Magnetic structure and orbital ordering in tetragonal and monoclinic

THE JOURNAL OF CHEMICAL PHYSICS 128, 164721 共2008兲

Magnetic structure and orbital ordering in tetragonal and monoclinic KCrF3 from first-principles calculations Yuanhui Xu,1 Xianfeng Hao,2,3,a兲 Minfeng Lv,2 Zhijian Wu,2 Defeng Zhou,1,b兲 and Jian Meng2,b兲 1

School of Biological Engineering, Changchun University of Technology, Changchun 130012, People’s Republic of China 2 State Key Laboratory of Rare Earth Resources Application, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People’s Republic of China 3 Graduate School, Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

共Received 1 February 2008; accepted 17 March 2008; published online 30 April 2008兲 KCrF3 has been systematically investigated by using the full-potential linearized augmented plane wave plus local orbital method within the generalized gradient approximation and the local spin density approximation plus the on-site Coulomb repulsion approach. The total energies for ferromagnetic and three different antiferromagnetic configurations are calculated in the high-temperature tetragonal and low-temperature monoclinic phases, respectively. It reveals that the ground state is the A-type antiferromagnetic in both phases. Furthermore, the ground states of the two phases are found to be Mott–Hubbard insulators with the G-type orbital ordering pattern. In addition, our calculations show the staggered orbital ordering of the 3dx2 and 3dy2 orbitals for the tetragonal phase and the 3dz2 and 3dx2 orbitals for the monoclinic phase, which is in agreement with the available data. More importantly, the relationship between magnetic structure and orbital ordering as well as the origin of the orbital ordering are analyzed in detail. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2908740兴 I. INTRODUCTION

Orbital, charge, spin, and lattice degrees of freedom play important roles in the electronic, magnetic, and transport properties of transition-metal oxides. Orbital degrees of freedom and, in particular, their ordering give rise to a very rich physics1,2 and lead to extraordinary ground states, lowenergy excitations, and phase transitions. A well-known example for such ordering phenomena is the perovskite-based manganites.3,4 These phenomena involve dramatic changes of the electronic and magnetic properties, which constitute a challenge for the understanding of strongly correlated electron systems.5–8 Most efforts in this field have centered almost exclusively on oxides; however, similar intriguing electronic and magnetic phenomena also occur in the nonoxide strongly correlated materials, for which their physical mechanism has not been investigated in detail. The unusual orbital degree of freedom in the nonoxide pervoskite KCrF3, which is the focus of our study, originates from the singly occupied degenerate eg states of the Cr2+ 3d electron with the high-spin electronic configuration 共t2g3eg1兲. This orbital degeneracy makes the Cr2+ ion Jahn–Teller 共JT兲 active: The degeneracy can be split via the lattice distortion of the surrounding fluorine octahedron, which is reminiscent of the Mn3+ 共3d4兲 and Cu2+ 共3d9兲 ions in LaMnO3 and KCuF3, respectively. Interestingly, the early work has reported that KCrF3 adopts a tetragonal structure at room temperature with an A-type antiferromagnetic 共A-AFM兲 magFAX: ⫹86-0431-85698041. Electronic mail: [email protected]. Authors to whom correspondence should be addressed. Electronic addresses: [email protected] and [email protected].

a兲

b兲

0021-9606/2008/128共16兲/164721/7/$23.00

netic order below 40 K.5 Very recently, Margadonna and Karotsis reinvestigated the structural and magnetic properties of KCrF3.6 Their results showed that KCrF3 in the tetragonal phase exhibits a strong cooperative JT distortion 共CJTD兲 in the lattice structure, which provides a signature of an antiferrodistortive orbital ordering at the Cr2+ sites. Below 250 K, KCrF3 undergoes a phase transition to a monoclinic structure characterized by the pronounced tilting of the CrF6 octahedron, giving rise to an increase in the magnitude of the CJTD. Nevertheless, the orbital ordering motifs are essentially identical to those in the tetragonal structure. Moreover, the magnetic measurement implied that KCrF3 possesses the long-range AFM order below the magnetic transition temperature of TN ⬃ 46 K. Although several experimental data are available, for the ternary transition-metal fluoride KCrF3, the theoretical calculations are highly desirable, since they can provide further details for better understanding of these systems. Therefore, it is worthwhile to perform theoretical studies on the KCrF3 system to get insight into the intriguing physical properties.

