First-principles study of the magnetic interactions in honeycomb $\mathrm{N}{\mathrm{a}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$

First-principles study of the magnetic interactions in honeycomb $N a_{2} Ir O_{3}$

Y. S. Hou, J. H. Yang, H. J. Xiang, and X. G. Gong

Phys. Rev. B 98, 094401 – Published 4 September 2018

Abstract

Honeycomb iridate $N a_{2} Ir O_{3}$ , a $J_{eff} = 1 / 2$ magnet, is a potential platform for realizing quantum spin liquid. Many experiments have shown that its magnetic ground state is a zigzag antiferromagnetic order. However, there is still a lack of consensus on the theoretical model explaining such order, since its second-nearest-neighbor (NN) and long-range third-NN magnetic interactions are highly unclear. By properly considering the orbital moments achieved through constraining their directions in first-principles calculations, we obtain that the relative angle between orbital and spin moments is fairly small and in the order of several degrees, which thus validates the $J_{eff} = 1 / 2$ state in $N a_{2} Ir O_{3}$ . Surprisingly, we find that the long-range third-NN Heisenberg interactions are sizable, whereas the second-NN magnetic interactions are negligible. Furthermore, we show that sizable long-range third-NN Heisenberg interactions closely correlate with the appreciable distribution of Wannier orbitals of $J_{eff} = 1 / 2$ states over the three NN Ir atoms. Based on our study, we propose a minimal $J_{1} − K_{1} − Γ_{1} − J_{3}$ model in which the magnetic excitations have an intensity peak at 5.6 meV, consistent with the inelastic neutron-scattering experiment [Phys. Rev. Lett. 108, 127204 (2012)]. The present work demonstrates again that constraining orbital moments in first-principles calculations is a powerful way to investigate the intriguing magnetism in $J_{eff} = 1 / 2$ magnets, and it paves the way toward gaining a deep insight into the novel magnetism discovered in the honeycomb $J_{eff} = 1 / 2$ magnets.

Received 7 August 2017
Revised 20 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.094401

Physics Subject Headings (PhySH)

Magnetism

Iridates

Density functional theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Y. S. Hou^1,2, J. H. Yang³, H. J. Xiang^1,2,*, and X. G. Gong^1,2,†

¹Key Laboratory of Computational Physical Sciences (Ministry of Education), State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China
²Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
³National Renewable Energy Laboratory, Golden, Colorado 80401, USA

^*hxiang@fudan.edu.cn
^†xggong@fudan.edu.cn

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Vol. 98, Iss. 9 — 1 September 2018

