## 3 Results and Discussion
### Thickness of Interfacial IMC Layer  
The thicknesses of both Cu\({}_{6}\)Sn\({}_{5}\) and Cu\({}_{3}\)Sn IMC layers increased with aging time in SAC387/Cu and SAC0387-0.05 Nd/Cu interface. Figure 7c shows the computing process of IMC layer's thickness. The average thickness of IMC layer t was calculated by t = s/d, where s represents the area of IMC layer, which was analyzed by Image Pro-plus software, and d was measured as the width of the  
Figure 5: Microstructure evolution of SAC387-0.05 Nd solder/Cu solder joint after aging treatment: (a) 140 h; (b) 360 h; (c) 720 h; (d) 1440 h.  
Figure 6: Mechanism of Nd atoms on hindering the growth of intermetallic compounds (IMCs): (a,b) IMCs in solder matrix; (c,d) interfacial IMCs; (e) effect of Nd in Sn-Zn-Nd solder [19]. (J\({}_{\rm I}\) and J\({}_{\rm R}\) stands for the interfacial reaction flux and the ripening flux of Cu atoms, respectively). interface. Table 1 summarized the thickness of IMC layers after 1440 h aging; it can be seen that the thickness of total interfacial IMC layer at SAC387/Cu interface increased to 8.3 \(\upmu\)m, while that at the SAC0387-0.05 Nd/Cu was 6.8 \(\upmu\)m. Moreover, the interfacial Cu\({}_{3}\)Sn IMC layer reached 3.4 and 3.3 \(\upmu\)m, respectively. As a result, trace amount of Nd could effectively inhibit the growth of interfacial IIMCs. Generally, the growth of interfacial IMC layer is controlled by atom diffusion. The relationship between the thickness of interfacial IMC layer and aging time coincides with the following diffusion formula [13]:  
\[T_{t}=T_{0}+\sqrt{Dt} \tag{1}\]  
where \(T_{t}\) is the thickness of interfacial IML after a certain time \(t\), \(T_{0}\) is the initial thickness, \(D\) is the growth coefficient, as can be obtained from the slope of linear fitted curve of interfacial IMC layer thickness with aging time, as shown in Figure 7. Here below are the fitted equations:  
\[T_{(SAC387-Total)}=3.024+0.137\sqrt{t};\ T_{(SAC387-Cu_{3}Sn)}=0.023+0.090\sqrt{t} \tag{2}\]  
\[T_{(SAC387-0.05 Nd-Total)}=2.801+0.105\sqrt{t};\ T_{(SAC387-0.05 Nd-Cu_{3}Sn)}=- 0.103+0.084\sqrt{t} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**IMC Layer**} & \multirow{2}{*}{**Thickness (\(\upmu\)m)**} & \multicolumn{4}{c}{**Aging Time (h)**} \\ \cline{2-6}  & **0** & **144** & **360** & **720** & **1440** \\ \hline Cu\({}_{6}\)Sn\({}_{5}\) + Cu\({}_{3}\)Sn in SAC387 & 3.3 & 4.4 & 5.3 & 6.9 & 8.3 \\ \hline Cu\({}_{3}\)Sn in SAC387 & 0 & 1.0 & 1.9 & 2.4 & 3.4 \\ \hline Cu\({}_{6}\)Sn\({}_{5}\) + Cu\({}_{3}\)Sn in SAC387-0.05 Nd & 2.9 & 4.1 & 4.4 & 5.9 & 6.8 \\ \hline Cu\({}_{5}\)Sn in SAC387-0.05 Nd & 0 & 0.9 & 1.5 & 1.9 & 3.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thickness of IMC layers in the interface of solder/Cu during the aging process. Figure 7: Thickness of the interfacial IMC layers (IMLs) aged at 150 \({}^{\circ}\)C. (**a**): Thickness of Cu\({}_{6}\)Sn\({}_{5}\) + Cu\({}_{3}\)Sn layer; (**b**): Thickness of Cu\({}_{3}\)Sn layer; (**c**): The computing process of IMC layer’s thickness. It was calculated that the growth constant of total interfacial IMCs (\(D_{T}\)) and Cu\({}_{3}\)Sn (\(D_{Cu3}\)) in the SAC387/Cu joint was 1.88 \(\times\) 10\({}^{-10}\) cm\({}^{2}\)/s and 0.81 \(\times\) 10\({}^{-10}\) cm\({}^{2}\)/s, respectively, while in SAC0387-0.05 Nd/Cu joint the growth constant was 1.10 \(\times\) 10\({}^{-10}\) cm\({}^{2}\)/s and 0.71 \(\times\) 10\({}^{-10}\) cm\({}^{2}\)/s, respectively. Obviously, doping with 0.05 wt. % Nd can distinctly decrease the growth constant of total IMCs growth, while slightly decrease the growth constant of Cu\({}_{3}\)Sn (\(D_{Cu3}\)). The decrease in growth constant of total IMCs was mainly caused by the pinning effect on the growth of Cu\({}_{6}\)Sn\({}_{5}\) exerted by Nd atoms.