Implanted neural network potentials: Application to Li-Si alloys

Modeling the behavior of materials composed of elements with different bonding and electronic structure character for large spatial and temporal scales and over a large compositional range is a challenging problem. Cases in point are amorphous alloys of Si, a prototypical covalent material, and Li, a prototypical metal, which are being considered as anodes for high-energy-density batteries. To address this challenge, we develop a methodology based on neural networks that extends the conventional training approach to incorporate pre-trained parts that capture the character of different components, into the overall network; we refer to this model as the “implanted neural network” method. We show that this approach works well for the Si-Li amorphous alloys for a wide range of compositions, giving good results for key quantities like the diffusion coefficients. The method is readily generalizable to more complicated situations that involve two or more different elements.

Figure 1

Schematic representations of the construction of the ANN and INN models for Si: The descriptors consist of radial Si-$X$ terms (circles) and angular Si-$X-Y$ terms (inverted triangles), where $X, Y$ = Si or Li; the ANN model contains $N$ nodes in each of the two hidden layers all trained simultaneously, whereas the INN model contains $N_1$ nodes for c-Si, $N_2$ nodes for a-Si, and $N_3$ nodes for the Si-Li interactions, trained in three separate stages, labeled a (red arrows), b (green arrows), c (cyan arrows).

Figure 2

The errors between the energy predictions of INN and the results of DFT calculations for Li, Si, and its amorphous alloys. The colored areas show the Li concentration change where the lightest color does not include any Li and darkest color indicate pure Li structures. While blue dots show the error at structures in the training database, green dots show validation data errors.

Figure 3

Root-mean-square errors (in eV/atom) for the total energies of Li-Si structures, from the ANN model (blue curves) and the INN model (red curves), for the training and validation data.

Figure 4

Comparison of the results from ANN (blue) and INN (red) calculations for the total energy of crystalline bulk Si, in eV/atom relative to the equilibrium energy ${\mathrm{E}}_0$, as a function of volume, normalized by the equilibrium volume ${\mathrm{V}}_0$; the DFT energies are shown as black circles.

Figure 5

Features of the crystalline and amorphous bulk phases of Si at 300 K, and of the liquid phase at 2000 K, as obtained from the INN potential.

Figure 6

Formation energy of a-${\mathrm{Li}}_x\mathrm{Si}$ structures for $0\le x\le 0.75$, as obtained from DFT calculations and from the ANN and INN models. Each point in the ANN and INN calculations is the average of five different configurations at a given value of $x$. The two insets, at $x=0$ and $x=0.75$ show representative atomic structures (yellow spheres: Si atoms, purple spheres: Li atoms).

Figure 7

Diffusion coefficients of Li and Si elements in a-Li-Si alloys. The results of present INN Li-Si potential is compared with the results of MEAM in Ref. [34], DFT calculations in Ref. [35], and the experimental coefficients taken from Ref. [36].

Figure 8

Logarithm of the probability $P_D$ of a Li atom to diffuse more than 1 Å in the next 200 fs as a function of its number of neighbors, ${\mathrm{nn}}_x$, for $x=\mathrm{Li}$ (other Li neighbors), $x=\mathrm{Si}$ (Si neigbhors), and $x=\mathrm{Li}$+Si (either kind of neighbors). The data was taken from an MD simulation of a-${\mathrm{Li}}_{3.75}\mathrm{Si}$ at 300 K and the analysis was carried out over 261 diffusion events. The dashed lines are linear fits to $log{P}_D$.

Figure 9

Atomic structure of a hard (top row) and soft (bottom row) site, each one viewed from five different angles for better perspective. The atom at the center (black) is a Li atom, its nearest neighbor Si (yellow) and Li (purple) atoms are shown linked to it by bonds, and the next neighbors are shown unlinked.