After illustrating the differences among various treatments of dislocation energy and their impact on the equilibrium critical thickness , the Freund and Steeds+Willis et al. models, including the dislocation core energies estimated in Section 3.6, are used to calculate the equilibrium critical thickness as a function of composition, x, for the three different alloys. The two models, Freund and Steeds+Willis et al., are chosen as they are based on opposite hypotheses (see Table 4.1).
The 1∕3⟨113⟩{101} and 1∕3⟨113⟩{112} slip systems for AlxGa1-xN film on GaN substrate and InxGa1-xN film on GaN substrate, and the 60∘ misfit dislocation with the ⟨110⟩{111} slip system for Si1-xGex film on Si substrate are considered. The calculated critical thickness results obtained from the Freund and Steeds+Willis et al. models are compared with experimental observations and data available in literature (Figures 4.14, 4.15, and 4.16). The difference between the Freund and Steeds+Willis et al. results is small, with the Steeds+Willis et al. model yielding lower values as compared to F. In all cases the experimental data are close to the theoretical curves, suggesting that the experimental critical thickness values were obtained from epitaxial depositions close to the thermodynamical equilibrium.
Regarding the AlxGa1-xN/GaN and InxGa1-xN/GaN systems, the Steeds+Willis et al. model provides a more severe condition for the onset of the misfit dislocation at the interface than the Freund model. It is important to realize that the critical thickness reported here is the so-called equilibrium critical thickness . That means that it corresponds to the configuration where it for the first time becomes energetically favorable to relieve the misfit strain by introducing misfit dislocations. However, any mechanism for the creation of the misfit dislocations, which may require certain extra activation energy, is not considered in the model. Similarly, parameters influencing the kinetics of the epitaxial deposition, such as temperature and deposition rate, are also not considered by the equilibrium critical thickness models. Finally, the current experimental techniques are unable to detect the exact onset of the appearance of misfit dislocations. It therefore follows that no misfit dislocations are expected below the predicted critical thickness values, however, the detection of misfit dislocations may be (sometimes significantly) higher than the theoretical critical thickness.
| Figure 4.14: | The equilibrium critical thickness hc as a function of the AlN mole fraction x calculated through Freund and Steeds+Willis et al. (S+WJB) models including the core energy for two different slip systems. The theoretical curves are compared to experimental data, 1- [41], 2- [77], 3- [6], 4- [20], 5- [15]. Empty and filled circles indicate the absence and the presence of misfit dislocations respectively. Crosses indicate the experimental value of the critical thickness hc. |
| Figure 4.15: | The equilibrium critical thickness hc as a function of the InN fraction x calculated through Freund and Steeds+Willis et al. (S+WJB) models including the core energy for two different slip systems. The theoretical curves are compared to experimental data, 1 - [44], 2- [71], 3- [46], 4- [33], 5- [60], 6- [62], 7- [28]. Empty and filled circles indicate the absence and the presence of misfit dislocations respectively. Crosses indicate the experimental value of the critical thickness hc. |
| Figure 4.16: | The equilibrium critical thickness hc as a function of the Ge fraction x calculated with the Freund (F – dashed line) and Steeds+Willis et al. (S+WJB – solid line) models including the core energy. The 60∘ misfit dislocation with the ⟨110⟩{111} slip system is considered. The theoretical curves are compared with experimental data from [30]. Empty and filled circles indicate the absence and the presence of misfit dislocations respectively. |
A closer inspection of Figures 4.14-4.16 reveals that the here refined Steeds+Willis et al. model fulfills this criterion (unlike the Freund model). Indeed for all the studied systems, the misfit dislocations are always experimentally detected above the critical thickness values predicted by the model Steeds+Willis et al. unlike the values given by the Freund model.