Isolated Fe-sites on silica substrate have recently been reported for direct and non-oxidativeconversion of gaseous methane with high selectivity. The activated catalyst was proposed to beFeC2 cluster embedded in silica. Using a combination of density-functional theoretic methodsand micro-kinetic modeling, we show that under the same reaction conditions (1223 K , 1 atm)FeC2 sites convert to FeC3 and the latter is instead responsible for the observed activity. Weinvestigate the detailed mechanism of conversion of methane to methyl radical and hydrogenon FeC3@SiO2 under different conditions of methane partial pressure. We find that methylradical evolution is the rate-determining step for the overall conversion. Our calculations alsoindicate that the conversion of embedded FeC3 to FeC4 competes with methyl radical evolutionfrom the active catalyst. However, due to the higher stability of FeC3 sites, we anticipate thatformation of higher carbides can be inhibited by controlling the hydrogen partial pressure.

Towards precisely controlled nanostructure growth, patterned substrates are used as templates to direct heteroepitaxial self-assembly. This affects the size, shape and ordering of nanostructures, which are formed as a consequence of the mismatch in strain. In the well-studied case of Si-Ge heteroepitaxy on Si, the lattice mismatch leads to spontaneous formation of quantum dots. On patterned substrates, the competition between the length scale of the pattern and the intrinsic quantum dot size leads to rich behavior, where the localization of dots can be modified with respect to the features of the patterns. We show by continuum modeling that, in cubic elastic materials such as silicon and germanium, there is also a competition between the pattern orientation and the elastically soft directions of the film, which affects the precise location of quantum dots on the surface. When the pattern is between the elastically soft directions, the quantum dots can form purely in the narrow region directly between two neighboring pits, referred to as the saddle region. On the other hand, when the pattern is along the elastically soft directions, the quantum dots prefer to form in the region at the centre of four pits, referred to as the crown region. This resolves a discrepancy between theory and experiments and gives another dimension to control quantum dot formation in strained nanocrystalline systems.

Surface stress, which is an inherent property of several crystals, is expected to play a role in morphology of nanostructures. Here the authors describe how surface stress effects can be explicitly incorporated in the computation of quantum dot structure and evolution in coherent heteroepitaxial thin films. In the usual continuum formulation for surface evolution, surface stress enters as an additional contribution to the local chemical potential. Additionally, it modifies the film–vacuum interface boundary condition for the set‐up of the elastic problem. The authors show how the resulting evolution can be computed order‐by‐order in the surface slope. The sign of the surface stress with respect to the mismatch affects the stability of the film to surface undulations. For the case of a positive‐mismatched heteroepitaxial system, a tensile surface stress leads to film stabilization and increases the critical thickness for quantum dot formation in Stranski–Krastanov growth.

The role of elastic anisotropy on quantum dot formation and evolution on a pre-patterned substrate is evaluated within the framework of a continuum model. We first extend the formulation for surface evolution to take elastic anisotropy into account. Using a small slope approximation, we derive the evolution equation and show how it can be numerically implemented up to linear and second order for stripe and egg-carton patterned substrates using an accurate and efficient procedure. The semi-infinite nature of the substrate is used to solve the elasticity problem subject to other boundary conditions at the free surface and at the film-substrate interface. The positioning of the quantum dots with respect to the peaks and valleys of the pattern is explained by a competition between the length scale of the pattern and the wavelength of the Asaro-Tiller-Grinfeld instability, which is also affected by the elastic anisotropy. The alignment of dots is affected by a competition between the elastic anisotropy of the film and the pattern orientation. A domain of pattern inversion, wherein the quantum dots form exclusively in the valleys of the patterns is identified as a function of the average film thickness and the elastic anisotropy, and the time-scale for this inversion as function of height is analyzed.

Using a continuum evolution equation, we model the growth and evolution of quantum dots in the heteroepitaxial Ge on Si(0 0 1) system in a molecular beam epitaxy unit. We formulate our model in terms of evolution due to deposition, and due to surface diffusion which is governed by a free energy. This free energy has contributions from surface energy, curvature, wetting effects and elastic energy due to lattice mismatch between the film and the substrate. In addition to anisotropy due to surface energy which favors facet formation, we also incorporate elastic anisotropy due to an underlying crystal lattice. The complicated elastic problem of the film-substrate system subjected to boundary conditions at the free surface, interface and the bulk substrate is solved by perturbation analysis using a small slope approximation. This permits an analysis of effects at different orders in the slope and sheds new light on the observed behavior. Linear stability analysis shows the early evolution of the instability towards dot formation. The elastic anisotropy causes a change in the alignment of dots in the linear regime, whereas the surface energy anisotropy changes the dot shapes at the nonlinear regime. Numerical simulation of the full nonlinear equations shows the evolution of the surface morphology. In particular, we show, for parameters of the [Formula: see text] [Formula: see text] on Si(0 0 1), the surface energy anisotropy dominates the shapes of the quantum dots, whereas their alignment is influenced by the elastic energy anisotropy. The anisotropy in elasticity causes a further elongation of the islands whose coarsening is interrupted due to [Formula: see text] facets on the surface.