First-principles study of the electronic structure and magnetism of CaIrO 3 Alaska Subedi Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany I study the electronic structure and magnetism of postperovskite CaIrO3 using first-principles calculations. The density functional calculations within the local density approximation without the combined effect of spin-orbit coupling and on-site Coulomb repulsion show the system to be metallic, which is in disagreement with the recent experimental evidences that show CaIrO3 to be an antiferromagnetic Mott insulator in the J = 1/2 state. However, when spin-orbit coupling is eff takeninto account, theIrt2g bandssplit into fully filled Jeff = 3/2 bandsand half-filled Jeff = 1/2 2 bands. I find that spin-orbit coupling along with a modest on-site Coulomb repulsion opens a gap 1 leading to a Mott insulating state. The ordering is antiferromagnetic along the c axis with total 0 momentsaligned antiparallel along thec axis and canted along theb axis. 2 PACSnumbers: 71.30.+h,75.25.Dk,75.50.Ee n a J I. INTRODUCTION also found that spin-orbit coupling plays an important 7 role in the electronic properties even for a 4d5 system 2 such as Sr RhO .9,10 2 4 Transition-metal oxides (TMOs) in perovskite and ] It has been suggested that systems in the Jeff = 1/2 l related structures exhibit myriad interesting proper- e state, depending on bond geometry, lead to interest- ties. These include unconventional superconductivity - ing varieties of low-energy Hamiltonians, including the r in cuprates,1 colossalmagnetoresistancein manganites,2 t isotropic Heisenberg model and the highly anisotropic s and ferroelectricity in Pb(Zr,Ti)O .3 In these materials, . 3 quantumcompassorKitaevmodelsrelevantforquantum at thetransition-metalionissituatedinsideanoxygenocta- computing.11 Therefore, it is important to investigate hedralcage,whichmaybe arrangedinacorner-oredge- m materials that exhibit the J = 1/2 state in different eff sharedmanner. TheinterestingpropertiesofTMOsarise structuresinordertostudytheeffectofdifferentlocalen- - d becauseofthecompetitionbetweenthecrystal-fieldsplit- vironments and spin-exchange pathways. Recently, Oh- n ting (which arises because of covalency between transi- gushi et al. reported resonant x-ray diffraction study of o tionmetaldandoxygenpstates),on-siteCoulombrepul- CaIrO that indicates this material also exhibits a Mott [c sionU,Hund’scoupling,spin-orbit(SO)couplingdueto insulat3ing Jeff = 1/2 state.12 CaIrO3 exists in the post- orbital degeneracy (leading to unquenched angular mo- perovskite structure with space group Cmcm as shown 2 ment in the ground state), and different spin-exchange in Fig. 1. The Ir4+ (5d5) ions are situated inside the O v pathways. octahedra,buttheseoctahedrashareanedgealongthec 6 8 In this regard, the discovery of a spin-orbital Mott axis,unlikethecaseofSr2IrO4. Thus,CaIrO3 isanother 3 state in Sr2IrO4 by Kim et al.4,5 is significant because ideal material to investigate the interplay between spin- 0 it enables us to study the case where spin-orbit cou- orbit coupling and on-site Coulomb repulsion that may . pling and its interplay with the Coulomb repulsion is an help in understanding the unique properties that might 0 1 important ingredient in determining the electronic and be exhibited by Jeff =1/2 systems. 1 magnetic properties of the system. Sr2IrO4 exists in a Tsuchiya et al. have reported first-principles density 1 layeredperovskitestructure. The Ir4+ (5d5) ions are sit- functionalcalculationsthatshowthismaterialisametal : uatedinsidecorner-sharedOoctahedralcages,whichare within the local density approximation.13 This is con- v i themselves arranged in a square lattice in the ab plane. trary to the experimental evidence that shows this ma- X As there are an odd number of electrons per formula terial is a Mott insulator that undergoes an antiferro- r unit, one might expect this material to be a metal in magnetic transition at T = 115 K.14 A recent reso- a N the band picture. However, Sr IrO is experimentally