Microscopic study of spin-orbit-induced Mott insulator in Ir oxides
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Microscopic study of spin-orbit-induced Mott insulator in Ir oxides
Hiroshi Watanabe , , ∗ Tomonori Shirakawa , , and Seiji Yunoki , , Computational Condensed Matter Physics Laboratory, RIKEN ASI, Wako, Saitama 351-0198, Japan CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan Quantum Systems Materials Science Research Team, RIKEN AICS, Kobe, Hyogo 650-0047, Japan (Dated: October 17, 2018)Motivated by recent experiments of a novel 5 d Mott insulator in Sr IrO , we have studied the two-dimensional three-orbital Hubbard model with a spin-orbit coupling λ . The variational Monte Carlomethod is used to obtain the ground state phase diagram with varying a on-site Coulomb interaction U as well as λ . It is found that the transition from a paramagnetic metal to an antiferromagnetic(AF) insulator occurs at a finite U = U MI , which is greatly reduced by a large λ , characteristic of5 d electrons, and leads to the “spin-orbit-induced” Mott insulator. It is also found that the Hund’scoupling induces the anisotropic spin exchange and stabilizes the in-plane AF order. We have furtherstudied the one-particle excitations using the variational cluster approximation, and revealed theinternal electronic structure of this novel Mott insulator. These findings are in agreement withexperimental observations on Sr IrO , and qualitatively different from those of extensively studied3 d Mott insulators.
PACS numbers: 71.30.+h, 75.25.Dk, 71.20.-b
Transition metal oxides have been one of the most fas-cinating classes of materials in recent years [1]. For thepast decays, tremendous amount of efforts have been de-voted to explore the nature of 3 d transition metal ox-ides where various exotic states and phenomena haveemerged such as high- T c cuprate superconductors, colos-sal magneto-resistant manganites, multiferroics, and var-ious magnetic orders. It has been established that thesestates and phenomena are caused by strong Coulombinteractions along with cooperative interactions of spin,charge, and orbital degrees of freedom, which are basi-cally separable in 3 d electrons [2].Very recently, 5 d transition metal oxides have attractedmuch attention as a candidate of a novel Mott insulator.Because of the extended nature of 5 d orbital, Coulombinteractions are expected to be smaller for 5 d electrons( ∼ d electrons ( ∼ λ is estimated to be con-siderably larger in 5 d ( ∼ d ( ∼ d transition metal oxides, inherentlyspin and orbital degrees of freedom are strongly entan-gled.One of such examples is the layered 5 d transition metaloxide Sr IrO . In Sr IrO , t g and e g orbitals are wellseparated by the large crystal field, and the lower t g orbital is filled with five electrons, ( t g ) . In spite ofthe large band width and small on-site Coulomb interac-tion U , Sr IrO is an antiferromagnetic insulator with aweak ferromagnetic moment [4, 5]. Neutron diffractionpatterns do not detect any superlattice structure that in-dicates charge order or charge density wave states [6]. Itis proposed that the strong SOC is responsible for theinsulating mechanism [7]. Indeed, the 4 d counterpart ofSr RhO , which has a larger U and a smaller λ than ∗ Electronic address: [email protected] Sr IrO , is metallic [8].The proposed picture of this “spin-orbit-induced”Mott insulator in Sr IrO is as follows. The SOCsplits the t g orbitals into J eff = 1 / J z eff = ± /
2, twofold degenerate) and J eff = 3 / J z eff = ± / , ± /
2, fourfold degenerate). Here J eff denotes the“effective” total angular momentum derived from thelarge SOC with the large crystal field [7]. When the SOCis large enough, the lower J eff = 3 / J eff = 1 / W eff ) is much smaller than the original one without theSOC ( W ) as shown in Fig. 1(a), and thus even small U can lead the system into a Mott insulator. This pictureof “ J eff = 1 / d Mott insulators where the effectof the SOC is only perturbative and thus the spin andorbital are essentially separate objects. Although thispicture is supported by first-principle calculations basedon the density functional theory [7, 11], a study treat-ing many-body effects beyond the mean-filed level is stilllacking. The main purpose of this paper is to understandthe nature of Mott insulator induced in spin-orbital en-tangled 5 d systems.Here, we study the novel Mott transition and mag-netic order induced by the SOC for Sr IrO . The groundstate properties of the three-orbital Hubbard model witha SOC term are investigated with the variational MonteCarlo (VMC) method. We show that the large SOCworks cooperatively with U and leads the system intoa novel Mott insulating state with an in-plane AF order.We also calculate the one-particle excitation spectrum us-ing the variational cluster approximation (VCA) [12] todiscuss the difference between the “spin-orbit-induced”Mott insulator and the 3 d Mott insulator.
