First-principles study of the electronic structure and magnetism of CaIrO 3
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n First-principles study of the electronic structure and magnetism of CaIrO Alaska Subedi
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
I study the electronic structure and magnetism of postperovskite CaIrO using first-principlescalculations. The density functional calculations within the local density approximation withoutthe combined effect of spin-orbit coupling and on-site Coulomb repulsion show the system to bemetallic, which is in disagreement with the recent experimental evidences that show CaIrO to bean antiferromagnetic Mott insulator in the J eff = 1/2 state. However, when spin-orbit coupling istaken into account, the Ir t g bands split into fully filled J eff = 3/2 bands and half-filled J eff = 1/2bands. I find that spin-orbit coupling along with a modest on-site Coulomb repulsion opens a gapleading to a Mott insulating state. The ordering is antiferromagnetic along the c axis with totalmoments aligned antiparallel along the c axis and canted along the b axis. PACS numbers: 71.30.+h, 75.25.Dk, 75.50.Ee
I. INTRODUCTION
Transition-metal oxides (TMOs) in perovskite andrelated structures exhibit myriad interesting proper-ties. These include unconventional superconductivityin cuprates, colossal magnetoresistance in manganites, and ferroelectricity in Pb(Zr,Ti)O . In these materials,the transition-metal ion is situated inside an oxygen octa-hedral cage, which may be arranged in a corner- or edge-shared manner. The interesting properties of TMOs arisebecause of the competition between the crystal-field split-ting (which arises because of covalency between transi-tion metal d and oxygen p states), on-site Coulomb repul-sion U , Hund’s coupling, spin-orbit (SO) coupling due toorbital degeneracy (leading to unquenched angular mo-ment in the ground state), and different spin-exchangepathways.In this regard, the discovery of a spin-orbital Mottstate in Sr IrO by Kim et al. is significant becauseit enables us to study the case where spin-orbit cou-pling and its interplay with the Coulomb repulsion is animportant ingredient in determining the electronic andmagnetic properties of the system. Sr IrO exists in alayered perovskite structure. The Ir (5 d ) ions are sit-uated inside corner-shared O octahedral cages, which arethemselves arranged in a square lattice in the ab plane.As there are an odd number of electrons per formulaunit, one might expect this material to be a metal inthe band picture. However, Sr IrO is experimentally acanted antiferromagnetic insulator. As explained lucidlyby Kim et al. , a state with an effective total angularmomentum J eff = 1 / IrO , which arises due to the com-bined influence of strong spin-orbit coupling and mod-erate on-site Coulomb repulsion. A similar conclusionon Sr IrO has been reached through the study of athree-orbital Hubbard model with spin-orbit coupling and combined density functional theory and dynamicalmean field theory (LDA+DMFT) calculations, althoughArita et al. suggest that Sr IrO is a Slater insulatorbased on their LDA+DMFT study. Interestingly, it was also found that spin-orbit coupling plays an importantrole in the electronic properties even for a 4 d systemsuch as Sr RhO . It has been suggested that systems in the J eff = 1/2state, depending on bond geometry, lead to interest-ing varieties of low-energy Hamiltonians, including theisotropic Heisenberg model and the highly anisotropicquantum compass or Kitaev models relevant for quantumcomputing. Therefore, it is important to investigatematerials that exhibit the J eff = 1 / et al. reported resonant x-ray diffraction study ofCaIrO that indicates this material also exhibits a Mottinsulating J eff = 1 / CaIrO exists in the post-perovskite structure with space group Cmcm as shownin Fig. 1. The Ir (5 d ) ions are situated inside the Ooctahedra, but these octahedra share an edge along the c axis, unlike the case of Sr IrO . Thus, CaIrO is anotherideal material to investigate the interplay between spin-orbit coupling and on-site Coulomb repulsion that mayhelp in understanding the unique properties that mightbe exhibited by J eff = 1 / et al. have reported first-principles densityfunctional calculations that show this material is a metalwithin the local density approximation. This is con-trary to the experimental evidence that shows this ma-terial is a Mott insulator that undergoes an antiferro-magnetic transition at T N = 115 K. A recent reso-nant x-ray diffraction study shows that the ordering isof stripe-type antiferromagnetism along the c axis, withtotal moments aligning parallel along the a axis andantiparallel along the c axis. The inverse susceptibil-ity 1/ χ deviates from the linear behavior at a temper-ature ∼