a nant x-ray diffraction study shows that the ordering is 2 4 cantedantiferromagneticinsulator.5Asexplainedlucidly of stripe-type antiferromagnetism along the c axis, with by Kim et al., a state with an effective total angular total moments aligning parallel along the a axis and momentum J = 1/2 that has a complex wave func- antiparallel along the c axis.12 The inverse susceptibil- eff tion is realized in Sr IrO , which arises due to the com- ity 1/χ deviates from the linear behavior at a temper- 2 4 bined influence of strong spin-orbit coupling and mod- ature ∼350 K that is considerably higher than T and N erate on-site Coulomb repulsion.4 A similar conclusion a Curie-Weiss fit to χ above 400 K gives a Curie-Weiss on Sr IrO has been reached through the study of a temperature of 3900 K.14 This indicates that the anti- 2 4 three-orbital Hubbard model with spin-orbit coupling6 ferromagnetic correlations arise much before the anti- and combined density functional theory and dynamical ferromagnetic transition, and magnetic ordering is sup- meanfieldtheory(LDA+DMFT)calculations,7although pressed by low dimensionality or competing ordering in- Arita et al. suggest that Sr IrO is a Slater insulator teractions. Jang et al. have studied the electronic struc- 2 4 basedontheirLDA+DMFT study.8 Interestingly,itwas tureofmeta-stableperovskiteCa1−xSrxIrO3 (x=0,0.5, 2 14 total Ir d 12 O(1) p O(2) p 10 8 E) N( 6 4 2 FIG.1: (Coloronline)CrystalstructureofCaIrO3. Thelarge 0 (cyan)ballsareCa,small (red)ballsareO,andtheIratoms -8 -6 -4 -2 0 2 4 reside inside the (brown) octahedra. E (eV) FIG. 2: (Color online) Non-spin-polarized LDA DOS of and 1) thin films using transport measurements, optical CaIrO3(instates/eV).Theprojectionsareontotherespective spectroscopy, and pseudopotential-based first-principles muffin-tinspheresandareonlyindicativeofthecontribution calculations.15 They find that perovskite CaIrO thin to thetotal DOS.The Fermi energy is at 0 eV. 3 films are semimetallic andnearthe metal-insulatortran- sition. Their calculations with spin-orbit coupling and on-site Coulomb repulsion U found that the spin-orbit the electronic and magnetic properties of CaIrO us- 3 coupling splits the Ir t2g states into Jeff = 3/2 and 1/2 ing density functional calculations. The calculations states,andU =2.0eVfurthersplitstheJeff=1/2states, were performed within the local density approximation althoughthevalenceandconductionbandsstilltouchthe (LDA) using the general full-potential linearized aug- Fermi level, resulting in a semimetallic state. mented plane-wave method as implemented in the ELK The experimental evidences that have so far been ac- software package.16 Muffin-tin radii of 2.2, 2.0, and 1.6 cumulated suggest that calculations that include the ef- a.u. forCa,Ir,andO,respectively,wereused. A8×8×8 fectofspin-orbitcouplingandon-siteCoulombrepulsion k-pointgridwasusedto performthe Brillouinzoneinte- wouldbehelpfulinclarifyingtheelectronicandmagnetic gration,andtheconvergenceofmomentswascheckedon properties of CaIrO3. In this paper, I report the results a 10×10×10grid. The effect ofspin-orbitcouplingwas of density functional calculations that show CaIrO3 is treated using a second-variational scheme, and the fully in a Mott insulating state that is induced by the com- localized limit17 is used to take into account the double bined effect of spin-orbit coupling and on-site Coulomb countinginLDA+U calculations. Avalueforthe on-site repulsion. This state arises out of spin-orbit split Ir t2g CoulombrepulsionU =2.75eV(whichgivesabandgap bands that get separated into lower lying fully filled and close to the experimental value) was used unless other- higherlyinghalf-filledbandsthathaveeffectivetotalan- wise mentioned. gular momenta J = 3/2 and 1/2, respectively, in the eff I used the experimental lattice parameters a = 3.145 strong spin-orbit coupling limit. The half-filled Jeff = ˚A,b=9.855˚A,andc=7.293˚A,18butrelaxedtheatomic 1/2bandsarenarrow,soevenamodeston-siteCoulomb