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- ItemAnisotropic surface energy formulations and their effect on stability of a growing thin film(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Korzec, Maciek D.; Münch, Andreas; Wagner, BarbaraIn this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies. We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature. Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov growth that include surface energies also with large anisotropy as well as the effect of wetting energy, elastic energy and a randomly perturbed atomic deposition flux. A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions. These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slow-down of the coarsening of the dots.
- ItemFrom bell shapes to pyramids . continuum model for self-assembled quantum dot growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Korzec, Maciek D.; Evans, Peter L.A continuum model for the growth of self-assembled quantum dots that incorporates surface diffusion, an elastically deformable substrate, wetting interactions and anisotropic surface energy is presented. Using a small slope approximation a thin film equation for the surface profile that describes facetted growth is derived. A linear stability analysis shows that anisotropy acts to destabilize the surface. It lowers the critical height of flat films and there exists an anisotropy strength above which all thicknesses are unstable. A numerical algorithm based on spectral differentiation is presented and simulation are carried out. These clearly show faceting of the growing islands and a logarithmically slow coarsening behavior.
- ItemEquilibrium shapes of poly-crystalline silicon nanodots(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Korzec, Maciek D.; Roczen, Maurizio; Schade, Martin; Wagner, Barbara; Rech, BerndThis study is concerned with the topography of nanostructures consisting of arrays of poly-crystalline nanodots. Guided by transmission electron microscopy (TEM) measurements of crystalline Si (c-Si) nanodots that evolved from a dewetting process of an amorphous Si (a-Si) layer from a SiO2 coated substrate, we investigate appropriate formulations for the surface energy density and transitions of energy density states at grain boundaries. We introduce a new numerical minimization formulation that allows to account for adhesion energy from an underlying substrate. We demonstrate our approach first for the free standing case, where the solutions can be compared to well-known Wulff constructions, before we treat the general case for interfacial energy settings that support partial wetting. We then use our method to predict the morphologies of poly-crystalline silicon nanodots.
- ItemStability analysis of non-constant base states in thin film equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dziwnik, Marion; Korzec, Maciek D.; Münch, Andreas; Wagner, BarbaraWe address the linear stability of non-constant base states within the class of mass conserving free boundary problems for degenerate and non-degenerate thin film equations. Well-known examples are the finger-instabilities of growing rims that appear in retracting thin solid and liquid films. Since the base states are time dependent and do not have a simple travelling wave or self-similar form, a classical eigenvalue analysis fails to provide the dominant wavelength of the instability. However, the initial fronts evolve on a slower time-scale than the typical perturbations. We exploit this time-scale separation and develop a multiple-scale approach for this class of stability problems. We show that the value of the dominant wavelength is rapidly attained once the base state has entered an approximately self-similar scaling. We note that this value is different from the one obtained by the linear stability analysis with "frozen modes", frequently found in the literature. Furthermore we show that for the present class of stability problems the dispersion relation behaves linear for large wavelengths, which is in contrast to many other instability problems in thin film flows.
- ItemStationary solutions of driven fourth- and sixth-order Cahn-Hilliard type equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Korzec, Maciek D.; Evans, P.L.; Münch, A.; Wagner, B.New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function. Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.