Disentangling orbital and spin exchange interactions for ${\mathrm{Co}}^{2+}$ on a rocksalt lattice

Disentangling orbital and spin exchange interactions for ${Co}^{2 +}$ on a rocksalt lattice

P. M. Sarte, R. A. Cowley, E. E. Rodriguez, E. Pachoud, D. Le, V. García-Sakai, J. W. Taylor, C. D. Frost, D. Prabhakaran, C. MacEwen, A. Kitada, A. J. Browne, M. Songvilay, Z. Yamani, W. J. L. Buyers, J. P. Attfield, and C. Stock

Phys. Rev. B 98, 024415 – Published 17 July 2018

Abstract

Neutron spectroscopy was applied to study the magnetic interactions of orbitally degenerate ${Co}^{2 +}$ on a host MgO rocksalt lattice where no long-range spin or orbital order exists. The paramagnetic nature of the substituted monoxide ${Co}_{0.03} {Mg}_{0.97} O$ allows for the disentanglement of spin exchange and spin-orbit interactions. By considering the prevalent excitations from ${Co}^{2 +}$ spin pairs, we extract seven exchange constants out to the fourth coordination shell. An antiferromagnetic next-nearest-neighbor $180^{\circ}$ exchange interaction is dominant; however, dual ferromagnetic and antiferromagnetic interactions are observed for pairings with other pathways. These interactions can be understood in terms of a combination of orbital degeneracy in the $t_{2 g}$ channel and the Goodenough-Kanamori-Anderson rules. Our work suggest that such a hierarchy of exchange interactions exists in transition-metal-based oxides with a $t_{2 g}$ orbital degeneracy.

Received 23 October 2017
Revised 16 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.024415

Physics Subject Headings (PhySH)

Antiferromagnetism Exchange interaction Magnetism Spin-orbit coupling

Oxides

Inelastic neutron scattering Magnetization measurements

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

P. M. Sarte^1,2, R. A. Cowley^3,*, E. E. Rodriguez⁴, E. Pachoud^1,2, D. Le⁵, V. García-Sakai⁵, J. W. Taylor⁵, C. D. Frost⁵, D. Prabhakaran³, C. MacEwen⁶, A. Kitada⁷, A. J. Browne^1,2, M. Songvilay^2,6, Z. Yamani⁸, W. J. L. Buyers^8,9, J. P. Attfield^1,2, and C. Stock^2,6

¹School of Chemistry, University of Edinburgh, Edinburgh EH9 3FJ, United Kingdom
²Centre for Science at Extreme Conditions, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
³Department of Physics, Clarendon Laboratory, University of Oxford, Park Road, Oxford OX1 3PU, United Kingdom
⁴Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA
⁵ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom
⁶School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
⁷Department of Materials Science and Engineering, Kyoto University, Yoshida-honmachi, Sakyo, Kyoto 606-8501, Japan
⁸National Research Council, Chalk River, Ontario K0J 1JO, Canada
⁹Canadian Institute of Advanced Research, Toronto, Ontario M5G 1Z8, Canada

^*Deceased.

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Vol. 98, Iss. 2 — 1 July 2018

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Figure 1
(a) Cubic (room-temperature) rocksalt $F m \bar{3} m$ crystal structure of CoO [25]. The pair distances between first shell (nearest) neighbors, second shell (next-nearest) neighbors, etc. are denoted by $m = 1, 2$ , etc., respectively. (b) The effective pair Hamiltonian ${\hat{H}}_{pair}$ for ${Co}_{0.03} {Mg}_{0.97} O$ . (c) The energy eigenvalues of the single-ion Hamiltonian including a molecular field from magnetic order with Kanamori's estimate [16] of $J_{2}$ shown by the solid red line.
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Figure 2
(a) Background (using pure and nonmagnetic MgO) subtracted powder-averaged neutron-scattering intensity maps of ${Co}_{0.03} {Mg}_{0.97} O$ measured on (a, top left) MARI at 5 K with an $E_{i} = 30$ meV, (a, middle left) MARI at 5 K with an $E_{i} = 10$ meV and (a, bottom left) IRIS at 11 K with an $E_{f}$ of 1.84 meV revealing seven low-energy bands of dispersionless magnetic excitations. The right column shows $| Q |$ -integrated cuts. Labels denote the coordination shell $m$ and the type of coupling present with label $n$ , both of which are determined in Fig. 3. (b) The black curve denotes the pair energy splitting as a function of the normalized exchange $Δ E (|\frac{J}{λ}|)$ . The points are measured energy positions from (a). The gray line is the same relationship derived using the projection theorem in the large- $λ$ limit [19, 20].
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Figure 3
(a) Constant- $E$ cut $(Δ E = [12, 14]$ meV) from MARI at 5 K with an $E_{i} = 30$ meV. The green curve is a fit to Eq. (5) with $| R | = 4.2 (3) Å (m = 2$ pairs). The red curve is with $| R |$ fixed as 2.98 Å $(m = 1$ pairs). (b) Scaled and form-factor-corrected $| Q |$ dependence of the intensities for all magnetic excitations with $| R |$ calculated from the fitting routine described in (a). The solid black curve is $1 - \frac{sin (| Q | | R_{n} |)}{| Q | | R_{n} |}$ . (c) Constant- $| Q |$ cut (MARI, $E_{i} = 10$ meV) showing a different temperature dependence for the two peaks despite both being from $m = 1$ pairs. (d) Normalized temperature dependence of the Bose-factor-corrected integrated intensity for all seven excitations (Fig. 2) showing two universal curves calculated (dashed lines) for antiferromagnetic and ferromagnetic coupling. Both the integrated intensities and the calculated behavior of antiferromagnetic or ferromagnetically coupled pairs were normalized by $I_{F} (T)$ , as described in the main text. The inset is a pictorial representation of the sign of $J$ as predicted by the GKA rules [40, 41, 42]—antiferromagnetism (left) is a result of exchange between two half-filled and completely filled $t_{2 g}$ orbitals, while weaker ferromagnetism (right) is a result of exchange between a half-filled and completely filled $t_{2 g}$ orbitals. Yellow arrows denote local $t_{2 g}$ spin configurations and teal arrows denote total spin configurations on each ${Co}^{2 +}$ .
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Physical Review B

covering condensed matter and materials physics

Disentangling orbital and spin exchange interactions for ${Co}^{2 +}$ on a rocksalt lattice

P. M. Sarte, R. A. Cowley, E. E. Rodriguez, E. Pachoud, D. Le, V. García-Sakai, J. W. Taylor, C. D. Frost, D. Prabhakaran, C. MacEwen, A. Kitada, A. J. Browne, M. Songvilay, Z. Yamani, W. J. L. Buyers, J. P. Attfield, and C. Stock

Phys. Rev. B 98, 024415 – Published 17 July 2018

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Physical Review B

covering condensed matter and materials physics

Disentangling orbital and spin exchange interactions for Co2+ on a rocksalt lattice

P. M. Sarte, R. A. Cowley, E. E. Rodriguez, E. Pachoud, D. Le, V. García-Sakai, J. W. Taylor, C. D. Frost, D. Prabhakaran, C. MacEwen, A. Kitada, A. J. Browne, M. Songvilay, Z. Yamani, W. J. L. Buyers, J. P. Attfield, and C. Stock

Phys. Rev. B 98, 024415 – Published 17 July 2018

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Disentangling orbital and spin exchange interactions for ${Co}^{2 +}$ on a rocksalt lattice