II. STRUCTURE AND MAGNETIC ORDERS

The crystal structures of the two KCrF3 polymorphs are given in Fig. 1, namely, the tetragonal phase at high temperature and the monoclinic phase at low temperature. The tetragonal KCrF3 共space group I4 / mcm兲共left兲 is built from CrF6 octahedral chains parallel to the c axis. The CrF6 octahedron is axially distorted, in which short Cr–F bonds

128, 164721-1

© 2008 American Institute of Physics

Downloaded 23 Dec 2010 to 159.226.166.22. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

164721-2

J. Chem. Phys. 128, 164721 共2008兲

Xu et al.

FIG. 1. 共Color online兲 Crystal structures for KCrF3 in tetragonal 共left兲 and monoclinic 共right兲 perovskite lattices. Red 共big兲, green 共middle兲, and yellow 共small兲 spheres represent K, Cr, and F atoms, respectively.

共2.005 Å兲 along the c axis and alternating long 共2.294 Å兲 and short 共1.986 Å兲 Cr–F bonds in the ab plane are observed, thus giving rise to the CJTD. As aforementioned, KCrF3 shows a phase transition to a monoclinic structure 共space group I112/ m兲 below 250 K, as displayed in Fig. 1 共right兲. The main features of the crystal structure are preserved for the monoclinic structure. Due to the pronounced tilting of the CrF6 octahedra, however, the alternating shortenings and elongations of the Cr–F bond lengths now occur in the plane defined by the c axis and the 具1-10典 base diagonal. The monoclinic structure is characterized by short and long distances alternating both within the ab plane and along the c axis. It suggests that the further deformation of the CrF6 octahedra in the monoclinic structure increases the CJTD in magnitude and leads to two crystallographically inequivalent Cr sites with a similar coordinate environment.6 To explore the electronic and magnetic properties in the ground state, we compare the total energy of KCrF3 for different magnetic configurations in the tetragonal and monoclinic phases. Calculations are performed for the FM order and three different kinds of AFM ones, including A-AFM, C-AFM, and G-AFM. For the FM structure, both interplane and intraplane interactions are ferromagnetic. While for the AFM structure, AFM interplanar and FM intraplanar couplings lead to A-type, the opposite situation 共AFM intraplanar and FM interplanar coupling兲 gives C type. In case where both interactions are AFM, the G-type arises. III. COMPUTATIONAL DETAILS

Our calculations are carried out in the framework of density-functional theory 共DFT兲 using the high accurate fullpotential linearized augmented plane wave plus local orbital method9,10 implemented in the WIEN2K package.11,12 In this method, the space is divided into nonoverlapping muffin-tin 共MT兲 spheres centered at the atomic sites and an interstitial region between the spheres. In the MT region, the basis sets are described by radial solutions of the one-particle Schrödinger equation 共at fixed energy兲 and their energy derivatives multiplied by spherical harmonics. The values of the atomic sphere radii 共RMT兲 were chosen as 2.5, 1.96, and 1.74 a.u for K, Cr, and F, respectively. To achieve energy

convergence, the wave functions in the interstitial region min were expanded in plane waves with a cutoff RMT Kmax = 7, min where RMT denotes the smallest atomic sphere radius and Kmax gives the magnitude of the largest K vector in the planewave expansion. The valence wave functions inside the spheres are expanded up to lmax = 10, while the charge density was Fourier expanded up to Gmax = 14. A large number of 3000 k points were sampled in the total Brillouin zone. Selfconsistency was considered to be achieved when the totalenergy difference between succeeding iterations is less than 10−5 Ry/ f.u. The present setup has been checked to ensure a sufficient accuracy of the calculations. As for the exchange-correlation potential, we adopt the standard generalized gradient approximation 共GGA兲 by using the Perdew–Burke–Ernzerhof scheme.13 To properly describe the strong electron correlation associated with the Cr 3d states, a so-called LSDA+ U 共U denotes coulomb repulsion兲 method14–16 is employed which combines the local spin density approximation 共LSDA兲 with the Hubbard model approach that is especially suited for treating strongly correlated transition-metal systems. In the present work, we take the experimental crystal structure parameters as reported in Ref. 6. IV. RESULTS AND DISCUSSIONS A. GGA calculations

