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 (epsilon*(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 epsilon*(IP) and out-of-plane lattice mismatch (epsilon*(IP)), 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 epsilon*(IP) and epsilon*(OP), the predi(c)ted 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 similar to 10.8 nm when fully embedded in GaAs (epsilon*(IP) = epsilon*(OP) =-0.072); 16.4 nm when the particle is grown on GaAs, but capped with InSb (epsilon*(IP) =-0.072 and epsilon*(OP)= 0.065); and a maximum of 18.4 nm if capped with an alloy corresponding to epsilon*(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 epsilon*(OP).