## 2 Analytical Model  
The overall solar cell efficiency can be calculated using a single diode model and by applying the Green empirical expression [4]. Short-circuit current density, \(J_{sc}\), can be calculated using Equation (1) by assuming that an incident photon will produce an electron [9].  
\[J_{sc}=\frac{q}{hc}\int I(\lambda)A(\lambda)\lambda\ d\lambda \tag{1}\]  
where \(q\) is the electron charge, \(h\) is the Planck constant, \(c\) is the speed of light, \(I(\lambda)\) is the standard air mass 1.5 (AM1.5) spectral irradiance, and \(A(\lambda)\) is the optical absorption. On the other hand, the open-circuit voltage, \(V_{\infty}\), of silicon nanowire pillars is given by Equation (2) [24].  
\[V_{\text{OC}}=V_{Th}\ln[\frac{\Delta n(N_{A}+\Delta nt)}{n_{t}^{2}}] \tag{2}\]  
where \(V_{Th}\) is the thermal voltage which equals 25.8 mV, \(N_{A}\) is the doping concentration (\(N_{A}=3.3\times 10^{17}\) cm\({}^{-3}\)), \(\Delta n\) is the excess carrier concentration for \(n\)-type (\(\Delta n=10^{15}\) cm\({}^{-3}\)) [11],and \(n_{i}\) is the intrinsic carrier concentration. The open-circuit voltage of the silicon nanowires can be calculated and considered as a fixed value equal to 0.626 V as it depends on the energy level of the material and for \(n_{i}=10^{10}\) cm\({}^{-3}\) at room temperature.  
Hence, the fill factor can be calculated using Equation (3) [9].  
\[FF=\frac{\frac{V_{ox}}{V_{Th}}-\ln\left(\frac{V_{ox}}{V_{Th}}+0.72\right)}{\frac {V_{Th}}{V_{Th}}+1} \tag{3}\]  
The maximum output power, \(P_{max}\), can be calculated using Equation (4).  
\[P_{max}=J_{sc}\times V_{oc}\times FF \tag{4}\]  
Hence, the overall solar cell efficiency, \(\eta\), can be calculated as a ratio of maximum output power to solar input power.  
Gold, \(Au\), plasmonic nanoparticles distributed on the substrate layer will change the optical power absorbed in the active layer and depend on the maximum value of reflectivity. Nanoparticle shape has a great effect on the transmitted optical power as well as the relative permittivity of the gold nanoparticles and the dielectric function of the surrounding medium [9]. The maximum reflectivity can be calculated using Equation (5).  
\[\lambda_{max}=\frac{L}{g}\bigg{(}\frac{\varepsilon_{Au}\varepsilon_{m}(\lambda _{max})}{\varepsilon_{m}+\varepsilon_{Au}(\lambda_{max})}\bigg{)}^{1/2} \tag{5}\]  
where \(\varepsilon_{m}\) is the permittivity of the surrounding medium, \(\varepsilon_{Au}\) is a gold nanoparticle dielectric constant at corresponding \(\lambda_{max}\), \(g\) is an integer, and \(L\) is structural periodicity. Hence, the dielectric permittivity can be expressed by using a multioscillator Drude-Lorentz model [9] as given in Equation (6):  
\[\varepsilon_{Au}=\varepsilon_{\infty}-\frac{\omega_{D}^{2}}{\omega^{2}+j \omega\gamma_{D}}-\sum_{k=1}^{6}\frac{\delta_{k}\omega_{k}^{2}}{\omega^{2}- \omega_{k}^{2}+2j\omega\gamma_{k}} \tag{6}\]  
where \(\varepsilon_{\infty}\) is the gold high-frequency dielectric permittivity, \(\omega_{D}\) and \(\gamma_{D}\) are the plasma and collision frequencies of the free electron gas, \(\delta_{k}\) is the amplitude of Lorentz oscillator, \(\omega_{k}\) is the resonance angular frequencies, and \(\gamma_{k}\) is the damping constants for k value from 1 to 6.