Spin-orbital ordering in alkali superoxides (2024)

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  Figure 1

  Summary of the structural phase transitions and magnetic properties in ${\mathrm{CsO}}_2$, ${\mathrm{RbO}}_2$, and ${\mathrm{KO}}_2$ [5, 7, 8, 9].

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  Figure 2

  (a)Structure of tetragonal ${\mathrm{CsO}}_2$ ($I4/mmm$ space group). (b)Wannier functions of oxygen ${\pi }_g^{*}$ orbitals in $({\pi }_a$, ${\pi }_b)$ basis and in $({\pi }_{a+b}$, ${\pi }_{a-b})$ basis.

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  Figure 3

  (a)Band structure and density of states of ${\mathrm{CsO}}_2$ in $I4/mmm$ space group ($T=300\mathrm{K}$ structure). The $\mathit{k}$ points are defined in [30]. (b)Wannier fit of the two bands near $E_{\mathrm{F}}$ with weights of the ${\pi }_a$ and ${\pi }_b$ Wannier orbitals.

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  Figure 4

  (a)Band structure and density of states of ${\mathrm{CsO}}_2$ in $Immm$ space group ($T=40\mathrm{K}$ structure). The $\mathit{k}$ points are defined in [33]. (b)Wannier fit of the two bands near $E_{\mathrm{F}}$ with weights of the ${\pi }_a$ and ${\pi }_b$ Wannier orbitals.

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  Figure 5

  Relevant transfer integrals between ${\pi }_g^{*}$ orbitals on the ${\mathrm{O}}_2^{-}$ ions.

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  Figure 6

  Parameterization of the bond dependence of the transfer integrals using polar coordinates. The symbols indicate the DFT estimates for ${\mathrm{CsO}}_2$, ${\mathrm{RbO}}_2$, and ${\mathrm{KO}}_2$ (only results without the optimization are shown).

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  See Also

  5.1.2: Molecular Orbitals from p Orbitals

  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 $T^z$ with the eigenvalue $+1/2$ ($-1/2$). The orbital state is rotated around the $c$ axis by $\phi /2$ as the operator is rotated by $\phi$ in the pseudospin space.

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  Figure 8

  The orbital-orbital interactions ${\tau }_i^l{\tau }_j^l$ described by the bond-dependent orbital operator ${\tau }_i^l$.

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  Figure 9

  Configurations of the ordered states. The ordering vector $\mathit{q}$ 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 $z=0$ ($z=1/2$), respectively. States are labeled F for ferro-orbital/ferromagnetic and AF for antiferro-orbital/antiferromagnetic.

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  Figure 10

  The ground-state phase diagram of the tetragonal model in the $(\varphi ,\theta )$ plane for $J_{\mathrm{H}}/U=0.10$. The background colors distinguish orbital states. The diagonally shaded areas indicate phases having stripe-AFM order. The symbols indicate the DFT estimates for ${\mathrm{CsO}}_2$, ${\mathrm{RbO}}_2$, and ${\mathrm{KO}}_2$ (see Table2). The open symbols are for the optimized O $z$ 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 $r^l$ were set to (iv) in Table2.

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  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 $\mathit{q}=(1/2,0,0)$ and $\mathit{q}=(0,1/2,0)$. The one stabilized under orthorhombic distortion with $a<b$ is shown.

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  Figure 12

  (a)The orbital and (b)spin-order parameters in the ground state as functions of $\theta$ for $\varphi ={46}^{\circ }$ and $J_{\mathrm{H}}/U=0.1$. The labels such as ${\langle S^z\rangle }_{\text{F}}$ and ${\langle T^x\rangle }_{\text{3D-AF}}$ 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.

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  Figure 13

  Schematic diagrams of (a)the orthorhombic distortion with $a<b$ and (b)the monoclinic distortion with $\gamma >{90}^{\circ }$, and the resultant CEF splitting of the ${\pi }_g^{*}$ orbitals.

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  Figure 14

  (a)The orbital and (b)spin-order parameters in the orthorhombic model as a function of ${\mathrm{\Delta }}_{\mathrm{ortho}}$ defined in Eq.(15). The hopping parameters (ii) in Table2 were used. The arrow represents the DFT estimate for the orthorhombic ${\mathrm{CsO}}_2$, ${\mathrm{\Delta }}_{\mathrm{ortho}}/(t^2/U)=0.60$. (c)Schematic ordering patterns in the orthorhombic model.

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  Figure 15

  The ground-state phase diagram of the orthorhombic model in the $(\varphi ,\theta )$ plane. The symbols indicate the DFT estimates for ${\mathrm{CsO}}_2$. See the caption of Fig.10 for more details. The parameter set (ii) in Table2 were used with ${\mathrm{\Delta }}_{\mathrm{ortho}}/(t^2/U)=0.60$.

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  Figure 16

  (a)The orbital- and (b)spin-order parameters in the monoclinic model as a function of ${\mathrm{\Delta }}_{\mathrm{mono}}$ with fixed ${\mathrm{\Delta }}_{\mathrm{ortho}}/{\mathrm{\Delta }}_{\mathrm{mono}}=0.232$. The hopping parameter set (vi) in Table2 was used. The arrow represents the DFT estimate for monoclinic ${\mathrm{RbO}}_2$, ${\mathrm{\Delta }}_{\mathrm{mono}}/(t^2/U)=2.0$. (c)Schematic ordering patterns in the monoclinic model.

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  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 ${\mathrm{\Delta }}_{\mathrm{ortho}}/(t^2/U)=0.60$ were used.

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  Figure 18

  Values of the coupling constants $J_{ij}$ in units of $t^2/U$ 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 ${\mathrm{CsO}}_2$ in (ii) of Table2 was used.

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  Figure 19

  Comparison between the experimental and theoretical finite-$T$ phase diagrams.

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  Figure 20

  The ratios $J_2^l/J_1^l$ and $J_3^l/J_1^l$ as a function of $J_{\mathrm{H}}/U$.

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  Figure 21

  The ground-state phase diagram and orbital configuration in the orbital-only model. The parameters are the same as in Fig.10.

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  Figure 22

  The ground-state phase diagram in $(\varphi ,J_{\mathrm{H}}/U)$ plane with $\theta =12.0^{\circ }$. The parameter set for the tetragonal ${\mathrm{CsO}}_2$ was used [(iv) in Table2]. The vertical dashed lines indicate the DFT estimates of the $\varphi$ value for ${\mathrm{CsO}}_2$ (see Table2).

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