The orbital degree of freedom is integral to many exotic phenomenain condensed matter, including colossal magnetoresistance and unconventional superconductivity. The standard model of orbital physics is the Kugel–Khomskii model1, which first explained the symmetry of orbital and magnetic order in KCuF3 and has since been applied to virtually all orbitally active materials2. Here we present Raman and X-ray scattering measurements showing that KCuF3 exhibits a previously unidentified structural phase transition at T=50 K, involving rotations of the CuF6 octahedra. These rotations are quasi-ordered and exhibit glassy hysteresis, but serve to stabilize Néel spin order at T=39 K. We propose an explanation for these effects by supplementing the Kugel–Khomskii model with a direct, orbital exchange term that is driven by a combination of electron–electron interactions and ligand distortions3. The effect of this term is to create a near degeneracy that dynamically frustrates the spin subsystem but is lifted at low temperature by subdominant, orbital–lattice interactions. Our results suggest that direct orbital exchange may be crucial for the physics of many orbitally active materials, including manganites, ruthenates and the iron pnictides.
It has long been believed that the prototypical orbital ordering material KCuF3 exhibits Kugel–Khomskii (KK)-type orbital order1, which causes Jahn–Teller distortions of the CuF6 octahedra, at temperatures below TJT∼800 K (refs 4, 5, 6, 7). However, several properties of KCuF3 have never fully been explained by the KK model. First, whereas the orbitals order at temperatures of the order of 800 K, the spins order below a much lower Néel temperature, TN=40 K (ref. 5). The KK model can account for, at most, a factor of 10 difference between these two energy scales1,8. Second, whereas the KK model correctly accounts for the relative signs of the in-plane and out-of-plane spin superexchange, it does not explain their disparate magnitudes8,9,10,11. Finally, several authors have recently reported unexplained structural fluctuations associated with the F−ions at temperatures far below TJT, where, according to the KK model, structural fluctuations should be frozen out12,13,14,15.
To gain further insight into these discrepancies, we carried out Raman scattering, as well as hard- and soft-X-ray scattering, measurements of the lattice and magnetic degrees of freedom in KCuF3 (see Methods). At T=10 K the Raman spectrum (Fig. 1) exhibits several modes associated with vibrations of the F− ions, with A1g, B1g and Eg symmetry, consistent with previous studies16. We have found that several of these modes exhibit anomalous temperature dependencies. Whereas the A1g mode behaves in a manner expected from normal anharmonic effects, shifting to higher frequency with decreasing temperature, the B1g and Eg modes soften anomalously in the range 50 K<T<300 K (Fig. 1, middle and lower panels). At T=50 K—just above the Néel temperature—the frequencies of the modes stabilize and the higher Eg mode splits into two, distinct, non-degenerate modes (Fig. 1). These observations indicate a reduction in crystal symmetry from tetragonal to orthorhombic, and may be understood as the stabilization of GdFeO3-type rotations of the CuF6 octahedra at TR=50 K. This interpretation is further supported byX-raymeasurements, described below. The anomalous structural fluctuations observed in refs 12, 13, 14, 15 can, then, be understood as arising from critical, precursor fluctuations of this transition. The close coincidence of TR and TN suggests that freezing in of CuF6 rotations contributes to the stabilization of Néel spin order.
To explain the relationship between octahedral rotations and Néel order in KCuF3, we suggest a refined version of the KK model. Its main new ingredient is a direct-orbital exchange term proposed in ref. 3, described by the Hamiltonian
where the hole orbital on site i, which lies in or orbitals, is described by pseudospin operators τia, τib and τic (see Supplementary Information; ref. 17), and J1α is the nearest-neighbour orbital coupling along the a, b and c directions. Equation (1) was argued3 to be the dominant coupling between orbitals in a charge-transfer insulator, and can arise from either on-ligand interactions or cooperative Jahn–Teller distortions18,19.
