Prediction of Born effective charges using neural network to study ion migration under electric fields: applications to crystalline and amorphous Li$_3$PO$_4$

Understanding ionic behaviour under external electric fields is crucial to develop electronic and energy-related devices using ion transport. In this study, we propose a neural network (NN) model to predict the Born effective charges of ions along an axis parallel to an applied electric field from atomic structures. The proposed NN model is applied to Li$_3$PO$_4$ as a prototype. The prediction error of the constructed NN model is 0.0376 $e$/atom. In combination with an NN interatomic potential, molecular dynamics (MD) simulations are performed under a uniform electric field of 0.1 V/angstrom, whereby an enhanced mean square displacement of Li along the electric field is obtained, which seems physically reasonable. In addition, the external forces along the direction perpendicular to the electric field, originating from the off-diagonal terms of the Born effective charges, are found to have a nonnegligible effect on Li migration. Finally, additional MD simulations are performed to examine the Li motion in an amorphous structure. The results reveal that Li migration occurs in various areas despite the absence of explicitly introduced defects, which may be attributed to the susceptibility of the Li ions in the local minima to the electric field. We expect that the proposed NN method can be applied to any ionic material, thereby leading to atomic-scale elucidation of ion behaviour under electric fields.