Variational models for materials science: Epitaxial growth and geometrically constrained walls

Materials science is a field that incorporates many branches of science, such as physics, chemistry, engineering, and, last but not least, mathematics. This dissertation specifically focuses on crystal growth and magnetism. We develop variational models to describe these two phenomena.

The first line of research concerns thin films, whose relevance and production have been growing exponentially due to the increasing demand from electronic devices and many other applications. Among the countless ways a crystal can form, we investigate the case in which a crystalline material is deposited layer by layer onto a fixed crystalline substrate. If the atoms of the substrate at the interface occupy the natural lattice positions of the thin film, such crystal growth is called epitaxial.

The second topic of this research concerns magnetic domain walls, which are regions where the magnetisation of a material transitions from one orientation to another. In particular, we study changes in magnetisation in domains with extreme geometries, such as dumbbell-shaped domains. The analysis of magnetisation behaviour in such geometries is relevant in micro- and nano-electronics applications, where the neck of the dumbbell serves as a model for magnetic contact points.