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Spin-orbital ordering in alkali superoxides
Kohei Shibata, Makoto Naka, Harald O. Jeschke, and Junya Otsuki
Phys. Rev. B 109, 235115 – Published 7 June 2024
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Abstract
Alkali superoxides , due to an open shell of the oxygen ion with degenerate orbitals, have spin and orbital degrees of freedom. The complex magnetic, orbital, and structural phase transitions observed experimentally in this family of materials are only partially understood. Based on density functional theory, we derive a strong-coupling effective model for the isostructural compounds from a two-orbital Hubbard model. We find that has highly frustrated exchange interactions in the plane, while the frustration is weaker for and . We solve the resulting Kugel-Khomskii model in the mean-field approximation. We show that exhibits an antiferro-orbital (AFO) order with the ordering vector and a stripe antiferromagnetic order with , which is consistent with recent neutron scattering experiments. We discuss the role of the -orbital degrees of freedom for the experimentally observed magnetic transitions and interpret the as-yet-unidentified transition in as an orbital ordering transition.
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- Received 22 March 2024
- Accepted 24 May 2024
DOI:https://doi.org/10.1103/PhysRevB.109.235115
©2024 American Physical Society
Physics Subject Headings (PhySH)
- Research Areas
Frustrated magnetismOrbital order
- Physical Systems
Mott insulatorsOxides
- Techniques
Density functional theoryMean field theory
Condensed Matter, Materials & Applied Physics
Authors & Affiliations
Kohei Shibata1, Makoto Naka2, Harald O. Jeschke3, and Junya Otsuki3
- 1Department of Physics, Okayama University, Okayama 700-8530, Japan
- 2School of Science and Engineering, Tokyo Denki University, Saitama 350-0394, Japan
- 3Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan
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Vol. 109, Iss. 23 — 15 June 2024
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Figure 1
Summary of the structural phase transitions and magnetic properties in , , and [5, 7, 8, 9].
Figure 2
(a)Structure of tetragonal ( space group). (b)Wannier functions of oxygen orbitals in , basis and in , basis.
Figure 3
(a)Band structure and density of states of in space group ( structure). The points are defined in [30]. (b)Wannier fit of the two bands near with weights of the and Wannier orbitals.
Figure 4
(a)Band structure and density of states of in space group ( structure). The points are defined in [33]. (b)Wannier fit of the two bands near with weights of the and Wannier orbitals.
Figure 6
Parameterization of the bond dependence of the transfer integrals using polar coordinates. The symbols indicate the DFT estimates for , , and (only results without the optimization are shown).
Figure 7
A diagram for the orbital (pseudospin) operators and the orbital states. The orbital on the right (left) represents the eigenstate of the operator with the eigenvalue (). The orbital state is rotated around the axis by as the operator is rotated by in the pseudospin space.
Figure 8
The orbital-orbital interactions described by the bond-dependent orbital operator .
Figure 9
Configurations of the ordered states. The ordering vector is indicated in units of the reciprocal lattice vectors of the conventional unit cell. The circles with the same color represent the same spin or orbital state. The gray sites are not involved in the two-dimensional ordered states in (c)–(e). The sites connected by the solid (dashed) lines represent the layer at (), respectively. States are labeled F for ferro-orbital/ferromagnetic and AF for antiferro-orbital/antiferromagnetic.
Figure 10
The ground-state phase diagram of the tetragonal model in the plane for . The background colors distinguish orbital states. The diagonally shaded areas indicate phases having stripe-AFM order. The symbols indicate the DFT estimates for , , and (see Table2). The open symbols are for the optimized O positions, and the filled symbols are for the experimental values. The spin-orbital configuration of each phase is shown in Fig.11. The values of were set to (iv) in Table2.
Figure 11
Schematic diagrams of the spin-orbital ordered states appearing in the phase diagram for the tetragonal model in Fig.10, The arrows represent spins. The stripe-AF orders have two degenerate states with and . The one stabilized under orthorhombic distortion with is shown.
Figure 12
(a)The orbital and (b)spin-order parameters in the ground state as functions of for and . The labels such as and stand for the Fourier components, where the subscript indicates the configuration in Fig.9. A, C, F, and G represent the labels of spin-orbital ordered phases listed in Fig.11. The hopping parameters are set to (iv) in Table2.
Figure 13
Schematic diagrams of (a)the orthorhombic distortion with and (b)the monoclinic distortion with , and the resultant CEF splitting of the orbitals.
Figure 14
(a)The orbital and (b)spin-order parameters in the orthorhombic model as a function of defined in Eq.(15). The hopping parameters (ii) in Table2 were used. The arrow represents the DFT estimate for the orthorhombic , . (c)Schematic ordering patterns in the orthorhombic model.
Figure 15
The ground-state phase diagram of the orthorhombic model in the plane. The symbols indicate the DFT estimates for . See the caption of Fig.10 for more details. The parameter set (ii) in Table2 were used with .
Figure 16
(a)The orbital- and (b)spin-order parameters in the monoclinic model as a function of with fixed . The hopping parameter set (vi) in Table2 was used. The arrow represents the DFT estimate for monoclinic , . (c)Schematic ordering patterns in the monoclinic model.
Figure 17
Temperature dependence of the order parameters in phase C. (a)The tetragonal parameter set (iv) in Table2 and (b)the orthorhombic parameters (ii) with were used.
Figure 18
Values of the coupling constants in units of in the effective Heisenberg model in Eq.(17). The orange and blue indicate AFM and FM interactions, respectively. (a)3D-AFO ordered state, (b)disordered state. The parameter set for the orthorhombic in (ii) of Table2 was used.
Figure 19
Comparison between the experimental and theoretical finite- phase diagrams.
Figure 21
The ground-state phase diagram and orbital configuration in the orbital-only model. The parameters are the same as in Fig.10.
Figure 22
The ground-state phase diagram in plane with . The parameter set for the tetragonal was used [(iv) in Table2]. The vertical dashed lines indicate the DFT estimates of the value for (see Table2).