Simulations of strained films evolution: extending accessible timescales through Convolutional Neural Networks

In recent years Neural Networks (NN) have proven to be a flexible and effective tool to accelerate materials simulations [1]. Here we present a proof of concept, integrating this class of methods to traditional models describing the evolution of thin heteroepitaxial films driven by the (generalized) chemical potential. This term comprises surface energy, strain effects and possibly interactions with the underlying substrate. For simplicity, the 2D, isotropic case [2] is considered. In this context, the surface elastic energy density is one of the most time-consuming chemical potential contributions to calculate. While for small slope configurations this term can be obtained via semi-analytical approaches [3], for general profiles numerical solution via Finite Element Method (FEM) is usually required. Since tracking a full evolution may require several hundred thousand integration steps [4], FEM calls represent one of the main computational bottlenecks in simulating large systems over long timescales. We show that using only a moderate dataset (~70’000 examples) of FEM evaluations as examples, an efficient and accurate NN model which directly maps the free surface profile to the elastic energy density can be obtained. Approximation errors are discussed and quantified on several different free surface morphologies. Once the NN model is trained, it is then used to run simulations using a simple, forward-Euler integration scheme. Importantly, computational costs are reduced by several orders of magnitude, allowing for the simulation of large systems (up to hundreds of times the training examples) at time scales hard to access via full FEM approaches. As a technologically relevant case, applications to simulations of the morphological evolution of epitaxial Ge films on Si will be discussed. Considered cases include corrugation formation and growth, island formation and coarsening, and material deposition. [1] P. Mehta et al., Physics Reports vol. 810 (2019), p. 1-124 [2] R. Bergamaschini et al. Advances Physics X 1, 331 (2016). [3] D. Srolovitz, Acta Metallurgica, vol. 37, no. 2, (1989): 621–625 [4] F. Rovaris et al, Physical Review B 94.20 (2016): 205304.