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 quantum 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 misfit dislocation nucleation in SK and VW growth islands whereby they predict a power-law relation between misfit strain, epsilon(m), and the minimum island size to nucleate a misfit dislocation, R(c): R(c) proportional to epsilon(1/(lambda-1))(m) where lambda <1 is a function of the island-substrate contact angle. This problem is treated here in three dimensions as an application of a numerical dislocation simulation using the finite element method to take proper boundary conditions 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.