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SEPTEMBER 2024 I SMT007 MAGAZINE 79 der joint resistance due to increased joule heat- ing raising the solder temperature followed by a gradual parabolic increase in resistance that can be explained by the shape of the solvus line on the Sn-rich side of the phase diagram. Let us consider the room temperature to corre- spond to the g-f-e line and a higher tempera- ture to the d-c-b line on the phase diagram of Figure 8b. Notice that the Bi content of the Sn- rich phase increases from point f to point c on the phase diagram because of the temperature rise, thereby correspondingly increasing the solder joint resistance. e increase in the Bi content of the Sn-rich phase is diffusion con- trolled and therefore time delayed, that is the increase of the Bi content of the Sn-rich phase is not instantaneous but gradual as the Bi-rich phase particles dispersed in the Sn-rich phase dissolve with time into the Sn-rich phase. e Bi atoms given off by the Bi-rich phase diffuse with time in the Sn-rich phase matrix, raising the resistance of the Sn-rich phase matrix in which the electric current prefers to flow. e resistance of the solder thus rises with time in a parabolic fashion when there is step increase in current flowing through the solder joint. e same resistance increase phenomenon will occur if there is a step rise in the solder tem- perature due to the raising of the oven temper- ature as shown in Figure 8a at the two-hour mark. When the temperature of a solder joint is decreased in a step manner as shown at the 4.5-, six-, seven- and eight-hour marks, due to either the decrease in joule heating or the low- ering of the oven temperature, the solder joint resistance has a sharp drop due to a decrease in solder joint temperature followed by a grad- ual parabolic decrease in resistance due to the decrease of the Bi content of the Sn-rich phase. If the higher temperature corresponds to the line d-c-b and the lower temperature to the line g-f-e in the phase diagram of Figure 8b, the lowering of temperature lowers the equilib- rium solubility of Bi in the Sn-rich phase from point c to point f. Bismuth atoms in the Sn- rich phase diffuse to and deposit on the Bi-rich phase particles dispersed in the Sn-rich phase, thus gradually lowering the Bi content of the Sn-rich phase and the overall resistance of the solder joint. e above-discussed gradual change in sol- der joint resistance due to step change in cur- rent and/or solder temperature was reported in 2024 by a team from Purdue University that stated that the cause for such a change was yet to be determined 6 . Black's Equation vs. the Physics Approach A common engineering approach to deter- mining the electromigration life of solder joints is to determine the electromigration lives of a statistically significant number of similar sol- der joints at a contact temperature and electric current density and plot the times to failure on a Weibull plot. e process can be repeated at other temperatures and current densities and the mean-time-to-failure (MTTF) results used to calculate the constants in the following Black's equation: where B is a constant, j is the electric current density, Q is the activation energy for electro- migration, k is the Boltzmann constant and T is the temperature in kelvin. Once the con- stants in the equation are known, Black's equa- tion can be used to predict the electromigra- tion lifetime under any current density and temperature condition. Another approach to determining the elec- tromigration life of solder joints is to take a purely physics approach based on the Nernst- Einstein equation: where D is the diffusion coefficient of the elec- tromigrating ions, and F is the electrostatic force on the ions. e equation can be trans-