## 2 Optoelectronic Modeling Methods
### Electrical Modeling  
Electrical modeling is applied based on the drift-diffusion model, which is widely used in the electrical modeling of CIGS solar cells [9]. The drift-diffusion model consists of coupled Poisson and continuity equations for the electrons and holes. First, the electron current density \(J_{n}(\overrightarrow{r})\) and hole current density \(J_{p}(\overrightarrow{r})\), composed of the drift and diffusion components, were  
\[J_{n}(\overrightarrow{r})=-q\mu_{n}n(\overrightarrow{r})\
u V(\overrightarrow {r})+qD_{n}\
abla n(\overrightarrow{r}), \tag{9}\]  
\[J_{p}(\overrightarrow{r})=-q\mu_{p}p(\overrightarrow{r})\
u V(\overrightarrow {r})-qD_{p}\
abla p(\overrightarrow{r}), \tag{10}\]  
where \(q\) is the basic electric charge and \(V(\overrightarrow{r})\) is the electric potential. The terms \(n(\overrightarrow{r})\) and \(p(\overrightarrow{r})\) indicate the electron and hole concentrations, respectively. When \(\mu_{n}\) and \(\mu_{p}\) represent the electron and hole mobilities, \(D_{n}\) and \(D_{p}\) indicate the diffusion coefficients for the electrons and holes, which were obtained based on the [PERSON] relation \(D_{n(p)}=\mu_{n(p)}k_{B}T/q\). Here, \(k_{B}\) is the Boltzmann constant and \(T\) is the absolute temperature. The Poisson equation is expressed as  
\[\varepsilon_{0}\
abla\cdot\left[\varepsilon_{r}\
u V(\overrightarrow{r}) \right]=q[n(\overrightarrow{r})-p(\overrightarrow{r})+N_{A}^{-}( \overrightarrow{r})-N_{D}^{+}(\overrightarrow{r})], \tag{11}\]  
where \(\varepsilon_{0}\) is the electric permittivity in a free space and \(\varepsilon_{r}\) is the dielectric constant, whereas the acceptor and donor densities were \(N_{A}^{-}(\overrightarrow{r})\) and \(N_{D}^{+}(\overrightarrow{r})\). Finally, the continuity equation is given by  
\[\frac{1}{q}\
abla\cdot J_{n}(\overrightarrow{r})=-\left[G(\overrightarrow{r })-R(\overrightarrow{r})\right], \tag{12}\]  
\[\frac{1}{q}\
abla\cdot J_{p}(\overrightarrow{r})=G(\overrightarrow{r})-R( \overrightarrow{r}), \tag{13}\]  
where \(G(\overrightarrow{r})\) indicates the generation rate of the electron-hole pair, which is determined through Equations (7) or (8) depending on whether the thick cover glass is considered to be coherent (\(G(\overrightarrow{r},t_{b})\)) or incoherent (\(G^{ETAM}(\overrightarrow{r})\)), respectively. The term \(R(\overrightarrow{r})\) represents the recombination rate of the electron-hole pair, where we include the non-radiative Shockley-Read-Hall (SRH) and radiative recombination processes through \(R(\overrightarrow{r})=R_{SRH}(\overrightarrow{r})+R_{md}(\overrightarrow{r})\) in this calculation. The non-radiative SRH recombination rate is expressed as  
\[R_{SRH}(\overrightarrow{r})=\frac{n(\overrightarrow{r})p(\overrightarrow{r}) -n_{i}^{2}(\overrightarrow{r})}{\tau_{p}(\overrightarrow{r})\left[n( \overrightarrow{r})+n_{1}(\overrightarrow{r})\right]+\tau_{n}( \overrightarrow{r})\left[p(\overrightarrow{r})+p_{1}(\overrightarrow{r}) \right]}, \tag{14}\]  
where \(n_{i}(\overrightarrow{r})\) is the intrinsic carrier concentration when \(\tau_{p}(\overrightarrow{r})\) and \(\tau_{n}(\overrightarrow{r})\) represent the carrier lifetimes of the electron and hole, respectively. The intrinsic carrier concentration is different in each material and can be obtained by  
\[n_{i}(\overrightarrow{r})=\sqrt{N_{c}(\overrightarrow{r})N_{v}( \overrightarrow{r})}\exp[-\frac{\varepsilon_{g}(\overrightarrow{r})}{2k_{B }T}], \tag{15}\]where \(N_{c}(\overrightarrow{r})\) and \(N_{v}(\overrightarrow{r})\) were the effective density of states in the conduction and valence bands, respectively, when \(E_{g}(\overrightarrow{r})\) represents the band gap energy of the material.