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Self-assembled quantum dot growth Below a growing surface h(x,y,t) developing humps on a nano-scale, so called quantum dots, is visualized. This can happen in different systems such as germanium deposited by molecular beam epitaxy onto a silicon substrate. Detailed information about modeling, analysis and numerical simulation of thin crystalline surfaces growing due to self-assembly will be given soon in my PhD thesis. Also at the Matheon project C-10 homepage information are given. Tekalign and Spencer [1] derived a nonlinear partial differential equation from a surface diffusion equation with a chemical potential that includes isotropic surface energy, stress and wetting effects. The resulting evolution can be simulated with a pseudospectral method. Below an animated gif shows how the surface evolves. The highest points of the surface at each time is given a white color and the lowest (slightly above zero - a thin layer) is black. [1] W. T. Tekalign and B. J. Spencer, ''Evolution equation for a thin epitaxial film on a deformable substrate'', J. Appl. Phys. 96(10), 2004.

Extension to anisotropy We extended the model by an anisotropy term, so that the behavior is more realistic to real processes (see the recent work From bell shapes to pyramids: A reduced continuum model for self-assembled quantum dot growth M.D. Korzec and P. L. Evans, Physica D 239(8), 2010. In the below picture shows a small part of a domain at a fixed time. On the left the isotropic dots (G=0) are shown, on the right (G=0.25) we see clearly pyramidal faceting due to anisotropy. The next two animated gifs show how the surfaces evolve with time for two different anisotropy strengths. You see initially rounded dots that eventually obtain a squared form (pyramids) that grow in height and width and coarsen. However, here also a top view is used, such that high regions are only indicated by bright colors. The two figures use a different anisotropy strength, in the right evolution it is increased, as it is clearly visible by the more quadratic shapes of the dots.

Extension to deposition flux Furthermore the model is extended by a deposition flux, which is necessary to simulate realistic quantum dot self-assembly. Here three figures show the deposition, starting from a flat film of 2ML thickness that is randomly perturbed and ending at an average thickness of 10ML. While for 0.00213 nm/s the wetted regions are still very pronounced, it is visible for the rates 0/0053 nm/s and 0.01 nm/s (from left to right the rates are growing) that the island density increases as in experiments. The animations have different time-scales and a different amount of frames per second.

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