Mathematical Modeling of the Formation of Surface Nanostructures in Thin Solid Films

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The self-assembly of quantum dots (QDs) in thin solid films is an important area of nanotechnology with many relevant applications. In the present thesis, three problems related to the growth and self-assembly of QDs are investigated. In Chapter 1, a new instability mechanism for the formation of QDs associated with strong surface energy anisotropy coupled with wetting interactions between the film and the substrate is proposed. A nonlinear anisotropic evolution equation describing the shape of a thin solid film deposited on a solid substrate is derived and the stability analysis of a planar film is performed. The wetting interactions are found to change the instability spectrum from long-wave to short-wave, leading to the possibility of the formation of stable regular arrays of QDs. Near the short-wave instability threshold, it is found that the formation of stable hexagonal arrays of QDs is possible. In Chapter 2, the effects of wetting interactions on another mechanism of QD formation are investigated. This mechanism is associated with the Asaro-Tiller-Grinfeld instability that releases epitaxial stress caused by the lattice mismatch between the film and the substrate. The elasticity problem in the long-wave approximation is solved and a nonlocal integro-differential equation governing the evolution of the film surface is derived. It is shown that wetting interactions can change instability spectrum from the spinodal decomposition type to the Turing type leading to the possibility of pattern formation. For typical semiconductor systems, hexagonal arrays of QDs are found to be unstable as a result of a subcritical bifurcation. It is shown that the QDs coarsen after formation and the coarsening dynamics are studied by numerical simulations. In Chapter 3, the formation of an epitaxial film by molecular beam epitaxy (MBE), which precedes the formation of QDs, is investigated. The Burton-Cabrera-Frank theory for the growth of a stepped crystal surface is studied when the adatom diffusion is anomalous (Levy flights). The step-flow velocity is obtained as an eigenvalue of the corresponding superdiffusion problem described by a fractional partial differential equation. The crystal surface growth rate is found as a function of the terrace length and the anomalous diffusion exponent.

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  • 08/07/2018
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