Accelerating materials discovery using integrated deep machine learning approaches

We present an integrated deep machine learning (ML) approach that combines crystal graph convolutional neural networks (CGCNN) for predicting formation energies and artificial neural networks (ANN) for constructing interatomic potentials....

1. Artificial neural network machine learning interatomic potential for La-Si-P Fig. S1 shows the developed ANN-ML potential wells reproduce the ab initio calculated formation energies for a wide range of La-Si-P ternary compounds, even many of these compounds that are not included in the training data set.The La2SiP3 phase discovered in this paper was not included in the training data set.In order to demonstrate the reliability of the developed ANN-ML potential for MD simulations of the La2SiP3 phase at finite temperatures, 200 snapshots from AIMD simulations of La2SiP3 at 1200 K and 2500 K are randomly selected to validate the ANN-ML model for the interatomic force prediction in comparison with the ab initio calculation data.Fig. S2 and S3 show the comparisons between the forces obtained by ab initio simulations and the ANN-ML model prediction in La2SiP3 crystal at 1200K and La2SiP3 liquid at 2500K, respectively.The root mean square (RMS) force error for x, y, and z components is shown in the figures.Fig. S4 shows a comparison of partial pair-correlation functions between AIMD and ANN-ML MD simulations for La2SiP3 liquid at T=2500K.It shows the ANN-ML MD simulation can well reproduce the atomic trajectory in AIMD simulation.These results indicate that the ANN-ML potential is reliable for MD simulation of La2SiP3 phase.    5. Gibbs free energies (at zero pressure) as a function of the temperature for relevant La-P, Ba-P binary, Ba-Si-P tenary, and Ba-La-Si-P quaternary compounds Fig. S8.Gibbs free energy (at p=0 kbar) as a function of temperature for (a) Ba-P binary system, (b) Ba-Si-P ternary system, and (c) Ba-La-Si-P quaternary system.

Structure information of the known and predicted low-energy binary and ternary compounds in La-Si-P system
Table S1.Structure database of the La-Si-P system, including existing phases, newly predicted stable phases and metastable phases.We list the space group, formula unit per cell, lattice parameters and the formation energies above the convex hull (Ed) of the elements (La,Si,P), binary, and all La-Si-P ternary compounds either existing or obtained from our ML-guided ab initio calculation prediction.

Fig. S1 .
Fig. S1.The formation energies (Ef) of La-Si-P ternary compounds calculated by the ANN-ML interatomic potential are compared with the results calculated by DFT-PBE.(a) 806 La-rich La-Si-P ternary compounds from Ref (1), and (b) 353 P-rich La-Si-P ternary compounds from the present paper.

Fig. S2 .
Fig. S2.Comparisons between the forces obtained by ab initio simulations and the ANN-ML model prediction in La2SiP3 crystal at 1200K.

Fig. S3 .
Fig. S3.Comparisons between the forces obtained by ab initio simulations and the ANN-ML model prediction in La2SiP3 liquid at 2500K.

Fig. S4 .
Fig. S4.Comparison of partial pair-correlation functions between AIMD and ANN-ML MD simulations for La2SiP3 liquid at T=2500K.

2.
Fig. S5.The band structure and density of states of the two predicted La2SiP3 phases with the formation energies above convex hull (a) 1 meV/atom and (b) 33 meV/atom, respectively.

Fig. S9 .
Fig. S9.Calculated Ghull as a function of temperature for the BaLaSiP3 quaternary phase.

Table S2 .
The phases in bold are thermodynamically stable.The results are based on GGA calculations.Crystallographic data of La2SiP3 (space group of Pnma), which has the formation energy above convex hull Ed = 1 meV/atom.

Table S3 .
Crystallographic data of BaLaSiP3 (space group of Pnma), which is energetically stable.