A general perturbation method for inhomogeneities in anisotropic and piezoelectric solids with applications to quantum-dot nanostructures

By introducing a homogeneous piezoelectric material and its Green’s function, we present a new semi-analytical three-dimensional perturbation method for general inhomogeneity problems in anisotropic and piezoelectric solids. This method removes the limitations associated with previous analytical methods, which often ignore the anisotropic properties or the difference between the material properties of the inhomogeneity and its surrounding matrix. As an important application, the proposed theory is employed to calculate the elastic and electric fields in a truncated pyramidal InAs/GaAs quantum-dot (QD) nanostructure. Numerical results demonstrate that the anisotropy of the materials and the difference between the material constants of the QD and the matrix have a significant influence on the strain and electric fields. The relative differences of the strain and electric field inside the QD between the simplified isotropic and homogeneous model and the real anisotropic and heterogeneous one may reach 22% and 53%, respectively. The accuracy of the calculated elastic strain and electric fields is improved greatly by a second order approximate solution (OAS). Since the third OAS nearly coincides with the second one, good convergence of the iteration procedure is demonstrated. Moreover, contours of the hydrostatic strain and electric potential within and around the QD are also presented and analyzed.