Analysis of quantum-dot-induced strain and electric fields in piezoelectric semiconductors of general anisotropy

Characteristics of the self-organized quantum dots (QDs) such as electron and hole energy levels and wave functions are dependent to the state of strain and electric field produced during the growing process of QDs in a semiconductor substrate. The calculation of the strain and electric field is one of the most challenging components in the QDs simulation process. It involves material anisotropy induced coupling between the elastic and electric fields and it must include the full three-dimensional and usually intricate shapes of the QDs. Numerical simulations are often performed by finite difference, finite element, or atomistic techniques, all require substantial computational time and memory. In this paper, we present a new Green’s function approach which takes into account QDs of arbitrary shape and semiconductor substrates with the most general class of anisotropy and piezoelectricity. Following the literature of micromechanics, the problem is formulated as an Eshelby inclusion problem of which the solution can be expressed by a volume-integral equation that involves the Green’s functions and the equivalent body-force of eiegenstrain. The volume integral is subsequently reduced to a line integral based on exploiting a unique structure of the Green’s functions. The final equations are cast in a form that most of the computational results can be repeatedly used for QDs at different locations—a very attractive feature for simulating large systems of QD arrays. The proposed algorithm has been implemented and validated by comparison with analytical solutions. Numerical simulations are presented for pyramidal QDs in the substrates of gallium arsenide (GaAs) (001