## 3 Results and Discussion
### Deep Levels  
In the dark, the plot of the J-V curve on a log-log graph exhibits three distinct linear regions [47; 48; 49; 50]:  
* Ohmic region (\(J\propto V\));
* Trap-filling-limit region (\(J\propto V^{n>2}\));
* Child region (\(J\propto V^{2}\)). In the presence of traps, increasing the applied voltage results in an increased injected current, which will fill empty traps. The trap density (\(N_{t}\)) can be determined from the trap-filling-limit (V\({}_{\rm{ff}}\)) voltage, which is the intersection between the linear fits of the ohmic and trap-filling-limit regions of Figure 12. Once the intersection is known, the trap density can be calculated from the following relation [21; 51]:  
\[N_{t}=\frac{2e\epsilon_{0}V_{tfl}}{qL^{2}} \tag{11}\]  
where \(\varepsilon\) is the relative dielectric constant (assumed equal to 15.1 [45; 46]), \(\epsilon_{0}\) is the vacuum dielectric constant, \(q\) is the electron charge, and \(L\) is the antimony selenide thickness [52]. With an estimated trap density of \(\rm{N_{t}}\approx 1.1\cdot 10^{13}\) cm\({}^{-3}\), it can be then assumed that all traps in the Child region are filled. Consequently, the relationship governing the current-voltage trend in this region follows the Mott-Gurney law [48]. \[J_{C}=\frac{9}{8}\epsilon\epsilon_{0}\mu_{h}\frac{V_{C}-\rm{V_{bi}}}{L^{3}} \tag{12}\]  
Here, \(J_{C}\) and \(V_{C}\) are the values at the intercept between the linear fits of trap-filling-limit and Child regions. \(V_{bi}\) is the built-in voltage in the dark, and \(\mu_{h}\) denotes the hole mobility. Solving Equation (12) yields \(\mu_{h}\approx 8.9\) cm\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\). Figure 11: Plot of \(1/\rm{C^{3}}\) as a function of the bias voltage: (**a**) under light, (**b**) in dark conditions. Utilizing the resistivity measured in our previous work for antimony selenide in the dark [9], and applying the well-known relationship, we obtain the following:  
\[\sigma=\frac{1}{\rho}=q\rho\mu \tag{13}\]  
A free hole density \(p\) of 1.4 \(\times\) 10\({}^{14}\) cm\({}^{-3}\) is obtained. The difference of just one order of magnitude between the free hole density and the density of traps may explain the rather low performances of the solar cell. To assess the energy position of the traps, admittance spectroscopy measurements in the dark and under illumination at steps of 20 K from 193 K to 353 K were carried out. Notably, the curves at the lowest temperatures exhibit an inflection point for the capacitance at high frequency, which again confirms the presence of deep traps [53] as shown in Figure 13.  
By evaluating the frequency \(\omega_{0}\) at which the inflection points occur and considering the following relationship [54]:  
\[\frac{\omega_{0}}{T^{2}}=2\xi_{0}\exp\left(-\frac{E_{a}}{KT}\right) \tag{14}\]  
Figure 12: Plotting log(J) (unit: mA/cm\({}^{2}\)) against log(V) (unit: V) under dark conditions reveals three distinct regions, delineated by dotted vertical lines. Linear fittings of the data in the respective region are shown in red, blue and green for the Ohmic, TFL and Child region respectively. Figure 13: Admittance spectroscopy measurements (**a**) under light (**b**) in the dark. the activation energy of the deep levels can be determined, as shown in Figure 14.  
Two deep levels acting as traps were identified at (183 \(\pm\) 23) meV and (324 \(\pm\) 39) meV in the dark and under light, respectively. It is important to note that this is only an estimate since the C-\(\omega\) curves were obtained at relatively large d-factor values and that, with this procedure, it is not possible to determine the exact position of these traps, whether in the p-type or the n-type region. Anyway, from the curves shown in Figures 13 and 14, we can deduce that these energies represent the limits of a narrow energy band corresponding to a distribution of traps. This conclusion is supported by the absence of an evident inflection in each curve, as the decrease in capacitance is not sharp, which points to the overlapping effects of multiple trap levels close in energy.