Antiferromagnetic Slater insulator phase of Na₂IrO₃.

Using a hybrid density-functional theory (DFT) calculation including spin-orbit coupling (SOC), we predict that the zigzag antiferromagnetic (AFM) ground state of the honeycomb layered compound Na₂IrO₃ opens the observed insulating gap through a long-range magnetic order. We show that the effect of SOC and the correction of self-interaction error inherent in previous local or semilocal DFT calculations play crucial roles in predicting the band gap formation in Na₂IrO₃. It is revealed that the itinerant AFM order with a strong suppression of the Ir magnetic moment is attributed to a considerable hybridization of the Ir 5d orbitals with the O 2p orbitals. Thus, our results suggest that the insulating phase of Na₂IrO₃ can be represented as a Slater insulator driven by itinerant magnetism.

Recently, there have been intensive studies concerning the interplay between the spin-orbit coupling (SOC), the on-site Coulomb repulsion (U), and the bandwidth (W) in 5d transition metal oxides (TMO)1234567891011121314. Compared to 3d orbitals in 3d TMO, 5d orbitals in 5d TMO are spatially more extended to yield a relatively smaller (larger) value of U (W). On the other hand, the strength of the SOC in 5d TMO is enhanced due to the higher atomic numbers of transition metals. The resulting energy scales arising from SOC, U, and W in 5d TMO may become comparable to each other, leading to a broad spectrum of exotic quantum phases such as topological insulator23, spin liquid456, and Mott insulator7891011121314. In particular, the nature of the insulating phase in iridium (Ir) oxides has been under intense debate whether it is a Mott-type insulator78 or a Slater-type insulator1516. Here, the gap formation in the Mott-type insulator is driven by electron correlation while that in the Slater-type insulator is associated with magnetic ordering.

As a prototypical example of 5d TMO, we here focus on Na2IrO3, where Ir atoms form a honeycomb lattice and each Ir atom is surrounded by an octahedron of six O atoms [see Fig. 1(a) and 1(b)]. It was experimentally observed that Na2IrO3 has the antiferromagnetic (AFM) insulating ground state with a zigzag spin alignment [Fig. 1(c)] below the Néel temperature 14171819 but the insulating gap is preserved even at room temperature14. This separation between the insulating behavior and the onset of AFM ordering may imply that Na2IrO3 can be regarded as a Mott insulator driven by electron correlations12131417. To address the nature of the insulating phase in Na2IrO3, it was suggested that the Ir4+ t2 states locating around the Fermi energy would be treated in terms of relativistic atomic orbitals with the effective angular momentum jeff = 1/2 and jeff = 3/22121314. Here, the upper jeff = 1/2 band decoupled from the lower jeff = 3/2 bands was presumed to be half-filled, and therefore Ir atoms can have localized magnetic moments corresponding to an effective spin one-half Ir4+ ion69. This jeff scenario caused by strong SOC leads not only to a novel magnetic structure suggested from the Kitaev-Heisenberg model61120 but also a spin-orbit Mott insulator where the half-filled jeff = 1/2 band splits into two Hubbard bands by taking into account on-site Hubbard U121314. However, contrasting with such highly localized jeff = 1/2 orbitals at Ir atoms, a recent ab initio density functional theory (DFT) calculation together with the tight-binding model analysis21 showed that the t2 bands can be described by quasimolecular orbitals (QMOs) which are fully delocalized over six Ir atoms forming a honeycomb lattice. Despite this rather itinerant character of QMOs, the explicit treatment of U was required to obtain the observed insulating gap of ~0.34 eV131421. It is thus likely that the effect of electron correlations plays an indispensable role in describing the insulating phase of Na2IrO3, thereby being represented as a Mott insulator121314.

Figure 1

Crystal structure of Na2IrO3.

(a) projection on the xz plane and (b) projection on the xy plane. a1, a2, and a3 denote unit vectors of the unit cell. The large, medium, and small circles represent Na, Ir, and O atoms, respectively. Three different AFM structures with the zigzag, stripy, and Néel spin orders are schematically shown in (c), where the solid and open circles indicate up and down spins.

