## 3 Simulation
Figure 3: The time-domain signal graph of ultrasound. The change process of the relative position between the solder ball and the ultrasonic probe was simulated by modifying the position of the solder ball. At the beginning of the simulation, the virtual probe is located directly above the center of the micro defect, and then the position of the micro defect moves in a step of 1 \(\upmu\)m to simulate the whole process of the ultrasonic probe scanning solder ball. When a solder ball moves 1 micron, it will be simulated once. The simulation moved a total of 40 \(\upmu\)m and obtained 41 transient simulation results. Combining all of the transient simulation results, the A-scan signal, in a similar B-scan mode, results in a B-scan image of the typical defects of the solder balls, as shown in Figure 4.  
Comparing all the four different defect diagrams, it can be seen that the energy distribution of the four diagrams is obviously different between 75 and 95 ns, as shown in Figure 5. This phenomenon also denotes that the ultrasonic echo of typical defects of solder balls is within the time threshold. At the same time, comparing the four B-scan diagrams in Figure 4, we can see that the acoustic pressure energy near 50 ns is the highest, which is consistent with the phenomenon that the highest reflection energy occurs at the interface of water and silicon, which is caused by the reflection and transmission of the ultrasonic energy at each acoustic impedance interface. Figure 5a shows the B-scan image of a defect-free solder ball. Within the time threshold 75-110 ns of the defect, it can be seen that the ultrasound echo amplitude in the area of 0-15 microns at the position of P0 is obviously less than that in the area of 15-40 microns. This is because the acoustic impedance of the silicon and tin-silver-copper interface is smaller than that of silicon and the water interface, which results in the ultrasonic energy of transmission and refraction occurring in the region of the radius of the solder ball within 15 microns, and the ultrasonic energy and amplitude of reflection are smaller. As can be seen from the B-scan image of the ball missing in Figure 5b, the energy expressed at 0-40 microns is the same, because the interface passed by ultrasound is the silicon and dielectric water interface at the position of 0-40 microns. Figure 4: B-scan image of solder ball defects ((**a**) defect-free, (**b**) ball missing, (**c**) void, and (**d**) crack). For the B-scan image of void defect in Figure 5c, it can be seen that the ultrasonic amplitude (yellow-blue-yellow) in 87-90 ns time domain is arc-shaped, because the void defect in solder balls is oval. However, the void in solder balls is very small, and the difference between the B-scan image and the image without defects is not particularly obvious. For the crack defect in Figure 5d, because the crack is located on the middle cross-section of the solder ball and the filling medium is water, it can be seen that within the time threshold 80-85 ns, the ultrasonic reflection amplitude is less than the ultrasonic amplitude within 87-92 ns at the position of 0-15 microns. This is because the ultrasound experiences two interfaces during this time, namely the interface of silicon and tin-silver-copper and the interface of tin-silver-copper and water. The ultrasonic amplitude at the position of 15-40 is consistent with the amplitude in Figure 5a,c. To observe the sound wave transmission, reflection, and refraction during the sound propagation process, the sound wave propagation process in the entire time domain was integrated into one image, the acoustic propagation map, which can realize better analysis of the acoustic energy distribution around the micro defects. In order to highlight the changes in the energy of sound waves of the micro defects caused by ultrasound, we further extract and focus the scan results. The sampling area is shown in Figure 6. The adopted area was under the probe in the different probe detection positions.