Unusual Structural Relaxation for Rare-Earth Impurities in Sapphire:Ab InitioStudy of Lanthanum

A first-principles density-functional local density approximation study of substitutional La, using 120-atom supercells of $\alpha -\mathrm{Al}{}_2{\mathrm{O}}_3$, shows that La assumes a highly displaced position, moving over $0.5Å$ from the original Al site. The La-O nearest neighbor distance is much smaller than the sum of standard $\mathrm{La}{}^{+3\mathrm{}}$ and ${\mathrm{O}}^{-2\mathrm{}}$ ionic radii. This is due to La being more ionic than in its own oxide, and to the role of the $5d\mathrm{shell}$, as the electron density within the displaced La is quite anisotropic. We discuss the implications for a systematic description of rare-earth ions implanted in sapphire.