By contrast, we here propose a different mechanism for the observed insulating phase of Na2IrO3 based on a long-range magnetic order. This magnetically driven insulating phase through an itinerant single-particle approach can be represented as a Slater insulator22. It is noticeable that a rather delocalized character of the Ir t2 states, as described by QMOs21, may be associated with the self-interaction error (SIE) inherent to the conventional DFT calculations with the local density approximation (LDA)23 or the generalized gradient approximation (GGA)24. Note that the SIE causes the electron density to artificially spread out because delocalization reduces the spurious self-repulsion of electron2526. This so-called delocalization error tends to give an inaccurate estimation of the ionization energy and the electron affinity, resulting in the underestimation of band gap26. In this regard, previous LDA and GGA calculations13141721 may not adequately describe the insulating phase of Na2IrO3. Therefore, it is very challenging to examine how the electronic properties of Na2IrO3 can be changed by the correction of SIE with an exchange-correlation functional beyond the LDA or GGA.

In this paper, we present a new theoretical study of Na2IrO3 based on the hybrid DFT scheme including SOC. We find that the effect of SOC and the correction of SIE with the screened hybrid exchange-correlation functional of Heyd-Scuseria-Ernzerhof (HSE)2728 opens the observed14 insulating gap of ~0.34 eV for the zigzag AFM ground state. Compared with the effective spin S = 1/2 moments (M = 1 μ) within the Kitaev-Heisenberg model611, the calculated magnetic moment per Ir atom is much reduced to be 0.37 μ, close to that (~0.22 μ) measured by a combined neutron and x-ray diffraction (XRD) experiment18. Such an itinerant character of magnetism is revealed to be due to a considerable hybridization of the Ir 5d orbitals with the O 2p orbitals.

Calculation details

Our hybrid DFT calculations including SOC were performed using the Vienna ab initio simulation package (VASP) with the projector augmented wave (PAW) method2930. For the exchange-correlation energy, we employed the HSE functional2728, which is given by Here, the mixing factor α controls the amount of exact Fock exchange energy and the screening parameter ω ( = 0.20 Å−1) defines the separation of short range (SR) and long range (LR) for the exchange energy. Note that the HSE functional with α = 0 becomes identical to the Perdew-Burke-Ernzerhof (PBE)24 functional. We used the experimental lattice constants a = 5.427, b = 9.395, and c = 5.614 Å with the C2/m monoclinic crystal structure, obtained by an XRD study19. The k-space integration was done with the 6 × 3 × 6 uniform meshes in the Brillouin zone. All atoms were allowed to relax along the calculated forces until all the residual force components were less than 0.02 eV/Å.

Results and discussion

We begin to study the experimentally observed171819 zigzag AFM ground state of Na2IrO3 using the PBE calculation. The optimized structural parameters such as the Ir-Ir bond lengths [d12 and d23 in Fig. 1(b)] and the Ir-O-Ir bond angles [θ12 and θ23 in Fig. 1(b)] are given in Table I. The calculated values of θ12 = 99.84° and θ23 = 98.51° (greater than the ideal 90° Ir-O-Ir bond angle) show a sizable trigonal distortion of the IrO6 octahedra, consistent with an XRD analysis19. Figure 2(a) shows the calculated band structure and density of states (DOS) of the zigzag AFM structure, which exhibit the presence of partially occupied Ir t2 states at the Fermi level (E), indicating a metallic feature. Note that there are twelve t2 bands which originate from four different Ir atoms within the unit cell of the zigzag AFM structure. It is noticeable that, for the t2 states locating near E, the partial DOS projected onto the O 2p orbitals amounts to ~50% of that projected onto the Ir 5d orbitals [see Fig. 2(a)], indicating a considerable hybridization between the two orbitals. Indeed, the spin characters of the t2 state at E, as shown in Fig. 2(a), reveal the electron delocalization over IrO6 octahedra on each zigzag chain side. These aspects of the t2 states lead not only to a smaller magnetic moment of ~0.53 μ for an Ir atom compared to the effective spin S = 1/2 moment assumed in the Kitaev-Heisenberg model611 but also a slightly induced magnetic moment of ~0.10 μ for the O1 atom [see Fig. 2(b) and Table II]. It is noteworthy that the PBE calculation may involve the over-delocalization of the t2 states due to the SIE, therefore incorrectly predicting the zigzag AFM structure to be metallic rather than insulating.

