Is the ground state of 5d4 double-perovskite Iridate Ba2YIrO6 magnetic or nonmagnetic?

Accepted Manuscript Is the ground state of 5 d 4 double-perovskite iridate Ba6YIrO6 magnetic or nonmagnetic? Hoshin Gong, Kyoo Kim, Beom Hyun Kim, Bongjae Kim, Junwon Kim, B.I. Min PII: DOI: Reference:

S0304-8853(17)33233-X https://doi.org/10.1016/j.jmmm.2018.01.051 MAGMA 63629

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

12 October 2017 16 January 2018 18 January 2018

Please cite this article as: H. Gong, K. Kim, B.H. Kim, B. Kim, J. Kim, B.I. Min, Is the ground state of 5 d 4 doubleperovskite iridate Ba6YIrO6 magnetic or nonmagnetic?, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/j.jmmm.2018.01.051

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Is the ground state of 5d4 double-perovskite iridate Ba2 YIrO6 magnetic or nonmagnetic? Hoshin Gong,1 Kyoo Kim,2 Beom Hyun Kim,3, 4 Bongjae Kim,1 Junwon Kim,1 and B. I. Min1, ∗ 1

4

Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea 2 M P P C CP M , Pohang University of Science and Technology, Pohang, 37673, Korea 3 Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan Interdisciplinary Theoretical Science (iTHES) Research Group, RIKEN, Wako, Saitama 351-0198, Japan We have investigated electronic structures and magnetic properties of double perovskite Iridate Ba2 YIrO6 with 5d4 configuration, employing the exact diagonalization method for multi-site clusters. We have considered a many-body Hamiltonian for all d states (eg and t2g ) including all relevant physical parameters such as the Coulomb correlation, spin-orbit coupling, crystal-field effect, and Hund coupling. We have found that the ground state of Ba2 YIrO6 is nonmagnetic and that the Hund coupling plays an important role in the magnetic properties of the 5d4 systems, unlike the well-studied 5d5 systems. PACS numbers: 75.25.Dk,75.30.Et,75.30.Kz

I.

INTRODUCTION

The 5d Ir-oxides with perovskite structure exhibit many novel ground states, which arise from the delicate interplay between relevant interactions such as Coulomb correlation, spin-orbit coupling (SOC), crystal-field effect, and so on.1–6 Of particular interest is the well-studied 5d5 layered Iridates, Sr2 IrO4 , which has a unique jef f =1/2 Mott insulating ground state arising from the combined effect of strong SOC and weak Coulomb correlation.1 Recently, intriguing experiment results have been reported for 5d4 double perovskite systems, Sr2 YIrO6 and Ba2 YIrO6 , having Ir5+ valence states (see Fig. 1(a)).7–13 In the susceptibility measurements, paramagnetic CurieWeiss behaviors having seemingly nonvanishing Ir magnetic moments were observed. In fact, Ir5+ ion would be in the Jef f = 0 singlet ground state in the strong SOC limit, which does not possess any finite magnetic moment (see Fig. 1(b)).14 Therefore, recent attentions have been paid to explore the origin of the nonvanishing magnetic moments in these 5d4 double perovskites.7–13,15,16 Khaliullin14 proposed an excitonic magnetism model, in which unconventional magnetism emerges from exciton condensation rather than from the preexisting local moments. For Sr2 YIrO6 , Cao et al.7 argued that the local magnetic moment could be induced by the strong crystalfield effect arising from distorted octahedra, and claimed that magnetism in Sr2 YIrO6 arises from the excitonic mechanism. This argument, however, seems not plausible, because the energy scale corresponding to the lattice distortion is much smaller than the spin-gap between the Jef f =0 and Jef f =1 states. Moreover, this argument is not applicable to Ba2 YIrO6 . Note that, even though both Ba2 YIrO6 and Sr2 YIrO6 have 5d4 double perovskite structure, Ba2 YIrO6 has ideal cubic IrO6 octahedra, unlike Sr2 YIrO6 that has the tilted and rotated IrO6 octahedra. Based on the DFT (density functional theory) band calculations for Sr2 YIrO6 and Ba2 YIrO6 , Bhowal et al.15 claimed that they would have antiferromagnetic ground

