Accounting for the vibrational contribution to the configurational entropy in disordered solids with machine learned forcefields: a case study of garnet electrolyte Li ₇ La ₃ Zr ₂ O ₁₂

Accounting for lattice vibrations to accurately determine the phase stabilities of site-disordered solids is a long-standing challenge in computational material designs, due to the high computational cost associated with sampling the vast configurational space to obtain the converged thermodynamic quantities. One example is the garnet electrolyte Li₇La₃Zr₂O₁₂, the high-temperature and high-ion-mobility cubic phase of which is disordered in its Li⁺ site occupations, such that both the vibrational and configurational entropic contributions to its phase stability cannot be ignored. Understanding the subtle interplay between vibrational and configurational entropies in this material will therefore play a critical role in the rational manipulation of dopants and defects to stabilise cubic Li₇La₃Zr₂O₁₂ at room temperature for practical applications. Here, by developing machine learned forcefields based on an equivariant message-passing neural network SO3KRATES, we follow a strict statistical thermodynamic protocol to quantify the phase stability of cubic Li₇La₃Zr₂O₁₂ through structural optimisations, as well as molecular dynamics simulations at 300 and 1500 K, for a total of 70 120 configurations of cubic Li₇La₃Zr₂O₁₂. Although this only covers a tiny fraction of the configurational space (∼7 × 10³⁴ configurations in total), we are able to deterministically show that the vibrational contributions to the total configurational free energy at 1500 K are significant (on the order of 1 eV per atom) in correctly ordering the stability of the cubic Li₇La₃Zr₂O₁₂ over its tetragonal counterpart, thanks to the high data efficiency, accuracy, stability and good transferability of the transformer-based equivariant network architecture behind SO3KRATES. Therefore, our work opens up new avenues to accelerate the accurate computational designs of disordered solids, such as solid electrolytes, for technologically important applications.