Energetics of misfit- and threading-dislocation arrays in heteroepitaxial films.

Abstract

A theory relating the separation of misfit dislocations to lattice mismatch and film thickness in heteroepitaxial thin films is presented. From this, the energy as a function of dislocation spacing is calculated and is shown to include an attractive and repulsive region. The dislocation-formation energy and Peierls barrier to network ordering are shown to be estimable on the basis of measured dispersions in dislocation spacings. The spacing is predicted to be more uniform as the mismatch increases. Thermodynamic functions, such as the compressibility of the dislocation network, can be calculated from the energy--dislocation-spacing relationship. A formula relating the equilibrium dislocation spacing to film thickness, mismatch, and misfit-dislocation character is also derived. Finally, the density of threading dislocations is calculated both at the heterojunction and at the film surface, by assuming a threading-dislocation reaction process. The results are shown to be in good agreement with the experimental data for ${\mathrm{Si}}_{\mathit{x}}$${\mathrm{Ge}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$/Si, InSb/GaAs, and ${\mathrm{In}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As/GaAs structures.