## 1 Introduction
\[J=-D\frac{\partial c}{\partial x}+\Sigma_{i}CM_{i}F_{i} \tag{3}\]  
where  
\(D\) is the rate of diffusion,  
\(C\) is the atomic concentration,  
\(M\) is the atom mobility,  
\(F\) is the atomic diffusion driving force,  
\(-D\frac{\partial c}{\partial x}\) represents the chemical potential gradient,  
\(\Sigma_{i}CM_{i}F_{i}\) represents the sum of all atomic diffusion driving forces. When two metals with different properties and chemical potentials are in contact for a long time, a metal interface reaction occurs; that is, their atoms diffuse with each other. Depending on the characteristics of the interface and the resulting interface reaction, a new equilibrium state can be achieved under specific conditions, producing intermetallic compounds (IMCs). At present, the mainstream soldering materials for electronic packaging processes are copper (Cu) and tin (Sn). When they are in contact for a long time, the change in temperature produces IMCs, such as Cu\({}_{6}\)Sn\({}_{5}\) and Cu\({}_{3}\)Sn. [PERSON] et al. studied the diffusion phenomenon of Cu-Sn alloy and found that the IMC formation of Cu\({}_{6}\)Sn\({}_{5}\) is mainly due to the diffusion of Cu atoms to the Sn layer [17]. Further, [PERSON] et al. studied the effect of current crowding in solder bumps on electromigration [18] and found that gaps and voidswill be produced first at a current crowded area; these gaps will start to accumulate, initially resulting in failure at the current crowded area. This study also confirmed that high current density makes solder bumps prone to failure. In addition, [PERSON] and [PERSON] et al. studied the effect of joule heating and current crowding on IC circuits and found that when the current is kept constant and the circuit sectional area is reduced, the current density will increase and joule heating and temperature will increase [7]. Previously, [PERSON] et al. studied the electromigration effects of Cu-Sn IMC formation from different current inputs on a circuit with a fixed sectional area and found that a relatively high current will generate a high current density. This will cause the circuit to produce more holes, resulting in faster circuit failure [18]. However, because of the vast variety of modern integrated circuits in the market, the interconnecting IMC of ICs may experience electrical circuits with different sectional area. In order to understand the electromigration effect and IMC formation with different current stress caused by geometrical differences, this study designed an experiment that uses different cross-sectional settings resulting in changing current density and current crowded area of the IMC; thus, a circuit that with fixed input currents passing through different cross-sectional areas was employed to explore the effects of electromigration. As previously mentioned, the present-day IC circuit encounters reliability problems after long-term use; thus, this study designed three samples with different cross-sectional areas, subjected them to the same current, and examined the resulting current density and temperature change. Two of the samples were trapezoidal in shape, and the other was rectangular. As the current passed through each sample at a certain length of time, joule heating, current density, and resulting electromigration effect were measured, analyzed, and compared with the other samples.