350 K that is considerably higher than T N anda Curie-Weiss fit to χ above 400 K gives a Curie-Weisstemperature of 3900 K. This indicates that the anti-ferromagnetic correlations arise much before the anti-ferromagnetic transition, and magnetic ordering is sup-pressed by low dimensionality or competing ordering in-teractions. Jang et al. have studied the electronic struc-ture of meta-stable perovskite Ca − x Sr x IrO ( x = 0, 0.5, FIG. 1: (Color online) Crystal structure of CaIrO . The large(cyan) balls are Ca, small (red) balls are O, and the Ir atomsreside inside the (brown) octahedra. and 1) thin films using transport measurements, opticalspectroscopy, and pseudopotential-based first-principlescalculations. They find that perovskite CaIrO thinfilms are semimetallic and near the metal-insulator tran-sition. Their calculations with spin-orbit coupling andon-site Coulomb repulsion U found that the spin-orbitcoupling splits the Ir t g states into J eff = 3/2 and 1/2states, and U = 2.0 eV further splits the J eff = 1/2 states,although the valence and conduction bands still touch theFermi level, resulting in a semimetallic state.The experimental evidences that have so far been ac-cumulated suggest that calculations that include the ef-fect of spin-orbit coupling and on-site Coulomb repulsionwould be helpful in clarifying the electronic and magneticproperties of CaIrO . In this paper, I report the resultsof density functional calculations that show CaIrO isin a Mott insulating state that is induced by the com-bined effect of spin-orbit coupling and on-site Coulombrepulsion. This state arises out of spin-orbit split Ir t g bands that get separated into lower lying fully filled andhigher lying half-filled bands that have effective total an-gular momenta J eff = 3/2 and 1/2, respectively, in thestrong spin-orbit coupling limit. The half-filled J eff =1/2 bands are narrow, so even a modest on-site Coulombrepulsion induces a Mott insulating state that is topolog-ically different from the metallic state given by the lo-cal density approximation, without taking into accountthe spin-orbit coupling and on-site Coulomb repulsion.This is a Mott insulating state in the sense that a single-particle theory such as the density functional theory im-plemented using Kohn-Sham formalism cannot explainthe insulating state, and an explicit treatment of on-siteCoulomb repulsion is needed. The Mott insulating statethus obtained is antiferromagnetically ordered along the c axis with total moments aligned antiparallel along the c axis and canted along the b axis. II. APPROACH
The purpose of this paper is to elucidate the role ofspin-orbit coupling and on-site Coulomb repulsion on N ( E ) E (eV) totalIr dO(1) pO(2) p
FIG. 2: (Color online) Non-spin-polarized LDA DOS ofCaIrO (in states/eV). The projections are onto the respectivemuffin-tin spheres and are only indicative of the contributionto the total DOS. The Fermi energy is at 0 eV. the electronic and magnetic properties of CaIrO us-ing density functional calculations. The calculationswere performed within the local density approximation(LDA) using the general full-potential linearized aug-mented plane-wave method as implemented in the ELKsoftware package. Muffin-tin radii of 2.2, 2.0, and 1.6a.u. for Ca, Ir, and O, respectively, were used. A 8 × × k -point grid was used to perform the Brillouin zone inte-gration, and the convergence of moments was checked ona 10 × ×
10 grid. The effect of spin-orbit coupling wastreated using a second-variational scheme, and the fullylocalized limit is used to take into account the doublecounting in LDA+ U calculations. A value for the on-siteCoulomb repulsion U = 2.75 eV (which gives a band gapclose to the experimental value) was used unless other-wise mentioned.I used the experimental lattice parameters a = 3.145˚A, b = 9.855 ˚A, and c = 7.293 ˚A, but relaxed the atomicpositions. The calculated atomic positions Ca (0, 0.2498,0.25), Ir (0, 0, 0), O(1) (0.5, 0.4253, 0.25), and O(2) (0.5,0.1230, 0.0485) agree well with the experimental valuesCa (0, 0.2498, 0.25), Ir (0, 0, 0), O(1) (0.5, 0.4331, 0.25),and O(2) (0.5, 0.1296, 0.0553). The results presented inthis paper are for the relaxed atomic positions, but I alsoperformed calculations with experimental atomic posi-tions and came to the same physical conclusions. Thereare two formula units per primitive unit cell in the Cmcm structure. The Ir ions make a two-dimensional rectan-gular lattice in the ac plane (not shown) and the Ir-Olayer is separated by a layer of Ca atoms along the b axis. For the relaxed atomic positions, the O octahe-dra are tilted by an angle of 22 ◦ . The O octahedra areslightly compressed along the corner-shared O direction(from left to right in the left figure of Fig. 1) with abond-length ratio of 0.97. The Ir-O distances along thecorner-shared c axis and edge-shared a axis are 1.97 and2.02 ˚A, respectively. III. RESULTS