positions. ThecalculatedatomicpositionsCa(0,0.2498, repulsioninducesaMottinsulatingstatethatistopolog- 0.25),Ir(0,0,0),O(1)(0.5,0.4253,0.25),andO(2)(0.5, ically different from the metallic state given by the lo- 0.1230, 0.0485) agree well with the experimental values cal density approximation, without taking into account Ca(0,0.2498,0.25),Ir(0,0,0), O(1)(0.5,0.4331,0.25), the spin-orbit coupling and on-site Coulomb repulsion. and O(2) (0.5, 0.1296, 0.0553). The results presented in This is aMottinsulating state inthe sense thatasingle- thispaperarefortherelaxedatomicpositions,butIalso particle theory such as the density functional theory im- performed calculations with experimental atomic posi- plemented using Kohn-Sham formalism cannot explain tions and came to the same physical conclusions. There the insulating state, and an explicit treatment of on-site aretwoformulaunitsperprimitiveunitcellintheCmcm Coulomb repulsion is needed. The Mott insulating state structure. The Ir4+ ionsmakeatwo-dimensionalrectan- thus obtained is antiferromagnetically ordered along the gular lattice in the ac plane (not shown) and the Ir-O c axis with total moments aligned antiparallel along the layer is separated by a layer of Ca atoms along the b c axis and canted along the b axis. axis. For the relaxed atomic positions, the O octahe- ◦ dra are tilted by an angle of 22 . The O octahedra are slightly compressed along the corner-shared O direction II. APPROACH (from left to right in the left figure of Fig. 1) with a bond-length ratio of 0.97. The Ir-O distances along the The purpose of this paper is to elucidate the role of corner-sharedc axis and edge-shareda axis are 1.97 and spin-orbit coupling and on-site Coulomb repulsion on 2.02 ˚A, respectively. 3 III. RESULTS 1 (a)LDA V) 0 Letusfirstconsiderthenon-spin-polarized(NSP)LDA e calculations. Even though they are inadequate to de- E( -1 scribetheground-stateproperties,thesecalculationsgive -2 adecentdescriptionofthebandstructureofCaIrO that 3 provide a playground for the interplay between SO cou- 1 (b)LDA+U pling and U. The electron density of states (DOS) and V) 0 band structure for this case are shown in Figs. 2 and e 3(a), respectively. Most of the bands along the b direc- E( -1 tion (Γ-Y) have low dispersion, which suggests that the -2 physics related to two dimensionality might be relevant inthissystem. Thereare24bandsbetween−9.0and1.0 1 (c)LDA+SO eV.The18bandsbetween−9.0and−2.3eV(notshown) have a dominant O p characterand thus derive from the V) 0 e porbitalsofthesixOatomsintheunitcell. Thesebands E( -1 also show Ir d character, which implies significant cova- -2 lency between the O p and Ir d states as the unoccupied IrdstatesaboveFermilevelalsocontainsomeadmixture 1 (d)LDA+SO+U of O p states. There is a very small gap of ∼0.05 eV at −2.36eV,beyondwhichliesixbandswithadominantIr V) 0 d character. These are the Ir t2g states that areformally E(e -1 antibonding,andthesebandscorrespondinglyshowsome O p contribution. A gap of ∼1.5 eV separates these t -2 2g states from a group of four bands that have a mostly Ir Γ Y T Γ S R Γ Z d character, which are the Ir e states. The Ir e also g g have some O p characterdue to Ir d–O p covalency. The FIG. 3: NSP LDA, LDA+U, LDA+SO and LDA+SO+U Ir 5d states are quite delocalized and the edge-sharing bandstructures,respectivelyfrom toptobottom,of CaIrO3. ◦ compressed IrO6 octahedra are rotated by 22 , and this ThebandstructuresareplottedalongthepathΓ(0,0,0)→Y leads to some hybridization the between Ir t and e (0,1,0) → T(0, 1, 1) → Γ (0,0,0) → S(1, 1,0) → R (1, 1, 2g g 2 2 2 4 4 4 4 levels. The Ca and Ir s states are high above the Fermi 1) →Γ (0,0,0) → Z(0,0,1). Here,U = 2.75 eV isused. The 2 2 level,andwithinanioniclimittheelectronicstructureis bandsare exchange split only for thecase of LDA+SO+U. consistent with the ionic states Ir4+ and O2−, although there is significantdeviation from this because of Ir d–O p hybridization. The two Ir4+ ions nominally have five Irt bandsintoalower-lyinggroupoffourandahigher- 2g electrons each in their d orbitals. As