FIG. 1: (a) Schematic picture of the splitting of density ofstates by the SOC. Non-interacting tight-binding energy band(b) without the SOC, (c) with the SOC ( λ/t = 1 . We take the three-orbital Hubbard model on a two-dimensional (2D) square lattice defined by the follow-ing Hamiltonian H = H kin + H SO + H I , where H kin = P k ασ ε α ( k ) c † k ασ c k ασ is the kinetic term, H SO = λ P i L i · S i is the SOC term with a coupling constant λ , and H I = U X i,α n iα ↑ n iα ↓ , + X i,α<β,σ [ U ′ n iασ n iβ − σ + ( U ′ − J ) n iασ n iβσ ]+ J X i,α<β,σ ( c † iα ↑ c † iβ ↓ c iα ↓ c iβ ↑ + c † iα ↑ c † iα ↓ c iβ ↓ c iβ ↑ + H . c . )(1)is the Coulomb interactions including intraorbital,interorbital, and spin-flip and pair-hopping interac-tions [13]. Here, σ indicates electronic spins and the in-dices α and β represent three t g orbitals, yz (1), zx (2),and xy (3).The kinetic and SOC terms can be combined ( H = H kin + H SO ) in the matrix form, H = X k σ (cid:16) c † k σ , c † k σ , c † k − σ (cid:17) × ε ( k ) i σλ/ − σλ/ − i σλ/ ε ( k ) i λ/ − σλ/ − i λ/ ε ( k ) c k σ c k σ c k − σ , (2)from which it is apparent that the SOC mixes the up- anddown-spin states, and new quasiparticles are obtainedsimply by diagonalizing H . The new quasiparticles arecharacterized by the pseudo-orbital α and pseudo-spin σ with a creation (annihilation) operator a † k ασ ( a k ασ ). Inthe atomic limit ( ε ( k ) = ε ( k ) = ε ( k ) = 0), sixfolddegenerate states are split into twofold degenerate J eff =1 / J eff = 3 / IrO , and also in Sr RhO , thereis a large hybridization between the xy and x − y orbitals originated from the tilting of IrO octahedra.Because of this hybridization, the xy orbital is pusheddown below the Fermi energy [10]. To take into accountthe effect of this hybridization, we introduce the next-nearest and third-nearest hopping integrals in the xy or-bital. The chemical potential µ is also introduced totake account of both the hybridization and tetragonalsplitting [11]. The resulting energy dispersion of the xy orbital is ε ( k ) = − t (cos k x +cos k y ) − t cos k x cos k y − t (cos 2 k x + cos 2 k y ) + µ . On the contrary, the yz and zx orbitals have almost one-dimensional character, ε ( k ) = − t cos k x − t cos k y and ε ( k ) = − t cos k x − t cos k y ( t ≫ t ). Using this form, the band dispersionof Sr IrO calculated by the LDA+SO (spin-orbit) [7] iswell reproduced by choosing the tight-binding parame-ters with ( t , t , t , t , t , µ , λ )=(0.36, 0.18, 0.09, 0.37,0.06, -0.36, 0.37) eV, as shown in Fig. 1(c). In thefollowing, we set t = t as an energy unit. To studythe Mott transition and the role of the SOC, we onlychange the value of λ and fix the other tight-binding pa-rameters. This assumption is justified by the fact thatthe LDA+SO band dispersion for the 4 d counterpart ofmetallic Sr RhO [14] is well reproduced by choosing λ/t ∼ . | Ψ i = P J c P G | Φ i . (3) | Φ i is the one-body part obtained