We calculate the total energies and magnetic moments by using the GGA scheme for considered magnetic configurations in the tetragonal and monoclinic phases. The calculated results are listed in Table I. It can be seen that the A-AFM structure in the tetragonal phase is the lowest in total energy among the considered magnetic configurations, in good agreement with the experimental report5 and the results calculated by Giovannetti et al.17 Note that there exists a small difference in total energy between the A-AFM and FM states, being only 4 meV/ f.u., while the energy differences between the A-AFM and the other two AFM configurations, namely, C-AFM and G-AFM, differ by about 89 and 77 meV/ f.u., respectively. It is recalled that A-AFM and FM structures have a ferromagnetic ab plane but different stacking along the c axis, whereas both the C-AFM and G-AFM states are antiferromagnetically ordered in the ab plane but have opposite stacking in the c direction. This reveals that FM exchange interactions in the ab plane are more favorable and robust than along the c direction, indicating a strong two-dimensional 共2D兲 character for KCrF3. In addition, the GGA results show that the calculated magnetic moment per Cr atom varies from 3.29 to 3.36␮B depending on the magnetic configuration considered in our calculations. Based on Hund’s rule, Cr atoms should take the high-spin configuration 共t2g3eg1兲, being consistent with the experimental measurement.6 Meanwhile, it is also found that the F atom carries a small magnetic moment 共0.01– 0.03␮B兲 induced by the hybridization with Cr atoms. We present the total electronic density of states 共DOS兲 obtained by the GGA calculations for the A-AFM configuration of the tetragonal phase in Fig. 2共a兲. Apparently, the A-AFM state turns out to be an insulator with a finite band gap at the Fermi level, which is


164721-3

J. Chem. Phys. 128, 164721 共2008兲

Magnetic structure and orbital ordering in KCrF3

TABLE I. The total energy 共relative to the lowest state in meV/f.u.兲 and magnetic moment 共in ␮B/atom兲 for KCrF3 with FM, A-AFM, C-AFM, and G-AFM magnetic configurations in both tetragonal and monoclinic phases from the GGA calculations. Magnetic moment Configuration

E

Cr

F

Interstitial

Total

Tetragonal

FM AFM-A AFM-C AFM-G

4 0 89 77

3.36 ⫾3.34 ⫾3.31 ⫾3.29

0.03/ 0.03 0 / ⫾ 0.03 ⫾0.01/ ⫾ 0.03 ⫾0.01/ 0

0.54 0 0 0

4.00 0 0 0

Monoclinic

FM AFM-A AFM-C AFM

13 0 82 87

3.34/ 3.38 ⫾3.33/ ⫾ 3.35 ⫾3.31/ ⫾ 3.32 ⫾3.29

0.03/ 0.03/ 0.04 0 / ⫾ 0.03/ ⫾ 0.04 ⫾0.01/ ⫾ 0.01/ ⫾ 0.03 0 / ⫾ 0.01

0.54 0 0 0

4.00 0 0 0

due to the sufficiently large exchange-splitting and crystalfield splitting effects. In addition, the other two AFM structures considered in our work also exhibit the insulating character 共not shown兲. However, for the FM one, a half-metal is achieved with the metal in the spin-up channel but the insulator behavior in the spin-down channel 共not shown兲. The total magnetic moment of FM per f.u. is 4.00␮B 共see Table I兲, implying that the Cr2+ ion plays a dominant role in determining the total magnetic moment in KCrF3. On the other hand, for the monoclinic phase, the A-AFM state with the lowest energy is found to be the ground state in accordance with the experimental observation6 and the result in Ref. 17. Compared to the FM, C-AFM, and G-AFM configurations, the total energies of the A-AFM order lower by about 13, 82, and 87 meV/ f.u., respectively. It should be pointed out that in the monoclinic phase, the FM interactions in the A-AFM state lie in the plane defined by the c axis and the 具1–10典 base diagonal, while the AFM interactions between these planes. As for the magnetic moment, similar arguments hold for the monoclinic phase, due to the similar features of two crystal structures. Figure 2共b兲 shows the total DOS for the ground-state A-AFM structure in the monoclinic phase by using the GGA scheme. Remarkably, the shape and energy distributions of the DOS corresponding to the