Table 1

Ir-Ir bond lengths [d12 and d23 in Fig. 1(b)] and Ir-O-Ir bond angles [θ12 and θ23 in Fig. 1(b)] obtained using the PBE and HSE calculations, in comparison with the experimental data

	d12 (Å)	d23 (Å)	θ12 (°)	θ23 (°)
PBE	3.129	3.139	99.84	98.51
HSEα = 0.05	3.130	3.138	100.23	98.86
HSEα = 0.1	3.109	3.180	99.51	99.73
Experiment (Ref. 19)	3.130	3.138	99.45	97.97

Figure 2

(a) Band structure and DOS of the zigzag AFM structure, obtained using the PBE functional. The band dispersions are plotted along the symmetry lines shown in the Brillouin zone of the unit cell (see the inset). The total DOS, Ir 5d partial DOS, and O 2p partial DOS are displayed with solid, dot-dashed, and dotted lines, respectively. The energy zero represents the Fermi level. The charge characters of the spin-up and spin-down t2 state near the Fermi level are shown with an isosurface of 0.004 e/Å3. In (b), the spin-up (spin-down) density is displayed in bright (dark) color with an isosurface of 0.004. (−0.004) e/Å3, and the two different species of O atoms are denoted as O1 (for O atoms on the same zigzag chain side) and O2 (for O atoms between two zigzag chains).

Table 2

Magnitude of the magnetic moments (in units of μ) of Ir, O1, and O2 atoms obtained using the PBE, HSE, PBE + SOC, and HSE + SOC calculations

	Ir	O1	O2
PBE	0.53	0.10	0.00
HSEα = 0.05	0.56	0.11	0.00
HSEα = 0.1	0.62	0.11	0.00
PBE + SOC	0.33	0.06	0.05
HSE + SOCα = 0.05	0.37	0.06	0.05
HSE + SOCα = 0.1	0.41	0.06	0.05

In order to correct the SIE, we use the HSE functional to optimize the zigzag AFM structure. We find that the structural parameters slightly depend on the magnitude of α in the HSE functional (see Table I): i.e., d12 and d23 change little by less than 0.01 Å in the range of 0 < α ≤ 0.08, but, as α increases to 0.1, d12 (d23) decreases (increases) by ~0.02 (0.04) Å. We also find a drastic variation of the band structure and DOS as a function of α. Especially, the DOS at E is found to decrease as α increases up to 0.08, giving rise to the creation of a pseudogap. For α > 0.08, the pseudogap is turned into an insulating gap whose magnitude increases with increasing α (see Fig. 3). Figure 4(a) shows the band structure and DOS of the zigzag AFM structure, obtained using the HSE calculation with α = 0.05. It is seen that, compared with the PBE result [Fig. 2(a)], (i) the DOS for the occupied t2 states is shifted to a lower energy and (ii) the DOS at E is much reduced. We note that, as α increases, the Ir magnetic moment increases compared to that obtained using the PBE calculation (see Table II), reflecting that the HSE functional corrects the over-delocalization of Ir 5d electrons due to the SIE of the PBE functional.

Figure 3

Calculated band gap of the zigzag AFM structure as function of the mixing factor α in the HSE functional.

Figure 4

Band structure and DOS of the zigzag AFM structure, obtained using (a) the HSE and (b) HSE + SOC calculations with α = 0.05. The magnetic moment (M, M, M), obtained using the HSE + SOC calculation with α = 0.05, is drawn in (c). Here, M is calculated by integrating the corresponding component of magnetic moment inside the PAW sphere with a radius of 1.4 (0.8) Å for Ir (O). In (c), the circles represent Ir atoms.

According to the jeff scenario, the SOC splits the t2 states into the jeff = 1/2 and jeff = 3/2 states269, and the experimental 0.34 eV-gap was opened by taking into account an on-site interaction U of 1–3 eV, leading to a conclusion that the insulating phase of Na2IrO3 can be represented as a spin-orbit Mott insulator121314. In this study, we examine the effect of SOC on the electronic structure of the zigzag AFM structure using the HSE + SOC calculation. Here, we employ the optimized HSE structure because the effect of SOC changes little the Ir-Ir bond lengths and the Ir-O-Ir bond angles by less that 0.01 Å and 1°, respectively. As shown in Fig. 3, the inclusion of SOC opens the band gap E, which monotonically increases with increasing α. For α = 0 (equivalent to the PBE + SOC calculation), we obtain E = 0.05 eV, in good agreement with a previous PBE + SOC calculation31. On the other hand, as α increases to 0.05, E becomes 0.33 eV, close to the experimental value of ~0.34 eV14. The HSE + SOC band structure and DOS computed with α = 0.05 are displayed in Fig. 4(b). It is seen that (i) the t2 bands just below and above E become almost dispersionless, thereby possibly correcting the over-delocalization of the t2 states, and (ii) there are six separated energy regions for the t2 bands, consistent with the experimental observation of the five d-d interband transitions in optical conductivity31. Since the HSE + SOC calculation with α = 0.05 adequately predicts the insulating electronic structure of the zigzag AFM ground state, we can say that the insulating phase of Na2IrO3 can be represented as a spin-orbit Slater insulator through itinerant magnetism. We note that the optimal HSE + SOC value of α = 0.05 reproducing the experimental 0.34 eV-gap14 is smaller than that (α ≈ 0.15) obtained from the HSE calculation (see Fig. 3). This indicates that the HSE calculation needs larger α to cure over-delocalization of the t2 states due to relatively larger SIE.

