Strain and piezoelectric effects in quantum-dot structures

A discussion of computational methods for calculating strain and piezoelectric fields in nanostructures is presented. Emphasis is on a comparison of continuum and valence force field atomistic models and the validity of the former in predicting, accurately, strain fields for nanostructures with dimensions down to a few nm. This is done on the experimentally relevant InAs/InGaAs quantum-dot wetting layer structures and on spherical quantum dot structures. We next address the influence of boundary conditions imposed at the computational domain for strain fields near and inside the quantum dot; a point largely missing in literature. Boundary conditions discussed include fixed, free, fixed-free, and periodic, and it is shown that the particular choice of boundary conditions is unimportant for the strain results; a conclusion that allows to choose the computationally most effective one being the fixed-free boundary conditions as it requires the smallest computational domain for obtaining convergent results. While this result is fortunate, it is not obvious from a mathematical point of view. A further important, and a priori not obvious conclusion, is that a continuum model captures well atomistic strain results; a fact that allows us to use a continuum formulation even in cases where structure dimensions are down to only a few lattice constants. In realistically grown structures, inhomogeneous concentration profiling exists. We present investigations for strain and piezoelectric results in the case where a spherical quantum dot region is gradually profiled from GaAs to InAs assuming the concentration is a function of the distance to the quantum dot sphere center. It is shown that quantum dot concentration profiling affects strain fields and biaxial strains in particular, electronic states and hence optical properties. We finally present some effective quasi-analytical studies of electronic states and strain fields in curved quantum dots based on applications of differential geometry and perturbation theory.