constants extracted from a fit to the magnon dis-persions, Jc= –3.1 meV and Jab= 2.6 meV, are about a factor of two smaller than in the low temperature phase (Fig. 35). This overall col-lapse of the magnon spectrum is not in ac-cord with the conventional superexchange mod-els. In addition, Jc is actually larger than
Jab, in striking contrast to the Goodenough-Kanamori rules according to which ferromag-netic superexchange interactions are generally significantly smaller than antiferromagnetic in-teractions. Finally, the splitting of the magnon spectrum into acoustic and optical branches along c is completely unexpected. In particu-lar, detailed spin wave calculations show that it is not a simple consequence of the non-collinear spin structure.
Figure 36: (a) Magnetic intensity measured in the C-type intermediate temperature phase (T = 85 K) of YVO3along (12,12,ξ). (b) Spin wave dispersion rela-tions ofYVO3in the C-type phase at the same tem-perature. The lines are results of a fit to the orbital dimer model described in the text.
For comparison, we have carried out an in-elastic neutron scattering experiment of LaVO3, which also has a C-type spin structure in its magnetically ordered phase. The spin wave spectrum extracted from these data is shown in Fig. 37, together with a fit to a standard spin wave dispersion. None of the unusual fea-tures characterizing the intermediate phase of YVO3 are present: The exchange parameters (Jc= –4.0 meV, Jab= 6.5 meV) are comparable to those of YVO3 at low temperatures; their magnitudes are close to the expected ratio; and neither a non-collinearity of the spin structure nor an optical-acoustic splitting of the magnon dispersions are found.
Figure 37: Spin wave dispersion relations ofLaVO3 in the C-type antiferromagnetic phase. The lines are results of a fit to a standard magnon dispersion as described in the text. The open symbols indicate phonons.
In order to explain the unusual magnetism of YVO3 in the intermediate phase, we follow a model suggested earlier by Ren et al. and as-sume that the xy-orbital dominates the antifer-romagnetic coupling within the layers. Along the c-axis, the superexchange Hamiltonian can be written to first order as
Ht2
U
∑
ij
SiSj12τiτi
1 2 where t is the hopping parameter, U is the intra-atomic Coulomb repulsion,S is a spin-1 op-erator andτ a pseudospin-12 operator acting in the subspace spanned by the xz and yz orbitals responsible for the superexchange along the c-axis direction. The standard ferromagnetic su-perexchange in a state with classical orbital or-der arises from additional terms in the Hamilto-nian that are reduced in magnitude by a factor of JHU, where JH is the intra-atomic Hund’s
rule interaction. A novel mechanism for ferro-magnetic superexchange arises if orbital fluc-tuations are present. An easy way to see this is to consider a pair of vanadium ions along c: An orbital singlet with 2τiτj=32 gives a large ferromagnetic exchange coupling. A full many body theory incorporating orbital sin-glets appears capable of explaining many of the experimentally observed features of the inter-mediate phase, including in particular the non-collinear spin structure and the optical-acoustic splitting. The latter could arise from dimer-ization of the pseudospin-12 orbital chain, anal-ogous to the spin-Peierls instability. Possible crystallographic manifestations of this instabil-ity are currently under investigation.
In summary, our experiments have established the insulating cubic vanadates as an interesting model system to explore the interplay between spin and orbital dynamics. The spin excitations probed by neutron scattering are highly sensi-tive to the details of the orbital state. The next step will be an attempt to explore the orbital ex-citations directly over the entire Brillouin zone by neutron scattering, as has been done recently by Raman scattering at the zone center. De-tailed predictions for the ‘orbiton’ dispersions were made on the basis of the model discussed above. Further, it will be interesting to con-tinue these investigations in doped systems, as both LaVO3 and YVO3 are known to undergo insulator-to-metal transitions when the number of d-electrons is changed by chemical substitu-tion.