FIG. 2. The total electronic density of states 共DOS兲 obtained by the GGA calculations for the ground-state A-AFM configuration in the 共a兲 tetragonal and 共b兲 monoclinic phases. Fermi energy is set to zero.

magnetic configuration of the monoclinic phase are almost the same with those of the tetragonal phase, including the half-metallic FM, the insulating C-, and G-AFM states 共not shown兲. It is commonly accepted that the DFT-GGA or LSDA approach can be inadequate to account for the electronic structure of compounds containing partially filled d or f valence states. Therefore, it is necessary to make further calculations by including an on-site Coulomb repulsion U, as have been done for Cr 3d shells in the previous theoretical studies.18,19

B. LSDA+ U calculations

To properly describe the strong electron correlation for the Cr 3d electrons in KCrF3, we employ the LSDA+ U method. This method is known to improve over GGA in these highly correlated transition-metal systems and to predict better values of the magnetic moments and energy gaps. As shown recently,20,21 the exact value of U depends on the computational method and the approximations made in the procedures used. In the present work, we assume the Coulomb parameter Ueff 共Ueff = U − J兲 to be 3.0 eV as interpolated in Refs. 18 and 19. To check how the results depend on Ueff values, we performed further calculations with different Ueff values 共0.0, 3.0, 4.0, 5.0, and 6.0 eV兲 and found that for Ueff = 6.0 eV, the lowest state of the total energy is the G-AFM configuration in the tetragonal phase, which is in disagreement with the experimental observations. Thus, we conclude that for the KCrF3 system, the reasonable range of the Ueff value is less than 6.0 eV and the calculated results for Ueff = 0.0, 3.0, 4.0, and 5.0 eV are listed in Table II. One can see from Table II that the A-AFM solution remains stable in both phases upon increasing the Ueff values to 4.0 and 5.0 eV. In addition, the energy gap enhances and the magnetic moments get more localized inside the atomic spheres, as expected. Therefore, we want to stress that all our conclusions in this paper do not depend on Ueff values as long as it is chosen in a physically reasonable range.


164721-4

J. Chem. Phys. 128, 164721 共2008兲

Xu et al.

TABLE II. The total energy for the different magnetic configurations in both phases by means of the LSDA + U approach with different Ueff values. All the energies are given relative to that of the A-AFM state and the unit is meV/f.u. Total energy Configuration

Ueff = 0.0 eV

Ueff = 3.0 eV

Ueff = 4.0 eV

Ueff = 5.0 eV

Tetragonal


4 0 89 77

7 0 35 13

7 0 27 6

6 0 24 1

Monoclinic


13 0 82 87

15 0 31 29

15 0 24 23

16 0 20 19

1. The electronic, magnetic, and orbital orderings for the tetragonal KCrF3

We investigate the FM and three different AFM configurations by using the LSDA+ U method, while the A-AFM state in total energy is still the lowest among the four configurations. To get further insight into the electronic properties for the ground state, we present the total and partial DOSs for the A-AFM in the tetragonal phase obtained by LSDA+ U 共Ueff = 3.0 eV兲 calculations in Fig. 3. One can observe that the energy gap in the total DOS plot significantly increases to 2.0 eV in comparison with 0.7 eV obtained by the GGA calculations 关Fig. 2共a兲兴. This remarkable increase in the energy gap is accompanied by the enhancement of the magnetic moment per Cr atom. As expected, LSDA+ U tends to enhance the localization of Cr d states 共thereby increasing the magnetic moment兲 and to push unoccupied states up in energy 共thereby increasing the band gap兲. It is obvious that the lower group of bands extending from −8.2 to − 5.0 eV is mainly from the F 2p states. In contrast, the states derived from K atoms are found mainly above 3.8 eV, indicating the striking ionic character for K atoms. Most of the Cr 3d states