The orbital–spin interactions are described by the usual KK superexchange Hamiltonian,
where Si denotes the true electron spin1. Following ref. 17, a Hund’s rule term has been included, where the parameter η=JH/U≪1is the ratio of the Hund coupling JH to the Hubbard U parameter for the d electrons. Crucially, as suggested in ref. 3, we assume J1≫J2, that is, that the orbital–spin interactions play a sub-dominant role and mainly determine the impact of orbital physics on the development of magnetic order1,3,17. This assumption is justified by the different influence of Jahn–Teller distortions on these parameters, discussed below.
Finally, we include a coupling of the orbitals and spins to the orthorhombic rotations of the octahedra,
where the dynamical variables Qi j describe fluctuations in the positions of the F− ion transverse to the line connecting adjacent in-plane Cu sites i and j, and μ describes the coupling between the orbital-spin fluctuations and the octahedral rotations. By symmetry, HOR depends on |Qi j|2, which we shall see favours the formation of a glassy state at low temperatures. For completeness, we also include a weak, tetragonal crystal-field term, ,with the crystal-field coupling λCF≪J1. Our complete Hamiltonian is the sum of the four terms Heff=Hτ τ+Hτ S+HOR+HCF. To analyse this model we take the usual variational approach1,11,17, optimizing the energy with respect to a hybrid orbital wavefunction, |i σ〉=cos(θi)|i z σ〉+sin(θi)|i x σ〉, which represents the orbital configuration on lattice site i. Here, |i z σ〉 represents a orbital, and |i x σ〉 a orbital, with spin σ and mixing angle θi as the variational parameters. The sign of θi is taken to alternate between the two sublattices, ensuring that the variational wavefunction has the appropriate space-group symmetry.
In our model, in the high-temperature phase (T≫800 K) the system is cubic, all coupling constants are isotropic and the physics is governed by the highest energy scale, Hτ τ. As the temperature is lowered, the system undergoes a transition at TJT∼800 K into a phase with cooperative Jahn–Teller distortions and orbital order, characterized by a mixing angle, θ≈π/4, that minimizes Hτ τ. We note that, in our model, this mixing angle depends only on the existence of a J1 term regardless of the microscopic geometry of the Jahn–Teller distortion. Despite caveats about the relationship between the orbital state and the geometry of Jahn–Teller distortions20, this result suggests that the octahedra exhibit three—rather than just two—distinct bond lengths, in agreement with the known structure4,6.
In the intermediate-temperature regime (50 K<T<800 K), according to our model, the system resides in a free-energy minimum corresponding to the Jahn–Teller distorted state. To understand what happens at lower energy scales, it is necessary to use effective coupling constants that describe small excursions around this minimum, that is, that apply to the Jahn–Teller-distorted structure. The primary effect of the Jahn–Teller distortion is to introduce anisotropy in the exchange parameters J1 and J2. The anisotropy in J2, which arises purely from superexchange and depends exponentially on the bond lengths, should be more pronounced than that in J1, which has a crystal-field component and power-law dependence. We therefore anticipate that J2c>J2a (=J2b) and J1a=J1b≈J1c.
Surprisingly, minimizing Heff reveals a ground state that is nearly degenerate. Two distinct hybrid orbital states—called |HO1〉 and |HO2〉—were found, which exhibit G-type and A-typeorder, respectively (see Supplementary Information). Both of these states are characterized by mixing angles close to θ≈π/4 (see Fig. 2a,b), and their energies are separated by the smallest couplings in the problem:
Note that, in our model, the KK ground state1, characterized by a mixing angle θ=π/6and corresponding to alternating and orbitals, is higher than |HO1〉 and |HO2〉by an energy of the order of 3J1/8.
