Abstract
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 model^{1}, which first explained the symmetry of orbital and magnetic order in KCuF_{3} and has since been applied to virtually all orbitally active materials^{2}. Here we present Raman and Xray scattering measurements showing that KCuF_{3} exhibits a previously unidentified structural phase transition at T=50 K, involving rotations of the CuF_{6} octahedra. These rotations are quasiordered 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 distortions^{3}. 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.
Main
It has long been believed that the prototypical orbital ordering material KCuF_{3} exhibits Kugel–Khomskii (KK)type orbital order^{1}, which causes Jahn–Teller distortions of the CuF_{6} octahedra, at temperatures below T_{JT}∼800 K (refs 4, 5, 6, 7). However, several properties of KCuF_{3} 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, T_{N}=40 K (ref. 5). The KK model can account for, at most, a factor of 10 difference between these two energy scales^{1,8}. Second, whereas the KK model correctly accounts for the relative signs of the inplane and outofplane spin superexchange, it does not explain their disparate magnitudes^{8,9,10,11}. Finally, several authors have recently reported unexplained structural fluctuations associated with the F^{−}ions at temperatures far below T_{JT}, where, according to the KK model, structural fluctuations should be frozen out^{12,13,14,15}.
To gain further insight into these discrepancies, we carried out Raman scattering, as well as hard and softXray scattering, measurements of the lattice and magnetic degrees of freedom in KCuF_{3} (see Methods). At T=10 K the Raman spectrum (Fig. 1) exhibits several modes associated with vibrations of the F^{−} ions, with A_{1g}, B_{1g} and E_{g} symmetry, consistent with previous studies^{16}. We have found that several of these modes exhibit anomalous temperature dependencies. Whereas the A_{1g} mode behaves in a manner expected from normal anharmonic effects, shifting to higher frequency with decreasing temperature, the B_{1g} and E_{g} 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 E_{g} mode splits into two, distinct, nondegenerate modes (Fig. 1). These observations indicate a reduction in crystal symmetry from tetragonal to orthorhombic, and may be understood as the stabilization of GdFeO_{3}type rotations of the CuF_{6} octahedra at T_{R}=50 K. This interpretation is further supported byXraymeasurements, 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 T_{R} and T_{N} suggests that freezing in of CuF_{6} rotations contributes to the stabilization of Néel spin order.
To explain the relationship between octahedral rotations and Néel order in KCuF_{3}, we suggest a refined version of the KK model. Its main new ingredient is a directorbital exchange term proposed in ref. 3, described by the Hamiltonian
where the hole orbital on site i, which lies in $3{d}_{3{z}^{2}{r}^{2}}$ or $3{d}_{{x}^{2}{y}^{2}}$ orbitals, is described by pseudospin operators τ_{i}^{a}, τ_{i}^{b} and τ_{i}^{c} (see Supplementary Information; ref. 17), and J_{1}^{α} is the nearestneighbour orbital coupling along the a, b and c directions. Equation (1) was argued^{3} to be the dominant coupling between orbitals in a chargetransfer insulator, and can arise from either onligand interactions or cooperative Jahn–Teller distortions^{18,19}.
The orbital–spin interactions are described by the usual KK superexchange Hamiltonian,
where S_{i} denotes the true electron spin^{1}. Following ref. 17, a Hund’s rule term has been included, where the parameter η=J_{H}/U≪1is the ratio of the Hund coupling J_{H} to the Hubbard U parameter for the d electrons. Crucially, as suggested in ref. 3, we assume J_{1}≫J_{2}, that is, that the orbital–spin interactions play a subdominant role and mainly determine the impact of orbital physics on the development of magnetic order^{1,3,17}. 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 Q_{i j} describe fluctuations in the positions of the F^{−} ion transverse to the line connecting adjacent inplane Cu sites i and j, and μ describes the coupling between the orbitalspin fluctuations and the octahedral rotations. By symmetry, H_{OR} depends on Q_{i j}^{2}, which we shall see favours the formation of a glassy state at low temperatures. For completeness, we also include a weak, tetragonal crystalfield term, ,with the crystalfield coupling λ_{CF}≪J_{1}. Our complete Hamiltonian is the sum of the four terms H_{eff}=H_{τ τ}+H_{τ S}+H_{OR}+H_{CF}. To analyse this model we take the usual variational approach^{1,11,17}, optimizing the energy with respect to a hybrid orbital wavefunction, i σ〉=cos(θ_{i})i z σ〉+sin(θ_{i})i x σ〉, which represents the orbital configuration on lattice site i. Here, i z σ〉 represents a ${d}_{3{z}^{2}{r}^{2}}$ orbital, and i x σ〉 a ${d}_{{x}^{2}{y}^{2}}$ orbital, with spin σ and mixing angle θ_{i} as the variational parameters. The sign of θ_{i} is taken to alternate between the two sublattices, ensuring that the variational wavefunction has the appropriate spacegroup symmetry.