by diagonalizing H with renormalized (variational) tight-binding pa-rameters, ¯ H (˜ t i , ˜ µ , ˜ λ αβ ). Note that by the effect ofCoulomb interaction and tetragonal splitting, the “ef-fective” coupling constant of the SOC has orbital de-pendence: λ → ˜ λ αβ . To consider magnetically orderedstates, a term with the different magnetic order param-eter is added to ¯ H . Here, we study the out-of-planeAF order (along z axis) and in-plane AF order (along x axis), described by ∆ z P e i Q · r i ( a † iα ↑ a iα ↑ − a † iα ↓ a iα ↓ )and ∆ x P e i Q · r i ( a † iα ↑ a iα ↓ + a † iα ↓ a iα ↑ ), respectively. Here Q = ( π, π ) is an ordering vector. Note that the staggeredfield is applied to newly formed quasiparticles in the realspace ( a iασ , a † iασ ) and σ represents the pseudo-spin, notthe original spin. The matrix to be diagonalized at each k becomes 12 ×
12 for the AF state (not shown).The operator P G = Q i,γ [1 − (1 − g γ ) | γ i h γ | i ] inEq. (3) is a Gutzwiller factor extended for the three-orbital system [15]. Here, i is a site index and γ runsover possible electron configurations at each site. Forthe three-orbital system, there are 4 = 64 electron con-figurations, namely, | i = | i , | i = | ↑i , · · · , | i = |↑↓ ↑↓ ↑↓i . We control the weight of each elec-tron configuration by varying g γ from 0 to 1. The setof { g γ } is a variational parameter and optimized so as FIG. 2: (a) Ground state phase diagram of the three-orbitalHubbard model in a 2D 10 ×
10 square lattice with n = 5, U ′ /U = 0 .
7, and
J/U = 0 .
15. PM denotes the paramagneticmetal and x -AFI denotes the AF insulator with an in-plane(along x axis) magnetic moment. The solid line indicatesthe first-order phase boundary U MI . The expected locationsof Sr IrO (Ir) and Sr RhO (Rh) are also indicated in thephase diagram. (b) J/U dependence of U MI for different λ . to give the lowest energy. In this study, we classify thepossible 64 patterns into 12 groups by the Coulomb in-teraction energy, and g γ ’s in the same group are set tobe the same.The remaining term P J c = exp h − P i = j v ij n i n j i inEq. (3) is a charge Jastrow factor that controls the long-range charge correlation. The long-range charge correla-tion is known to be important for describing Mott tran-sition [16]. In this study, we assume that v ij dependsonly on the distances, v ij = v ( | r i − r j | ). For instance,we consider up to 19th-neighbor correlation for a 10 × v ij is 19.The ground state energy and other physical quan-tities are calculated with the VMC method. Thevariational parameters mentioned above are simultane-ously optimized by using the stochastic reconfigurationmethod [17]. This method makes it possible to optimizemany parameters efficiently and stably.Fig. 2(a) shows the obtained ground state phase di-agram for a 2D 10 ×
10 square lattice with the electrondensity n = 5, corresponding to ( t g ) , and the Coulombinteractions U ′ /U = 0 . J/U = 0 .
