4.4  The critical thickness according to Willis, Jain and Bullough

Willis, Jain, and Bullough  [82] derived a procedure to calculate the energy d_d∕dy of a misfit dislocation at the interface between a film with a finite thickness and a semi-infinite substrate. The film and the substrate are both supposed to be isotropic but with different elastic properties.

Figure 4.9 shows two sets of curves representing d_d∕dy as a function of the film thickness for three different systems: Al_0.2Ga_0.8N film on a GaN substrate, In_0.2Ga_0.8N on GaN, and a GaN film grown on a GaN substrate. One set of the d_d∕dy curves is calculated with the Willis et al. model, while the second uses the Freund treatment. Therefore, the difference stems from including (Willis et al.) or neglecting (F) the free surface effects and the difference in the elastic constants of the film and the substrate. In both cases the isotropic elasticity framework is used. The curves within each set are very close to each other, meaning that the impact of different elastic properties of the film and the substrate is negligible for these material systems. Therefore it can be concluded that the difference between the two sets originates predominantly from the impact of free surface. This factor significantly increases the d_d∕dy term and, as a consequence, also the critical thickness value.

Figure 4.10 shows a similar analysis for two different systems: a Si_0.2Ge_0.8 film on a Si substrate and a Si film grown on a Si substrate. As in the case of the III-nitrides, the variation caused by the different elastic properties of the film and the substrate is negligible. On the other hand, the difference between the Willis et al. and Freund formalism, which is now related predominantly to the inclusion of the free surface, is significant. Therefore, also for silicon as a substrate the free surface increases d_d∕dy and, as a consequence, the critical thickness value.

Figure 4.9:

dd∕dy is a function of the film thickness for the material systems AlxGa1-xN/GaN and InxGa1-xN/GaN. The ⟨1123⟩{1101} slip system is considered. The two sets of curves are calculated through the Willis et al. (WJB) and Freund (F) procedures, respectively.

Figure 4.10:

dd∕dy is a function of the film thickness for Si1-xGex/Si. The ⟨110⟩{111} slip system of a 60∘ dislocation is considered. The two sets of curves are calculated through the Willis et al. (WJB) and Freund (F) procedures, respectively.