4.5  Overall comparison of various effects on the predicted critical thickness

In order to consider the anisotropy, the difference between the elastic properties of the film and the substrate as well as the impact of the free surface, the Steeds and the Willis et al. approaches are combined. Based on the conclusions of Section 4.3, in the following the contribution of the integral along the core surface S₃ is neglected. The detailed mathematical derivation of this treatment (indicated with Steeds+Willis et al.) is given in 4.2.3. The model Steeds+Willis et al. is used to compute d_d∕dy as a function of the film thickness for three different alloys: an Al_0.2Ga_0.8N film on a GaN substrate, In_0.2Ga_0.8N on GaN and a Si_0.8Ge_0.2 film on a Si substrate. The results are shown and compared with corresponding Freund, Steeds, and Willis et al. results in Figures 4.11, 4.12, and 4.13. The critical thickness values of the systems are listed in Table 4.2. Considering all the three systems, the Willis et al. model increases the critical thickness by ≈ 300% with respect to the Freund model. The critical thickness value of the theoretically most complete scheme, Steeds+Willis et al., is always between the Steeds and the Freund values. In particular, the Steeds+Willis et al. critical thickness is lower by ≈ 50% relative to the Freund critical thickness for Al_0.2Ga_0.8N and In_0.2Ga_0.8N, while it is the same in the case of Si_0.8Ge_0.2.

Figure 4.11:

dd∕dy-d∕dy is a function of the film thickness. dd∕dy is calculated according to the Freund, Steeds, Willis et al., and Steeds+Willis et al. approaches. d∕dy is calculated according to equation (4.9). The critical thickness values an Al0.2Ga0.8N film grown on a GaN substrate are indicated by a circle.