## 2 Theory  
### Detailed Balance Equations of the MEG  
The DB equations of the MEG are represented by Equations (1) to (5). QY is the most crucial parameter for determining all the characteristics of the MEG [5; 6; 7; 8; 9]. Both the IQY (QY = 14; Figure 1a) and NQY (E\({}_{\text{th}}\), from 2E\({}_{\text{g}}\) to 4E\({}_{\text{g}}\); Figure 1b) are shown in Equation (1) and Figure 2. The IQY follows a step function with a full generation of multiple EHPs per photon. The actual QY (the NQY case) extracted from the pump-probe measurements shows a deviation from the IQY after reaching the E\({}_{\text{th}}\)[12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 32]. In the pump-probe measurements, the NQY is induced from the scale of the transient absorption signals (peak intensities and their decay signals). While pump intensities between multiple EHPs (high peak) and a single EHP (bleach signal) are compared, certain phenomena can be analyzed at the point of reaching E\({}_{\text{th}}\) in MEG, the potential number of generated EHPs, and MEG efficiency (the increment of additional EHPs by applied intensities) [11; 12; 33]. The modeling and related equations are reported in [12; 34; 35; 36; 30]. In NQY, E\({}_{\text{th}}\) is a significant parameter that explains the carrier extraction of MEGSCs (E\({}_{\text{th}}\): the onset point of QY over 100%). The delayed E\({}_{\text{th}}\) (\(>\)2E\({}_{\text{g}}\)) requires a higher photon energy for MEG and describes the status of MEG materials. Typically, E\({}_{\text{th}}\) depends on the effective mass of electrons and holes in a material. The surface state has a large impact on the MEG process because of fast carrier decay, which creates other decay paths at trap states [12]. Therefore, creating a near-perfect MEGSC is the first priority for maintaining its idealities. To ensure minimal or no mathematical errors, the open-circuit voltage (V\({}_{\text{OC}}\)) must be less than the bandgap energy. QY in Equation (5) relates to the generated number of EHPs for MEG, the excited high-energy state, and its photon energy emission through radiative recombination without losses. For instance, the optimum bandgap is 0.05 eV with QY = 200 and C = 46,200 suns. When 199 electrons recombine into holes, the energy is transferred to the 200 th electron. This excites the electron to an energy state of 10 eV and causes the emission of photon energy without losses under blackbody radiation [6]. However, conventional AR is the nonradiative recombination in actual materials and  
Figure 2: Ideal quantum yield (IQY) (maximum QY = 14) and nonideal quantum yields (NQYs) with three different threshold energies (2E\({}_{\text{g}}\), 3E\({}_{\text{g}}\), and 4E\({}_{\text{g}}\)), where Eg = 0.5 eV. uses an alternative DB in silicon solar cells [10; 31]. Thus, the MEGSC theory only considers blackbody radiation [6; 7; 8]. \[ \begin{array}{ll}\text{Ideal}&\text{QY(E)}=\left\{\begin{array}{cc}0&0< \text{E}<\text{E}_{\text{g}}\\ \text{m}&\text{m}\cdot\text{E}_{\text{g}}<\text{E}<(\text{m}+1)\cdot\text{E}_{ \text{g}}\ \ \text{m}=1,2,3,\cdots\\ \text{M}&\text{E}\geq\text{M}\cdot\text{E}_{\text{g}}\end{array}\right.\\ \text{Non}&\text{Ideal}&\text{QY(E)}==\left\{\begin{array}{cc}0&0<\text{E}< \text{E}_{\text{g}}\\ 1&\text{E}_{\text{g}}<\text{E}<\text{E}_{\text{th}}\cdot\text{E}_{\text{g}}\\ 1+\cdot\text{A}\cdot(\frac{\text{E}-\text{E}_{\text{th}}}{\text{E}_{\text{g}}}) &\text{E}\geq\text{E}_{\text{th}}\cdot\text{E}_{\text{g}}\end{array}\right.\\ \end{array} \right.\end{array} \tag{1}\]  
where m is the number of multiple EHPs generated, M is the maximum number of EHPs, E\({}_{\text{g}}\) is the bandgap, and E is the photon energy. A (=1) is the slope of the linearized QY, and E\({}_{\text{th}}\) is the threshold energy for an MEG event.