A serious issue for the reliability and performance of heterostructure devices is the deteriorating impact of dislocations, mostly originating from epitaxy, when a thin layer is grown on a substrate with significantly different lattice parameters. Below a certain layer thickness, called the critical thickness, the layer is grown pseudomorphically on a substrate, i.e., the layer is grown with the same lattice parameter as the substrate. Consequently, the layer is strained, leading to large strain energy. When the critical thickness is reached, a relaxation of the strain occurs via plastic flow. The most common mechanism of plastic relaxation is an introduction of misfit dislocations along the interface between the film and the substrate.
Several models have been proposed to calculate the critical thickness. In 1974 Matthews and Blakeslee [50] compared the force exerted by strain on the extension of the misfit dislocation line with the tension in the dislocation line acting against its elongation. Their model assumes that both the film and the substrate have the same isotropic elastic properties, the film has a finite thickness (however, neglecting free surface effects) and the substrate is semi-infinite. Starting from the same hypotheses, Freund [17,19] compared the energy necessary to create a misfit dislocations with the energy inside the fully strained thin film. He arrived at the same critical thickness formula as Matthews and Blakeslee. However, the two models differ conceptually: Matthews and Blakeslee assumed a pre-existing dislocation in the substrate which creates an misfit dislocation segment along the heteroepitaxial interface due to the action of the misfit stress; Freund assumed a freshly generated dislocation at the film surface which glides into the interface between the film and the substrate. The impact of the free surface of the film and the difference between the elastic constants of the film and the substrate on the critical thickness was discussed by Willis, Jain and Bullough [82], yet they still worked in the framework of isotropic elasticity.
The anisotropy of the heterostructure can be considered in the critical thickness calculation using the methodology developed by Steeds [72] for the energy of an infinitely long straight dislocation inside an anisotropic medium. This has been done in the works by Holec et al. [26–28], however, the free surfaces and differences in elastic response of the film and the substrate were ignored.
Here, the gap is closed by evaluating the impact of different effects – elastic anisotropy (Section 4.3), difference between the elastic constants of the film and the substrate and the free surface of the film (Section 4.4) – on the critical thickness values. After reviewing the methodology in Sections 4.2-4.4, this method is applied to several material systems, namely, AlxGa1-xN/GaN, InxGa1-xN/GaN, and Si1-xGex/Si, in Section 4.5, and compared to available experimental data from literature in Section 4.6.