FIG. 3. 共Color online兲 The total and partial DOSs for the A-AFM state in the tetragonal phase from the LSDA+ U 共Ueff = 3.0 eV兲 calculations. Fermi level is set to zero.

are located in the vicinity of the Fermi level, suggesting that the ground state of the tetragonal phase is a Mott–Hubbard insulator.22,23 In an ideal perovskite structure of KCrF3, the 3d orbitals of Cr atoms are split into triply degenerate t2g and doubly degenerate eg levels by the octahedral crystal-field effects. Owing to the electronic configuration of Cr2+ 共t2g3eg1兲 in KCrF3, the degeneracy of the eg state is easily broken by the strong CJTD. This will affect the eg orbital population and may develop the orbital ordering at Cr2+ sites, similar to the case in LaMnO3.24 In Fig. 4, we display the orbitaldecomposed DOS for Cr atoms in the A-AFM state. It is evident that the Cr d state exhibits a large exchange-splitting energy of ⬃5.2 eV, which should be responsible for the high-spin electronic configuration of Cr2+ ions. Moreover, the spin-up channels of the dxy, dxz, and dyz states are fully occupied and lie in a relative lower energy range, while the dx2−y2 and dz2 bands are partially occupied close to the Fermi level. This means that the Cr ion is essentially in the d4 共t2g3eg1兲 configuration with the spin-up t2g state fully occupied and eg state partially occupied. Note that the Cr eg bands

FIG. 4. The orbital-decomposed DOS 共only d orbitals兲 of Cr1 in the A-AFM state of the tetragonal phase within LSDA+ U 共Ueff = 3.0 eV兲. Fermi energy is set to zero. Majority and minority spins are shown above and below the axes. The distribution of d orbitals for the Cr2 atom is the same to that of the Cr1 one but with completely opposite spin direction due to the antiferromagnetic structure 共not shown兲.


164721-5

Magnetic structure and orbital ordering in KCrF3

J. Chem. Phys. 128, 164721 共2008兲

FIG. 5. 共Color online兲 Spin density plot 共isosurface at 0.2 e / Å3, mapped by XcrySDen兲 within the energy interval −0.8– 0 eV from the result of LSDA + U 共Ueff = 3.0 eV兲 calculations for the A-AFM state in the tetragonal phase, to illustrate the staggered orbital ordering of the 3dx2 and 3dy2 orbitals 共the local coordinate of the orbitals is defined by rotating 45° around the c axis in the present coordinate system兲. Red and blue denote spin up and spin down, respectively.

are more delocalized than the t2g ones, due to the stronger pd␴-type hybridization for the former than the pd␲-type one for the latter. To clarify the existence of the orbital ordering in tetragonal KCrF3, we show the spin density plot for the A-AFM state in Fig. 5. The staggered orbital ordering is clearly seen in the ab planes, comprising the 3dx2 and 3dy2 orbitals 共the local coordinate of the orbitals is defined by rotating 45° around the c axis in the present coordinate system兲, which demonstrates the existence of the preferential occupation of the orbitals in the quarter-filled eg state. Consequently, we may conclude that KCrF3 in the tetragonal phase possesses a G-type orbital ordering pattern 共the antiferrodistortive ordering orbitals are in the ab plane but along the c axis, the orbitals form angles of 90° to each other兲, which is consistent with the local distortions of the fluorine octahedra surrounding the Cr sites, as well as the experimental observations6 and the results reported in Ref. 17. 2. The electronic, magnetic, and orbital orderings for the monoclinic KCrF3

