## 3 Results and Discussion
### Thickness Changes of Interfacial IMC Layer during Isothermal Aging  
For interfacial IMC layers at both interfaces of SAC0307/Cu and SAC0307-0.05 Pr/Cu, the thickness of Cu\({}_{6}\)Sn\({}_{5}\) and Cu\({}_{3}\)Sn IMC layers raised with aging time, as summarized in Table 2. With 840 h aging, the total interfacial IMC layer thickness at SAC0307/Cu interface grew to 10.2 \(\upmu\)m, whereas that at the SAC0307-0.05 Pr/Cu increased only to 6.5 \(\upmu\)m. In addition, Cu\({}_{3}\)Sn IMC layers' thicknesses at these two types of interfaces reached 4.9 \(\upmu\)m and 3.9 \(\upmu\)m, respectively. Obviously, a minute amount of Pr effectively hindered the growth of interfacial IMCs. Commonly, the growth and development of  
Figure 4: SEM images of the interfacial morphology at SAC0307-0.05 Pr solder/Cu interfaces aged at 150 \({}^{\circ}\)C for different hrs: (**a**) 72 h; (**b**) 240 h; (**c**) 528 h; (**d**) 840 h.  
Figure 5: Illustration of changes in interfacial IMCs at (**a**) SAC0307 solder/Cu interface and (**b**) SAC0307-0.05 Pr solder/Cu interface during aging process. Figure 3: SEM images of the interfacial morphology at SAC0307 solder/Cu interfaces aged at 150 \({}^{\circ}\)C for different hrs: (**a**) 72 h; (**b**) 240 h; (**c**) 528 h; (**d**) 840 h.  
interfacial IMC layer is diffusion-controlled. The relationship between interfacial IMC layer thickness and aging time meets the following classic diffusion formula [16]:  
\[x_{t}=x_{0}+\sqrt{Dt} \tag{1}\]  
where \(x_{t}\) is interfacial IMC layer's thickness change after one period of aging time \(t\), \(x_{0}\) is its initial thickness, \(D\) is the growth coefficient, as calculated from the slope of the fitted curve of interfacial IMC layer thickness with aging time, as shown in Figure 6. The following lists the final results. \[ \begin{array}{l}T_{(SAC0307\text{-}Total)}=5.04+0.18\sqrt{t};\ T_{(SAC0307 \text{-}Cu_{6}Sn_{5})}=4.28+0.03\sqrt{t};\\ T_{(SAC0307\text{-}Cu_{5}Sn)}=0.76+0.14\sqrt{t};\\ T_{(SAC0307\text{-}0.05 Pr\text{-}Total)}=2.67+0.13\sqrt{t};\ T_{(SAC0307\text{-}0.05 Pr \text{-}Cu_{5}Sn_{5})}=2.31+0.005\sqrt{t};\\ T_{(SAC0307\text{-}0.05 Pr\text{-}Cu_{5}Sn)}=0.35+0.12\sqrt{t};\\ \end{array}  \tag{2}\]  
\[ \begin{array}{l}T_{(SAC0307\text{-}0.05 Pr\text{-}Total)}=2.67+0.13\sqrt{t};\ T_{(SAC0307\text{-}0.05 Pr\text{-}Cu_{5}Sn_{5})}=2.31+0.005 \sqrt{t};\\ T_{(SAC0307\text{-}0.05 Pr\text{-}Cu_{5}Sn)}=0.35+0.12\sqrt{t};\\ \end{array}  \tag{3}\]  
It can be calculated that the growth constant of total interfacial IMC layer (\(D_{T}\)), Cu\({}_{6}\)Sn\({}_{5}\) (\(D_{Cu6}\)), Cu\({}_{3}\)Sn (\(D_{Cu3}\)) for the SAC0307/Cu joint was \(3.11\times 10^{-10}\) cm\({}^{2}\)/s, \(0.12\times 10^{-10}\) cm\({}^{2}\)/s and \(2.02\times 10^{-10}\) cm\({}^{2}\)/s, respectively, while that for SAC0307-0.05 Pr/Cu joint was \(1.65\times 10^{-10}\) cm\({}^{2}\)/s, \(0.0021\times 10^{-10}\) cm\({}^{2}\)/s (approximate to 0), and \(1.53\times 10^{-10}\) cm\({}^{2}\)/s, respectively. Obviously, doping 0.05 wt.% Pr can evidently lower the growth constant of Cu\({}_{6}\)Sn\({}_{5}\) (\(D_{Cu6}\)), while the growth constant of Cu\({}_{3}\)Sn (\(D_{Cu3}\)) was decreased slightly. The decreased growth constant of Cu\({}_{6}\)Sn\({}_{5}\) (\(D_{Cu6}\)) was mainly due to Pr atoms pinning on the growth of Cu\({}_{6}\)Sn\({}_{5}\). The suppressed mechanism for interfacial Cu\({}_{3}\)Sn was mainly associated with a decreased gradient of Sn atom concentration at Cu\({}_{6}\)Sn\({}_{5}\)/Cu\({}_{3}\)Sn interface. In addition, it can be observed that the decreased level for the growth constant of interfacial Cu\({}_{6}\)Sn\({}_{5}\) (\(D_{Cu6}\)) is much larger than that of D\({}_{Cu3}\). This can be explained as follows. Generally, there are two ways to form interfacial Cu\({}_{6}\)Sn\({}_{5}\). One is forming directly through Sn and Cu atom diffusion. \[6\text{Cu}+5\text{Sn}\rightarrow\text{Cu}_{6}\text{Sn}_{5} \tag{4}\]