Revised model for thermopower and site inversion in ${\mathrm{Co}}_{3}{\mathrm{O}}_{4}$ spinel

Revised model for thermopower and site inversion in ${Co}_{3} O_{4}$ spinel

Taylor D. Sparks, Aleksander Gurlo, Michael W. Gaultois, and David R. Clarke

Phys. Rev. B 98, 024108 – Published 30 July 2018

Abstract

In recent years, the fundamental understanding of thermopower in strongly correlated systems as it relates to spin and orbital degeneracy has advanced, but the consequences for determining cation distribution have not been considered. In this work we compare measurements of the electrical conductivity and thermopower in ${Co}_{3} O_{4}$ spinel with different models for the thermopower based on different assumptions of the cation distributions. Using thermopower measurements we calculate the in situ cation distribution according to three different models: case (i) the original Heikes equation, case (ii) Koshibae and co-workers' modified Heikes equation, and case (iii) a new model, using the modified Heikes equation but with contributions from both octahedral and tetrahedral sites. We find that only the modified Heikes equation that includes contributions from both octahedral and tetrahedral sites satisfies the constraints of stoichiometry in the spinel structure. The findings suggest either complete $(\sim 100 %)$ inversion of the spinel structure if no change in spin state, or a combination of a minimum 40% inversion with a change in spin state of the octahedral ${Co}^{3 +}$ cation.

Received 16 May 2018
Revised 9 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.024108

Physics Subject Headings (PhySH)

Spin state transition Thermopower Transport phenomena

Spinels

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Taylor D. Sparks^*

Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, USA

Aleksander Gurlo

Institute for Material Science and Technologies, Technische Universität Berlin, D-10587 Berlin, Germany

Michael W. Gaultois

Leverhulme Research Centre for Functional Materials Design, The Materials Innovation Factory, Department of Chemistry, University of Liverpool, 51 Oxford Street, Liverpool, L7 3NY, United Kingdom

David R. Clarke

School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

^*sparks@eng.utah.edu

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Vol. 98, Iss. 2 — 1 July 2018

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Figure 1
An Arrhenius plot of electrical conductivity for ${Co}_{3} O_{4}$ measured in air shows three characteristic regions, each with a different activation energy. Above 1188 K, ${Co}_{3} O_{4}$ decomposes to CoO, a band conductor.
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Figure 2
${Co}_{3} O_{4}$ has a maximum thermopower of $\sim 600 μ V$ /K near 850 K before the thermopower decreases linearly with temperature until transformation to CoO. This decrease in thermopower with increasing temperature is attributed to inversion of the spinel structure. In the range 900 to 1100 K the CoO sample likely begins to transform to ${Co}_{3} O_{4}$ during the thermopower measurement before returning to the CoO phase at high temperatures.
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Figure 3
Possible occupation of tetrahedral and octahedral sites in spinel-type ${Co}_{3} O_{4}$ at elevated temperatures due to the inversion and spin unpairing processes and the corresponding electronic degeneracies $(g = g_{spin} g_{orbital})$ . Spin degeneracy is $g_{spin} = 2 S + 1$ , where $S$ is the spin. Total orbital degeneracy is calculated as a product of orbital degeneracies for each orbital type $(t_{2 g}, e_{g}$ , ...): $\prod \frac{(n_{orbital})!}{(n_{orbital} - n_{e})! n_{e}!}$ , where $n_{orbital}$ is the number of orbitals of one type and $n_{e}$ denotes the number of unpaired electrons on this specific orbital type.
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Figure 4
Fractional occupancy of ${Co}^{3 +}$ ions on the octahedral sublattice in ${Co}_{3} O_{4}$ as a function of temperature calculated from the experimental thermopower measurements of ${Co}_{3} O_{4}$ in air according to case (i) (squares), case (ii) (circles), and case (iii) (diamonds) described in detail in the text. The solid line is the distribution calculated thermodynamically from free energies according to the procedure given by Chen and co-workers [14]. Good agreement with Chen and co-workers' prediction was found for case (iii), proposed in this work, with no variable parameters. Case (i) also had good agreement, but only when an unphysical value of $β = 2$ was used.
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Figure 5
Thermopower plotted against fractional occupancy of ${Co}^{3 +}$ ions on the octahedral sublattice according to cases (i), (ii), and (iii) described in detail in the text. The vertical dotted line represents the minimum possible value for ${Co}_{3} O_{4}$ . As thermopower decreases due to inversion, both cases (i) and (ii) approach the limit of $[{Co}_{O}^{3 +}] = 0$ , in violation of stoichiometry in the spinel structure. Meanwhile, case (iii) remains consistent with stoichiometry requirements.
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Figure 6
Fractional occupancy of ${Co}^{3 +}$ ions on the octahedral site in ${Co}_{3} O_{4}$ as a function of temperature accounting for different octahedral ${Co}^{3 +}$ spin states in case (ii) (left) and case (iii) (right) The solid line is the distribution calculated from free energies according to the procedure given by Chen et al. [14]. The dashed horizontal line represents the minimum ${Co}^{3 +}$ octahedral occupancy based on stoichiometry.
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Physical Review B

covering condensed matter and materials physics

Revised model for thermopower and site inversion in ${Co}_{3} O_{4}$ spinel

Taylor D. Sparks, Aleksander Gurlo, Michael W. Gaultois, and David R. Clarke

Phys. Rev. B 98, 024108 – Published 30 July 2018

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Physical Review B

covering condensed matter and materials physics

Revised model for thermopower and site inversion in Co3O4 spinel

Taylor D. Sparks, Aleksander Gurlo, Michael W. Gaultois, and David R. Clarke

Phys. Rev. B 98, 024108 – Published 30 July 2018

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Revised model for thermopower and site inversion in ${Co}_{3} O_{4}$ spinel