The near degeneracy of |HO1〉 and |HO2〉 explains the existence of both the extended fluctuational regime and the second structural transition in KCuF3. For illustration, we choose the parameter values J1a=600 K, J1c=630 K, J2c=200 K, J2a/b=30 K, λCF=50 K and η=0.1 (see refs 1, 9, 10, 11, 17). These choices reflect the dominant role of J1, which arises partly from lattice effects, as well as the expected larger anisotropy in J2. For these values, we find the energy difference between |HO1〉 and |HO2〉 to be of the order of a few Kelvin. Hence, throughout the intermediate-temperature regime (50 K<T<800 K), the system undergoes rapid fluctuations between the two hybrid orbital configurations. Each is coupled to the octahedral rotations through equation (3), resulting in the extended regime of dynamic, orthorhombic distortions reported in several studies6,14,15.
Although the size of the orbital fluctuations (as reflected in Fig. 2) is small, in our model their consequences for the in-plane magnetic order are severe. As |HO1〉 and |HO2〉 have opposite in-plane spin exchange, even small orbital fluctuations disrupt the in-plane magnetic order17. This is consistent with the experimental observation that, in the intermediate-temperature regime, the in-plane spin correlations are shorter ranged than those along the c axis.
At the lower structural transition, TR=50 K, the orthorhombic fluctuations freeze into a static pattern of GdFeO3-type rotations of the CuF6 octahedra, resulting in a non-zero value of the variable |Q|2. This distortion, in accordance with equation (4), lifts the near-degeneracy, lowering the energy of the |HO2〉state when compared with |HO1〉, resulting in A-type antiferromagnetic order.
The emergence of in-plane spin correlations, in our model, has pronounced effects on the stiffness of the lattice, explaining the anomalous phonon shifts observed in KCuF3 (see Fig. 1). According to equation (3), aligned neighbouring spins cause a softening of the lattice, whereas anti-aligned neighbours cause a hardening. This implies that, as the ferromagnetic (A-type) in-plane correlations develop with decreasing temperature, the spring constant decreases, leading to the phonon softening shown in Fig. 1 (see Supplementary Information).
Our model also provides some explanation for the extreme anisotropy in the spin exchange, Γ, in KCuF3, which neutron-scattering experiments have suggested is as large as Γc/Γa/b∼−100 below the Néel temperature9,10. Using our parameter values, we find Γc=180 K and =−2 K, or Γc/=−90 (see Supplementary Information). Some of this anisotropy arises from our chosen anisotropy J2c/J2a=6.67, which is expected from the symmetry of the Jahn–Teller distortion. We emphasize, however, that our orbital pattern provides the correct sign of Γc/, and further amplifies its magnitude above J2c/J2a/b by a factor of 13.5. This amplification is an outcome of the small value of the Cu2+ Hund parameter, η, and the small in-plane overlap of our orbitals (Fig. 2b) due to their staggered arrangement.
To confirm the relationship between tilting of the CuF6 octahedra and the magnetic ordering transition, we also studied KCuF3 with conventional and magnetic X-ray scattering (see Methods). Magnetic-scattering studies, summarized in Fig. 3a, show that long-ranged, A-type antiferromagnetic order sets in at TN=40 K, in agreement with previous studies5,9,10. Above TN, diffuse magnetic scattering is observed, which arises from critical fluctuations in the magnetic order. The momentum linewidth is highly anisotropic, indicating quasi-one-dimensional magnetism, as has been observed above TN (ref. 10). The correlation length in all three directions diverges simultaneously, indicating a single, three-dimensional Néel transition.
Hard-X-ray measurements are summarized in Fig. 3b, showing scans through the (1,0,5) Bragg position, which corresponds to the wave vector of both the orbital order and the CuF6 rotations, and is symmetry equivalent to the orbital reflections studied in ref. 7. The (1,0,5) reflection is present at room temperature, consistent with the existence of Jahn–Teller-distorted octahedra at T<800 K. As the temperature is lowered the intensity of this reflection increases, reaching a maximum at T∼100 K, that is, in the middle of the fluctuational regime identified in Raman measurements (Fig. 1). This observation supports the interpretation that the fluctuations arise from CuF6 rotations, which amplify the (1,0,5) structure factor. Finally, for T<TN, the (1,0,5) reflection abruptly drops in intensity and acquires a series of diffuse sidebands that exhibit hysteresis when cycling the temperature through TN. This suggests that the CuF6 rotations become static below TN, but are glassy and short-range ordered.
