Analysis of the Thermal Stress for Combined Electrode of Soldered Crystalline Silicon Solar Cells under Temperature Field

Based on the theory of material mechanics and thermal stress analysis, the stress distribution of combined electrode for crystalline silicon solarmodulewas studied for the first time.The shear stress and normal stress distribution of soldered structure for crystalline silicon solar cells under the thermal field were discussed. And the results show that the stress distribution is not simply linear relationship as some results found. But there is a stress concentration at the edge, which was considered as the true reason that caused microcracks at the edge of soldered solar cells.The conclusions we got in this paper provide a theoretical basis for deceasing the breakage rates of soldered crystalline silicon solar cells and improving the reliability of crystalline silicon solar modules.


Introduction
Crystalline silicon solar cells have become the main force of the photovoltaic industry.Crystalline silicon solar module is the critical component of photovoltaic (PV) generation system.Effective controlling of power degradation and increasing the conversion efficiency have the same importance for the investment income and reliability of PV power plant [1][2][3].Manufacturers usually guarantee that the lifetime of the solar cell module is expected to be 25 years with 20% reduction in its power output over this period.Installed and operated in hot and humid climate, the PV modules' typical operating temperature will change in the range −30∼+100 ∘ C. In the process of crystalline silicon solar modules production, the cells of PV modules are electrically connected in series by interlacing tin-coated copper ribbon.Due to the large difference of the coefficient of thermal expansion (CTE) between metal and silicon, thermal stress of such combined electrode will appear and accumulate in the metallization interconnection system when the PV module is suffering from a temperature cycle.The cyclic thermal stress will cause microcracks and voids, which will increase the contact resistance of the electrode and reduce PV modules' lifetime.Although many researches on module reliability have been done [4][5][6][7], there was no quantitative research for the thermal stress distribution of combined electrode inside the crystalline silicon solar module.
In this paper, a model of the thermal stress distribution of combined electrode for crystalline silicon solar module is established for the first time.The shear stress and normal stress distribution of soldered structure for crystalline silicon solar cell under the thermal field were discussed.And the results show that the stress distribution is not simply linear relationship as some results found [8].But there is a stress concentration at the edge, which was considered as the true reason that caused microcracks at the edge of the soldered solar cell.

Structure Model of Combined Electrode for
Crystalline Silicon Solar Cells As we know, glass layer is the main binder phase at Ag paste/Si contact interface, and the peeling off of cell electrodes always occurred in Ag paste layer.The coefficients of thermal expansion of metal materials are quite different from Si substrate, while the glass frit is between metal and Si.As shown in Figure 1, a simplified structure model with only three layers was established (Cu ribbon, Ag paste layer, and Si substrate).The assumption that each layer satisfies the homogeneous, isotropic, and linear conditions is made to simplify the model.
The thickness, width, and Young's modulus of each layer are   ,   , and   for ribbon,   ,   , and   for Si substrate layer, and   ,   , and   for thin glass layer, respectively. is the bond length.Considering the element shown in Figure 1, the equations of equilibrium for metal layer and Si substrate layer can be written as and   represent the forces caused by thermal stresses for metal layer and Si substrate layer.  is shear stress in the middle layer.