states with finite magnetic moments resulting from the band effects. However, for both systems, no evidences of long-range magnetic ordering have been observed at finite temperature(T ) in various experiments.8–12 As for the paramagnetic behavior of Ba2 YIrO6 , there are two interpretations. First one is that the nonzero local magnetic moment intrinsically originates from Ir5+ of the 5d4 system.7,13 Cao et al. claimed that the experiments captured the magnetic transition between the paramagnetic and antiferromagnetic phases, which is theoretically explained by the superexchange interaction between singlet-triplet excitons.14 Others claimed that the unexpected local moments come from extrinsic reasons like oxygen deficiency and/or anti-sites disorder in the 5d4 double perovskite structure.8–10,12,13 Obviously, the theoretical approaches based on DFT, DMFT (dynamical mean field theory), and the effective Hamiltonian model failed to reproduce paramagnetic behavior for the 5d4 double perovskite system. Furthermore, the effective Hamiltonian models often produce inaccurate results for the phase transitions because they deal with only limited physical conditions. Hence it is necessary to investigate the magnetic properties of the 5d4 double perovskite system with more realistic Hamiltonian models than the existing effective Hamiltonian models.15,16 In this paper, employing the ED (exact diagonalization) method for the multi-site clusters,17 we have investigated the ground state of 5d4 double perovskite. To exclude the redundant effects from octahedra distortions, we have focused on the magnetic properties of Ba2 YIrO6 . We considered a many-body Hamiltonian taking into account essential physical parameters: Coulomb correlation U , Hund coupling JH , SOC λ, and crystal-field splitting ∆. The hopping parameters are obtained from the Wannierization of the DFT band structures of Ba2 YIrO6 .18,19 We have demonstrated the robustness of the nonmagnetic ground state of this system. To further investigate the magnetic properties of the system, we have extended our model to a model incorporating the magnetic field. We have found that the Van Vleck magnetization decreases

2

FIG. 1: (a) Double perovskite crystal structure of Ba2 YIrO6 with octahedral symmetry. For the two-site cluster calculations, we used hopping paths (denoted by dotted line) between Ir1 and Ir2 on the ab-plane with the Wannierized hopping parameters (tef f ). The hopping paths for the four-site calculations are also given in the inset. (b) Multiplet states of Ir5+ (5d4 ) ion in the octahedral symmetry for ∆  JH  λ, where ∆, JH and λ represent the crystal-field, Hund, and SOC parameter, respectively. The ground state of the 5d4 is the singlet state, Jef f = 0 (Lef f = 1, S = 1).

with increasing the hopping strength, while it increases with increasing the Hund coupling.

HInter =

inter X

X

† (t˜αβ ij ci,ασ cj,βσ + H.c.),

(5)

i,j α,β,σ

II.

COMPUTATIONAL DETAILS

We have carried out the multiplet calculations with the ED method. The first step in the ED is to define a basis set, which can be specified by the occupation of electron for each orbital, |xy σ , yz σ , zxσ , (z 2 )σ , (x2 − y 2 )σ , xy −σ , ...i = |T, T, F, T, F, T, ...i,

The local part of the Hamiltonian includes the Coulomb correlation interaction (HC ), crystal-field effect(HCF ), SOC (HSOC ), Zeeman term (HZ ), HL = HC + HCF + HSOC + HZ , where, HC =

1X 2 i

(2)

The nonlocal part can be decomposed into intra-cluster and inter-cluster terms, HN L = HIntra + HInter

(3)

where HIntra =

intra X

X

i,j

α,β,σ

† (tαβ ij ci,ασ cj,βσ + H.c.),

α,β,γ,δ,σ,σ 0

(7) HCF =

(4)

XX i

HSOC = λ

HZ =

X

B·(gs

X α,σ,σ 0

∆α c†i,ασ ci,ασ ,

(8)

α,σ

X X i

i

HT = HN L + HL .

0

σσ (Uαβγδ −Jαβγδ )c†i,ασ c†i,βσ0 ci,γσ0 ci,δσ ,

X

(1)

where {xy,yz,zx,z 2 ,x2 −y 2 } and σ refer to orbitals and spin, respectively. T /F indicates occupied/unoccupied state. We used the entire d-orbitals to investigate magnetic behavior more accurately. Namely, we used more generalized Hamiltonian, compared with the previous theoretical studies. The second step is to construct the Hamiltonian matrix. The total Hamiltonian is composed of nonlocal and local part.

