A CIGS solar cell is a p-n heterojunction (p-type CIGS absorber, n-type buffer layer, and TCO window). Its electrical behavior is classically modeled by a two-diode equivalent circuit comprising a photogenerated current source ( $I_{ph}$), a series resistance ( $R_s$), and a shunt resistance ( $R_{sh}$), as shown in Figure 1 [9] [10] .

In this framework, $R_s$ aggregates ohmic losses due to the sheet resistance of the transparent conducting oxide (i-ZnO/ZnO:Al), interfacial resistances, the back contact (Mo/CIGS), and the bulk resistivity of the active layers, whereas $R_{sh}$ accounts for leakage currents that partially short the junction (pinholes, percolation along grain boundaries, deep defects, metallic impurities) [9] [11] . The two diodes represent distinct recombination pathways (space-charge region vs. quasi-neutral regions), which enables faithful reproduction of the dark and illuminated J-V characteristics and of the impact of $R_s$ and $R_{sh}$ on the fill factor (FF) and the power conversion efficiency (PCE) [8] - [10] .

The current density delivered by a CIGS thin-film solar cell under load, within the two-diode model, is written as:

$\begin{array}{c}J=J_{ph}-J_{01}\left\{\exp \left[\frac{q\left(V+R_{s}J\right)}{n_1{k}_{B}T}\right]-1\right\}\\ -J_{02}\left\{\exp \left[\frac{q\left(V+R_{s}J\right)}{n_2{k}_{B}T}\right]-1\right\}-\frac{V+R_{s}J}{R_{sh}}\end{array}$ (1)

where $J_{01}$ and $J_{02}$ are the saturation current densities (A·cm⁻²), $n_1$ and $n_2$ are the diode ideality factors, $R_s$ and $R_{sh}$ are the (area-normalized) series and shunt resistances (Ω·cm²), J is the current density (A·cm⁻²), q the elementary charge (C), $k_B$ the Boltzmann constant (J·K⁻¹), T the temperature (K), V the terminal voltage, and $J_{ph}$ the photogenerated current density (A·cm⁻²) [11] . The J-V characteristic is measured under standard test conditions (STC), typically AM1.5G spectrum, $T=25˚\text{C}$, and irradiance $G=1000\text{W}\cdot {\text{m}}^{-2}$ [12] .

The short-circuit current density, measured at $V=0$, ideally equals the photocurrent ( $J_{sc}\approx J_{ph}$). It depends on irradiance and spectrum, optical absorption and reflection, device thickness/quality, and minority-carrier diffusion lengths. A general expression is:

$J_{sc}=q{\displaystyle {\int }_0^{{\lambda }_g}{I}_O\left(\lambda \right)\frac{\lambda }{hc}EQE\left(\lambda \right)\text{d}\lambda }\approx q{\displaystyle {\int }_0^{{\lambda }_g}{I}_O\left(\lambda \right)\frac{\lambda }{hc}\text{d}\lambda }\left(\text{if}EQE\simeq 1\right)$ (2)

where $I_O\left(\lambda \right)$ is the spectral irradiance (W·cm⁻²·nm⁻¹), $h$ is Planck’s constant (J·s), $c$ is the speed of light in vacuum (m·s⁻¹), $\lambda$ is the wavelength (m), and ${\lambda }_g$ is the cutoff wavelength associated with the absorber’s optical bandgap [11] .

Here $EQE\left(\lambda \right)$ is the external quantum efficiency (electrons collected per incident photon; $0\le EQE\le 1$), i.e., the fraction of incident photons at wavelength $\lambda$ that generate carriers collected at the external circuit.

In (2), we assume $EQE\left(\lambda \right)\approx 1$ and negligible front-side reflection $R_{front}\left(\lambda \right)\approx 0$. This first-order simplification is reasonable for an optically optimized device (e.g., low-reflectance TCO window with an anti-reflective coating and/or surface texturing), for which nearly all above-bandgap photons generate carriers that are actually collected. Formally, writing $EQE\left(\lambda \right)=IQE\left(\lambda \right)\left[1-R_{front}\left(\lambda \right)\right]A\left(\lambda \right)$ we adopt the upper-bound approximation $IQE\to 1$, $R_{front}\to 0$, and $A\left(\lambda \right)\to 1$ for $\lambda <{\lambda }_g$.

Where:

$IQE\left(\lambda \right)$: the internal quantum efficiency,

$R_{front}\left(\lambda \right)$: the front-side spectral reflectance,

$A\left(\lambda \right)$: the spectral absorptance of the device,

${\lambda }_g$: the bandgap wavelength of the absorber.

