## 2 Theory
### Numerical Analysis of the Nonradiative Recombination of an MEGSC  
The deviation in QY depends on the material properties, such as effective mass, surface states, or defects of QD materials [37]. Thus, NQY describes the loss of EHPs from m\(\cdot\)E\({}_{\text{g}}\) to (m + 1)\(\cdot\)E\({}_{\text{g}}\). To account for this, we reconfigured the DB of the MEGSC to include nonradiative recombination [1]. In the rate-based calculations, the DB of MEG is shown in Equation (6) [1]:  
\[\text{F}_{\text{s,MEG}}-\text{F}_{\text{c,MEG}}(\text{V})+\text{R}_{\text{MEG }}(0)-\text{R}_{\text{MEG}}(\text{V})-\text{J}_{\text{BB}}/\text{q}=0 \tag{6}\]  
where F\({}_{\text{S,MEG}}\) and F\({}_{\text{C,MEG}}(\text{V})\) are the generation and recombination for the radiative term, respectively, and R\({}_{\text{MEG}}(0)\) and R\({}_{\text{MEG}}(\text{V})\) are the nonradiative generation and recombination, respectively [1]. Equation (6) is reordered to show the net rate of generation and recombination in Equation (7) [1]. \[ \begin{array}{ll}\text{F}_{\text{s,MEG}}-\text{F}_{\text{c0,MEG}}+\\ \left[\text{F}_{\text{c0,MEG}}-\text{F}_{\text{c,MEG}}(\text{V})+\text{R}_{\text{ MEG}}(0)-\text{R}_{\text{MEG}}(\text{V})\right]-\text{J}_{\text{BB}}/\text{q}=0\end{array}  \tag{7}\]To show the radiative and nonradiative limits, the following change is made to the part of Equation (7) inside brackets [1]:  
\[ \begin{array}{l}\mathrm{F_{c0,MEG}-F_{c,MEG}(V)}\\ =\mathrm{f_{NR}}\cdot\left[\mathrm{F_{c0,MEG}-F_{c,MEG}(V)+R_{MEG}(0)-R_{MEG}(V )}\right]\end{array}  \tag{8}\]  
where \(\mathrm{f_{NR}}\) denotes the ratio between radiative recombination and nonradiative recombination, which indicates the contribution of radiative recombination in the MEGSC [1]. If the nonradiative recombination fits the ideal rectifying equation, \(\mathrm{f_{NR}}\) can show the ideal reverse saturation current (\(\mathrm{J_{0}}\)) (Equation (9)) [1]:  
\[\mathrm{f_{NR}}=\frac{\mathrm{F_{c0,MEG}}}{\mathrm{F_{c0,MEG}+R_{MEG}(0)}}= \frac{\mathrm{F_{c0,MEG}}}{\mathrm{J_{0}}} \tag{9}\]  
where \(0<\mathrm{f_{NR}}\leq 1\) and \(\mathrm{J_{0}}=(\mathrm{F_{C0,MEG}}+R_{MEG}(0))\). In an ideal rectifying diode, \(\mathrm{J_{0}}\) is a voltage-dependent parameter. Therefore, the term for the recombination current, \(\mathrm{exp(qQY\cdot V/k\cdot T_{C})}\), is included as shown in Equation (10):  
\[ \begin{array}{l}\mathrm{J_{BB}}=\mathrm{q}\cdot\left(\mathrm{F_{s,MEG}-F_{ c0,MEG}}\right)\\ \phantom{\mathrm{q}\cdot\left(\mathrm{F_{c0,MEG}/R_{N}}\right)}\left[1-\mathrm{ exp}\left(\frac{qQY\cdot V}{k\cdot T_{C}}\right)\right]\\ \phantom{\mathrm{q}\cdot\left(\mathrm{F_{s,MEG}-F_{c0,MEG}}+J_{0}\cdot\left[1- \mathrm{exp}\left(\frac{qQY\cdot V}{k\cdot T_{C}}\right)\right]\right.}\end{array}  \tag{10}\]  
where \(\mathrm{F_{S,MEG}}=\mathrm{C\cdot f_{S}\cdot\phi_{MEG}(E_{g},\infty,T_{S},0)+ C\cdot(1-f_{S})\phi_{MEG}(E_{g},\infty,T_{C},0)}\), and \(\mathrm{F_{C0,MEG}}=\phi_{MEG}(E_{g},\infty,T_{C},0)\). In the DB equations above, \(\mathrm{V_{OC}}\) depends on \(\mathrm{f_{NR}}\), which decreases as \(\mathrm{J_{0}}\) increases. Its equation is shown in Equation (11). This leads to reduced theoretical efficiencies by increased nonradiative recombination for producing a smaller fraction of radiative recombination [1]. \[\mathrm{V_{OC}}=\frac{\mathrm{k\cdot T_{C}}}{\mathrm{q}}\cdot\mathrm{ln}\left[ \mathrm{f_{NR}}\cdot\frac{\mathrm{F_{S,MEG}}}{\mathrm{F_{C0,MEG}}-f_{NR}+1}\right] \tag{11}\]  
The correlated theoretical efficiency is shown in Equation (12):  
\[\eta(\mathrm{Efficiency})=\frac{\mathrm{J_{BB}(mpp)\cdot V(mpp)}}{\mathrm{P_{ in}}} \tag{12}\]  
where mpp is the maximum power point, \(\mathrm{P_{in}}\) is the input power, and \(\sigma\) is the Stefan-Boltzmann constant (\(=2\pi^{5}\mathrm{k}^{4}/(15c^{2}\mathrm{h}^{3})=5.670373\times 10^{-8}\mathrm{ Wm}^{-2}\mathrm{K}^{-4}\)). A similar approach has been discussed for nonradiative recombination in QD solar cells. This involves investigating the external radiative efficiency (ERE) and \(\mathrm{V_{OC}}\)[38].