Enthalpy of vacancy migration in Si and Ge

Values for the enthalpy of vacancy migration ${\Delta H}_m$ in Si and Ge of 0.33 and 0.2 eV, respectively, have been deduced from EPR and ir absorption experiments at low temperature. However, at high temperature, quenching and diffusion experiments yield values for ${\Delta H}_m$ of 1.2 and 1.0 eV. There are corresponding discrepancies in the values for the entropy of vacancy migration ${\Delta S}_m$, although ${\Delta S}_m$ values are not accurately determined in these experiments. It is here demonstrated that both empirical conclusions are correct and, in particular, that various suggestions of a complex species to account for the high-temperature migration are inconsistent with any reasonable thermodynamic analysis. It is concluded that the reason for this large change in ${\Delta H}_m$ and ${\Delta S}_m$ with temperature is that the predominant mode of single-vacancy migration, i.e., the transition state, at high temperatures is different than that at low temperatures. One may explain this change in the dominant transition state for migration and account for both sets of values for ${\Delta H}_m$ by resort to the macroscopic cavity model of the vacancy which was introduced by Phillips and Van Vechten. In that model there are two contributions to the enthalpy of a vacancy—a contribution due to the breaking of covalent bonds (short-ranged forces) and a larger contribution due to long-ranged forces. Both of these contributions have previously been determined from data completely independent of the vacancies. The high-temperature values of ${\Delta H}_m$ can be explained by assuming that, in addition to the bond-breaking term, there is a contribution from long-ranged forces. The latter is calculated by assuming that the lattice is undistorted beyond the nearest-neighboring atoms to the vacancy and that the anisotropy of these long-ranged forces at the vacancy is the same as is observed for the bulk crystal. For this mode ${\Delta S}_m$ is large. The low-temperature value of ${\Delta H}_m$ is accurately given as just the energy required to break the additional bonds necessary for vacancy migration. Thus, the contribution from long-ranged forces observed at high temperatures is avoided at low temperatures by a correlated motion of the atoms around the vacancy which minimizes the enthalpy of the vacancy at the saddle point and thus also minimizes ${\Delta H}_m$. For this mode ${\Delta S}_m$ is small and apparently even negative. The macroscopic model also provides a description of the atomic displacements and Jahn-Teller distortion about the vacancy when it is at a lattice site.