(6)

(lαβ · sσσ0 )c†i,ασ ci,βσ0 ,

(9)

α,β,σ,σ 0

sσσ0 c†i,ασ ci,ασ0 +gl

X

lαβ c†i,ασ ci,βσ ).

α,β,σ

(10) Here i(j), α(β, γ, δ), and σ(σ 0 ) refer to site, orbital, and spin, respectively. c†i,ασ (ci,ασ ) operator creates (annihilates) an electron with spin σ in orbital α of site i. s is the spin matrix and l is the orbital angular momentum ˜αβ matrix (l=2). tαβ ij (tij ) is the intra(inter)-cluster hopping parameter of an electron0 from orbital β of j site to orbital σσ α of i site. Uαβγδ , Jαβγδ , ∆α , λ, and B correspond to the Coulomb correlation,20 exchange correlation, crystalfield effect, SOC, and external magnetic field parameters,

3

p FIG. 2: Inter-site magnetization correlation function hM1p· M2 i/ |hM1 · M2 i| with varying the SOC parameter λ and the Hund coupling JH at zero temperature (the denominator, |hM1 · M2 i|, is used to distinguish the different magnetic phases more clearly). We have performed the calculations for two sets of hopping parameters: (a) t = original Wannierized tef f and (b) t = 2tef f . Other parameters used are U = 2.0 eV, ∆ = 3.0 eV. Regime C covers the realistic physical parameter regime for Ba2 YIrO6 , (JH ∼ 0.25 eV, λ ∼ 0.42 eV), which is in the regime of nonmagnetic phase.

respectively. gs and gl are g-factors of spin and orbital angular momentum, respectively (we used gs = 2 and gl = 1). Reasonable physical parameters for Ba2 YIrO6 are σσ U (≡Uαααα ) ∼ 2.0 eV, JH (≡Jαββα ) ∼ 0.25 eV, λ ∼ 0.42 eV, and ∆ ∼ 3.0 eV, respectively. These parameters were employed from the existing computational and numerical studies for 5d oxides.6,9,15,21–28 The Coulomb parameter U was compared with previous constrained random phase approximation (cRPA) results, and the CF parameter, ∆, was double checked with our DFT calculation. Hund JH and λ parameters, which were fit to the experimental data in a single-site model, are adopted from Ref.28 . We confirmed that our selected parameters give reasonable values for the charge-gap.9 Wannierized hopping matrices are used as the effective hopping matrices (tef f ). Band Lanczos method is used to solve the extensive Hamiltonian matrix. In the two-site cluster calculation, we implemented the entire multiplet basis states to consider the finite crystal-field effect. The number of basis in the two-site calculation is 104,580, where we used 44,100 bases for d4 -d4 configurations and 60,480 bases for d3 d5 and d5 -d3 configurations. Meanwhile, in the four-site cluster, we employed the truncated multiplet states and considered only t2g basis due to the computational limit. For the two-site cluster calculations, we used hopping paths between Ir1 and Ir2 shown in Fig. 1(a) with the following Wannierized parameters (t = tef f ): txy,xy = 12 zx,zx z 2 ,z 2 − 0.131 eV, tyz,yz = t = 0.023 eV, t = 0.001 12 12 12 2 2 2 2 eV, tx12 −y ,x −y = − 0.011 eV, tyz,zx = 0.019 eV, and 12 xy,z 2 t12 = − 0.034 eV. Arbitrary selections of the hopping paths will affect the calculation results, since the cluster does not have the same symmetry as that of the lattice. This problem may cause odd physical values, and so we have double-checked the results with the four-site cluster calculations too (inset of Fig. 1(a)).

III.

