Energetics of transient enhanced diffusion of boron in Ge and SiGe

We study the energetics and migration of boron in Ge and ordered, Si-epitaxial ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ via first-principles calculations, considering specifically the interstitial-mediated mechanism previously associated with B transient enhanced diffusion in Si. The temperature dependence of the migration length $\lambda$ of a B-interstitial complex is calculated from migration barriers and dissociation energies. In Ge, the migration length hardly depends on temperature, while in SiGe it is similar to that in Si, due to the preference of B for Si-like equilibrium sites and diffusion paths. The calculated solubility of B in Ge is similar to that in Si, about $1\times {10}^{19}{\mathrm{cm}}^{-3}.$ In Si-epi strained SiGe the solubility is instead enhanced by two orders of magnitude, and in free-standing SiGe by one order of magnitude. The calculated activation energy for B diffusion in Ge $\left(\sim 4.5\mathrm{eV}\right)$ is considerably higher than in Si $\left(\sim 3.6\mathrm{eV}\right)$ and in our model SiGe, in accordance with recent experiments.