a result, the six Ir lying group of two spin-degenerate bands. In the limit t2g bands are not fully filled, and the system is a metal of strong spin-orbit coupling, the lower and higher sets within LDA with a t hole on each Ir4+ ion. of bands within the t manifold would correspond to 2g 2g ThemetallicstateobtainedwithinLDAduetoincom- effective total angular momenta Jeff of 3/2 and 1/2, re- pletefillingofIrt2g statesiscontrarytotheexperimental spectively. This is similar to the case of Sr2IrO4 where evidence that indicates CaIrO3 is a Mott insulator. This thespin-orbitcouplingsplitstheIrt2g bandsintoalower suggeststhatspin-orbitcouplingand/oron-siteCoulomb lying quartet of Jeff = 3/2 and a higher lying doublet of repulsionplaycrucialrolesintheelectronicandmagnetic Jeff = 1/2 bands.4 In the case of CaIrO3, the Jeff = 1/2 properties of CaIrO . Let us now consider the effect of bands are narrowwith a width of ∼1 eV. These are sep- 3 SOcouplingandU ontheelectronicstructureofCaIrO3. aratedfrom the Jeff = 3/2 bands by ∼0.15 eV, although The NSP LDA, LDA+U, LDA+SO, and LDA+SO+U the gapisindirect. The tenIrdelectronsinthe unitcell (with U = 2.75 eV) Ir t2g bands are shown in Fig. 3. completely fill the Jeff = 3/2 bands, while the Jeff = 1/2 Let us first note that a value for U of 2.75 eV without bands are only half filled. As a result, the system is still the spin-orbit coupling has very little effect on the band metallic. structure (I did the calculation with U up to 5 eV with- Even though on-site Coulomb repulsion U and spin- outgettinganinsulatingstate). Thisisnotsurprisingas orbit coupling acting alone do not make the system an the Ir t manifold is spread over a bandwidth of ∼2.8 insulator, it is likely that their combined effect can in- 2g eV,anditwouldrequireasubstantiallylargerU toopen duce a Mott insulating state by splitting the narrow J eff upagap. However,turningonspin-orbitcouplingmakes = 1/2 bands. The LDA+SO+U calculations with U = a significant difference in the electronic structure. The 2.75 eV reveal that this scenario is realized in CaIrO . 3 system is non-magnetic, so the bands are spin degener- As shown in Fig. 3(d), the U in the presence of SO cou- ate as they are not exchange split. However, the spin- pling makes only minor modifications to the J = 3/2 eff oribt coupling splits the manifold of six spin-degenerate bands. The J = 3/2 bands get exchange split and a eff 4 TABLEI:ThehL~iandhS~iexpectationvaluescomputedoverIrmuffin-tinspheresandthebandgapEgap (eV)forsomevalues of on-site Coulomb repulsion U (eV) and Hund’scoupling J (eV). Themoments are in units of Bohr magneton. Site hL~i hS~i Ir(1) (0.00,0.06,−0.28) (0.00,0.03,−0.19) U =2.75, J =0.0 ′ Ir(1) (0.00,0.06,0.28) (0.00,0.03,0.19) Egap =0.33 Ir(1) (0.00,0.06,−0.27) (0.00,0.03,−0.18) U =2.75, J =0.3 ′ Ir(1) (0.00,0.06,0.27) (0.00,0.03,0.18) Egap =0.30 Ir(1) (0.00,0.04,−0.18) (0.00,0.02,−0.13) U =2, J =0.0 Ir(1′) (0.00,0.04,0.18) (0.00,0.02,0.13) Egap =0.03 degeneracy at the point T (0,0.5,0.5) is lifted, but oth- of the IrO octahedra causes the e bands to get mixed 6 g erwise the bandwidth and topology of the bands do not with the t states. These two effects should reduce the 2g change substantially. However, the half-filled J = 1/2 orbitalmomentbutmightenhancethespincontribution. eff bands, in addition to being exchange split by ∼0.1 eV, It is interesting to note that Sr IrO also has distortion 2 4 are split into the upper (UHB) and lower (LHB) Hub- of the IrO octahedra with a bond-length ratio of 1.04 6 bard bands, yielding a Mott insulating state. The fully and a tilting angle of 11◦,20 and it has a calculated or- occupied J = 1/2 LHB has a small bandwidth of ∼0.5 bital moment of 0.26 µ and spin moment of 0.10 µ .4 eff B B eV and is separated from the unoccupied UHB by a gap As the tilting angle in CaIrO is twice that of Sr IrO , 3 2 4 of ∼0.33 eV for U = 2.75 eV (the gap is ∼0.03 eV for it might be reasonable to expect that CaIrO deviates 3 U = 2 eV). This agrees well with a band gap of 0.34 further fromthe