Spin-orbit coupled systems in the atomic limit: rhenates, osmates, iridates

Arun Paramekanti, David J. Singh, Bo Yuan, Diego Casa, Ayman Said, Young-June Kim, and A. D. Christianson

Phys. Rev. B 97, 235119 – Published 11 June 2018

Abstract

Motivated by RIXS experiments on a wide range of complex heavy oxides, including rhenates, osmates, and iridates, we discuss the theory of RIXS for site-localized $t_{2 g}$ orbital systems with strong spin-orbit coupling. For such systems, we present exact diagonalization results for the spectrum at different electron fillings, showing that it accesses “single-particle” and “multiparticle” excitations. This leads to a simple picture for the energies and intensities of the RIXS spectra in Mott insulators such as double perovskites which feature highly localized electrons, and yields estimates of the spin-orbit coupling and Hund's coupling in correlated $5 d$ oxides. We present new higher resolution RIXS data at the Re $L_{3}$ edge in ${Ba}_{2} {YReO}_{6}$ which finds a previously unresolved peak splitting, providing further confirmation of our theoretical predictions. Using ab initio electronic structure calculations on ${Ba}_{2} M {ReO}_{6}$ (with $M = Re$ , Os, Ir) we show that while the atomic limit yields a reasonable effective Hamiltonian description of the experimental observations, effects such as $t_{2 g} - e_{g}$ interactions and hybridization with oxygen are important. Our ab initio estimate for the strength of the intersite exchange coupling shows that, compared to the $d^{3}$ systems, the exchange is one or two orders of magnitude weaker in the $d^{2}$ and $d^{4}$ materials, which may partly explain the suppression of long-range magnetic order in the latter compounds. As a way to interpolate between the site-localized picture and our electronic structure band calculations, we discuss the spin-orbital levels of the $M O_{6}$ cluster. This suggests a possible role for intracluster excitons in ${Ba}_{2} {YIrO}_{6}$ which may lead to a weak breakdown of the atomic $J_{eff} = 0$ picture and to small magnetic moments.

Received 5 April 2018
Revised 1 June 2018

DOI:https://doi.org/10.1103/PhysRevB.97.235119

Physics Subject Headings (PhySH)

Electronic structure Magnetism Spin-orbit coupling

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Arun Paramekanti^1,2,*, David J. Singh^3,†, Bo Yuan¹, Diego Casa⁴, Ayman Said⁴, Young-June Kim¹, and A. D. Christianson^5,6,7

¹Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7
²Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8
³Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211-7010, USA
⁴Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA
⁵Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
⁶Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
⁷Department of Physics & Astronomy, University of Tennessee, Knoxville, Tennessee 37966, USA

^*arunp@physics.utoronto.ca
^†singhdj@missouri.edu

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Vol. 97, Iss. 23 — 15 June 2018

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Images

Figure 1
Schematic picture of the $L_{3}$ edge inelastic RIXS processes (for the $5 d^{3}$ osmates), with dashed (purple) lines indicating photon-induced transitions and solid (green) lines indicating interaction-induced transitions. (a) Noninteracting case, where the two-photon process consists of a core electron getting excited into the $j_{eff} = 1 / 2$ manifold followed by a de-excitation transition from $j_{eff} = 3 / 2$ back into the core level. This would lead to a RIXS peak at $3 λ / 2$ . (b)–(d) Interacting case, where the local Hund's coupling leads to visible multiparticle excitations. Within perturbation theory, interactions can scatter electrons into higher energy states as depicted by the solid (green) lines for the (b) initial, (c) intermediate, and (d) final states. For $J_{H} < λ$ , this would lead to two-particle peaks near $3 λ$ but with suppressed intensity $\sim {(J_{H} / λ)}^{2}$ .
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Figure 2
Theoretically computed RIXS spectrum for (a) $d^{2}$ rhenates ( $L_{2}$ edge), (b) $d^{3}$ osmates ( $L_{3}$ edge), and (c) $d^{4}$ iridates ( $L_{3}$ edge) broadened by instrumental resolution. We have used shown best fit values for $λ$ (SOC) and $J_{H}$ (Hund's coupling). In (a), dashed arrows indicate two closely spaced peaks, which are not resolved in $L_{2}$ edge RIXS; see Fig. 3 for new $L_{3}$ edge RIXS which detects this splitting. In all panels, solid arrows indicate two-particle transitions due to Hund's coupling.
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Figure 3
RIXS intensity as a function of energy transfer $ℏ ω$ in ${Ba}_{2} {YReO}_{6}$ . The RIXS spectrum was obtained near the Re $L_{3}$ edge with incident energy $E_{i} = 10.537$ keV. A scattering geometry with $2 θ = 90^{\circ}$ was used to minimize the elastic background. The two indicated peaks at $ℏ ω = 0.40$ and 0.50 eV were unresolved in previous $L_{2}$ edge measurements.
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Figure 4
$5 d$ projections of the electronic density of states onto the Ir, Os, and Re LAPW spheres of majority and minority spin character on a per ion basis. The Os values are offset and the energy zero is at the highest occupied state.
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Figure 5
Schematic single-particle level diagram for Ir and O orbitals in the ${IrO}_{6}$ cluster of ${Ba}_{2} {YIrO}_{6}$ , with filled, partially filled, and empty boxes indicating electron filling. (a) Ir and O orbitals showing small crystal field splitting in the absence of hybridization; dashed black lines show intersite hybridization via hopping matrix elements $t_{σ}, t_{π}$ . Numbers indicate degeneracies of the orbitals including spin. (b) Hybridization leads to a large splitting of the $σ$ -hybridized orbitals, and to a smaller splitting of the $π$ -hybridized orbitals, together with a set of unbonded O orbitals. The notation Ir-O or O-Ir indicates, respectively, that the Ir or O states are the dominant contribution to the hybridized wave function. (c) SOC (thin blue lines) on Ir leads to a splitting of the hybridized Ir-O and O-Ir $t_{2 g}$ states. Nominally filled and unfilled levels are indicated by filled and empty boxes in this final schematic, which would lead to a $J_{eff} = 0$ insulator. Arrows shows possible impact of interactions which might cause a partial occupancy of the $j_{eff} = 1 / 2$ hybridized Ir-O level; such “intracluster excitons” may lead to a Mott insulator with weak magnetic moments.
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Figure 6
$5 d$ projections of the electronic density of states onto the Ir, Os, and Re LAPW spheres of majority and minority spin character on a per ion basis for the FM state. The Os values are offset and the energy zero is at the highest occupied state.
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