Thermal Stress Model of Combined Electrode for Crystalline Silicon Solar Cells.
As shown in Figure 2, this paper takes the expansive behavior of solar cell under a high temperature as an example to analyze shear stress distribution.Figure 3 shows the structure deformation of an element in the location .As the thickness of the middle thin glass layer is too thin, the shear stress in the glass layer can be considered as the same value   along thickness, and the shear stress in metal layer and Si substrate layer changes linearly along thickness.
The shear stresses in the middle layer, metal layer, and Si substrate layer can be written as follows: ,   , and   are the shear stresses in the middle layer, metal layer, and Si substrate layer, respectively.   and    are the shear stresses of the upper surface and lower surface.  () and   () are the coordinates of the two interfaces of the element in the location .   and    are the coordinates of the upper surface and lower surface. and  are the shear modulus and shear strain, respectively.
According to the stress-strain relationship, the coordinate functions in metal layer and Si substrate layer can be expressed as The displacement functions (, ) can be expressed as follows: (, ) =  (, ) − .
Substituting ( 4) into (5), the displacement functions in metal layer and Si substrate layer can be expressed as differential equations: The normal stresses of element  for the metal layer and Si substrate layer under temperature field are expressed as in which  is the coefficient of thermal expansion (CTE).Compared to the thickness of the metal layer and Si layer, the middle layer is so thin that it can be reasonably ignored.The forces in metal layer and Si layer can be expressed as They can also be written in the form of differential equation.Substituting (1) gives (9) Equation ( 2) can be written in the form of the following differential equation: Substituting (9) into (10) gives where As shown in Figure 3, the shear stresses of the upper surface and lower surface were very small and as well assumed to be 0.That is to say, coefficient  is 0. Hence, By solving differential equation (13), we can get the shear stress of the middle layer in the following form: The unknown constants  and  can be obtained by substituting the boundary conditions.4 gives out the analysis of the normal stress and strain.According to the thermal stress caused by expansion and contraction of materials under temperature Δ, the shear stress of location  ( = ) can be calculated.

Boundary Conditions. Figure
In the interface of metal layer and middle layer, two thin layers are analyzed.An assumption that the two thin layers are a rigid body and have the same cross section () is made to simplify the problem.After the expansion behavior, the final length of this interface layer is   .
The final length of this interface can be expressed as For the two thin layers, According to the normal stress definition, there are Similarly, by analyzing the interface of the middle layer and Si substrate layer, we got in which The final length of this interface can be expressed as Therefore, With the boundary conditions  = 0,   = 0,  = , The unknown constants in (14) can be obtained by substituting the boundary conditions

Solution and Analysis
The following results can be obtained by substituting the values of  and  into (14): Substituting the material parameters (shown in Table 1) into (25), curves of the shear stress in the middle layer were obtained by MATLAB.Figure 5 shows the shear stress distribution in the middle layer when the temperature is 55 ∘ C. Taking the derivatives of (26) with respect to  and substituting them into (6), the following relationships were obtained: By ( 27), when  is a fixed value, the relationship of normal stress and  is a quadratic function.When the value of  is 0.03, the distribution of upper and lower normal stress can be simulated by MATLAB.The result of simulation is shown in Figure 7.

Experimental Verification for the Thermal Stress Model of Combined Electrode for Soldered Crystalline Silicon Solar Cells
According to the above analysis, the stress of the soldered solar cells' side increases with increasing solder temperature, so microcracks induced by high stress begin from the side of the soldered solar cells.Figure 8 gives out the electroluminescence images of crystalline silicon solar module with microcracks at the edge of the soldered solar cell induced by thermal stress.From Figure 8, the microcracks always emerge on the side of the combined electrode for crystalline silicon solar cells.This proved the validity of the thermal stress model for soldered crystalline silicon solar cells.From the results, the way to decrease the breakage rates of soldered crystalline silicon solar cells is to reduce the thermal stress of combined electrode during the production of crystalline silicon solar modules.

Conclusions
In this paper, the stress distribution of combined electrode for crystalline silicon solar module was studied for the first time.The shear stress and normal stress distribution of soldered crystalline silicon solar cell under the thermal field were discussed.Studies have shown that the value of shear stress at the midpoint is the smallest and the shear stress along the direction of welding strip increases by function of cosh.The concentrated stress of edges leads to microcracks of the soldered solar cells.The results laid a theoretical basis for reducing the breakage rates of soldered crystalline silicon solar cells and improving the reliability of crystalline silicon solar modules during the production of solar modules.

Figure 2 :Figure 3 :
Figure 2: Simplification of cell structure.(a) Coordinate system with the origin at the center of the metallization.(b) Thermal expansion of the simplified structure.

Figure 5 :Figure 6 :
Figure 5: The shear stress distribution of the middle layer.

Figure 7 :
Figure 7: The normal stress distribution in Si substrate (a) and ribbon (b).

Figure 8 :
Figure 8: The electroluminescence images of solar modules with microcracks at the edge of soldered solar cell induced by thermal stress.

Table 1 :
Structural parameters of different material layers.