ideal J = 1/2 state due to the mixing eff eV obtained experimentally.14,19 I also performed calcu- of the e states. g lations with U = 1.0, 1.5, 2.0, and 2.5 eV. The system is metallic within LDA+SO+U for U up to 1.5 eV, but it becomes an insulator by U = 2.0 eV. To see the effect of the Hund coupling J, I did LDA+SO+U calculations IV. CONCLUSIONS with U = 2.75 eV and J = 0.1, 0.2, and 0.3 eV. I find that these values of Hund coupling J do not change the qualitativepicture—theIrt levelsarestillsplitintoJ 2g eff In summary, the electronic structure and magnetic = 3/2 and 1/2 bands and the J = 1/2 bands are fur- eff properties of CaIrO has been studied using first- 3 ther splitintofully occupiedLHB andunoccupiedUHB. principles calculations. The system is metallic within The inclusionofHundcoupling mainlyreducesthe band theLDAbecausetheIrt statesareincompletelyfilled. 2g gap (for J = 0.3 eV, the band gap is 0.30 eV) and the Modest values of on-site Coulomb repulsion alone have magnetic moment. very little effect on the LDA electronic structure as the The LDA+SO+U calculations give an antiferromag- Ir t states have a broad bandwidth. The introduction 2g netic groundstate for CaIrO3 alongthe c axis with total of spin-orbit coupling splits the Ir t2g states into fully moments aligning antiparallel along the c axis. The or- filled J = 3/2 bands and half-filled J = 1/2 bands. eff eff bital and spin moments are parallel to each other along Thehalf-filledbandshaveasmallbandwidthof∼1.0eV, the c axis and ferromagneticallycanted along the b axis. which is split into fully filled lower and unfilled upper The angular and spin expectation values computed over Hubbard bands by a modest value of on-site Coulomb ′ thetwoIrmuffin-tinspheresIr(1)(0.0,0.0,0.0)andIr(1) repulsion. This topologically non-trivial modification of (0.0, 0.0, 0.5) for different U and J values are given in theLDAelectronicstructureduetothecombinedeffects TableI.ForU =2.75eV,thetotalmomentis0.67µB/Ir of spin-orbit coupling and on-site Coulomb repulsion re- with an orbital moment of 0.29 µB and a spin moment sults in a Mott insulating state that is antiferromagnet- of0.38µ (=2|hS~i|). The cantingangle is 10◦, approxi- ically ordered along c axis with total moments aligning B ◦ mately half the octahedraltilting angle of 22 and twice antiparallelalong the c axis and canted along the b axis. ◦ the value of 4 reported in Ref. 12. The calculated val- For U = 2.75 eV, the total magnetic moment is 0.67 uesdifferconsiderablyfromwhatisexpectedfortheideal µ with an orbital contribution of 0.28 µ and a spin B B J =1/2state. Intheioniclimit,oneexpectsanorbital contributionof0.38µ . Thesevaluesdiffer fromwhatis eff B moment of 0.67 µ and a spin moment of 0.33 µ for a expectedfortheidealJ =1/2state,andthisdeviation B B eff J =1/2state.4 Incontrast,Iobtainanorbitalmoment mightbeexplainedbythemixingofJ =1/2bandswith eff eff that is lower than the spin moment. The reason for this Ir e bands due to the tilting of IrO octahedra. There g 6 deviation from the J = 1/2 may be the compression hasbeengreatinterestinfindingdifferentmaterialswith eff and tilting of the IrO octahedra, in addition to the co- unique magnetic properties. The results presented here 6 valencybetweenIrdandOpstates. The compressionof give strong support to the claim made by Ohgushi et al. theIrO octahedrawillquenchtheorbitalmomentasthe inRef.12thatCaIrO hasauniquespin-orbitintegrated 6 3 degeneracybetweent statesarelifted. Also,thetilting magnetic ground state. 2g 5 V. ACKNOWLEDGEMENTS L.Boeriforencouragement,helpfuldiscussions,andsug- gestions in improving the manuscript, and to A. Avella IamgratefultoG.Jackeli,D.I.Khomskii,D.J.Singh, for critical reading of the manuscript. and I. I. Mazin for helpful discussions. I am thankful to 1 J.G.BednorzandK.A.Muller,Z.Phys.B64,189(1986). 12 K. Ohgushi, J.-I. Yamaura, H. Ohsumi, K. Sugimoto, S. 2 R. von Helmolt, J. Wecker, B. Holzapfel, L. Schultz, and Takeshita, A. Tokuda, H. Takagi, M. 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