Size Effects and Scaling in Misfit Dislocation Formation in Self-Assembled Quantum Dots Lawrence H. Friedman, D. M. Weygand and E. van der Giessen Netherlands Institute for Metals Research, Netherlands U. of Groningen, Applied Physics, Micromechanics Group Nijenborgh 4, 9747 AG Groningen, Netherlands {L.Friedman, D.M.Weygand, E.van.der.Giessen}@phys.rug.nl ABSTRACT 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, ǫ m , and the minimum island size to nucleate a misfit dislocation, R c : R c ∝ ǫ 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. Keywords: dislocations, modelling, stress-concentration, quantum dots, self-assembly 1 Introduction Stranski-Krastanow (SK) and Volmer-Webber (VW) epitaxial growth modes are being used as a tool to pro- duce self-assembled quantum dots. Typical systems for this method of dot production include Ge/Si, GeSi/Si, InAs/GaAs and InGaAs/GaAs [1],[2]. The self-assembled dots form due to the lattice mismatch between the de- posited material and the substrate. As a mismatch strain relief mechanism, the deposited material grows in the SK or VW mode (Fig. 1) where islands form, rather than in the F. van der Merwe mode, monolayer by monolayer. The same mismatch strain that drives three dimensional growth also drives the formation of defects such as misfit dislocations. Motivated by the two-dimensional model of Johnson and Freund [2], a three-dimensional model of misfit dislocation nucleation at the island corners is presented here as an applica- tion of a three-dimensional dislocation dynamics simu- lation with treatment of arbitrary boundary conditions using the finite element method [3], [4]. A key result of the two-dimensional model was a power-law relating the transition island radius at which a coherent growth island will form a misfit dislocation, R c , to the island’s mismatch strain, ǫ m , R c = Kǫ 1 λ−1 m , (1) where the coefficient and the exponent are obtained an- alytically. A major goal of the three-dimensional mod- elling is to verify that the power-law holds in three di- mensions, R c = K 2D ǫ 1 λ−1 m ⇐⇒ R c = K 3D ǫ 1 λ−1 m . (2) In addition to the technological interest in self-assem- bled quantum dots, there is a more general interest in developing models of dislocations with full treatment of boundary conditions. In the development of such models, it is useful to have a simplified quantitatively treatable example that is also scientifically or techno- logically significant, and misfit dislocation formation in epitaxial islands is such an example. It is also of inter- est to improve micro-scale models or develop multi-scale models. The example presented here can be treated by both micro-level models such as dislocation dynamics and atomistic models, thus making an important con- nection between these two scales. An additional point of interest is the use of compu- tationally more efficient two-dimensional models [5] in place of three-dimensional models. Therefore, in addi- tion to making a connection between scales, it is im- portant to make a connection between dimensions. The three-dimensional model of misfit dislocation formation is designed to be easily compared with the two-dimension- al Johnson and Freund model. 2 Two-Dimensional Model 2.1 Growth and Dislocation Formation The formation of misfit dislocations in coherent epi- taxial islands is a transition driven by changing the scale of the system, and it fits inside the larger context of island growth which is also characterized by changing