Three-dimensional lattice matching of epitaxially embedded nanoparticles

For a given degree of in-plane lattice mismatch between a two-dimensional (2D) epitaxial layer and a substrate (ϵIP*), there is a critical thickness above which interfacial defects form to relax the elastic strain energy. Here, we extend the 2D lattice-matching conditions to three-dimensions in order to predict the critical size beyond which epitaxially encased nanoparticles, characterized by both ϵIP* and out-of-plane lattice mismatch (ϵOP*), relax by dislocation formation. The critical particle length (Lc) at which defect formation proceeds is determined by balancing the reduction in elastic energy associated with dislocation introduction with the corresponding increase in defect energy. Our results, which use a modified Eshelby inclusion technique for an embedded, arbitrarily-faceted nanoparticle, provide new insight to the nanoepitaxy of low dimensional structures, especially quantum dots and nanoprecipitates. By engineering ϵIP* and ϵOP*, the predicted Lc for nanoparticles can be increased to well beyond the case of encapsulation in a homogenous matrix. For the case of truncated pyramidal shaped InAs, Lc ~ 10.8 nm when fully embedded in GaAs (ϵIP*=ϵOP*=-0.072); 16.4 nm when the particle is grown on GaAs, but capped with InSb (ϵIP*=-0.072 and ϵOP*=+0.065); and a maximum of 18.4 nm if capped with an alloy corresponding to ϵOP*=+0.037. The effect, which we term “3D Poisson-stabilization” provides a means to increase the epitaxial strain tolerance in epitaxial heterostructures by tailoring ϵOP*.