At open circuit, $J=0$. In the single-diode effective approximation one obtains:

$V_{oc}=\frac{n{k}_{B}T}{q}\cdot \ln \left(\frac{J_{SC}}{J_o}+1\right)$ (3)

where n is the (effective) ideality factor, typically $1\le n\le 2$ [13] .

The fill factor is defined by

$FF=\frac{P_m}{J_{sc}{V}_{oc}}=\frac{J_m{V}_m}{J_{sc}{V}_{oc}}$ (4)

with ( $J_m,V_m$) the current density and voltage at the maximum-power point. The fill factor can be written as:

$FF=F{F}_{id}\cdot \left(1-\frac{V_{OC}}{R_{Sh}{J}_{SC}}-\frac{J_{SC}{R}_S}{V_{OC}}+\frac{R_S}{R_{Sh}}\right)$ (5)

where $F{F}_{id}$ is the ideal fill factor of the cell [11] [12] .

The efficiency is the ratio of the maximum electrical power density to the incident optical power density:

$\eta =\frac{P_m}{P_{in}}=\frac{J_{sc}{V}_{oc}FF}{P_{in}}$ (6)

where $P_{in}$ is the incident irradiance (e.g., 100 mW·cm⁻² under STC) [11] [14] .

The thin-film solar cell under study consists of a polyethylene terephthalate (PET) substrate, a molybdenum (Mo) back contact, a Cu(In, Ga)Se₂ (CIGS) absorber, a CdS buffer layer, a ZnO window (transparent conducting oxide, TCO), and two aluminum metal fingers deposited on the TCO to collect the front-side current [3] [4] [15] .

The stack is Al/ZnO/CdS/CIGS/Mo/PET, in a substrate configuration (front-side illumination through the TCO window and Al grid). The structural schematic is shown in Figure 2 [3] [5] [16] .

We investigate, by numerical simulation, the influence of two key absorber parameters, optical bandgap $E_g$ and electron affinity $\chi$ on the photovoltaic figures of merit (short-circuit current density $J_{sc}$, open-circuit voltage $V_{oc}$, fill factor $FF$, and efficiency $\eta$). Calculations are performed using SILVACO ATLAS, by solving Poisson’s equation and the carrier continuity equations coupled with the drift-diffusion transport model [7] [8] .

Layer thicknesses follow the reference device specifications from our laboratory when available; the remaining parameters are taken from the literature [3] [7] [8] . A parametric analysis is conducted over the following ranges:

Unless stated otherwise, the absorber doping is fixed at 1 × 10¹⁶ cm⁻³, a value retained from our previous simulations.

Numerical convergence. We verified convergence by refining the mesh (halving $\Delta x$ and $\Delta y$ in ZnO/CdS/CIGS and near the electrodes), reducing the voltage sweep step (vstep: 20 mV → 10 mV → 5 mV), and doubling the spectral discretization (120 → 240 wavelengths). Deviations in $V_{oc}$, $J_{sc}$, FF, and $J_0$ remained <0.5% (or <0.3 mA·cm⁻² for $J_{sc}$ and <0.3 percentage point for FF. The time per bias point reported by ATLAS ranges from 0.01 to 0.05 s with the final setup.

Table 1 and Table 2 present the optoelectronic and geometric parameters ( Table 1 ), as well as the defect parameters in the various layers ( Table 2 ).

The optical bandgap of the CIGS absorber is given by:

$E_g=1.04+0.64x-0.15x\left(1-x\right)$ (7)

Figure 3 shows the current-density evolution as a function of voltage for different absorber bandgap values.

As outlined above, Figure 4 reports $J_{sc}$ versus $E_g$ for several absorber thicknesses. Over our parameter range, $J_{sc}$ is nearly invariant with $E_g$, with minor scatter attributable to thickness-dependent optical losses [17] .

Figure 5 shows the variation of the open-circuit voltage as a function of the optical bandgap of the absorber layer, for different thickness values.

Figure 6 shows the variation of the fill factor as a function of the optical bandgap of the absorber layer, for different thickness values.

Figure 7 depicts how the optical bandgap of the CIGS absorber influences the solar cell efficiency for different values of layer thickness.

The electron affinity of a semiconductor is the energy required to lift an electron from the conduction band edge to the vacuum level:

$\chi =E_0-E_c$ (8)

where $E_0$ is the vacuum energy and $E_c$ the conduction-band edge.

We now examine how $\chi$ affects the $J\left(V\right)$ characteristic, short-circuit current, open-circuit voltage, fill factor, and efficiency.

Experimentally, the target electron-affinity range can be reached by tuning the Ga fraction $x=\text{Ga}/\left(\text{Ga}+\text{In}\right)$ during co-evaporation/three-stage selenization (Ga-grading), since for $\text{Cu}\left({\text{In}}_{1-x}{\text{Ga}}_x\right){\text{Se}}_2$ the electron affinity follows $\chi \left(x\right)\approx 4.61-1.162x+0.034x^2\text{eV}$, and recent work demonstrates such control via engineered Ga profiles [21] [22] .