15. The transitionfrom the paramagnetic metal to the AF insulator occursat U = U MI , which depends sensitively on the value of λ .This metal-insulator transition (MIT) is found to be first-order, indicating that the nesting of the Fermi surface isnot perfect and the Coulomb interaction is essential indriving the transition. The insulating mechanism is un-derstood as follows. When the staggered (AF) field isapplied, the degeneracy along the edge of the AF Bril-louin zone (along M-X in Fig. 1) is lifted and the highestband is split off by the AF gap [Fig. 1(d)]. As can beseen in Figs. 1(b) and 1(c), the SOC lifts upward the twobranches of the energy bands from the rest, which makesit easier to fully open the AF gap once the Coulomb in-teractions are considered. Namely, the larger the SOC is, the easier the system becomes the AF insulator. In-deed, Fig. 2(a) shows that U MI monotonically decreaseswith increasing λ . Moreover, we found that the effectivecoupling constant ˜ λ αβ increases with U , indicating that U and λ work cooperatively for insulating.We expect that this insulating mechanism found aboveis applied for Sr IrO . Because the Coulomb interaction U Ir is much smaller than the band width, the metallicstate is naively expected for Sr IrO . However, Sr IrO is experimentally found insulating with a canted AF or-der [9]. This counter-intuitive observation can be ex-plained if the SOC in Sr IrO , λ Ir , is large enough toreduce U MI smaller than U Ir . Indeed, λ Ir is estimated aslarge as 0.4–0.5 eV, which is much larger than the val-ues for 3 d and 4 d electron systems. We consider that U Ir > U MI is satisfied and the “spin-orbit-induced” Mottinsulator is realized in Sr IrO . On the other hand, inSr RhO , we consider that U Rh < U MI is satisfied andthus the metallic state is realized. The expected loca-tions of Sr IrO and Sr RhO in the phase diagram areindicated in Fig. 2(a), where both of them are locatednear the MIT point. Note that the observed insulatinggap of Sr IrO is very small ( ∼ ∼ RhO leads to the MIT [18],indicating that the system is located not far from theMIT point.The J/U dependence of U MI is also investigated andthe results are shown in Fig. 2(b). It is clearly seen inFig. 2(b) that U MI monotonically increases with increas-ing J/U , indicating that the Hund’s coupling is unfa-vorable for the spin-orbit-induced Mott insulator. Thisbehavior is naturally understood since the Hund’s cou-pling competes with the SOC and works destructivelyfor the formation of the quasiparticles originated fromthe SOC. It is, however, found that the overall shape ofthe phase diagram does not change qualitatively with in-creasing
J/U except for U MI shifting to a larger value.For J/U =0.15–0.25, we estimate U MI =1.2–1.6 eV forSr IrO and 1.6–2.4 eV for Sr RhO .In the insulating region, we found that the in-planeAF order ( x -AFI) is more stable than the out-of-planeAF order ( z -AFI). If there is no Hund’s coupling [ J = 0in Eq. (1)], the rotational symmetry in pseudo-spin spaceis preserved and z -AFI and x -AFI are energetically de-generate. However, the introduction of Hund’s couplinginduces the anisotropy and the in-plane AF order is morefavored than the out-of-plane AF order. This result isconsistent with the study of an effective strong SOC spinmodel by Jackeli and Khaliullin [19]. The magnetic x-ray diffraction experiment [9] also supports the in-planemagnetism in Sr IrO .We have also estimated the local magnetic momentas large as 0 . . µ B for the parameters appropriatefor Sr IrO . This value is comparable to the results ofmagnetic susceptibility measurements [5, 20] and muchsmaller than the atomic value of 1 µ B in the strong-SOC -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 ω (eV) ρ ( ω ) All J eff =1/2 J eff =3/2, | J eff |=1/2 J eff =3/2, | J eff |=3/2(e) zz FIG. 3: (color online). (a) One-particle excitation spectrumfor the Mott insulating state. Spectra projected onto (b) J eff = 1 /