Let us first consider the non-spin-polarized (NSP) LDAcalculations. Even though they are inadequate to de-scribe the ground-state properties, these calculations givea decent description of the band structure of CaIrO thatprovide a playground for the interplay between SO cou-pling and U . The electron density of states (DOS) andband structure for this case are shown in Figs. 2 and3(a), respectively. Most of the bands along the b direc-tion (Γ- Y ) have low dispersion, which suggests that thephysics related to two dimensionality might be relevantin this system. There are 24 bands between − − − p character and thus derive from the p orbitals of the six O atoms in the unit cell. These bandsalso show Ir d character, which implies significant cova-lency between the O p and Ir d states as the unoccupiedIr d states above Fermi level also contain some admixtureof O p states. There is a very small gap of ∼ − d character. These are the Ir t g states that are formallyantibonding, and these bands correspondingly show someO p contribution. A gap of ∼ t g states from a group of four bands that have a mostly Ir d character, which are the Ir e g states. The Ir e g alsohave some O p character due to Ir d –O p covalency. TheIr 5 d states are quite delocalized and the edge-sharingcompressed IrO octahedra are rotated by 22 ◦ , and thisleads to some hybridization the between Ir t g and e g levels. The Ca and Ir s states are high above the Fermilevel, and within an ionic limit the electronic structure isconsistent with the ionic states Ir and O − , althoughthere is significant deviation from this because of Ir d –O p hybridization. The two Ir ions nominally have fiveelectrons each in their d orbitals. As a result, the six Ir t g bands are not fully filled, and the system is a metalwithin LDA with a t g hole on each Ir ion.The metallic state obtained within LDA due to incom-plete filling of Ir t g states is contrary to the experimentalevidence that indicates CaIrO is a Mott insulator. Thissuggests that spin-orbit coupling and/or on-site Coulombrepulsion play crucial roles in the electronic and magneticproperties of CaIrO . Let us now consider the effect ofSO coupling and U on the electronic structure of CaIrO .The NSP LDA, LDA+ U , LDA+SO, and LDA+SO+ U (with U = 2.75 eV) Ir t g bands are shown in Fig. 3.Let us first note that a value for U of 2.75 eV withoutthe spin-orbit coupling has very little effect on the bandstructure (I did the calculation with U up to 5 eV with-out getting an insulating state). This is not surprising asthe Ir t g manifold is spread over a bandwidth of ∼ U to openup a gap. However, turning on spin-orbit coupling makesa significant difference in the electronic structure. Thesystem is non-magnetic, so the bands are spin degener-ate as they are not exchange split. However, the spin-oribt coupling splits the manifold of six spin-degenerate -2-101 E ( e V ) -2-101 E ( e V ) -2-101 E ( e V ) (c) LDA+SO-2-101 Γ Y T Γ S R Γ Z E ( e V ) (d) LDA+SO+ U (a) LDA(b) LDA+ U FIG. 3: NSP LDA, LDA+ U , LDA+SO and LDA+SO+ U band structures, respectively from top to bottom, of CaIrO .The band structures are plotted along the path Γ (0,0,0) → Y(0, ,0) → T (0, , ) → Γ (0,0,0) → S ( , , 0) → R ( , , ) → Γ (0,0,0) → Z (0,0, ). Here, U = 2.75 eV is used. Thebands are exchange split only for the case of LDA+SO+ U . Ir t g bands into a lower-lying group of four and a higher-lying group of two spin-degenerate bands. In the limitof strong spin-orbit coupling, the lower and higher setsof bands within the t g manifold would correspond toeffective total angular momenta J eff of 3/2 and 1/2, re-spectively. This is similar to the case of Sr IrO wherethe spin-orbit coupling splits the Ir t g bands into a lowerlying quartet of J eff = 3/2 and a higher lying doublet of J eff = 1/2 bands. In the case of CaIrO , the J eff = 1/2bands are narrow with a width of ∼ J eff = 3/2 bands by ∼ d electrons in the unit cellcompletely fill the J eff = 3/2 bands, while the J eff = 1/2bands are only half filled. As a result, the system is stillmetallic.Even though on-site Coulomb repulsion U and spin-orbit coupling acting alone do not make the system aninsulator, it is likely that their combined effect can in-duce a Mott insulating state by splitting the narrow J eff = 1/2 bands. The LDA+SO+ U calculations with U =2.75 eV reveal that this scenario is realized in CaIrO .As shown in Fig. 3(d), the U in the presence of SO cou-pling makes only minor modifications to the J eff = 3/2bands. The J eff = 3/2 bands get exchange split and a TABLE I: The h ~L i and h ~S i expectation values computed over Ir muffin-tin spheres and the band gap E gap (eV) for some valuesof on-site Coulomb repulsion U (eV) and Hund’s coupling J (eV). The moments are in units of Bohr magneton.Site h ~L i h ~S i Ir(1) (0 . , . , − .
