3.1. Theoretical Models

Three lanthanide ions (Ln³⁺ = Ce³⁺, Pr³⁺, Tb³⁺) were selected as doping ions to dope Sr₂Si₅N₈ and Sr₂Si₅N₈:Eu²⁺, respectively. As shown in Figure 9a, there are two kinds of Sr doping sites, namely, eight-coordinate Sr₁ [SrN₈] (0.5000, 0.8734, 0.9997) and ten-coordinate Sr₂ [SrN₁₀] (0.7500, 0.1158, 0.8683). Since in the doped ions, Ln are all positively trivalent and Sr is bivalent to neutralize the entire system’s charge, every time a positive trivalent lanthanide ion is introduced to replace Sr²⁺, a Li⁺ is introduced to replace Sr²⁺ to keep the entire system electrically neutral. The chemical formula of Sr₂Si₅N₈ doped with Ln³⁺/Li⁺ is Ln_Sr1/Sr2Li_Sr1/Sr2Sr₂Si₅N₈, while the chemical formula of Sr₂Si₅N₈ doped with Ln³⁺/Li⁺/Eu²⁺ is Ln_Sr1/Sr2Eu_Sr1/Sr2Li_Sr1/Sr2-Sr₂Si₅N₈. Figure 9b has established a 2 × 2 × 1 supercell (60 atoms) with a doping concentration of 12.5% for Ln³⁺, Eu²⁺, and Li⁺. If we continue to expand the unit cell to 3 × 2 × 1 to reduce the doping concentration, the calculation requires more K points than 5 × 8 × 6, and it is complicated for the structure to converge. Notably, 2 × 2 × 1 is the largest supercell structure that can be established under the premise that the structure can converge.

(a) Molecular structure of Sr₂Si₅N₈; (b) 2 × 2 × 1 supercell of Sr₂Si₅N₈: The selected atoms are Sr₂.