A machine learning approach for studying low-energy dislocation distributions: methodology and applications to Ge/Si(001) films

Controlling the density of dislocations and their spatial distribution in semiconductor heterostructures is an important ingredient of microelectronics device design. Computer simulations can greatly help to shed light on the complex behavior of such line defects, also in view of the severe challenges posed by high-resolution, time-dependent experimental observations. In bulk systems and in ideal corrugation-free (i.e. perfectly flat) films, the dislocation stress fields can be obtained analytically. If one is interested in non-trivial nanostructures or more complex morphologies, numerical solutions via, e.g., Finite Element Methods (FEM) are needed, increasing the computational demand. Here we present a convenient Machine Learning-based approach which allows one to significantly speed up calculations of both forces and total elastic energies, while maintaining the accurate description of the FEM solution. We decompose both energy and forces for an arbitrary dislocation distribution in one and two body terms which can be obtained by simpler and computationally cheaper evaluations. For an assigned morphology, this permits one to use FEM on a reduced configuration space, exploited to build a training set. We then train on such set an artificial neural network (NN) to approximate dislocation energy functions, which allow for a faster evaluation of both energies and forces acting on a system of an arbitrary number of dislocations. This new approach is here tested in simulations of SiGe films on Si(100), a system of direct interest for the microelectronic industry. Specifically, we focus on the influence of surface undulations on the minimum-energy distribution for dislocations. This is achieved by standard Monte Carlo (MC) implementations where time-consuming FEM energy calculations are replaced by faster and cheaper NN approximations. Results are consistent with previous experimental and theoretical investigations. In particular, the formation of interfacial arrays of edge dislocations has been obtained for flat films, while isolated 60° dislocations are predicted for films with strong undulations. Importantly, the overall computation speed-up obtained using the NN functions exceeded three orders of magnitude. The NN functions have also been exploited to run dislocation dynamics simulations based on the fitted Peach-Koehler forces. Results for graded SiGe layers show the presence of a thermodynamic driving force towards the formation of dislocation pile-ups and are here illustrated. Our method is currently implemented in 2D only. Extensions to full 3D problems, however, are envisioned and could provide a valuable tool to explore large distribution of dislocations in complex geometries.