It is found that the ground state of the monoclinic phase is the A-AFM order irrespective of the choice of approximation as GGA or LSDA+ U, which is in good agreement with the experimental data.6 Figure 6 shows the total and sitedecomposed DOS for the A-AFM state in the monoclinic phase by the LSDA+ U 共Ueff = 3.0兲 scheme. As a striking feature, the energy gap for the ground state is drastically increased to 2.1 from 0.6 eV obtained by the GGA calculations. One can see that the K states centered above the Fermi level show strong ionic behavior, while the bands lying in the lower energy range mainly derive from the F atoms. At the same time, Cr 3d states are found in the energy range of −1.9– 5 eV, which determines the band distributions around the Fermi level, indicating the d-d Mott–Hubbard character. The same as in the A-AFM case of the tetragonal phase, the inclusion of U enhances the localization of Cr d states and reduces the covalency between the Cr ions and the F ligands, which is also reflected by the increase in the Cr moment and the decrease in the F moment inside the atomic spheres. For an explicit view, we also plot the DOS projected onto the five 3d orbitals of the Cr1 and Cr2 atoms for the A-AFM state shown in Fig. 7. Note that the dz2 and dx2−y2

FIG. 6. 共Color online兲 The total and site-decomposed DOSs for the A-AFM order in the monoclinic phase obtained by the LSDA+ U 共Ueff = 3.0 eV兲 method. Fermi energy is set to zero.

orbitals near the Fermi level oriented toward the F ligands form the eg manifold, while the dxy, dxz, and dyz orbitals span the t2g manifold. It can be seen that the spin-up t2g orbitals are fully occupied and the eg orbitals are partially occupied, confirming the Cr2+ ions with the t2g3eg1 configuration in the monoclinic phase. More importantly, the degenerate eg level is split into the fully occupied dz2 orbital and nearly unoccupied dx2−y2 orbital in Cr1 atoms and vice versa for the eg states in Cr2 atoms. To some extent, this implies the existence of orbital ordering in the monoclinic KCrF3. To better distinguish the orbital shapes and distributions in the monoclinic phase, Fig. 8 displays the spin density plot for the A-AFM state from the LSDA+ U 共Ueff = 3.0 eV兲 results. The antiferrodistortive ordering of the 3dz2 and 3dx2 orbitals is clearly seen, which illustrates the existing of the preferential occupation in the doubly degenerate eg orbitals. Therefore, it turns out that there exists a G-type antiferrodis-

FIG. 7. The orbital-resolved DOS 共only d orbitals兲 of Cr1 and Cr2 atoms in the A-AFM state of the monoclinic phase within LSDA+ U 共Ueff = 3.0 eV兲. Fermi energy is set to zero. Majority and minority spins are shown above and below the axes. The distributions of d orbitals for Cr3 and Cr4 atoms are identical to those of Cr1 and Cr2 ones, respectively, but with completely opposite spin direction 共not shown兲.


164721-6

J. Chem. Phys. 128, 164721 共2008兲

Xu et al.

FIG. 8. 共Color online兲 Spin density plot 共isosurface at 0.2 e / Å3, mapped by XcrySDen兲 at the region extending from −0.7 to 0 eV using the result of LSDA+ U 共Ueff = 3.0 eV兲 calculations for the A-AFM state in the monoclinic phase, to illustrate the pattern of the 3dz2 and 3dx2 orbital ordering.

tortive orbital ordering for KCrF3 in the monoclinic phase, in excellent agreement with the suggestion derived from the experiment6 and the calculations carried out by Giovannetti et al.17 In addition, it should be pointed out that KCrF3 in the monoclinic phase shows weak FM, which can be ascribed to singly ion anisotropy effects and the Dzyaloshinsky–Moriya interaction. C. Discussions

