Computational methodology

Theoretical calculations were performed by using density functional theory (DFT) implemented in the Vienna ab initio simulation package (Vasp)^41,42. The exchange correlation potential was approximated by generalised gradient approximation (GGA) with the PBE functional⁴³. The cutoff energy Ecut and k-point mesh were set to 400 eV and the 1 × 1 × 1 Monkhorst-Pack grid, respectively, which are sufficient for energy convergence. The convergence criterion for the electronic energy was 10⁻⁵ eV and the structures were relaxed until the Hellmann–Feynman forces were smaller than 0.02 eV Å⁻¹. The formation energy (E_f) of the Mn²⁺ doped Li₂ZnSiO₄ can be calculated by:

where E(doped) and E(perfect) are the total energy of the doped and perfect (undoped) crystal, respectively. The μ_i and n_i represent the chemical potential and the number of the added (n_i > 0) or removed (n_i < 0) i-type atoms, respectively.