Transition-metal impurities and intercalation in Bi$_2$Se$_3$

The prototype topological insulator Bi$_2$Se$_3$ consists of 5-layer (QL) units. Using first-principles calculations, we show that even for large (20\%) elongations along the c-axis, the in-plane lattice constant remains essential unchanged and the nearest neighbor bond lengths within a QL vary by only $\sim$0.02\,\AA. These results suggest that impurities may preferentially intercalate between the QLs, possibly leading to $\delta$-doped topological insulator superlattices. For Cu-intercalated Bi$_2$Se$_3$, the calculated separation between QLs slightly contracts ($\sim$2\%), and the Cu intercalation layer provides the internal surfaces necessary for the material to exhibit a Dirac cone. The competition between substitutional impurities and intercalation layers for Cu and Mn will be discussed and compared to experiment.