In general, the orbital ordering takes place at higher temperatures than that of the magnetic order in the case of an eg electron with a strong JT distortion,25 such as LaMnO3,26 in which the orbital interactions between the eg electrons are much stronger 共the long-range orbital ordering temperature TOO ⬇ 800 K兲 than the magnetic interactions 共the magnetic transition temperature TN ⬇ 150 K兲. To gain insight into the relations between the orbital ordering and the magnetic order in KCrF3, we also looked at the spin density plots for the FM, C-, and G-AFM configurations in both phases by using the LSDA+ U results 共not shown兲. Take the FM configuration of the tetragonal phase as an example, the orbital plot looks very similar to that of the ground-state A-AFM, but the color 共denoting spin up and spin down兲 is equal for every Cr atom. Based on these observations, a preliminary remark can be concluded: For a fixed crystal structure of KCrF3, a change in the magnetic configuration does not arise a significant effect on the orbital ordering pattern; i.e., the orbital ordering is more robust than the magnetic order. In particular, we noticed that in the spin density plots, some spin densities on the F atoms induced by the hybridizations with Cr atoms clearly illustrate the exchange interaction paths. Following the Goodenough–Kanamori– Anderson rules,27,28 the eg-eg exchange interaction within the plane is FM due to a virtual electron hopping between an occupied and an empty orbital via the intermediate F ligand, while the half-filled orbitals of t2g-t2g exchange through the medial F atoms between the planes is AFM. This indicates that the orbital ordering pattern in both phases prefers the A-AFM magnetic configuration. In addition, the overlap of the eg orbitals between the Cr planes brings out the 2D character for this system, which also implies that the coupling between the Cr planes is relatively weak. Finally, a crucial issue is to clarify the origin of the orbital ordering in KCrF3. We investigate the spin density plots for the considered magnetic configurations of the two phases

by using the GGA calculations 共not shown兲. It is found that all plots exhibit the G-type orbital ordering patterns. However, due to the less localization of Cr d electrons within GGA, the preferential occupation between the 3dz2 and 3dx2−y2 orbitals lying in the quarter-filled eg level is not very complete compared to that obtained from LSDA+ U results. Therefore, it is suggested that the on-site Coulomb repulsion U in KCrF3 system plays an important role in forming the orbital ordering, although it is not the determining factor. There is a specific structural distortion in KCrF3, that is, the large CJTD of the CrF6 octahedron, which is strong enough to lift the doubly degenerate eg states and to open up an energy gap between the occupied and unoccupied bands. This promotes the A-AFM state in both phases to be a Mott– Hubbard insulator even by means of the GGA method without considering the electronic correlation effect for Cr 3d states. Thus, it turns out that the cooperative JT effect in KCrF3 should be responsible for the formation of the orbital ordering in the two phases. This is in strong contrast to the cases of LaMnO3 and KCuF3, where the orbital ordering in both compounds is purely electronic in origin.24 Moreover, it is worthwhile to note that more recently, Margadonna and Karotsis29 found that the static cooperative distortion of the structure 共which is CJTD兲 in KCrF3 disappears at TJT = 973 K, at which the orbital ordering at the Cr2+ sites abruptly melts and is accompanied by a phase transition to the cubic structure. This provides experimental support for our argument that the orbital ordering in KCrF3 mainly originates from the cooperative JT effect in the crystal structure. V. CONCLUSION

In summary, we have systematically investigated the electronic, magnetic properties and orbital ordering for the ternary perovskite-type fluoride KCrF3 in the tetragonal and monoclinic phases by using the GGA and LSDA+ U methods within the density-functional framework. Total-energy calculations show that the ground state is the A-type AFM configuration in both phases, which is in good agreement with the available data. The LSDA+ U 共Ueff = 3.0 eV兲 calculated results suggest that the ground states of two phases are strongly correlated Mott–Hubbard insulators with the G-type orbital ordering pattern, in accordance with the experimental observations, as well as the results calculated by Giovannetti et al. More importantly, we found that in the present system, the orbital ordering is more robust than the magnetic order and the cooperative JT effect should be responsible for the formation of the orbital ordering, but the on-site Coulomb U is also important. We hope that our results will stimulate more work both from theories and experiments. ACKNOWLEDGMENTS

The authors would like to thank Dr. F. M. Gao, Professor Jeroen van den Brink, and Dr. Gianluca Giovannetti for invaluable discussions and help. This work was supported by the National Natural Science Foundation of China for financial support 共Grant Nos. 20331030, 20671088, 20661026, 20571073, and 20771100兲.


