Origin of Spinel Nanocheckerboards via First Principles

Mordechai Kornbluth and Chris A. Marianetti

Phys. Rev. Lett. 114, 226102 – Published 3 June 2015

Abstract

Self-organizing nanocheckerboards have been experimentally fabricated in Mn-based spinels but have not yet been explained with first principles. Using density-functional theory, we explain the phase diagram of the ${ZnMn}_{x} {Ga}_{2 - x} O_{4}$ system and the origin of nanocheckerboards. We predict total phase separation at zero temperature and then show the combination of kinetics, thermodynamics, and Jahn-Teller physics that generates the system’s observed behavior. We find that the ${011}$ surfaces are strongly preferred energetically, which mandates checkerboard ordering by purely geometrical considerations.

Received 15 January 2015

DOI:https://doi.org/10.1103/PhysRevLett.114.226102

Authors & Affiliations

Mordechai Kornbluth and Chris A. Marianetti^*

Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA

^*chris.marianetti@columbia.edu

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Vol. 114, Iss. 22 — 5 June 2015

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Figure 1
Sketch of experimental results presented in Ref. [5]. For $x_{Mn} \leq 0.25$ a single-phase cubic structure ( $C$ ) appears, while for $x_{Mn} \geq 0.85$ it is a single-phase tetragonal structure ( $T$ ). For intermediate concentrations, a single-phase tetragonal structure ( $T^{'}$ ) is observed upon rapid quenching (the high-temperature phase), while slower cooling generates phase-separated nanocheckerboards with (011) and $(01 \bar{1})$ interfaces. Arrows in the tetragonal regions show the tetragonally elongated direction. The concentrations on the abscissa are those for which data were reported in Ref. [5]. The concentrations within the nanocheckerboards were not reported as being measured directly.
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Figure 2
(a) Energies of formation of 192 ${ZnMn}_{x} {Ga}_{2 - x} O_{4}$ supercells. (b) Phase diagram of mixed ZGO and ZMO, using ATAT’s EMC2 code. The stable phases (with respect to a semi-grand-canonical ensemble, where $x_{Mn}$ may vary) are shown at varying temperatures. The phases are marked: cubic ( $C$ ), tetragonal ( $T$ ), phase-separated ( $C + T$ ), and high-temperature fully mixed tetragonal ( $T^{'}$ ).
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Figure 3
Phase diagram for the noncooperative Jahn-Teller transition in spinels. DFT calculates the distortion (left axis) of our ZMGO structures [as in Fig. 2] at 0 K. The critical temperature for the tetragonal-to-cubic transition (right axis) is obtained based on the fitting $T_{c} \approx 9260 (c / a - 1) K$ , empirically accurate for a variety of spinels [40]. Therefore, at room temperature, $x_{Mn} ≲ 0.25$ is cubic.
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Figure 4
(a) Formation energies of ZGO/ZMO slabs layered in the given direction, normalized per $B$ site, plotted against the average thickness $t$ of the ZGO and ZMO slabs. The $t \to \infty$ limit corresponds to the contribution of $B$ sites far from the interface; finitely valued $t$ averages the contribution of all sites, including those near the interface. See also Figs. 5 and 5. (b) The same, multiplied by the number of $B$ sites ( $N$ ) and divided by the interface area ( $A$ ), giving a per-area normalization. (c) Fitting cubic ( $C$ ) and tetragonal ( $T$ ) domains with a coherent interface along ${011}$ surfaces. Arrows show the tetragonally elongated direction in corresponding domains. Geometrically, coherent ${011}$ $C$ - $T$ interfaces require a slight angular rotation (shown) between adjacent $C$ and $T$ regions. The consequent zigzag interface can match either an extended equivalent domain (illustrated on the left of the subfigure) or a commensurate domain (on the right), forming a checkerboard structure. This generates tetragonal domains rotated by 90° and cubic domains rotated by $θ$ (see the text).
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Figure 5
(a) Three types of ZMGO structures used to decompose slab formation energies. Black arrows represent cross-sectional biaxial strain, calculated by fully relaxing the heterostructure. All calculations use periodic boundary conditions. (b), (c) Slab formation-energy decomposition, as defined in the text. Motivation for unit choice is described in the text; for comparison, note that a cubic unit cell has 16 $B$ sites and a (100)-cross-sectional area of approximately $74 Å^{2}$ . Negative contact energy, as in the ${011}$ surface, indicates energetic preference for the layered-slab heterostructure over the strained bulk.
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