RESULTS AND DISCUSSIONS

To examine the phase diagram of 5d4 double perovskite system, we evaluated the inter-site magnetization correlation-function, hM1 · M2 i,29 where M1 and M2 are P magnetizations given by Miµ = gs ασσ0 sµσσ0 c†iασ ciασ0 + P µ † gl αβσ lαβ ciασ ciβσ (i = 1, 2 ; µ = x, y, z). Figure 2 shows hM1 · M2 i as functions of λ and JH for two sets of hopping parameters. In Fig. 2(a), A, B, and C regimes correspond to those of AFM (antiferromagnetic), FM (ferromagnetic), and NM (nonmagnetic) phases, respectively. In the AFM regime (A in Fig. 2), where both λ and JH are weak, hM1 · M2 i is negative. For FM regime (B in Fig. 2), where λ is weak but JH is strong, hM1 · M2 i is positive. This difference arises directly from the variation of JH . For weak SOC, the energy gain from the superexchange interaction is αt2 /(U − βJH ) in the AFM phase and α0 t2 /(U − β 0 JH ) in the FM phase. (α (α0 ) and β (β 0 ) depend on the hopping channels and spin configurations in the AFM (FM) phase, respectively.) α is larger than α0 due to the Pauli exclusion principle. β is smaller than β 0 since the number of parallel spin pairs in the intermediates states (d3 -d5 and d5 -d3 ) for FM phase is larger than that for AFM phase. In the weak JH limit, α(α0 ) plays a more important role than β(β 0 ) and the AFM phase is more stable than the FM phase. On the other hand, in the strong JH limit, β(β 0 ) is predominant over α(α0 ) and so the FM phase becomes stabilized. Evidently, in the regime of physical parameters for Ba2 YIrO6 as denoted by C, hM1 · M2 i is very small. This result shows that the ground state of Ba2 YIrO6 is nonmagnetic (not paramagnetic) with vanishing magnetic moment. The spin-gap due to a strong SOC suppresses the mixing between the nonmagnetic ground state and magnetic excited states. Also, because of this large spin-

4

FIG. 3: Spin-gap obtained in the two-site cluster calculations as functions of Hund coupling JH , SOC parameter λ, and the hopping strength t. For λ=0, the spin-gap exists only in the AFM phase, which comes from the anisotropic hopping parameters. For non-zero λ, as the hopping strength increases, the spin-gap decreases.

gap as much as 250 ∼ 400 meV, the nonmagnetic phase will not be transformed into the magnetic phase by the finite temperature effect.12,16 According to Fig. 2(a) and (b), as the hopping strength increases in the weak SOC limit, hM1 · M2 i in the AFM phase decreases. This is because the spin-gap increases in the weak JH and λ limit, as shown in Fig. 3. We have found that the superexchange with the assistance of anisotropic hopping parameters (txy,xy 6= tyz,yz = tzx,zx ) 12 12 12 leads to a non-degenerate ground state and increases the spin-gap only in the AFM phase. Meanwhile, we have identified that the isotropic hopping parameters (txy,xy = tyz,yz = tzx,zx ) make the ground states to be 12 12 12 degenerate in this limit. In the FM and NM phases with finite spin-gap, the spin-gap decreases as the hopping

FIG. 4: Total magnetization per site(Mz = Lz + 2Sz ) as a function of the Hund coupling JH at B = 10 T along the z-direction with varying the SOC strength λ.

FIG. 5: The spectral function and its projections to jef f = 1/2 and jef f = 3/2 states calculated by the VCA method for the four-site cluster (U = 2.0 eV, JH =0.4 eV, λ=0.4 eV). It is seen that the mixing between jef f = 1/2 and jef f = 3/2 states is minor.

strength increases, as shown in Fig. 3(b). The changes in the spin-gap for the FM and NM phases are determined by the superexchange interaction between the Jef f states. For the Jef f = 0 (Lef f = 1, S = 1) ground state, four electrons almost fully occupy the jef f = 3/2 single electron states (|Jef f = 0i ≈ |jef f = 3/2, +3/2i⊗|jef f = 3/2, +1/2i ⊗ |jef f = 3/2, −1/2i ⊗ |jef f = 3/2, −3/2i), while for the Jef f = 1 (Lef f = 1, S = 1) excited states, four electrons partially occupy both the jef f = 3/2 and jef f = 1/2 single electron states.31 Due to this occupation difference, the hopping amplitude for the Jef f = 1 state is greater than the hopping amplitude for the Jef f = 0 state. Hence, the energy gain from the superexchange interaction for the Jef f = 1 state is greater than the one for the Jef f = 0 state, and the spin-gap decreases with increasing hopping strength. Figure 4 shows the magnetic response to the external magnetic field, obtained by both the one-site and the four-site cluster calculations. It shows that only the Van Vleck-type susceptibility (Mz /B) is induced upon the very strong magnetic field of 10 Tesla. The obtained Van Vleck susceptibility has the same order of magnitude as the observed T -independent susceptibility.9 The induced magnetization from one-site calculation is similar to the one from the four-site one, which suggests that the hopping effect on the magnetization is not large even in the presence of the magnetic field. In Fig. 5, the spectral function for the 5d4 system is displayed, which is obtained by using the cluster perturbation theory (CPT)32 with chemical potential optimaization via the variational cluster approximation (VCA).33 The mixing between two different jef f states is shown to be not so large, indicating that the magnetic moment in the 5d4 double perovskites will not be induced from the mixing of the jef f states. Note that this feature distinguishes 5d4 systems from 5d3 and 5d5 ones, where the mixing between two jef f states plays a significant role.

