Christian Ratsch

Physical processes during epitaxial growth and many other problems in computational physics and computational materials sciences span length and time scales of many orders of magnitude. For example, on the microscopic level, atoms move several Angstroms (the atomic lattice constant), and vibrate with a frequency of approx. 1013 s-1. On the other hand, phenomena and applications of practical interest occur on a timescale of seconds, with system sizes that can be microns or larger. The grand challenge in computational materials sciences and applied mathematics is to link those vastly different time and length scales. In this talk, I will discuss different theoretical techniques that I have worked with. As an example, I will focus on the description of epitaxial growth. Here, the structures of interest are up to microns in size, but the quality of the growing film is determined by the motion of individual atoms. I will show that the different theoretical techniques that are valid on different time and length scales are in fact complementary to each other, and form a hierarchy of growth models. Density-functional theory (DFT) is a fully quantum-mechanical approach that allows us to calculate the energetics of microscopic processes with very high accuracy. I will show how DFT can be used to obtain microscopic parameters, such as diffusion constants of surface atoms. These microscopic parameters are indispensable input for more coarse grained models, which can be either stochastic models or completely analytic models. I will describe a kinetic Monte Carlo method as well as the newly developed level-set method. The level-set method is a general method to simulate the motion of a boundary that separates the phases of any two-phase problem. It is based on the solution of a set of coupled partial differential equations. I will discuss how this method can be adapted to study the formation of islands and the motion of their boundaries during epitaxial growth.