## 3 Size Effects on Mechanical Properties of Solder Joints
### Tensile Behavior of Solder Joint  
With the decreasing bump height, the transition of the failure mode from ductile fractures in solder to brittle fractures along IMCs was reported in tensile tests [46]. It is noted that Cu pillar micro-bump solder joints are an important type of interconnection in 3D electronic packaging. Researches usually employ the line-type specimen to investigate the properties of Cu pillar micro-bump solder joints, with different solder heights which range from 5 to 60 \(\upmu\)m, such as in [PERSON] et al. 's study [46]. When the solder height of the micro-bump solder joint drops to 10 \(\upmu\)m or less, the whole solder alloy has the potential to transform into IMCs. Then, Cu/IMC/Cu or Ni/IMC/Cu solder joints will occur. Apart from the brittle IMCs, micro-cracks due to Kirkendall voids will bring reliability challenges. More elaborate properties on Cu pillar micro-bump solder joints were summarized by [PERSON] et al. [1]. [PERSON] et al. [47] investigated the joint gap size effects on the tensile properties of Cu/Sn0.7 Cu0.05 Ni/Cu solder joints by experiments and simulations. [PERSON] et al. [31] conducted tensile tests on Cu/Sn3.0 Ag0.5 Cu/Cu solder joints with gap sizes ranging from 50 to 2000 \(\upmu\)m, and they found that the tensile strength increased as the joint gap decreased, and the stress triaxiality was used to explain the higher strength of joints with smaller gaps. [PERSON] et al. [48] performed microscale tensile tests of Ni/Sn3.0 Ag0.5 Cu/Ni and Cu/Sn3.0 Ag0.5 Cu/Cu solder joints with thickness-to-diameter ratios (\(R\)) from 1/3 to 1/12 by a dynamic mechanical analyzer (DMA). Both tensile strength and fracture mode were dependent on \(R\), and the damage-equivalent stress, which is deemed as a suitable criterion in evaluating the strength and damage of the joints by the finite element method, showed a decreasing trend with the decrease of the \(R\) value. In addition to the typical gap size effects on solder joint behavior, void formation in the solder joints during flux outgassing was found to enhance the ductility of the Cu/Sn3.5 Ag0.75 Cu/Cu solder joints in tensile tests. This effect was significantly more obvious in joints with larger solder gaps [49]. For an ideal rigid plastic body, the geometry effect on fracture strength \(\sigma_{F}\) of the solder joint was suggested as:  
\[\sigma_{F}=\sigma_{yield}(1+d/6h) \tag{3}\]  
Figure 7: Eutectic structures and \(\beta\)-Sn dendrite phases in the Ni/Sn3.5 Ag/Cu solder joints with different bump heights [44]. (Reused with permission). where \(\sigma_{yield}\) is the yield strength of the solder material, and \(h\) and \(d\) are the solder thickness and substrate diameter, respectively. However, the ideal rigid plasticity is a strong assumption, especially when \(h\) approaches \(d\). Tensile strengths of different types of solder joints are summarized in Figure 8. A linear relationship between the logarithmic tensile strength and the logarithmic V/A is found. As noticed for the data from [PERSON] et al. and [PERSON] et al. 's studies [24; 26] which are not well-appropriated by linear fitting, this may be due to the significantly slower loading rate, which leads to creep failure [26], and also the excessive IMC growth in solder joints with small V/A ratios, which leads to failure mechanism transition [24]. [PERSON] et al. [50] evaluated the mechanical size effects of Cu/Sn3.5 Ag/Cu solder joints with gap sizes ranging from 25 to 850 \(\upmu\)m by the Orowan model. Based on a constraint mechanism that enhanced the shear strength and was different from that in tensile specimen, they also pointed out that the constraint effect which enhanced the ultimate tensile and yield strengths was only dominant for gap sizes below 200 \(\upmu\)m. The IMC effect on the mechanical strength, as found in [PERSON]'s research [50], does not seem obvious. However, when the IMCs occupy a significant portion of the whole solder joint, the IMC effect cannot be neglected. [PERSON] et al.