The thermodynamic stability region at the athermal limit of tetragonal LLZO was found by calculating possible competing phases. All structures were first geometrically optimised using the PBEsol functional to the same convergence criterion as bulk LLZO. Individual k-points and formation enthalpies for each phase are provided in Si Table , S4, S5, S6, S7 and S8. Thereafter, the Chemical Potential Limits Analysis Program (CPLAP) was used to determine the thermodynamic stability region and thus the constrained chemical potential limits for LLZO. The phases that make up the boundaries of the stability region were then subjected to a geometry optimisation using HSE06 and new enthalpies of formation were calculated the details or which are given in SI Table .