## 5 Assessment of Additional External Quantum Efficiency
The photoluminescence (PL) spectra were acquired by integrating emission angles in both the down-converted (DC) and solar cell (sc) materials, as illustrated in Figure 11. Through numerical calculations, the extraction efficiency \(\eta^{\text{-980}}_{\text{ext}}\) was determined to be 53%. The detailed expression of the term \(\eta^{\text{-DC}}_{\text{A}}\) in Equation (5) is:  
\[\eta^{\text{-DC}}_{\text{A}}=\frac{\int_{\lambda_{1}}^{\lambda_{2}}|\text{A}( \lambda)|\text{d}\lambda}{\Delta\lambda} \tag{5}\]  
where \(\Delta\lambda=\lambda_{2}-\lambda_{1}\) corresponds to DC wavelength range (250-500 nm). The second term \(\eta^{\text{-L}}_{\text{EQE}}\) in Equation (1) is the loss of the external quantum efficiency of the solar cell caused by the presence of the downconverter layer, which partially occults the cell and can be expressed as follows:  
\[\eta^{\text{-L}}_{\text{EQE}}=\eta^{\text{-400}}_{\text{sc}}\cdot\eta^{\text{ -DC}}_{\text{A}}\cdot\eta^{\text{-DC}\rightarrow\text{980}}_{\text{conv}}\cdot \eta^{\text{-980}}_{\text{ext}}\]  
where:  
\(\eta^{\text{-400}}_{\text{sc}}\) is the efficiency obtained at the middle of down-conversion range (250-550 nm) calculated in the detailed balance model by [PERSON]. Assuming that the DC  
Figure 10: The absorption spectra of the films for the 70 SiO\({}_{2}\)-30 HfO\({}_{2}\)-xTb\({}^{3+}-\text{y}\)Yb\({}^{3+}\) thin films. layer has no effect on the efficiency (\(\eta^{\text{DC}\to 980}=100\%\)) and that all downconverted photons are fully extracted by the solar cell (\(\eta^{\text{980}}_{\text{ext}}=100\%\)). The absorption efficiency, \(\eta^{\text{-DC}}_{\text{A}}\), is determined using Equation (5) using the same method. In Table 3, the results clearly highlight the significant advantage of incorporating a DC layer, resulting in an approximate 2% to 3% increase in external quantum efficiency (EQE) compared to a bare solar cell. To properly interpret these findings, it is crucial to compare them with previous studies. [PERSON] et al. estimated a maximum EQE of approximately 9% [23], whereas [PERSON] et al. reported a value of 7% [29]. In contrast, our study achieved an EQE that is approximately four times lower, indicating the need for further improvements. Nonetheless, it is worth noting that the EQE can be enhanced by optimizing various technical processes. By adjusting the size and shape of nanocrystals, the number of emission centers in the material can be increased, leading to higher light output. The dielectric constant of the silica hafnia can also be increased to promote energy transfer between the dopants, thus increasing its efficiency. Finally, optimizing the fabrication process can help decrease non-radiative recombination processes, thus improving the overall efficiency of the co-doped material. Figure 11: The normalized photoluminescence spectra of down-conversion layer and solar cell centered at 980 nm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **Effective Quantum Efficiency** & \(\eta^{\text{-with DC}}_{\text{EQE}}\) & \(\eta^{\text{-L}}_{\text{EQE}}\) & \(\eta^{\text{-Add}}_{\text{EQE}}\) \\ \hline Sample 1 & 123 & 3.35 & 0.71 & 2.64 \\ Sample 2 & 150 & 3.55 & 0.78 & 2.77 \\ Sample 3 & 146 & 4.16 & 0.715 & 3.44 \\ Sample 4 & 179 & 3.94 & 0.83 & 3.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Additional Effective quantum efficiency as a function of RE concentration for G and GC silica hafnia thin films.