Machine learning potential for interacting dislocations in the presence of free surfaces

Computing the total energy of a system of N interacting dislocations in the presence of arbitrary free surfaces is a difficult task, requiring Finite Element (FE) numerical calculations. Worst, high accuracy requires very fine meshes in the proximity of each dislocation core. Here we show that FE calculations can be conveniently replaced by a Machine Learning (ML) approach. After formulating the elastic problem in terms of one and two-body terms only, we use Sobolev training to obtain consistent information on both energy and forces, fitted using a feed-forward neural network (NN) architecture. As an example, we apply the proposed methodology to corrugated, heteroepitaxial semiconductor films, searching for the minimum-energy dislocation distributions by using Monte Carlo. Importantly, the presence of an interaction cutoff allows for the application of the method to systems of different sizes without the need to repeat training. Millions of energy evaluations are performed, a task which would have been impossible by brute-force FE calculations. Finally, we show how forces can be exploited in running 2D ML-based dislocation dynamics simulations.


Energy decomposition
We prove that the energy of an arbitrary configuration of N dislocations in an heteroepitaxial film can be written in the form: From the linear theory of elasticity, the energy of a (traction-free) deformed body Ω can be expressed as [1,2]: where σ tot and ε tot are the total stress and strain fields in the material. Expanding these terms with respect to heteroepitaxial field and dislocation contributions, Eq. (S2) can be rephrased as: where "het" subscript indicates fields generate by the heteroepitaxial mismatch and i and j refer to dislocations. Notice that we decomposed the stress/strain terms in heteroepitaxial and dislocation contributions for consistency with the specific application in the Paper. Eq. (S3) and the following statements still hold if an arbitrary external stress/strain field replaces the heteroepitaxial field and arbitrary stress/strain inducing defects replace dislocations.
Distributing the tensor product :, the total energy reads Physical interpretation of terms in Eq. (S4) is as follows. The first term represents the elastic energy stored in a non-dislocated film. In our work we disregarded its value as we were not interested in comparing the total energy of different film geometries, but it could be easily inserted in further developments of our method. The second and third terms represent the dislocationheteroepitaxial film interaction and the so-called dislocation self-energy, i.e. the distortion energy associated with the dislocation presence by its own existence. It can be clearly seen that, once surface morphology is fixed, the value of these terms only depends on the position and Burgers vector of dislocation i. The last term is the dislocation-dislocation interaction energy. Comparison between Eq. (S1) and Eq. (S4) yields:

Cutoff radius and interaction convergence
In order to extend our results to simulation cells bigger than those present in the TSs we exploited a cutoff scheme setting to zero the dislocation-dislocation interactions when their horizontal separation was greater than 600 nm. This procedure introduce an approximation error in energy evaluations. In order to quantify this error, we plotted the relative difference in energy at the cutoff value for different dislocation array (and simulation cell) periodicity in Figure S1. Dislocations were placed 150 nm below a flat free surface, as this profile allows an analytical expression of stress fields, and energies have been obtained by Peach-Koehler force integration. The reference value E ∞ is that of interaction between isolated dislocations without periodic images in an semi-infinite solid with flat free surface.
The periodicity chosen in our work is 1200 nm, which ensures a relative error of less than 1%. In order to reduce by another order of magnitude, a periodicity greater than 2000 nm would be required, corresponding to a cutoff radius of more than 1000 nm. Figure S1: Relative difference in dislocation interaction energies for dislocation 600 nm apart and 150 nm below a flat free surface as a function of the cell periodicity. The reference value E ∞ is that of an isolated dislocation without periodic images.

Regression plots
In this Section we report for completeness all the regression plots for the Soboelev training as discussed in Section 2.3 of the Paper. In Figure S2 we show regression plots for all the terms of Eq. (S5) regarding both Energies and Forces for the case of a perturbed free surface with A = 60 nm.