Surface nanostructures that possess a minimum in formation energy as a function of cluster size are promising systems for production of self-assembled quantum dots. However, it has turned out to be somewhat difficult to find many such systems, because the size dependent energetics requires either the presence of strain fields due to the lattice mismatch, or dot-dot interaction providing the energetics driving the size-selection. Recently it has been demonstrated that by using the energetic ion beams during cluster growth the cluster size distribution can be controlled [1]. Particularly interesting is the Ge/Si system where ion bombardment modifies the strain field leads to enhanced transition from two-dimensional (2D) to three-dimensional (3D) growth modes, and consequently to narrowing of island size distribution, as compared with conventional MBE experiments [1]. These effects of ion bombardment can be traced back to the modification on surface strain and generation of surface defects, which both affect the free energy of clusters.
In this report, the size selection of nanodots during the ion bombardment assisted growth is studied by using a reaction kinetic model, where reaction rates depend on the cluster size through the formation energy of 2D and 3D clusters. The characteristic feature of the reaction rates is assumed to be the bi-stable free energy function, which has two minima corresponding the 2D and 3D clusters [2]. We construct the rate equations describing the growth in such bi-stable system, which are solved by using a particle coalescence simulation method. We find that such system may become metastable against cluster growth by ripening either in state corresponding the 2D clusters or 3D clusters, and that by using ion beams, transformation from 2D to 3D metastable state can be promoted. In both states the size distribution is closely Gaussian, but the ion bombardment makes the final 3D cluster distribution narrower than the distribution corresponding smaller 2D-clusters. These results agree with the experimental findings of the effect of ion bombardment on growth of nanoclusters.
[1] A.V. Dvurechenskii et al., Surface & Coatings Technology 196, 25– 29 (2005).
[2] D. J. Vine et al. , Phys. Rev. B 72, 241304R (2005). |