Interatomic potential development for topological insulator Bi1-xSbx and its dislocation by force-following active learning

TL;DR

This work addresses the challenge of modeling dislocation mechanics in Bi1-xSb_x topological insulators by developing a force-following active-learning workflow that couples density functional theory (DFT) with Gaussian Approximation Potentials (GAP). Starting from an initial IP trained on strained unit-cell data, the method iteratively retrains the IP during dislocation relaxation to maintain low force uncertainty, yielding IP5 that reproduces the observed relaxation pathway and lattice properties across Sb concentrations. The final potential is used to compute dislocation properties, including the Peierls stress and dislocation formation under compression, with convergence with system size and agreement to DFT/experimental benchmarks. This approach enables efficient, large-scale atomistic simulations of dislocation physics in Bi-Sb binaries and highlights directions for enhancing robustness through multiple relaxation pathways and improved uncertainty quantification for advanced surrogate models.

Abstract

We introduce a force following active learning algorithm that integrates density functional theory DFT with the Gaussian Approximation Potential GAP framework to develop a robust interatomic potential IP for a dislocation in a topological insulator Bi1xSbx. Starting from an initial potential IP0 trained on unit cell data from strained Bi Sb binaries our active learning approach iteratively refines the IP during a structural relaxation. In each cycle if the force error uncertainty of any atom near the dislocation core exceeds a threshold value the IPi is efficiently retrained IPi to IPi1 by incorporating DFT computed forces and energies of atoms near the high uncertainty atom. This strategy ensures that the relaxation process maintains a low force error until full convergence is achieved. Consequently the final IP here IP5 has two capabilities 1 it reproduces the relaxation pathway observed during the active learning process unlike the initial IP0 which lacks prior dislocation core knowledge and 2 it captures the lattice and elastic properties of Bi Sb binaries across a range of Sb concentrations. We also evaluate dislocation properties Peierls stresses and dislocation generation by compression to assess the performance of the trained potential IP5.

Figures (7)

• Figure 1: Structural similarity between hexagonal Bi and monoclinic Bi$_7$Sb$_1$. (A) Hexagonal and rhombohedral structure of Bi. Black and gray atoms present alternating layers. (B) The two atomistic structures above represent the viewing direction of $\mathbf{r}^{\rm H}_3$ for Hexagonal Bi and the equivalent direction ($\mathbf{r}_{2}^{\rm H} \otimes \mathbf{r}_{3}^{\rm H}$) on monoclinic Bi$_7$Sb$_1$. The two structures below represent the viewing direction of $\mathbf{r}^{\rm H}_1$ on hexagonal Bi and the equivalent direction ($\mathbf{r}_{1}^{\rm H}$) on monoclinic Bi$_7$Sb$_1$. The Burgers vector $\mathbf{b}=[100]$ in rhombohedral notation is presented as a red arrow.
• Figure 2: The atomistic structure of a single dislocation ($\mathbf{b}=[100]$) in replicated Bi$_7$Sb$_1$ under fixed boundary conditions. In left figure, atoms in the yellow region are constrained during relaxation. The 2D maps on the right represent atom displacements along the $x$-axis ($u_x$) and $z$-axis ($u_z$).
• Figure 3: Flowchart of the active learning algorithm. (A) Two main iterative processes, Atomistic simulation and IP retraining, with the corresponding software packages presented. (B) Subprocess in Atomistic simulation for uncertainty evaluation and structural relaxation. (C) Subprocess in IP retraining, including the generation of representative structures, DFT calculations, and model retraining.
• Figure 4: Structural and elastic properties change of Bi$_{1-x}$Sb$_x$ as a function of Sb concentration. Both DFT and GAP results are shown, with experimental data (jain1959 and lichnowski1976) provided where available.
• Figure 5: Dislocation relaxation in Bi$_7$Sb$_1$ using the active learning. (A) Evolution of force error and force norm ($f_{\rm norm}$) during the relaxation via the active learning. The black dotted line indicates the drop in force error following the retraining of the interatomic potential (IP$_i\rightarrow$ IP$_{i+1}$), while the green-shaded region highlights the retraining phase (a zoomed-in view is provided in the inset). (B) Two-dimensional UMAP projection of the local atomic environments near the dislocation core. Relaxation trajectories for different IPs are shown: IP$_5$ (red), active learning relaxation (green), and IP$_0$ (black), depicted with arrows and circles. $\Delta E$ denotes the energy difference of each configuration relative to the initial structure as predicted by IP$_5$.