Role of atomic radius and $d$-states hybridization in the stability of the crystal structure of $M_2{\mathrm{O}}_3$ $(M=\text{Al}$, Ga, In) oxides

We study the stability of the corundum, gallia, and bixbyite structures of ${\mathrm{Al}}_2{\mathrm{O}}_3,{\mathrm{Ga}}_2{\mathrm{O}}_3$, and ${\mathrm{In}}_2{\mathrm{O}}_3$ with density functional theory calculations. To artificially control the relative position of the $d$ states within the band structure, we add a Hubbard-like on-site Coulomb interaction to the $d$ states. We quantitatively show that smaller (larger) atomic radii favor the corundum (bixbyte) structure, which supports empirical models based on the atomic radius ratio between the cation and anions and the spacing-filling condition. Thus, ${\mathrm{Al}}_2{\mathrm{O}}_3$ and ${\mathrm{In}}_2{\mathrm{O}}_3$ crystallizes in the corundum and bixbyite structures, which is consistent with experimental observations. The empirical models based on atomic radius and space filling would predict a corundum or bixbyite structure for ${\mathrm{Ga}}_2{\mathrm{O}}_3$. However, as expected from experimental observations, we find gallia to be the most stable structure for ${\mathrm{Ga}}_2{\mathrm{O}}_3$. Our results explain why ${\mathrm{Ga}}_2{\mathrm{O}}_3$ crystallizes in the gallia structures instead of the corundum or bixbyite as follows. The stability of gallia increases as the hybridization of the Ga $d$ states with the O $2s$ states grows and the $p\text{−}d$ splitting increases, which is maximized by the presence of fourfold cation sites. The presence of the fourfold cation sites in gallia is a key structural feature that increases its relative stability compared with the corundum and bixbyite structures for ${\mathrm{Ga}}_2{\mathrm{O}}_3$, which contain only sixfold cation sites, so that the effect of the $d$ states is unimportant.

Figure 1

Equilibrium PBE crystal structures for ${\mathrm{Al}}_2{\mathrm{O}}_3,{\mathrm{Ga}}_2{\mathrm{O}}_3$, and ${\mathrm{In}}_2{\mathrm{O}}_3$. The space group and the relative total energy per formula unit $(\Delta E_{\mathrm{tot}}^i=E_{\mathrm{tot}}^i-E_{\mathrm{tot}}^{\mathrm{bixbyite}})$ are shown above and below the structures, respectively. The numbers in parentheses are the results without $d$ states in the valence for the Ga and In atoms. The cation and anion atoms are indicated by the large and small spheres, respectively, and the dashed lines depict the unit cells in the calculations.

Figure 2

Relative total energy of ${\mathrm{Ga}}_2{\mathrm{O}}_3$ and ${\mathrm{In}}_2{\mathrm{O}}_3$ as functions of the effective Hubbard parameter, $U_{\mathrm{eff}}$.

Figure 3

Relative total energy as a function of the cation average radius $d_{\mathrm{av}}/2$ for Al, Ga, and In atoms in different crystal structures. The ${\mathrm{Al}}_2{\mathrm{O}}_3,{\mathrm{Ga}}_2{\mathrm{O}}_3$, and ${\mathrm{In}}_2{\mathrm{O}}_3$ systems are represented by a lone symbol, a continuous line, and a dashed line respectively.

Figure 4

(a) Local densities of states for the corundum, gallia, and bixbyite structures of ${\mathrm{Al}}_2{\mathrm{O}}_3,{\mathrm{Ga}}_2{\mathrm{O}}_3$, and ${\mathrm{In}}_2{\mathrm{O}}_3$ respectively. (b) Local densities of states for O-$s$, O-$p$, and Ga-$d$ for ${\mathrm{Ga}}_2{\mathrm{O}}_3$ in the $C2/m$ structure, for the indicated $U_{\mathrm{eff}}$. (c) Local densities of states for O-$s$ and Ga-$s$, O-$p$ and Ga-$p$, Ga-$d$ for ${\mathrm{Ga}}_2{\mathrm{O}}_3$ in the $C2/m$ (gallia) structure with different PAW projectors for Ga: Ga $\left(3d^{10}4s^{2}4p^1\right)$ and Ga $\left(4s^{2}4p^1\right)$. The dashed vertical line marks the Fermi level.