The derivation of the dislocation energy is described in this chapter assuming an isotropic and anisotropic elasticity. In order to show the different results of the two models, they were used to calculate the equilibrium configuration for dislocations in AlN, GaN and InN compounds, all [0001] oriented. Two cases were considered.
At first, dislocations were supposed 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 screw type - as predicted in the isotropic framework - but instead mixed type dislocations.
Secondly, dislocations were supposed to be close to the free surface, i.e., the actual length of the dislocation must be considered. Different crystallographic planes were considered as 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.
In the the case of the inclined facets {1122} and {1101}, the results differ depending on the compound. Regarding AlN, the preferred dislocation line is the [0001] direction independent from the dislocation type and crystallographic plane as 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 free surfaces when the island-growth mode is favored during the deposition of the compounds. The conclusion is that the island-growth mode favors the inclination of the dislocations. The importance of the inclination angle of dislocations will be shown in more detail in Chapter 5 in relation to the reduction of the dislocation density in real heterostructures.
In order to include the dislocation core energy in the evaluation of the equilibrium critical thickness (see Chapter 4), the pre-logarithmic terms of the analytical models are compared with the corresponding values obtained by atomistic simulations. Table 3.1 shows a good agreement between the continuum predictions based on anisotropic elasticity.