Interplay of many-body and single-particle interactions in iridates and rhodates

Natalia B. Perkins, Yuriy Sizyuk, and Peter Wölfle

Phys. Rev. B 89, 035143 – Published 29 January 2014

Abstract

Motivated by recent experiments exploring the spin-orbit-coupled magnetism in $4 d$ - and $5 d$ -band transition metal oxides, we study magnetic interactions in Ir- and Rh-based compounds. In these systems, the comparable strength of spin-orbit coupling, crystal-field splitting, and Coulomb and Hund's coupling leads to a rich variety of magnetic exchange interactions, leading to new types of ground states. Using a strong coupling approach, we derive effective low-energy superexchange Hamiltonians from the multiorbital Hubbard model by taking full account of the Coulomb and Hund's interactions in the intermediate states. We find that in the presence of strong SOC and lattice distortions the superexchange Hamiltonian contains various kinds of magnetic anisotropies. Here we are primarily interested in the magnetic properties of Sr $_{2}$ IrO $_{4}$ and Sr $_{2}$ Ir $_{1 - x}$ Rh $_{x}$ O $_{4}$ compounds. We perform a systematic study of how magnetic interactions in these systems depend on the microscopic parameters and provide a thorough analysis of the resulting magnetic phase diagram. Comparison of our results with experimental data shows good agreement. Finally, we discuss the parameter space in which the spin-flop transition in Sr $_{2}$ IrO $_{4}$ , experimentally observed under pressure, can be realized.

Received 4 November 2013
Revised 18 December 2013

DOI:https://doi.org/10.1103/PhysRevB.89.035143

Authors & Affiliations

Natalia B. Perkins¹, Yuriy Sizyuk¹, and Peter Wölfle^1,2

¹Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
²Institute for Condensed Matter Theory and Institute for Nanotechnology, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany

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Vol. 89, Iss. 3 — 15 January 2014

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Figure 1
Ir-O-Ir bond in the presence of octahedra rotations in one IrO $_{2}$ layer. $x$ and $y$ are the global axes adopted for the intermediate oxygen atoms. $x_{A (B)}$ and $y_{A (B)}$ are local axes on sublattices $A$ and $B$ . (a) Local $\tilde{Z}$ orbital on Ir ion overlaps with $p_{y}$ oxygen orbital in the global reference frame. (b) Local $\tilde{Y}$ orbital on Ir ion overlaps with $p_{z}$ oxygen orbital in the global reference frame.
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Figure 2
(a) Anisotropic exchange couplings $δ J_{x y}, δ J_{z}$ and the DM constant $D$ in meV (shown by red diamond, green circle, and blue square lines, respectively) as functions of Hund's coupling, $J_{H}$ . (b) The exchange couplings $J_{x}, J_{y}, J_{z}$ in meV (shown by gray square, cyan circle, and magenta diamond lines, respectively) as functions of Hund's coupling, $J_{H}$ . The microscopic parameters of the model are considered to be $α = 0$ rad, $Δ = 0.15$ eV, $U_{2} = 1.8$ eV, $λ = 0.4$ eV, and $t_{eff} = 0.13$ eV.
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Figure 3
(a) Anisotropic exchange couplings $δ J_{x y}, δ J_{z}$ and DM constant $D$ in meV (shown by red diamond, green circle, and blue square lines, respectively) and (b) the exchange couplings $J_{x}, J_{y}, J_{z}$ in meV (shown by gray square, cyan circle, and magenta diamond lines, respectively) as functions of the rotation angle $α$ . The other microscopic parameters are considered to be $Δ = 0.15$ eV, $U_{2} = 1.8$ eV, $J_{H} = 0.3$ eV, $λ = 0.4$ eV, and $t_{eff} = 0.13$ eV.
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Figure 4
(a) Anisotropic exchange couplings $δ J_{x y}, δ J_{z}$ and DM constant $D$ in meV (shown by red diamond, green circle, and blue square lines, respectively) as functions of the tetragonal CF splitting computed for $α = 0.2$ rad. (b) The dependencies of the isotropic exchange $J$ on the strength of the tetragonal CF splitting $Δ$ . Green square, orange circle, blue diamond, and magenta triangle lines correspond to $α = 0$ , 0.1, 0.2, and 0.3 rad, respectively. The other parameters are $λ = 0.4$ eV, $U_{2} = 1.8$ eV, $J_{H} = 0.3$ eV, and $t = 0.13$ eV.
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Figure 5
Mean field magnetic phase diagram in the parameter space of rotation angle $α$ and the SOC coupling $λ$ computed (a) in the absence of the tetragonal distortion, $Δ = 0$ , eV and (b) in the presence of the tetragonal distortion, $Δ = 0.15$ eV. In both parameter sets, the obtained magnetic structure is coplanar antiferromagnet with varying spin canting angle $ϕ = [π - (ϕ_{A} - ϕ_{B})] / 2$ , where $ϕ_{A}$ and $ϕ_{B}$ are the polar angles of spins on sublattices $A$ and $B$ . The color on the plots indicates the scale for which the angle $ϕ$ changes with dark blue being the smallest and gray being the highest value of the angle $ϕ$ . The canted spin order is stabilized by a staggered rotation of oxygen tetrahedra in the presence of the SOC. (c) The dependence of the spin canting angle $ϕ = 1 / 2 {tan}^{- 1} (D / J)$ (in units of $α$ ) on the strength of the tetragonal CF splitting $Δ$ computed for $λ = 0.22$ eV (orange square line) and $λ = 0.4$ eV (purple circle line). The other microscopic parameters are chosen to be $U_{2} = 1.8$ eV, $J_{H} = 0.3$ eV, $t_{eff} = 0.13$ eV, and $α = 0.2$ rad.
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Figure 6
(a) Anisotropic exchange couplings $δ J_{x y}, δ J_{z}$ and the DM constant $D$ (in meV) as functions of SOC constant $λ$ shown by red diamond, green circle, and blue square lines, respectively. (b) The DM constant $D$ (in meV) as function of rotation angle $α$ . Black square, cyan circle, and blue diamond lines correspond to $λ = 0.2$ , 0.3, and 0.4 eV, respectively. (a),(b) The tetragonal field is considered to be equal to $Δ = 0.2$ eV. (c) The isotropic exchange $J$ (in meV) and (d) the spin canting angle $ϕ$ (in units of $α$ ) as functions of SOC constant $λ$ . Gray square, cyan circle, and magenta diamond lines correspond to $Δ = 0.1$ , 0.2, and 0.3 eV, respectively. (a),(c),(d) The rotation angle is considered to be equal to $α = 0.2$ rad. The remaining parameters are $J_{H} = 0.5$ eV, $U_{2} = 2.5$ eV, and $t_{eff} = 0.1$ eV.
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