Strain Energy on the Surface of an Anisotropic Half-Space Substrate: Effect of Quantum-Dot Shape and Depth

摘要

Quantum-dot (QD) semiconductor synthesis is one of the most actively investigated fields in strain energy band engineering. The induced strain fields influence ordering and alignment, and the subsequent surface formations determine the energy bandgap of the device. The effect of the strains on the surface formations is computationally expensive to simulate, thus analytical solutions to the QD-induced strain fields are very appealing and useful. In this paper we present an analytical method for calculating the QD-induced elastic field in anisotropic half-space semiconductor substrates. The QD is assumed to be of any polyhedral shape, and its surface is approximated efficiently by a number of flat triangles. The problem is formulated as an Eshelby inclusion problem in continuum mechanics whose solution can be expressed by a volume-integral equation involving the Green's functions and the equivalent body-force of eigenstrain. By virtue of the point-force Green's function solution, this volume integral is subsequently reduced to a line integral over [0, π] which is numerically integrated by the Gaussian quadrature. Numerical examples are presented for cubic, pyramidal, truncated pyramidal and point QDs in GaAs (001) and (111) half-space substrates. The strain energy distribution on the surface of the substrate indicates clearly the strong influence of the QD shape and depth on the induced strain energy. This long-range strain energy on the surface has been found to be the main source for determining QD surface size and pattern.