In our model, in the hightemperature phase (T≫800 K) the system is cubic, all coupling constants are isotropic and the physics is governed by the highest energy scale, H_{τ τ}. As the temperature is lowered, the system undergoes a transition at T_{JT}∼800 K into a phase with cooperative Jahn–Teller distortions and orbital order, characterized by a mixing angle, θ≈π/4, that minimizes H_{τ τ}. We note that, in our model, this mixing angle depends only on the existence of a J_{1} 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 distortions^{20}, this result suggests that the octahedra exhibit three—rather than just two—distinct bond lengths, in agreement with the known structure^{4,6}.
In the intermediatetemperature regime (50 K<T<800 K), according to our model, the system resides in a freeenergy 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–Tellerdistorted structure. The primary effect of the Jahn–Teller distortion is to introduce anisotropy in the exchange parameters J_{1} and J_{2}. The anisotropy in J_{2}, which arises purely from superexchange and depends exponentially on the bond lengths, should be more pronounced than that in J_{1}, which has a crystalfield component and powerlaw dependence. We therefore anticipate that J_{2}^{c}>J_{2}^{a} (=J_{2}^{b}) and J_{1}^{a}=J_{1}^{b}≈J_{1}^{c}.
Surprisingly, minimizing H_{eff} reveals a ground state that is nearly degenerate. Two distinct hybrid orbital states—called HO_{1}〉 and HO_{2}〉—were found, which exhibit Gtype and Atypeorder, 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 state^{1}, characterized by a mixing angle θ=π/6and corresponding to alternating $3{d}_{{x}^{2}{z}^{2}}$ and $3{d}_{{y}^{2}{z}^{2}}$ orbitals, is higher than HO_{1}〉 and HO_{2}〉by an energy of the order of 3J_{1}/8.
The near degeneracy of HO_{1}〉 and HO_{2}〉 explains the existence of both the extended fluctuational regime and the second structural transition in KCuF_{3}. For illustration, we choose the parameter values J_{1}^{a}=600 K, J_{1}^{c}=630 K, J_{2}^{c}=200 K, J_{2}^{a/b}=30 K, λ_{CF}=50 K and η=0.1 (see refs 1, 9, 10, 11, 17). These choices reflect the dominant role of J_{1}, which arises partly from lattice effects, as well as the expected larger anisotropy in J_{2}. For these values, we find the energy difference between HO_{1}〉 and HO_{2}〉 to be of the order of a few Kelvin. Hence, throughout the intermediatetemperature 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 studies^{6,14,15}.
Although the size of the orbital fluctuations (as reflected in Fig. 2) is small, in our model their consequences for the inplane magnetic order are severe. As HO_{1}〉 and HO_{2}〉 have opposite inplane spin exchange, even small orbital fluctuations disrupt the inplane magnetic order^{17}. This is consistent with the experimental observation that, in the intermediatetemperature regime, the inplane spin correlations are shorter ranged than those along the c axis.
At the lower structural transition, T_{R}=50 K, the orthorhombic fluctuations freeze into a static pattern of GdFeO_{3}type rotations of the CuF_{6} octahedra, resulting in a nonzero value of the variable Q^{2}. This distortion, in accordance with equation (4), lifts the neardegeneracy, lowering the energy of the HO_{2}〉state when compared with HO_{1}〉, resulting in Atype antiferromagnetic order.