Such glassy order is a natural outcome of equation (3). As the spin correlations couple to the magnitude, |Q|2, and are insensitive to the sign of Q, the free-energy cost of forming domain walls in the CuF6 tilt pattern is small. We expect, therefore, the magnetically ordered state to be associated with a large number of nearly degenerate configurations of tilt patterns, resulting in glassy behaviour.
Overall, our model suggests a picture of KCuF3 in which the orbital order occurs not through a single transition near 800 K, as is commonly believed, but in two stages: the higher-temperature transition reduces the manifold of available orbital states, but leaves a pair of nearly degenerate (to within a few Kelvin) hybrid states that have similar orbital configurations but very different in-plane spin configurations. The near degeneracy of these states results in a large temperature range of structural and in-plane spin fluctuations, as well as anisotropy in spin correlation lengths. The lower structural transition corresponds to a freezing of the octahedral rotations, which splits the hybrid orbital states, completing the ordering of the orbitals and stabilizing A-type spin order.
If correct, our result indicates that direct, orbital–orbital exchange—the key new ingredient in our model—may be crucial to the physics of many orbitally active materials, including manganites2, ruthenates21,22 and the iron pnictides23. One curious consequence of equation (1) is that the orbital pattern in Fig. 2 differs somewhat from the KK pattern; such deviations have also been observed in some24, though not all19, ab initio calculations. The discrepancies probably originate in the different ways in which these approaches treat the interactions on, and dynamical motions of, the F− ligands.
The KCuF3 single crystals were grown from solution by techniques described previously25. The crystals selected were screened with X-ray measurements and consisted of a >90%volume fraction of polytype a.
Raman scattering measurements were made in a variable-temperature, continuous-flow cryostat using the 6,471 Å line from a Kr laser. An incident power of ∼10 mW and ∼50 μm spot size was used to minimize laser heating. The spectra were analysed with a modified subtractive triple-stage spectrometer using a liquid-nitrogen-cooled CCD (charge-coupled device) detector.
Soft-X-ray magnetic scattering measurements were conducted at the Cu L3/2 edge at beamline X1B at the National Synchrotron Light Source (NSLS). Hard-X-ray measurements were made at Sectors 4 (CARS) and 15 at the Advanced Photon Source and C Line at the Cornell High Energy Synchrotron Source (CHESS). We denote momenta in terms of Miller indices, that is, (H,K,L)indicates a momentum transfer q=(2π H/a,2π K/b,2π L/c), where a=b=5.85 Å and c=7.82 Å.
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We gratefully acknowledge discussions with D. I. Khomskii, M. V. Mostovoy, A. J. Millis, A. K. Sood and H. R. Krishnamurthy. This work was supported by the US Department of Energy through grant DE-FG02-07ER46453, with soft-X-ray studies supported by DE-FG02-06ER46285.The Advanced Photon Source was supported by DE-AC02-06CH11357 and the NSLS by DE-AC02-98CH10886. CHESS and ChemMatCARS are supported by National Science Foundation grants CHE-0822838 and DMR-0225180, respectively. S.L. gratefully acknowledges financial support from the Department of Science and Technology, Government of India, through a Ramanujam Fellowship.
The authors declare no competing financial interests.
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Lee, J., Yuan, S., Lal, S. et al. Two-stage orbital order and dynamical spin frustration in KCuF3. Nature Phys 8, 63–66 (2012). https://doi.org/10.1038/nphys2117
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