28) (0 . , . , − . U = 2 . J = 0 . ′ ) (0 . , . , .
28) (0 . , . , . E gap = 0 . . , . , − .
27) (0 . , . , − . U = 2 . J = 0 . ′ ) (0 . , . , .
27) (0 . , . , . E gap = 0 . . , . , − .
18) (0 . , . , − . U = 2, J = 0 . ′ ) (0 . , . , .
18) (0 . , . , . E gap = 0 . degeneracy at the point T (0 , . , .
5) is lifted, but oth-erwise the bandwidth and topology of the bands do notchange substantially. However, the half-filled J eff = 1/2bands, in addition to being exchange split by ∼ J eff = 1/2 LHB has a small bandwidth of ∼ ∼ U = 2.75 eV (the gap is ∼ U = 2 eV). This agrees well with a band gap of 0.34eV obtained experimentally. I also performed calcu-lations with U = 1.0, 1.5, 2.0, and 2.5 eV. The systemis metallic within LDA+SO+ U for U up to 1.5 eV, butit becomes an insulator by U = 2.0 eV. To see the effectof the Hund coupling J , I did LDA+SO+ U calculationswith U = 2.75 eV and J = 0.1, 0.2, and 0.3 eV. I findthat these values of Hund coupling J do not change thequalitative picture—the Ir t g levels are still split into J eff = 3/2 and 1/2 bands and the J eff = 1/2 bands are fur-ther split into fully occupied LHB and unoccupied UHB.The inclusion of Hund coupling mainly reduces the bandgap (for J = 0.3 eV, the band gap is 0.30 eV) and themagnetic moment.The LDA+SO+ U calculations give an antiferromag-netic ground state for CaIrO along the c axis with totalmoments aligning antiparallel along the c axis. The or-bital and spin moments are parallel to each other alongthe c axis and ferromagnetically canted along the b axis.The angular and spin expectation values computed overthe two Ir muffin-tin spheres Ir(1) (0.0, 0.0, 0.0) and Ir(1 ′ )(0.0, 0.0, 0.5) for different U and J values are given inTable I. For U = 2.75 eV, the total moment is 0.67 µ B /Irwith an orbital moment of 0.29 µ B and a spin momentof 0.38 µ B (= 2 |h ~S i| ). The canting angle is 10 ◦ , approxi-mately half the octahedral tilting angle of 22 ◦ and twicethe value of 4 ◦ reported in Ref. 12. The calculated val-ues differ considerably from what is expected for the ideal J eff = 1/2 state. In the ionic limit, one expects an orbitalmoment of 0.67 µ B and a spin moment of 0.33 µ B for a J eff = 1/2 state. In contrast, I obtain an orbital momentthat is lower than the spin moment. The reason for thisdeviation from the J eff = 1/2 may be the compressionand tilting of the IrO octahedra, in addition to the co-valency between Ir d and O p states. The compression ofthe IrO octahedra will quench the orbital moment as thedegeneracy between t g states are lifted. Also, the tilting of the IrO octahedra causes the e g bands to get mixedwith the t g states. These two effects should reduce theorbital moment but might enhance the spin contribution.It is interesting to note that Sr IrO also has distortionof the IrO octahedra with a bond-length ratio of 1.04and a tilting angle of 11 ◦ , and it has a calculated or-bital moment of 0.26 µ B and spin moment of 0.10 µ B . As the tilting angle in CaIrO is twice that of Sr IrO ,it might be reasonable to expect that CaIrO deviatesfurther from the ideal J eff = 1/2 state due to the mixingof the e g states. IV. CONCLUSIONS
In summary, the electronic structure and magneticproperties of CaIrO has been studied using first-principles calculations. The system is metallic withinthe LDA because the Ir t g states are incompletely filled.Modest values of on-site Coulomb repulsion alone havevery little effect on the LDA electronic structure as theIr t g states have a broad bandwidth. The introductionof spin-orbit coupling splits the Ir t g states into fullyfilled J eff = 3/2 bands and half-filled J eff = 1/2 bands.The half-filled bands have a small bandwidth of ∼ c axis with total moments aligningantiparallel along the c axis and canted along the b axis.For U = 2.75 eV, the total magnetic moment is 0.67 µ B with an orbital contribution of 0.28 µ B and a spincontribution of 0.38 µ B . These values differ from what isexpected for the ideal J eff = 1/2 state, and this deviationmight be explained by the mixing of J eff = 1/2 bands withIr e g bands due to the tilting of IrO octahedra. Therehas been great interest in finding different materials withunique magnetic properties. The results presented heregive strong support to the claim made by Ohgushi et al. in Ref. 12 that CaIrO has a unique spin-orbit integratedmagnetic ground state. V. ACKNOWLEDGEMENTS
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