2, (c) J eff = 3 / , J z eff = ± /
2, and (d) J eff =3 / , J z eff = ± / ρ ( ω ) arealso shown. The parameters used are U =1.44 eV, U ′ =1.008eV, J =0.216 eV, and λ =0.432 eV, and ω = 0 corresponds tothe Fermi energy. limit. This large reduction is due to the large itinerancyof 5 d electrons. If we assume that the magnetic momentexactly follows the tilting of IrO octahedra observed ex-perimentally ( ∼ ◦ ) [4], the expected ferromagnetic mo-ment is 0.05–0.07 µ B , which is comparable to the experi-mental estimation [4, 5, 20].Finally, to explore the internal electronic structure inthe insulating state, the one-particle excitation spectrumis calculated using the VCA [21]. The results are shownin Fig. 3 for a set of parameters appropriate for Sr IrO .As seen in Fig. 3(a), the Fermi energy is located insidethe gap, and thus the state is insulating. The validity ofthe physical picture of “ J eff = 1 / J eff = 1 / J eff = 3 / J eff = 1 / J eff = 1 / J eff = 3 / J eff = 3 / J eff = 1 / J eff = 1 / J eff = 3 / d electrons.Fig. 3(e) shows the total density of states calculatedusing the VCA. As indicated by the arrow in Fig. 3(e),the optical excitation is expected mainly at around 1 eVoriginating from ∼ − ∼ J eff = 1 / J eff = 1 / ∼ U MI as well as the nature of the resulting Mottinsulating phase in Sr IrO . This is because the 5 d Mottinsulator, not like for 3 d systems, is well described bythe novel quantum number J eff , which is due to the largeSOC along with the large crystal field, a generic featurefor the 5 d transition metal oxides. Therefore, we expectthat not only U but also λ are crucial factors to be con-sidered in describing MIT for the 5 d systems in general.Moreover, because of the orders of magnitude different λ ,other novel quantum phenomena, such as the anomalousmetallic state, multipole order, and unconventional su-perconductivity that are not observed in the 3 d systems,are expected to emerge in the 5 d systems.The authors thank Y. Yanase, M. Taguchi, and A.Rusydi for useful discussions. A part of this work issupported by CREST (JST). The computation in thiswork has been done using the RIKEN Cluster of Clusters(RICC) facility and the facilities of the SupercomputerCenter, Institute for Solid State Physics, University ofTokyo. [1] M. Imada et al ., Rev. Mod. Phys. , 1039 (1998).[2] Y. Tokura and N. Nagaosa, Science , 462 (2000).[3] M. Taguchi, private communications.[4] M. K. Crawford et al ., Phys. Rev. B , 9198 (1994). [5] G. Cao et al ., Phys. Rev. B , R11039 (1998).[6] Q. Huang et al ., J. Solid State Chem. , 355 (1994).[7] B. J. Kim et al ., Phys. Rev. Lett. , 076402 (2008).[8] F. Baumberger et al ., Phys. Rev. Lett. , 246402 (2006). [9] B. J. Kim et al ., Science , 1329 (2009).[10] It is a quite different band structure from the supercon-ducting counterpart of Sr RuO , where the xy -originated γ band greatly contributes the superconducting behavior.[11] H. Jin et al ., Phys. Rev. B , 075112 (2009).[12] M. Potthoff et al ., Phys. Rev. Lett. U = U ′ + 2 J is assumed. See, e.g., J. Kanamori, Prog.Theor. Phys. , 275 (1963).[14] G.-Q. Liu et al ., Phys. Rev. Lett. , 026408 (2008).[15] J. B¨unemann et al ., Phys. Rev. B , 6896 (1998).[16] M. Capello et al ., Phys. Rev. Lett. , 026406 (2005).[17] S. Sorella, Phys. Rev. B , 024512 (2001). [18] H. Yoshida, private communication.[19] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. ,017205 (2009).[20] N. S. Kini et al ., J. Phys.: Condens. Matter , 8205(2006).[21] A √ ×√×√