164721-7

K. I. Kugel and D. I. Khomskii, Sov. Phys. JETP 37, 725 共1973兲; Sov. Phys. Usp. 25, 231 共1982兲. 2 Y. Tokura and N. Nagaosa, Science 288, 462 共2000兲. 3 T. Mizokawa, D. I. Khomskii, and G. A. Sawatzky, Phys. Rev. B 60, 7309 共1999兲. 4 I. S. Elfimov, V. I. Anisimov, and G. A. Sawatzky, Phys. Rev. Lett. 82, 4264 共1999兲. 5 A. J. Edwards and R. D. Peacock, J. Chem. Soc. 1959, 4126; V. Scatturin, L. Corliss, N. Elliott, and J. Hastings, Acta Crystallogr. 14, 19 共1961兲; S. Yoneyama and K. Hirakawa, J. Phys. Soc. Jpn. 21, 183 共1966兲. 6 S. Margadonna and G. Karotsis, J. Am. Chem. Soc. 128, 16436 共2006兲. 7 Y. Okimoto, T. Katsufuji, T. Ishikawa, A. Urushibara, T. Arima, and Y. Tokura, Phys. Rev. Lett. 75, 109 共1995兲. 8 P. G. Radaelli, D. E. Cox, M. Marezio, and S. W. Cheong, Phys. Rev. B 55, 3015 共1997兲. 9 E. Sjöstedt, L. Nordström, and D. J. Singh, Solid State Commun. 114, 15 共2000兲. 10 G. K. H. Madsen, P. Blaha, K. Schwarz, E. Sjöstedt, and L. Nordström, Phys. Rev. B 64, 195134 共2001兲. 11 K. Schwarz and P. Blaha, Comput. Mater. Sci. 28, 259 共2003兲. 12 P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2k, An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties 共Vienna University of Technology Press, Austria, 2001兲. 13 J. Perdew, S. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 共1996兲. 14 V. I. Anisimov, I. W. Solovyev, M. A. Korotin, M. T. Czyzyk, and G. A. Sawatzky, Phys. Rev. B 48, 16929 共1993兲. 15 A. Lichtenstein, V. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 共1995兲. 1

J. Chem. Phys. 128, 164721 共2008兲

Magnetic structure and orbital ordering in KCrF3 16

A. Petukhov, I. Mazin, L. Chioncel, and A. Lichtenstein, Phys. Rev. B 67, 153106 共2003兲. 17 G. Giovannetti, S. Margadonna, and J. van den Brink, Phys. Rev. B 77, 075113 共2008兲. 18 M. A. Korotin, V. I. Anisimov, D. I. Khomskii, and G. A. Sawatzky, Phys. Rev. Lett. 80, 4305 共1998兲. 19 H. Weng, Y. Kawazoe, X. Wan, and J. Dong, Phys. Rev. B 74, 205112 共2006兲. 20 H. J. Kulik, M. Cococcioni, D. A. Scherlis, and N. Marzari, Phys. Rev. Lett. 97, 103001 共2006兲. 21 M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 共2005兲. 22 J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev. Lett. 55, 418 共1985兲. 23 J. Zaanen and G. A. Sawatzky, Can. J. Phys. 65, 1262 共1987兲. 24 J. E. Medvedeva, M. A. Korotin, V. I. Anisimov, and A. J. Freeman, Phys. Rev. B 65, 172413 共2002兲. 25 Y. Murakami, J. P. Hill, D. Gibbs, M. Blume, I. Koyama, M. Tanaka, H. Kawata, T. Arima, Y. Tokura, K. Hirota, and Y. Endoh, Jpn. J. Appl. Phys., Part 1 38, 360 共1999兲. 26 Y. Murakami, J. P. Hill, D. Gibbs, M. Blume, H. Kawata, M. Tanaka, T. Arima, Y. Moritomo, Y. Tokura, K. Hirota, and Y. Endoh, Phys. Rev. Lett. 81, 582 共1998兲. 27 P. W. Anderson, Magnetic Ions in Insulators. Their Interactions, Resonances, and Optical Properties of Magnetism, Exchange in Insulators: Superexchange, Direct Exchange, and Double Exchange Vol. 1 共Academic, New York, 1963兲. 28 J. B. Goodenough, Magnetism and the Chemical Bond 共Interscience, New York, 1963兲. 29 S. Margadonna and G. Karotsis, J. Mater. Chem. 17, 2013 共2007兲.