For context between 2023 and 2025, SCAPS-1D modeling studies project CIGS efficiencies above 24%, e.g., 24.43% with a p-Si BSF [23] , >31% with an Sb₂S₃ BSF [24] , and 24.61% with a CuAlO₂ BSF [25] . On the experimental side for single-junction devices, the best cells reported over the same period remain below 24% (~23% - 23.6% under STC), which underscores the value of the optimizations investigated here (increasing $E_g$, reducing $J_0$, and engineering the BSF/contacts) to bring real-world performance closer to the simulated potential.

Figure 8 reports the current-density as a function of voltage for several values of the absorber $\chi$ in CIGS.

Figure 8 shows that all J-V curves intersect the current axis at the same point () as the electron affinity $\chi$ is swept from 4.0 to 4.8 eV. In contrast, the voltage-axis intercept varies: about 0.69 V at $\chi =4.0\text{eV}$, 0.68 V at $\chi =4.2$ or 4.4 eV, and 0.67 V at $\chi =4.6$ or 4.8 eV. Hence, $\chi$ does not materially affect $J_{sc}$ in our range, whereas $V_{oc}$ generally decreases with increasing $\chi$ (with a slight leveling at the upper end). To confirm these trends, we analyze $J_{sc}$ and $V_{oc}$ versus $\chi$ separately [26] .

Figure 9 illustrates the variation of the short-circuit current density as a function of the electron affinity of the CIGS absorber layer, for different absorber thickness values.

Figure 9 shows that $J_{sc}$ remains unchanged for absorber thicknesses between 0.3 and 2 μm as $\chi$ varies from 4.0 to 4.8 eV. For a very thin absorber (0.1 μm), $J_{sc}$ is constant from 4.0 to 4.2 eV, then changes as $\chi$ increases from 4.2 to 4.8 eV. Thus, the influence of $\chi$ on $J_{sc}$ emerges only when the absorber is too thin (<0.3 μm), where band alignment and thickness-dependent optics weigh more heavily on carrier collection [9] .

Figure 10 shows the evolution of the open-circuit voltage as a function of the electron affinity of the absorber layer, for various thickness values.

As shown in Figure 10 , $V_{oc}$ decreases for all thicknesses (0.1 - 3 μm) as $\chi$ rises from 4.0 to 4.6 eV, followed by a slight recovery toward 4.8 eV. This behavior is consistent with an increase in effective saturation current $J_0$ when the band alignment deteriorates (e.g., a conduction-band cliff), which reduces $V_{oc}$ via Equation (3); when the offset approaches a more favorable regime (e.g., a small spike), interface recombination drops and $V_{oc}$ recovers [10] [27] .

Figure 11 shows the variation of the fill factor as a function of the electron affinity of the absorber layer, for various thickness values.

From Figure 11 , the fill factor improves for $\chi$ 4.0 → 4.2 eV and 4.6 → 4.8 eV, while it degrades within 4.2 → 4.6 eV, across all thicknesses (0.1 - 3 μm). This response reflects the combined evolution of $V_{oc}$ (setting $F{F}_{id}$) and resistive losses (first-order in $R_s$ and $R_{sh}$, Equation (4), Equation (5)). A less favorable offset in the 4.2 - 4.6 eV window enhances interface recombination (apparent rise of $J_0$), thereby lowering $FF$; near 4.0 - 4.2 and 4.6 - 4.8 eV, a better alignment mitigates losses and raises $FF$.

Figure 12 shows the evolution of the conversion efficiency as a function of the electron affinity of the absorber layer, for various thickness values.

Figure 12 shows that the efficiency $\eta$ increases when $\chi$ goes from 4.0 to 4.2 eV and from 4.6 to 4.8 eV, but declines between 4.2 and 4.6 eV, for all thicknesses (0.1 - 3 μm). This mirrors the behavior of $FF$ and $V_{oc}$: the central dip is consistent with an increase of $J_0$ (recombination/leakage) in the less favorable offset region, while the 4.0 - 4.2 and 4.6 - 4.8 eV windows provide a more favorable alignment that enhances $\eta$ [10] [27] .

Partial conclusion. The electron affinity affects $J_{sc}$ only for ultrathin absorbers (<0.3 μm), yet it systematically impacts $V_{oc}$, $FF$, and $\eta$. For good performance, $\chi \approx 4.0\text{-}4.2\text{eV}$ and $\chi \approx 4.6\text{-}4.8\text{eV}$ appear favorable [28] .