5 IV.

CONCLUSION

We have investigated the electronic structures and magnetic properties of 5d4 double perovskite Ba2 YIrO6 , employing the cluster multiplet calculations. We have shown that the ground state of Ba2 YIrO6 is the nonmagnetic state, namely, effective magnetic moment will not be induced for its physical parameters. The resulting phase diagram indicates that the magnetic phase of 5d4 system can exist only in the parameter regime of small λ and large JH . Also, in the presence of the external magnetic field, the hopping effect makes the magnetization of the ground state smaller. For this reason, we conclude that Ba2 YIrO6 without defects or anti-site disorder would have the nonmagnetic ground state.

working vividly with Prof. Freeman and magnetic theory group members. Prof. Freeman was not only a thesis adviser but also a mentor to me. It was my great honor to be a first Korean Ph.D student of him. I would like to dedicate this paper to late Prof. Arthur J. Freeman. This work was supported by the NRF of Korea (Grant No. 2017R1A2B4005175), and the KISTI (Grant No. KSC-2016-C3-0062). K. Kim acknowledges the support by the NRF (Grant No. 2016R1D1A1B02008461 and No. 2017M2A2A6A01071297) and the support by MPK (Grant No. 2016K1A4A4A01922028). B. H. Kim was supported by RIKEN iTHES project.

Acknowledgments

While preparing this manuscript with my students, I (BIM) could look back on good old days at Evanston

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C. Martins, M. Aichhorn and S. Biermann, Journal of Physics: Condensed Matter 29 263001 (2017) B. Yuan, J. P. Clancy, A. M. Cook, C. M. Thompson, J. Greedan, G. Cao, B. C. Jeon, T. W. Noh, M. H. Upton, D. Casa, T. Gog, A. Paramekanti, and Young-June Kim, Phys. Rev. B 95, 235114 (2017). The inter-site magnetization correlation function is defined Ras hM1 · M2 i ≡ hM1 (iw) · ∞ M2 i|iw→0 = hM1 (τ ) · M2 ieiwτ dτ |iw→0 = 0 P R∞ µ µ −(En −EG −iw)τ hG|M |nihn|M |Gie dτ | , iw→0 1 2 n,µ 0 where µ = x, y, z, and |Gi (|ni) and EG (En ) represent the ground state (excited states) and energy (excited energy levels), respectively. O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi, Phys. Rev. B 91, 054412 (2015). jef f states are defined as single electron states, as follows,

|jef f = 3/2, +3/2i= √12 |yz+i+ √i2 |zx+i, √

|jef f = 3/2, +1/2i= √23 |xy+i− √16 |yz−i − √i6 |zx−i, √

32

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|jef f = 3/2, −1/2i= √23 |xy−i+ √16 |yz+i − √i6 |zx+i, |jef f = 3/2, −3/2i= √12 |yz−i− √i2 |zx−i, |jef f = 1/2, +1/2i= √13 |xy+i+ √13 |yz−i + √i3 |zx−i, |jef f = 1/2, −1/2i= √13 |xy−i− √13 |yz+i + √i3 |zx+i, where xy(yz, zx) and +(−) represent real orbitals and spin-up(down), respectively. Jef f states are defined as many-electron states, which are expressed in terms of the product of single-electron states. D. Sˇenˇechal, D. Perez, and D. Plouffe, Phys. Rev. B. 66, 075129 (2002). M. Potthoff, M. Aichhorn, and C. Dahnken Phys. Rev. Lett. 91, 206402 (2003).