  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.
