Size Effects and Scaling in Misfit Dislocation Formation in Self-Assembled Quantum Dots

Growth islands due to large mismatch strain arising in Stranski-Krastanow (SK) and Volmer-Weber (VW) film growth can be used to produce large arrays of quan-tum dots. This same mismatch strain may also cause misfit dislocations to form, presenting a quality control problem. Johnson and Freund (J. Appl. Phys. 81(9), 1997, p6081) developed a two-dimensional model of mis-fit dislocation nucleation in SK and VW growth islands whereby they predict a power-law relation between mis-fit strain, m, and the minimum island size to nucleate a misfit dislocation, Rc: Rc ∝ 1/(λ−1)m, where λ < 1 is a function of the island-substrate contact angle. This problem is treated here in three dimensions as an appli-cation of a numerical dislocation simulation using the finite element method to take proper boundary condi-tions into account. The predictions are analyzed in the context of the Johnson–Freund model, and modification of the power-law is shown to be necessary