Machine-learning potentials enable predictive and tractable high-throughput screening of random alloys

We present an automated procedure for computing stacking fault energies in random alloys from large-scale simulations using moment tensor potentials (MTPs) with the accuracy of density functional theory (DFT). To that end, we develop an algorithm for training MTPs on random alloys. In the first step, our algorithm constructs a set of $\sim 10$ 000 or more training candidate configurations with 50–100 atoms that are representative for the atomic neighborhoods occurring in the large-scale simulation. In the second step, we use active learning to reduce this set to $\sim 100$ most distinct configurations—for which DFT energies and forces are computed and on which the potential is ultimately trained. We validate our algorithm for the MoNbTa medium-entropy alloy by showing that the MTP reproduces the DFT $\frac{1}{4}\left[111\right]$ unstable stacking fault energy over the entire compositional space up to a few percent. Contrary to state-of-the-art methods, e.g., the coherent potential approximation (CPA) or special quasi-random structures (SQSs), our algorithm naturally accounts for relaxation, is not limited by DFT cell sizes, and opens opportunities to efficiently investigate follow-up problems, such as chemical ordering. In a broader sense, our algorithm can be easily modified to compute related properties of random alloys, for instance, misfit volumes, or grain boundary energies. Moreover, it forms the basis for an efficient construction of MTPs to be used in large-scale simulations of multicomponent systems.

Figure 1

Typical supercell types for computing the $\frac{1}{4}$[111] unstable stacking fault energy.

Figure 2

Schematic illustration of the active learning algorithm presented in Sec. 2d.

Figure 3

SFE for MoTa as a function of the Mo content predicted by (a)–(c) level-16 MTPs trained on ${\mathcal{T}}_{\mathrm{MoTa}}^1$–${\mathcal{T}}_{\mathrm{MoTa}}^3$, and by (d) level-20 MTPs trained on ${\mathcal{T}}_{\mathrm{MoTa}}^3$. The average predictions (${\overline{\text{MTP}}}_X$) coincide with the DFT values up to a few percent (except for 100 % Mo in (b)). This accuracy is remarkable given the small size of the training set containing only 48 configurations.

Figure 4

SFE for MoNb as a function of the Mo content predicted by (a) level-16 MTPs, and by (b) level-20 MTPs (all trained on ${\mathcal{T}}_{\mathrm{MoNb}}$). The average predictions (${\overline{\text{MTP}}}_X$) coincide with the DFT values up to a few percent (except for the percentage of Mo $⪅$ 0.25 and Mo $⪆$ 0.75 for the level-16 MTPs) which is very good, given that ${\mathcal{T}}_{\mathrm{MoNb}}$ contains only 44 configurations.

Figure 5

SFE for NbTa as a function of the Nb content predicted by an ensemble of level-16 MTPs which reproduce the DFT values without any practical error.

Figure 6

Three curves going through the MoNbTa ternary diagram for which the SFE is analyzed in Sec. 3c.

Figure 7

SFEs along the three curves from Fig. 6 as predicted by level-16 MTPs in (a)–(c) and level-20 MTPs in (d)–(f) trained on ${\mathcal{T}}_{\mathrm{MoNbTa}}^1$ that has been generated by the active learning algorithm described in Sec. 2d. From (a)–(c), it can be observed that an MTP level of 16 is not sufficient to predict the SFE uniformly over the entire compositional space. The level-20 MTP is able to do so, except in regions with high Nb content in (e).

Figure 8

SFEs along the three curves from Fig. 6 as predicted by level-16 MTPs trained on ${\mathcal{T}}_{\mathrm{MoNbTa}}^2$ that has been generated by rerunning the active learning algorithm with a smaller minimum extrapolation grade than the one used to generate ${\mathcal{T}}_{\mathrm{MoNbTa}}^1$. The additional training data helps to improve the predictions heavily [compare with Figs. 7–7] but some discrepancy in the high-Nb regime is still present.

Figure 9

Number of configurations per composition at various stages of the active learning algorithm. (a) Initial training set randomly selected from the full training candidate set after step 2. (b) Training set ${\mathcal{T}}_{\mathrm{MoNbTa}}^2$ generated by the active learning algorithm using ${\gamma }_{\min }$ $=$ 1.4. (c) Training set ${\mathcal{T}}_{\mathrm{MoNbTa}}^3$ after adding unary and binary configurations to ${\mathcal{T}}_{\mathrm{MoNbTa}}^2$. The added unary and binary configurations are necessary so that the MTPs predict the SFE over the entire compositional space (cf. Fig. 10).

Figure 10

SFEs along the three curves from Fig. 6 as predicted by level-16 MTPs in (a)–(c) and level-20 MTPs in (d)–(f) trained on ${\mathcal{T}}_{\mathrm{MoNbTa}}^3$. Both types of MTPs are able to reproduce the SFE, the ${\overline{\text{MTP}}}_{20}$ practically without error. The ${\mathrm{MTP}}_{16}^{\mathrm{uit}}$ and the ${\mathrm{MTP}}_{20}^{\mathrm{uit}}$ refer to the potentials from Sec. 3c2 fitted with respect to the training set that has been generated with a uniform initialization [Fig. 12].

Figure 11

Relative errors in the stacking fault energy of the MTPs trained on ${\mathcal{T}}_{\mathrm{MoNbTa}}^3$ with respect to DFT for several samples of random ternary alloys; The DFT energies were computed by performing single-point calculations on the bulk and stacking fault configurations that have been relaxed with a level-20 MTP. The errors are not worse than for the unaries and binaries indicating that also the SFE of random ternary configurations (not included in the training set) is well-reproduced by the MTPs.

Figure 12

Initial training sets and corresponding final training sets obtained after an active selection of configurations from ${\mathcal{T}}_{\mathrm{MoNbTa}}^3$. Random initial sampling (a)–(e) raises the danger of undersampling one or more unaries, e.g., in (a). Uniform initial sampling (f) resolves this issue leading to a reliable final training set: an MTP trained on the final training set in (f) predicts the SFE over the entire compositional space (see Fig. 10).