Figure 4(c) shows the magnetic moment (M, M, M) for each atom of the zigzag AFM structure, obtained using the HSE + SOC calculation with α = 0.05. We find that Ir, O1, and O2 atoms have the three components of magnetic moment (±0.12, 0.00, ±0.36), (±0.02, ±0.02, ±0.06) or (±0.02, , ±0.06), and (0.00, ±0.05, 0.00) in units of μ, respectively. It is notable that, when compared with the HSE calculation with the same α = 0.05, the inclusion of SOC reduces the magnitude for Ir (O1) atom by ~34(46)% but creates M = ±0.05 μ for O2 atom [see Table II and Fig. 4(c)]. Thus, the HSE + SOC value of M = 0.37 μ for Ir atom is closer to that (~0.22 μ) measured by a combined neutron and XRD experiment18. These theoretical and experimental evidences for a strong suppression of the Ir magnetic moment support a Slater-type insulator via itinerant magnetism rather than localized magnetism proposed by the Kitaev-Heisenberg model with effective S = 1/2 pseudospins61120. We note that the HSE calculations for half-filled systems3334 have highly debated about the distinction between a Mott-type insulator and a Slater-type insulator. For instance, the HSE calculation for the Cs3C60 crystal predicted33 not only a highly localized half-filled state but also a magnetic moment of ~1 μ corresponding to a S = 1/2 state, thereby being characterized as a Mott-type insulator. On the other hand, the HSE calculation for the Sn/Ge(111) surface predicted34 a rather delocalized half-filled state with a much reduced magnetic moment of ~0.2 μ, suggesting a Slater-type insulator.

It was experimentally observed that the paramagnetic insulating phase1112 of Na2IrO3 exists between and room temperature14. However, the precise nature of the paramagnetic insulating phase above T is still open to question. The experimental evidence that T is much smaller than the Weiss temperature1112 (≈−120 K) may indicate a frustration of the underlying AFM interactions. We note that the stripy and Néel AFM structures [see Fig. 1(c)] can be feasible to model the paramagnetic phase because these AFM structures and the paramagnetic phase have the common features such as similar Ir magnetic moments and zero net magnetic moments in their unit cells. For the stripy and Néel AFM structures, we perform the HSE + SOC calculations with α = 0.05. The stripy (Néel) AFM structure is found to be not only less stable than the zigzag AFM structure by 4.0 (12.5) meV per Ir atom, but also insulating with Eg = 0.18 (0.30) eV. However, we were not able to obtain the nonmagnetic structure which was always converged to the Néel AFM structure. Since the energy differences among the zigzag, stripy, and Néel AFM structures are very small, not only is T much lower compared to the cases of other iridates but quantum fluctuations of such AFM structures are also likely present above T, leading to an effectively paramagnetic phase. Further experimental and theoretical studies for the paramagnetic insulating phase are demanded to resolve its nature.

Conclusion

We performed the HSE + SOC calculation to investigate the zigzag AFM ground state of Na2IrO3. We found that the observed insulating gap of ~0.34 eV14 is well predicted by not only taking into account the SOC but also correcting the SIE inherent in the LDA or the GGA. Thus, our results indicate a significant Slater-type character of gap formation through itinerant magnetism. As a matter of fact, we predicted a small itinerant magnetic moment of 0.37 μ per Ir atom, contrasting with a fully localized magnetic moment of 1 μ per Ir atom presumed within the Kitaev-Heisenberg model61120. Similar to the present case of Na2IrO3, we anticipate that the correction of SIE would be of importance to describe the insulating phases of other Ir oxides. We note that other 5d TMO such as Sr2IrO416 and NaOsO335 were recently reported to display a magnetically driven gap formation, supporting a Slater-type insulator.

Author Contributions

H.J.K. and J.H.L. carried out the DFT calculations and data analysis. J.H.C. was responsible for the planning and the management of the project. J.H.C. wrote the main manuscript text and H.J.K. prepared figures 1–4. All authors reviewed the manuscript.