Fig. 1. Experimental tools (paradigm I) are utilized to ascertain the phases at the material interface, and also to perform the validation work during inverse design of electromigration parameters. The theoretical equations (paradigm II) are solved computationally in the finite element method based computational model (paradigm III). The artificial neural network (ANN, paradigm IV) is used to predict the effective charge number. And finally, with all the quantities determined, the FEM is used to compute the electrical resistance in the multiphase system.

Fig. 2. Experimental setup at beamline BL13W1 of SSRF for obtaining in-situ synchrotron radiation (SR) radiography images of electromigration tests at T = 523.15 K, is schematically sketched in (i). Asymmetrical growth of anode based IMC phase at t = (a) 5, (b) 10, (c) 25, and (d) 45 min for Cu/Sn/Cu joint undergoing electromigration can be clearly noticed in (ii).

Fig. 3. Schematic sketch to classify the anode and cathode electrodes in the Cu/Sn/Cu joint undergoing electromigration (a), outline the relevant phase at the anode interfaces for computational model (b).

Fig. 4. The Gibbs free energies with parabolic composition dependence assumed in this work for the FCC, IMC and Liquid phases: (a) the numerical values of G varying with composition, (b) sketch of common tangents, determining the equilibrium compositions in neighboring phases.

Fig. 5. Graphs of the IMC thickness as a function of time for a current density: (a) j = 1.0 × 106 A/m2, and (b) j = 10.0 × 106 A/m2 for the 5 sets of effective charge numbers, ${{\bar{Z}}_{1}}$, ${{\bar{Z}}_{2}}$, ${{\bar{Z}}_{3}}$, ${{\bar{Z}}_{4}}$ and ${{\bar{Z}}_{5}}$ (see Eq. (26)). The graphs show the effect on Cu6Sn5 IMC thickness due to variations in the effective charge numbers of the phases and components. For the higher current density (graph (b)), the effect of variations in the effective charge numbers on the growth is more pronounced.

Fig. 6. Electromigration parameters are considered as the input features whereas the IMC growth rate constant is taken as the output feature of the neural network.

Fig. 7. Pairplot of the effective charge numbers of the Cu component in each phase (ZCufcc,ZCuimc,ZCuliq) and growth rate constant (knmps, kem in nm/s) for two ranges of current density (j in A/mm2).

Fig. 8. Pairplot for the effective charge numbers of the Sn component in each phase (ZSnfcc,ZSnimc,ZSnliq) and growth rate constant (knmps, kem in nm/s) for 2 ranges of current density (j in A/mm2).

Fig. 9. Evolution of the mean square error of the neural network over the first 1000 epochs.

Fig. 10. Optimized effective charge numbers for Cu and Sn in FCC (a), IMC (b), and LIQUID (c) phase. The growth rate constants as a function of current density are shown in (d), in which the red line represents the experimental values whereas black line corresponds to the values predicted from optimization of neural network.

Fig. 11. Evolution of IMC grains at two different magnitudes of current densities as obtained from 2D phase field simulations using the effective charge numbers derived with the neural network analysis. The structure was initialized (t = 0) by placing the brown colored Cu6Sn5 IMC grains (smaller grain denoted as Gr 1 and larger grain denoted as Gr 2) at the interface between the LIQUID (red color) and FCC phase (blue color).

Fig. 12. The distribution of electric field potential ? corresponding to IMC evolution profiles at j = 2 × 106 A/m2 at t = 0.59 s.

Fig. 13. Evolution of the IMC grains Gr 1 and Gr 2 for j=(a) 2 × 106 A/m2, and (b) 10 × 106 A/m2 visualized using a color plot of the functionΣi(ηi)2.

Fig. 14. The temporal changes in area (S) of FCC, IMC and LIQUID phases for current density of j = (a) 2 × 106, and (b) 10 × 106 A/m2 are presented in the figure.

Fig. 15. Greater current density, associated with increase in effective charge number of IMC phase, accelerates the rate of increase of electrical resistance.