Initial stages of misfit stress relaxation through the formation of prismatic dislocation loops in GaN–Ga$_{2}$O$_{3}$ composite nanostructures

The initial stages of misfit stress relaxation through the formation of rectangular prismatic dislocation loops in model composite nanostructures have been considered. The nanostructures are either spherical or cylindrical GaN shells grown on solid or hollow $\beta$-Ga$_{2}$O$_{3}$ cores or planar thin GaN films on $\beta$-Ga$_{2}$O$_{3}$ substrates. Three characteristic configurations of prismatic dislocation loops, namely, square loops, loops elongated along the GaN/Ga$_{2}$O$_{3}$ interface, and loops elongated along the normal to the GaN/Ga$_{2}$O$_{3}$ interface, have been analyzed. The generation of prismatic dislocation loops from the interface into the bulk of the GaN shell (film), from the free surface into the GaN shell (film), and from the interface into the $\beta$-Ga$_{2}$O$_{3}$ core (substrate) has been investigated. It has been shown that, for the minimum known estimate of the lattice misfit (2.6%) in some of the considered nanostructures, no any prismatic dislocation loops can be generated. If the generation of prismatic dislocation loops is possible, then in all the considered nanostructures, the energetically more favorable case corresponds to prismatic dislocation loops elongated along the GaN/Ga$_{2}$O$_{3}$ interfaces, and the more preferred generation of prismatic dislocation loops occurs from the GaN free surface. The GaN/Ga$_{2}$O$_{3}$ nanostructures that are the most and least resistant to the formation of prismatic dislocation loops have been determined. It has been found that, among the considered nanostructures, the planar two-layer GaN/Ga$_{2}$O$_{3}$ plate is the most resistant to the generation of prismatic dislocation loops, which is explained by the action of an alternative mechanism for the relaxation of misfit stresses due to the bending of the plate. The least resistant nanostructure is the planar three-layer GaN/Ga$_{2}$O$_{3}$/GaN plate, in which GaN films have an identical thickness and which itself as a whole does not undergo bending. The critical thicknesses of the GaN shells (films), which must be exceeded to ensure the growth of these shells (films) so as to avoid the formation of prismatic dislocation loops, have been calculated for all the studied nanostructures and three known estimates of the lattice misfits (2.6, 4.7, and 10.1%).