## 5 Assessment of Additional External Quantum Efficiency  
The efficiency of a solar cell that includes a downconverter (DC) layer can be measured by its Additional External Quantum Efficiency (AEQE) \(\eta^{\text{ -Add}}_{\text{EQE}}\). This parameter, introduced by [28], compares the External Quantum Efficiency (EQE) of a solar cell with a DC layer to the EQE of a solar cell without a DC layer. The EQE of a standard solar cell with a DC film is represented as \(\eta^{\text{ -with DC}}_{\text{EQE}}\). To calculate the AEQE, the loss of EQE caused by the partial occlusion of the solar cell by the DC layer \(\eta^{\text{ -L}}_{\text{EQE}}\) must be taken into account. The AEQE can be determined using Equation (1). \[\eta^{\text{ -Add}}_{\text{ EQE}}=\eta^{\text{ -with DC}}_{\text{ EQE}}-\eta^{\text{ -L}}_{\text{ EQE}} \tag{1}\]  
The first term in Equation (1) \(\eta^{\text{ -with DC}}_{\text{ EQE}}\) is related to the external quantum efficiency of solar cell combined with downconverter layer, which can be defined as the measure of the ratio between the flux of photons converted by the DC layer which is transmitted to the cell at 980 nm (Eg) and the incoming flux from the sun in the wavelength range of the DC layer (250-500 nm). This ratio is then multiplied by the conversion efficiency of the solar cell at 980 nm, as shown in the equation:  
\[\eta^{\text{ -with DC}}_{\text{ EQE}}=\eta^{\text{ -980}}_{\text{ SC }}\frac{\dot{f}_{\text{ SC}}}{\dot{f}_{\text{sun}}} \tag{2}\]  
To calculate the \(\eta^{\text{ -with DC}}_{\text{ EQE}}\), we will define the flux conversion process, as illustrated in Figure 9; in this system, the flux coming from the sun \(f_{\text{sun}}\) is transmitted through the DC layer, which will then be absorbed by the latter \(\Delta f_{\text{DC}}\) and is converted to 980 nm \(\dot{f}_{\text{ DC}}\) to directly reach the solar cell \(\dot{f}_{\text{ SC}}\). Therefore, the \(\eta^{\text{ -with DC}}_{\text{ EQE}}\) is given by the product of several efficiencies:  
\[\eta^{\text{ -with DC}}_{\text{ EQE}}=\eta^{\text{ -980}}_{\text{ SC }}\left(\frac{\dot{f}_{\text{ DC}}}{\dot{f}_{\text{sun}}}\right)\left(\frac{\dot{f}_{\text{ DC}}}{\dot{f}_{\text{DC}}}\right)\left(\frac{\dot{f}_{\text{ SC}}}{\dot{f}_{\text{ DC}}}\right) \tag{3}\]  
\[\eta^{\text{ -with DC}}_{\text{ EQE}}=\eta^{\text{ -980}}_{\text{ SC }}\cdot\eta^{\text{ -DC}}_{\text{ -A }}\cdot\eta^{\text{ -DC}\to 980}_{\text{ -A }}\cdot\eta^{\text{ -980}}_{\text{ -A }} \tag{4}\]  
In Equation (4), \(\eta^{\text{ -980}}_{\text{ SC}}\) is the efficiency of a 1.26 eV band gap silicon solar cell (with a wavelength of 980 nm) calculated using the Shockley-Queisser detailed balance model. Figure 9: Schematic diagram of the down-converter system. This represents an efficiency of approximately 32.9%. \(\eta^{\text{-DC}}_{\text{A}}\) is the absorption efficiency of the downconverter layer, defined as the ratio of the absorbed intensity to the incident intensity, which is determined by the absorption spectrum of the DC layer (Figure 10). \(\eta^{\text{-980}}_{\text{con}}\) is the product of the energy transfer efficiency from the host matrix to \(\text{Tb}^{3+}\) (\(\eta^{\text{-}}_{\text{host}\rightarrow\text{Tb}^{3+}}\)) and the effective quantum efficiency \(\eta_{\text{EQE}}\). It assumes that \(\eta^{\text{-}}_{\text{host}\rightarrow\text{Tb}^{3+}}=100\%\), meaning that all photons absorbed by the host matrix are transmitted to \(\text{Tb}^{3+}\) to participate in the downconversion process. \(\eta^{\text{-980}}_{\text{ext}}\) takes into account the extraction of low-energy photons that are confined within a layered medium formed by the down-converted (DC) layer stacked on the solar cell (SC). It was assumed that the Yb\({}^{3+}\) spectral line could be described by a Lorentzian function centered at 980 nm, having FWHM of 52 nm.