## 2 Generation Mechanism of Electroluminescence Polarization Characteristics  
### Electroluminescence  
The crystalline silicon photovoltaic cells are generally N+/P homogeneous junction structure, and the surface of a photovoltaic cell usually has a dense pyramid texture structure of micrometer size or is covered with an antireflection film to reduce light reflection energy loss [13].  
In the equilibrium state, there is a built-in electric field E in the PN junction of crystalline silicon from the P to N region. The carrier diffusion current and drift current cancel out. When the photovoltaic cell is connected to the forward bias voltage, the carrier balance is broken, so that the electrons and holes can pass through the PN junction and become unbalanced carriers. The unbalanced carriers compound with majority carriers and radiate outward; this is called electroluminescence [14], and is shown in Figure 1.  
When the positive bias voltage is applied externally, the crystalline silicon solar cell can be simplified to an ideal PN junction with a thickness of \(d\). It can be assumed that the voltage directly acts on the PN junction, as shown in Figure 2.  
Figure 1: Schematic diagram of electroluminescence.  
Figure 2: Ideal PN junction under forward bias.  
Under the conditions of small injection and abrupt depletion layer, ignoring the reabsorption of photons generated by radiation recombination [15; 16], the electroluminescence intensity generated at different positions \(x\) can be represented as:  
\[I_{L}(E,x)=\frac{2\pi}{h^{3}c^{2}}\alpha(E)E^{2}\exp(-\frac{E}{kT})\exp(\frac{ qV_{f}}{kT})[\frac{\exp(-\frac{x}{\sqrt{L_{n}}})}{1-\exp(-\frac{2d}{\sqrt{L_{n}}})}+ \frac{\exp(\frac{x}{\sqrt{L_{n}}})}{1-\exp(\frac{2d}{\sqrt{L_{n}}})}] \tag{1}\]  
where, \(h\) is Planck constant; \(c\) is the speed of light in vacuum; \(\alpha(E)\) is the absorption coefficient of photons; \(E\) is the energy of photons generated by radiation recombination; \(q\) is the unit charge; \(V_{f}\) is the positive bias voltage applied on the battery chip; \(k\) is Boltzmann constant; \(T\) is the absolute temperature; \(L_{n}\) is the electron diffusion length, and \(L_{n}=\sqrt{D_{n}\tau_{n}}\) (\(D_{n}\) is the diffusion coefficient, \(\tau_{n}\) is the minority carrier lifetime).  
By integrating Formula (1), the luminous intensity of the photovoltaic cell can be calculated:  
\[I_{L}=\int\limits_{E}\frac{2\pi}{h^{3}c^{2}}\alpha(E)E^{2}\exp(-\frac{E}{kT}) dE\bullet\int_{0}^{d}\left[\frac{\exp(-\frac{x}{\sqrt{L_{n}}})}{1-\exp(- \frac{2d}{\sqrt{L_{n}}})}+\frac{\exp(\frac{x}{\sqrt{L_{n}}})}{1-\exp(\frac{2d }{\sqrt{L_{n}}})}\right]d\gamma\bullet\exp(\frac{qV_{f}}{kT}) \tag{2}\]  
Generally, the thickness \(d\) of silicon wafer is much greater than the diffusion length \(L_{n}\), so Formula (2) can be simplified as:  
\[I_{L}=C\sqrt{L_{n}}\exp(\frac{qV_{f}}{kT}) \tag{3}\]  
where \(C\) is the constant coefficient after simplification.  
It can be seen from Formula (3) that the electroluminescence intensity of photovoltaic cells is related to the external positive voltage and minority carrier diffusion length (minority carrier lifetime). At room temperature and constant voltage, the non-equilibrium carrier density at the defect is relatively low, the minority carrier life is reduced, and the diffusion length is shortened; therefore, the radiation intensity is relatively weak. The defect area is darker in the image and shows different morphologies. Therefore, electroluminescence can be used for defect detection of crystalline silicon photovoltaic cells.