Predicted structure and stability of $A_4{B}_3{\text{O}}_{12}$ $\delta$-phase compositions

A combination of atomistic simulation techniques has been employed to predict ordered structures for a series of $A_4{B}_3{\text{O}}_{12}$ $\delta$-phase compounds, where $A$ is a $3+$ cation ranging in size from ${\text{Sc}}^{3+}$ to ${\text{Ho}}^{3+}$ and $B$ is a $4+$ cation ranging from ${\text{Ti}}^{4+}$ to ${\text{Zr}}^{4+}$. Experimentally, a fully ordered cation structure has yet to be resolved for any of these compounds. Monte Carlo energy-minimization calculations using short-range pair potentials identified three low-energy arrangements of $A^{3+}$ and $B^{4+}$ cations. The details of these three structures were analyzed with the layer motif method. To quantitatively determine the $\delta$-phase structure of each composition, the three configurations were reevaluated with density-functional theory. We also used special quasirandom structures to compare the ordered low-energy configurations to cation disorder. For all compositions considered, we find that at least one of the three ordered structures is lower in energy than the disordered structure, suggesting the thermodynamic stability of an ordered phase. Of the three ordered structures identified by this approach, one has not been identified previously in the literature for any composition. In addition, we discuss the stability of $\delta$-phase compounds with respect to other “$AB{\text{O}}_{4-x}$” fluorite-derivative compositions and predict the structure of compositions for which none has been reported.

Figure 1

(Color online) Structure map for $A^{3+}{B}^{4+}{\text{O}}_{4-x}$ compounds from reported structures. Hollow gray triangles denote the monoclinic $A_2{B}_2{\text{O}}_7$ structure (space group $P{2}_1$, No. 4) from Refs. 47, 48, 49, 50, 51, 52, hollow blue squares denote $A_2{B}_2{\text{O}}_7$ pyrochlore (space group $Fd\overline{3}m$, No. 227) from Refs. 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, and hollow orange circles denote the rhombohedral $A_4{B}_3{\text{O}}_{12}$ $\delta$ phase (space group $R\overline{3}$, No. 148) from Refs. 15, 16, 17, 18, 19, 20, 21, 22, 88, 89, 90, 91). Solid black squares denote disordered fluorite (space group $Fm\overline{3}m$, No. 225) from Refs. 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107. Solid symbols of the same shape as stable ordered structures denote that both the ordered structure and disordered fluorite have been observed.

Figure 2

(Color online) Representative layering sequences of cations and anions for the $\delta$-phase structure, where red atoms are oxygen, blue atoms are $A^{3+}$ cations, and green atoms are $B^{4+}$ cations. For simplicity, only the ${\delta }_1$ structure is shown.

Figure 3

(Color online) The structure of the oxygen layers in $\delta$ phase, where the $3^4.6$ Archimedean-tiling pattern is highlighted.

Figure 4

(Color online) Plan view of the three cation layers of ${\delta }_1$, ${\delta }_2$, and ${\delta }_3$ phases.

Figure 5

(Color online) Comparison of experimental to theoretical $c/a$ values for $\delta$-phase compounds (where $a$ is the average of $a$ and $b$ in Table ). Solid circles correspond to values for disordered structures and hollow circles correspond to the lowest-energy ordered structure. Horizontal lines denote specific compounds; along each horizontal line appears an ordered and disordered $c/a$ value for that compound. Also shown are the ideal $c/a$ $\left(=\sqrt{6/7}=0.9258\right)$ and the $\text{slope}=1$, denoting perfect agreement between theory and experiment.