Strain distribution in quantum dot of arbitrary polyhedral shape: Analytical solution in closed form

TL;DR

This paper presents a novel analytical expression for the strain distribution in quantum dots of arbitrary polyhedral shape within an isotropic elastic medium, simplifying previous solutions and applicable to various geometries.

Contribution

It introduces a closed-form analytical solution for strain in polyhedral quantum dots using electrostatic-elastic analogy, applicable to multiple shapes and potentially extendable to anisotropic media.

Findings

Derived a strain expression involving face solid angles and edge potentials.

Simplified calculations for pyramidal, truncated pyramidal, and hut-shaped quantum dots.

Discussed potential extension to anisotropic elastic media.

Abstract

An analytical expression of the strain distribution due to lattice mismatch is obtained in an infinite isotropic elastic medium (a matrix) with a three-dimensional polyhedron-shaped inclusion (a quantum dot). The expression was obtained utilizing the analogy between electrostatic and elastic theory problems. The main idea lies in similarity of behavior of point charge electric field and the strain field induced by point inclusion in the matrix. This opens a way to simplify the structure of the expression for the strain tensor. In the solution, the strain distribution consists of contributions related to faces and edges of the inclusion. A contribution of each face is proportional to the solid angle at which the face is seen from the point where the strain is calculated. A contribution of an edge is proportional to the electrostatic potential which would be induced by this edge if it is charged with a constant linear charge density. The solution is valid for the case of inclusion having the same elastic constants as the matrix. Our method can be applied also to the case of semi-infinite matrix with a free surface. Three particular cases of the general solution are considered--for inclusions of pyramidal, truncated pyramidal, and "hut-cluster" shape. In these cases considerable simplification was achieved in comparison with previously published solutions. A generalization of the obtained solution to the case of anisotropic media is discussed.