The emergence of inplane spin correlations, in our model, has pronounced effects on the stiffness of the lattice, explaining the anomalous phonon shifts observed in KCuF_{3} (see Fig. 1). According to equation (3), aligned neighbouring spins cause a softening of the lattice, whereas antialigned neighbours cause a hardening. This implies that, as the ferromagnetic (Atype) inplane 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 KCuF_{3}, which neutronscattering experiments have suggested is as large as Γ^{c}/Γ^{a/b}∼−100 below the Néel temperature^{9,10}. Using our parameter values, we find Γ^{c}=180 K and ${\text{\Gamma}}_{{\text{HO}}_{\text{2}}}^{a/b}$=−2 K, or Γ^{c}/${\text{\Gamma}}_{{\text{HO}}_{\text{2}}}^{a/b}$=−90 (see Supplementary Information). Some of this anisotropy arises from our chosen anisotropy J_{2}^{c}/J_{2}^{a}=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 Γ^{c}/${\text{\Gamma}}_{{\text{HO}}_{\text{2}}}^{a/b}$, and further amplifies its magnitude above J_{2}^{c}/J_{2}^{a/b} by a factor of 13.5. This amplification is an outcome of the small value of the Cu^{2+} Hund parameter, η, and the small inplane overlap of our orbitals (Fig. 2b) due to their staggered arrangement.
To confirm the relationship between tilting of the CuF_{6} octahedra and the magnetic ordering transition, we also studied KCuF_{3} with conventional and magnetic Xray scattering (see Methods). Magneticscattering studies, summarized in Fig. 3a, show that longranged, Atype antiferromagnetic order sets in at T_{N}=40 K, in agreement with previous studies^{5,9,10}. Above T_{N}, diffuse magnetic scattering is observed, which arises from critical fluctuations in the magnetic order. The momentum linewidth is highly anisotropic, indicating quasionedimensional magnetism, as has been observed above T_{N} (ref. 10). The correlation length in all three directions diverges simultaneously, indicating a single, threedimensional Néel transition.
HardXray 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 CuF_{6} 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–Tellerdistorted 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 CuF_{6} rotations, which amplify the (1,0,5) structure factor. Finally, for T<T_{N}, the (1,0,5) reflection abruptly drops in intensity and acquires a series of diffuse sidebands that exhibit hysteresis when cycling the temperature through T_{N}. This suggests that the CuF_{6} rotations become static below T_{N}, but are glassy and shortrange 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 freeenergy cost of forming domain walls in the CuF_{6} 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 KCuF_{3} in which the orbital order occurs not through a single transition near 800 K, as is commonly believed, but in two stages: the highertemperature 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 inplane spin configurations. The near degeneracy of these states results in a large temperature range of structural and inplane 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 Atype 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 manganites^{2}, ruthenates^{21,22} and the iron pnictides^{23}. 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 some^{24}, though not all^{19}, 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.
Methods
The KCuF_{3} single crystals were grown from solution by techniques described previously^{25}. The crystals selected were screened with Xray measurements and consisted of a >90%volume fraction of polytype a.
Raman scattering measurements were made in a variabletemperature, continuousflow 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 triplestage spectrometer using a liquidnitrogencooled CCD (chargecoupled device) detector.
SoftXray magnetic scattering measurements were conducted at the Cu L_{3/2} edge at beamline X1B at the National Synchrotron Light Source (NSLS). HardXray 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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Acknowledgements
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 DEFG0207ER46453, with softXray studies supported by DEFG0206ER46285.The Advanced Photon Source was supported by DEAC0206CH11357 and the NSLS by DEAC0298CH10886. CHESS and ChemMatCARS are supported by National Science Foundation grants CHE0822838 and DMR0225180, respectively. S.L. gratefully acknowledges financial support from the Department of Science and Technology, Government of India, through a Ramanujam Fellowship.
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S.Y. and J.C.T.L. grew the crystals. S.Y. carried out Raman experiments. J.C.T.L., Y.I.J., S.S., Y.F., Y.G., A.R. and K.F. carried out the Xray experiments. S.L., P.M.G. and P.A. developed the model. P.A. and S.L.C. wrote the paper.
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Lee, J., Yuan, S., Lal, S. et al. Twostage orbital order and dynamical spin frustration in KCuF_{3}. Nature Phys 8, 63–66 (2012). https://doi.org/10.1038/nphys2117
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