## 3 Results and Discussion  
Theoretical Analysis of the Recombination Parameters of the Diffusion Region and Depletion Region for Silicon Solar Cells  
The recombination parameter \(J_{01}\) of the diffusion region and the recombination parameter \(J_{02}\) of the depletion region are distinguished by the double-diode exponential model exactly [38; 39; 40; 41]. Figure 2 presents the experimental silicon solar cells and the double-diode model. According to the double-diode model's equivalent circuit, the expression of the current density-voltage (J-V) relationship of the solar cells is given by  
\[J=J_{ph}-J_{d1}-J_{d2}-\frac{V+JR_{s}}{R_{sh}} \tag{1}\]  
where  
\[J_{d1}=J_{01}\bigg{(}\exp\frac{V+JR_{s}}{n_{1}V_{T}}-1\bigg{)} \tag{2}\]  
\[J_{d2}=J_{02}\bigg{(}\exp\frac{V+JR_{s}}{n_{2}V_{T}}-1\bigg{)}, \tag{3}\]  
then  
\[J=J_{ph}-J_{01}\bigg{(}\exp\frac{V+JR_{s}}{n_{1}V_{T}}-1\bigg{)}-J_{02}\bigg{(} \exp\frac{V+JR_{s}}{n_{2}V_{T}}-1\bigg{)}-\frac{V+JR_{s}}{R_{sh}} \tag{4}\]  
where \(V_{T}\) is the thermal voltage, \(V_{T}\) = kT/q; \(k\) is the Boltzmann constant; \(q\) is the electronic charge in coulombs; \(T\) is the temperature in degrees Kelvin; \(n_{1}\) and \(n_{2}\) are quality factors in the quasi-neutral region and in the space-charge region, respectively; \(V\) is the applied forward bias in volts; and \(J_{ph}\) is the photo-generated current density, which is nearly equal to \(J_{sc}\). For calculating the recombination parameter \(J_{01}\) of the diffusion region and the recombination parameter \(J_{02}\) of the depletion region, we used the current density and voltage at the maximum power point, at the short-circuit point, and at the open-circuit point. Equations (2) and (3) can be obtained as follows:  
\[J_{01}=\frac{\left(J_{sc}-J_{m}-\frac{V_{m}+J_{m}R_{s}}{R_{sh}}\right)\bigg{(} \exp\frac{V_{oc}}{n_{2}V_{T}}-1\bigg{)}-\left(J_{sc}-\frac{V_{oc}}{R_{sh}} \right)\bigg{(}\exp\frac{V_{m}+J_{m}R_{s}}{n_{2}V_{T}}-1\bigg{)}}{\left(\exp \frac{V_{m}+J_{m}R_{s}}{n_{1}V_{T}}-1\right)\bigg{(}\exp\frac{V_{oc}}{n_{1}V_{ T}}-1\bigg{)}} \tag{5}\]  
\[J_{02}=\frac{\left(J_{sc}-J_{m}-\frac{V_{m}+J_{m}R_{s}}{R_{sh}}\right)\bigg{(} \exp\frac{V_{oc}}{n_{1}V_{T}}-1\bigg{)}-\left(J_{sc}-\frac{V_{oc}}{R_{sh}} \right)\bigg{(}\exp\frac{V_{m}+J_{m}R_{s}}{n_{1}V_{T}}-1\bigg{)}}{\left(\exp \frac{V_{m}+J_{m}R_{s}}{n_{2}V_{T}}-1\right)\bigg{(}\exp\frac{V_{oc}}{n_{1}V_{ T}}-1\bigg{)}}}. \tag{6}\]  
Figure 2: The experimental silicon solar cell and the double-diode model. The \(J_{01}\)/\(J_{02}\) ratio can be derived by Equations (5) and (6). \[\frac{J_{01}}{J_{02}}=\frac{\left(J_{sc}-J_{m}-\frac{V_{m}+I_{m}R_{s}}{R_{sh}} \right)\left(\exp\frac{V_{\infty}}{n_{2}V_{T}}-1\right)-\left(J_{sc}-\frac{V_{ \infty}}{R_{sh}}\right)\left(\exp\frac{V_{m}+I_{m}R_{s}}{n_{2}V_{T}}-1\right)}{ \left(J_{sc}-\frac{V_{\infty}}{R_{sh}}\right)\left(\exp\frac{V_{m}+I_{m}R_{s}}{ n_{1}V_{T}}-1\right)-\left(J_{sc}-J_{m}-\frac{V_{m}+I_{m}R_{s}}{R_{sh}}\right) \left(\exp\frac{V_{\infty}}{n_{1}V_{T}}-1\right)} \tag{7}\]  
For an eligible solar cell, \(R_{sh}\) is sufficiently large, and the \((V_{m}+I_{m}R_{s})/R_{sh}\) and \(V_{\infty}/R_{sh}\) terms can be neglected. Usually, \(V_{\infty}\) is large enough so that \(V_{\infty}/(n_{2}V_{T})\) and \(V_{\infty}/(n_{1}V_{T})\) are very much larger than unity at 300 K or so. Since the \(\exp[V_{\infty}/(n_{2}V_{T})]\) and \(\exp[V_{\infty}/(n_{1}V_{T})]\) variables are far greater than 1, the factor of unity in the brackets can be neglected.