Fig. 3: Theoretical analysis of the enhancement of the upconversion lasing efficiency in the amorphous phase of NaYF₄ combined with SiO₂.

a Schematic and TEM images of the microstructure of the amorphous phase after liquid quenching of hexagonal NaYF₄ with SiO₂. b Phonon DOS of Yb³⁺ in the Yb-doped amorphous system (Model 6 in Supplementary Fig. 12, which shows the largest difference from hexagonal crystal NaYF₄) and Yb-doped hexagonal crystal NaYF₄ (Supplementary Fig. 15b). c Phonon DOS of Er³⁺ in the Er-doped amorphous system (Model 5 in Supplementary Fig. 13, which shows the largest difference from hexagonal crystal NaYF₄) and Er-doped hexagonal crystal NaYF₄ (Supplementary Fig. 15c). d Total phonon DOS of the amorphous system (Model 2 in Supplementary Fig. 11) and hexagonal crystal NaYF₄ (Supplementary Fig. 15a). Note that the total phonon DOS are normalized by dividing by the total number of atoms in each system. e, f Mechanisms of energy back transfer (Er³⁺ → Yb³⁺) (e) and multiphonon relaxation (Er³⁺) (f). ∆E for energy back transfer: ~1373.4 cm⁻¹ for EBT I (between ²F_7/2 → ²F_5/2 (Yb³⁺) and ²H_11/2 → ⁴I_11/2 (Er³⁺)), ~1553.1 cm^-1 for EBT II (between ²F_7/2 → ²F_5/2 (Yb³⁺) and ⁴S_3/2 → ⁴I_13/2 (Er³⁺)) and ~1563.7 cm^-1 for EBT III (between ²F_7/2 → ²F_5/2 (Yb³⁺) and ⁴F_9/2 → ⁴I_13/2 (Er³⁺)). ∆E for multiphonon relaxation of the Er³⁺ ion: ⁴F_7/2 → ²H_11/2 = ~1483 cm⁻¹, ⁴F_7/2 → ⁴S_3/2: ~2195 cm⁻¹, and ⁴S_3/2 → ⁴F_9/2 = ~3117 cm⁻¹. Note that red solid arrows with red dashed lines indicate energy transfers by EBT. Red curved arrows indicate an interaction between Yb³⁺ and Er³⁺. Gray dashed, black dot-dashed, and green solid arrows indicate absorption, energy transfer, and multiphonon relaxation, respectively.