Prediction of perovskite-related structures in $A{\mathrm{CuO}}_{3-x}$ ($A$ = Ca, Sr, Ba, Sc, Y, La) using density functional theory and Bayesian optimization

Oxygen vacancy ordering in perovskite-type transition-metal oxides plays an important role in the emergence of exotic electronic properties, as typified by superconducting cuprates. In this study, we predict the stability of oxygen-deficient perovskite structures in $A{\mathrm{CuO}}_{3-x}$ ($A$ = Ca, Sr, Ba, Sc, Y, La) by density functional theory calculation. We introduce a combination of the cluster expansion method, Gaussian process, and Bayesian optimization to find stable oxygen-deficient structures among a considerable number of candidates. Our calculations not only reproduce the reported structures but suggest the presence of several unknown oxygen-deficient perovskite structures, some of which are stabilized at high pressures. This work demonstrates the great applicability of the present computational procedure for the elucidation of the structural stability of strongly correlated oxides with a large tolerance for oxygen deficiency.

Figure 1

A complete set of nonequivalent oxygen-deficient structures of cubic perovskite for supercell sizes of $n=1$ and $n=2$ in ${\mathrm{ACuO}}_{3-x}$ ($0\le x\le 1$).

Figure 2

Formation energies of nonequivalent oxygen-deficient perovskite structures with up to $n=4$. The green line shows the convex hull of formation energy.

Figure 3

Distribution of energies of nonequivalent oxygen-deficient structures with up to $n=4$ predicted using a CE model with 36 clusters up to four bodies (model 3 in Table 2) in ${\mathrm{SrCuO}}_{3-x}$. The Pearson correlation coefficient between the DFT and CE energies is $R=0.90$.

Figure 4

Convergence behavior of the convex hull of formation energy in Bayesian optimization for $A{\mathrm{CuO}}_{3-x}$ systems. The vertical axis indicates the area of the convex hull below the line indicating zero formation energy, schematically illustrated as the shaded area at the top of the right panel. The area is normalized by the area of the converged convex hull.

Figure 5

Formation energy distribution of oxygen-deficient perovskite structures with up to $n=8$ selected by Bayesian optimization. The dashed black line and black solid circles show the convex hull of formation energy and its vertices, respectively, obtained only from oxygen-deficient perovskite structures. The purple solid triangles show the formation energies of experimental prototype structures. The green line shows the convex hull obtained from the combination of oxygen-deficient perovskite structures and the experimental prototype structures. The oxygen-deficient perovskite structures reported in the literature the convex hulls are indicated by p-1 (${\mathrm{Ca}}_2{\mathrm{Mn}}_2{\mathrm{O}}_5$ type) and p-2 (${\mathrm{LaNiO}}_2$ type). Their crystal structures are also shown. Moreover, the experimental structures on the convex hulls are indicated as e-1 (${\mathrm{Sc}}_2{\mathrm{Cu}}_2{\mathrm{O}}_5$ type), e-2 (delafossite ${\mathrm{FeCuO}}_2$ type), and e-3 (${\mathrm{SrCuO}}_2$ type).

Figure 6

(a) Pearson correlation coefficients between oxygen-deficient perovskite structures on the convex hull shown by the solid circles in Fig. 5. (b) Pair clusters used in the CE method and Bayesian optimization. (c) Dependence of WC-SRO parameters on the group classified by a hierarchical clustering. A bar shows the WC-SRO parameter averaged over the oxygen-deficient perovskite structures belonging to each group. The cluster indexes correspond to those of pairs shown in (b).

Figure 7

(a) Formation enthalpy distribution at 10 GPa of the oxygen-deficient perovskite structures satisfying the condition that their energies measured from the convex hull of the oxygen-deficient perovskite structures are less than 0.1 eV/$A{\mathrm{CuO}}_{3-x}$. The dashed black line and solid circles show the convex hull of the oxygen-deficient perovskite structures and its vertices, respectively. The purple closed triangles show the formation enthalpy values of experimental prototype structures. The green line shows the convex hull obtained from the combination of the oxygen-deficient perovskite structures and the experimental prototype structures. (b) Crystal structures of the oxygen-deficient perovskites with composition $x\ge 0.4$ on the convex hull or almost on the convex hull. Only ${\mathrm{LaCuO}}_{2.6}$ (p-7) is experimentally reported [7]. It has a crystal structure type identical to that of ${\mathrm{SrMnO}}_{2.6}$ [49].

Figure 8

Pressure-enthalpy curves of the oxygen-deficient perovskite structure (p-1) and ${\mathrm{Sc}}_2{\mathrm{Cu}}_2{\mathrm{O}}_5$-type structure (e-1) in ${\mathrm{SrCuO}}_{2.5}$.

Figure 9

Crystal structures of the oxygen-deficient perovskites (a) on the convex hull and (b) almost on the convex hull at 10 GPa except those shown in Figs. 5 and 7. The dark blue, sky blue, light blue, and pink polyhedrons show sixfold, fivefold, planer fourfold, and tetrahedral fourfold Cu-O polyhedrons.