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 (ϵ_IP*), 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 ϵ_IP* and out-of-plane lattice mismatch (ϵ_OP*), relax by dislocation formation. The critical particle length (L_c) 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 ϵ_IP* and ϵ_OP*, the predi_cted L_c for nanoparticles can be increased to well beyond the case of encapsulation in a homogenous matrix. For the case of truncated pyramidal shaped InAs, L_c ~ 10.8 nm when fully embedded in GaAs (ϵ_IP*=ϵ_OP*=-0.072); 16.4 nm when the particle is grown on GaAs, but capped with InSb (ϵ_IP*=-0.072 and ϵ_OP*=+0.065); and a maximum of 18.4 nm if capped with an alloy corresponding to ϵ_OP*=+0.037. The effect, which we term “3D Poisson-stabilization” provides a means to increase the epitaxial strain tolerance in epitaxial heterostructures by tailoring ϵ_OP*.