Gallium nitride (GaN) and its alloys with aluminum or indium are the most important semiconductors because they have better material and electronic properties compared to silicon and other III-V compounds, like GaAs. These properties enable the use of III-nitride based devices in a broad range of applications, from electronic to optical devices.
Mass production of III-nitrides is already a reality, but despite this, many efforts are still placed on the further development of their technology and the overcoming of challenges posed by the lack of a native substrate, such as the large amount of dislocations.
This thesis provides a theoretical contribution to the definition of design rules for improving the crystalline quality of III-nitrides, in particular, the reduction of the dislocation density. Continuum theory of dislocations treated within the linear elasticity theory and the laws of thermodynamics are used to gain understanding and model the dislocation development in these materials.
Chapter 3 summarizes the calculation of the dislocation energy within the isotropic and anisotropic elasticity. The models were used to calculate the equilibrium configuration for dislocations in AlN, GaN, and InN compounds, all oriented along the [0001] direction.
In the first part of this work, dislocations were assumed to be in bulk, far from the free surface of the material. In this case, the two theories give different results: in monocrystals the a and (a + c) dislocations are not of the screw type – as predicted in the isotropic framework – but instead mixed type dislocations.
Secondly, dislocations were assumed to be close to the free surface. When the free surface is the (0001) plane, elastic anisotropic theory predicts that the dislocation line along the [0001] direction possesses the lowest energy configuration regardless of the dislocation type in both AlN and GaN. In InN only the c-type dislocations are screw dislocations, while the other two types are mixed dislocations.
When the free surfaces are inclined facets {1122} and {1101}, elastic anisotropic theory predicts different results depending on the compound. Regarding AlN, the preferred dislocation line is along the [0001] direction independent from the dislocation type and crystallographic plane as a free surface. The a-type dislocations in GaN and InN propagate almost perpendicular to the {1122} and {1101} facets. Regarding the other types, in GaN the dislocations are inclined by 20∘-30∘, while in InN they are aligned along the [0001] direction.
The inclined facets {1122} and {1101} are the free surfaces when the island-growth mode is favored during the deposition of the compounds. The conclusion is that the island-growth mode increases in general the inclination of the dislocations with respect to the growth direction. The same theoretical result was obtained by Holec [25] for dislocations in GaN single crystals. However, Mathis [49] and other authors [22] measured experimentally other values of the inclination angle. In addition, experimental observations show that the inclination angle is influenced not only by the anisotropic structure of the crystal but also by factors not evaluated in this work. For example, Cantu [7] and Li [43] demonstrated experimentally that the doping of the crystal can influence the inclination angle. The wide scatter of the experimental data of the inclination angle suggests also that an important source of the dislocation inclination is the stress/strain gradient of the structure, as shown by Follstaedt [16]. In order to calculate the stress/strain gradient, it would be necessary to evaluate the numerous factors which influence the mechanical stability of the epilayers. This would address primarily the thermal behavior of the III-nitrides.
The pre-logarithmic terms of the analytical models have been compared with the corresponding values obtained by atomistic simulations. Good agreement has been found with the continuum predictions based on anisotropic elasticity. Therefore both approaches have been combined into a kind of a multiscale approach for predicting the onset of misfit dislocations in thin film, as shown in Section 4.6.
In Chapter 4, different continuum-based approaches for calculating the energy of a straight infinitely long dislocation in an elastic medium have been evaluated. Motivated by the misfit dislocations in a heteroepitaxial interface, we evaluated separately the influence of (i) free surface, (ii) different elastic constants in the film and substrate, and (iii) elastic anisotropy. 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. A new model including elastic anisotropy of the film and the substrate, the difference of their elastic constants and the impact of the film free surface has been proposed. Recalling that this is a model for the equilibrium critical thickness, i.e., it provides a condition when it first becomes energetically favorable to start the relaxation of the mismatch strain via plastic flow, the refined model yields excellent agreement with the available experimental data in the sense that no misfit dislocations are detected below the here predicted threshold.
The calculations of the critical thickness are of a great interest for technologists. The wide scatter of experimental data of the critical thickness suggests that an important role for the generation of the misfit dislocations is played by the growth technique. Evaluating the impact of the growth technique would be helpful to explain the scatter of the experimental data, nevertheless, the theory of dislocations would not be sufficient alone to reach this goal.
Another possible way to improve the current models is by accounting for the dislocation generation mechanism(s) and thus providing a theory which goes beyond the equilibrium critical thickness as presented in this chapter.
In Chapter 5, a new treatment assuming a simplified 2D geometry of the dislocation describes the dislocation density in heterostructures based on III-nitrides. The new model considers both different threading dislocation types with their specific reactions and inclinations and also their glide along the interfaces. The results of the calculations show that (i) the glide increases the probability of reactions among dislocations, reducing their density; (ii) the glide of the dislocations along one interface happen in the first ten nanometers after the critical thickness while the effect of the inclination angle is more evident after hundreds of nanometers after the critical thickness; and (iii) (AlN/GaN)x superlattice structures reduce the dislocation density more efficiently than the Al1-xGaxN step-graded layer because the first ones reduce the dislocation density by nearly one order of magnitude when the thickness of the structure is half μm high, yielding a good agreement with experimental data.
The model of the dislocation density can be improved in several ways. In order to have a model capable of better quantitative prediction of the experimental data of the dislocation density, it would be necessary to consider the real 3D geometry of the threading dislocations. This would imply the assumption that the direction of the misfit dislocation is different from the direction of the upper point of the threading dislocation, i.e. f≠m (see Paragraph 5.4.1).
Another significant improvement toward realism would be to consider dislocation-dislocation interactions. This would involve calculation of dislocation stress fields and, in fact, employing some methods of dislocation dynamics.