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Figure 1
(a) Crystal structure of the honeycomb $N a_{2} Ir O_{3}$ . Na, Ir, and O atoms are represented by the small black, large purple, and dark green medium-sized spheres, respectively. The cubic $x$ -, $y$ -, and $z$ -axes are the same as those used in Ref. [14] and are shown by the red arrows. The zigzag AFM order with magnetic moments along the [110] direction is shown by the black arrows. (b) The NN, second-NN, and third-NN Ir-Ir paths in the Ir atoms sublattice are connected by the solid double-arrowed, dashed double-arrowed, and dashed-dotted lines, respectively. The $x$ -, $y$ -, and $z$ -bond Ir-Ir paths are distinguished by green, red, and blue. Black and white spheres have up and down spins, respectively. Dependences of the energy $Δ E$ are caused by the derivation between the orbital and the spin moments on the polar angle $θ$ of spin moment when the orbital moment is along the (c) [110], (d) $x$ , (e) $y$ , and (f) $z$ axes.
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Figure 2
(a) FM, Neél AFM, stripe AFM, zigzag AFM, armchair AFM, zigzag-2 AFM, stripe-2 AFM, and armchair-2 AFM magnetic orders. Black and white spheres have up and down spins, respectively. (b) The relative energies of the considered magnetic orders with magnetic moments along various directions. The zigzag AFM order with magnetic moments along the [110] direction (zigzag-[110]) has the lowest energy and is set as the energy reference, which is highlighted by the blue rectangle. (c) Energy dependence of the zigzag AFM order on the polar angle $θ$ and azimuthal angle $ϕ$ in the ( $x$ , $y$ , $z$ )-coordinate system [see Fig. 1]. The energy of the case ( $θ = 80^{\circ}$ , $ϕ = 225^{\circ}$ ) is set as the energy reference. Because the cases of azimuthal $θ$ ranging from $0^{\circ}$ to $60^{\circ}$ have relatively high energy, they are not shown here.
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Figure 3
(a) DFT+SOC calculated band structure (black line) of $N a_{2} Ir O_{3}$ and Wannier-interpolated four bands (the red dashed-dotted lines) near the Fermi level ( $E_{Fermi} = 0$ ). (b) Real parts of spin-up of the two $| J_{eff} = 1 / 2, 1 / 2 〉$ Wannier orbitals of the third-NN Ir-Ir pair (black and blue labels “Ir-0”) in the $3 \times 3 \times 1$ supercell of the $N a_{2} Ir O_{3}$ primitive cell. (c) Real part of spin-up, (d) imaginary part of spin-down, (e) real part of spin-down, and (f) imaginary part of spin-up of the calculated $| J_{eff} = 1 / 2, 1 / 2 〉$ Wannier orbital on the Ir-0 atom. The three NN Ir atoms of the Ir-0 atom are labeled by Ir-1, Ir-2, and Ir-3 as shown in (f). Wannier orbitals in (c)–(f) are viewed along the $z$ axis, but they are viewed along the crystallographic $c$ axis in (b). The maximum (minimum) grid values are 7.2 (−7.0) in (c)–(e) and 0.9 (−0.9) in (f). The isosurface value is 1.5 in (b)–(e) and 0.4 in (f).
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Figure 4
(a) 24-site periodic $U C - 3 \times 2$ and (b) $\sqrt{3} \times \sqrt{3} - 2 \times 2$ clusters. (c) ED calculated energy-dependent magnetic excitation intensity of the minimal $J_{1} − K_{1} − Γ_{1} − J_{3}$ model whose magnetic parameters are listed in Table 1. A Gaussian broadening of 0.67 meV (FWHM), same as the instrumental energy resolution in Ref. [11], has been adopted. The intensity peaks of the $U C - 3 \times 2$ (blue) and $\sqrt{3} \times \sqrt{3} - 2 \times 2$ (red) clusters are indicated by the black arrows. Classical (d) and quantum (e) phase diagrams of the minimal $J_{1} − K_{1} − Γ_{1} − J_{3}$ model in the $Γ_{1} − J_{3}$ plane. The NN Heisenberg $J_{1}$ and Kitaev interactions $K_{1}$ are 1.63 and −10.0 meV, respectively. The red stars highlight the specific position ( $J_{3} = 0.83 meV$ , $Γ_{1} = 0.90 meV$ ) in the $Γ_{1} − J_{3}$ plane. The out-of-plane (OP) incommensurate (IC) and the in-plane (IP) IC phase regions can be much richer than the standard coplanar helix states.
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Figure 5
Dependences of the energy $Δ E$ on the polar angle $θ$ and the azimuthal angle $ϕ$ of the spin moment when the orbital moment is along the (a) $x$ , (b) $y$ , (c) $z$ , and (d) [110] axes.
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Physical Review B

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First-principles study of the magnetic interactions in honeycomb $N a_{2} Ir O_{3}$

Y. S. Hou, J. H. Yang, H. J. Xiang, and X. G. Gong

Phys. Rev. B 98, 094401 – Published 4 September 2018

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First-principles study of the magnetic interactions in honeycomb Na2IrO3

Y. S. Hou, J. H. Yang, H. J. Xiang, and X. G. Gong

Phys. Rev. B 98, 094401 – Published 4 September 2018

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First-principles study of the magnetic interactions in honeycomb $N a_{2} Ir O_{3}$