In this chapter, the different continuum-based approaches for calculating the energy of a straight infinitely long dislocation in an elastic medium has been revisited. Motivated by the misfit dislocations in a heteroepitaxial interface, the influence of (i) the free surface, (ii) different elastic constants in the film and substrate, and (iii) elastic anisotropy have been evaluated separately. The results suggest that starting from a homogeneous infinite isotropic medium, the inclusion of a free surface increases the dislocation energy, and the difference in elastic constants of the film and substrate does not play any significant role (because it is typically an order of magnitude smaller than the impact of, e.g., the free surface), while the inclusion of elastic anisotropy decreases the dislocation energy.
Finally, the equilibrium critical thickness was calculated for three important heteroepitaxial material systems, namely an AlxGa1-xN film on a GaN substrate, an InxGa1-xN film on a GaN substrate, and a Si1-xGex film on a Si substrate. The models for the equilibrium critical thickness, described in Section 4.2, provide a condition when it first becomes energetically favorable to start the relaxation of the mismatch strain via plastic flow. The results suggest that the model [8,24], reported in Paragraph 4.2.3, including the elastic anisotropy of the film and the substrate, the difference of their elastic constants and the impact of the film free surface, yields an excellent agreement with the available experimental data in the sense that no misfit dislocations are detected below the here predicted threshold.