Revisiting the computation of Vacancy Formation Energy using first-principles density functional theory calculations for VLSI Interconnection reliability

Accurate prediction of vacancy formation energies is essential for the understanding of defect densities, which can affect material reliability. This work presents a revisit of the first-principles investigation on vacancy formation in metallic systems that influence vacancy energetics in face-centered cubic (FCC) aluminum (Al) and copper (Cu) interconnection technology, as well as a newly emerging hexagonal closed-packed (hcp) ruthenium (Ru) metal interconnects. Besides different computational choices for modeling of atomic defect structures, our density functional theory computation results show that, along with Local density approximation (LDA), meta-GGA with Strongly Constrained and Appropriately Normed (SCAN) functional are the most suited exchange correlation functionals for optimizing vacancy defects energetics in Al, whereas for Cu, Generalized Gradient Approximation (GGA) using Perdew-Burke-Ernzerhof (PBE), and meta-GGA SCAN functional are suitable. However, for Ru, we found that SCAN can lead to convergence failure, raising optimization complexity in the defect-containing structures, but regularized–restored SCAN (r²SCAN) can be an alternate choice for correctly predicting vacancy formation energy at 0 K, along with GGA-PBE. Our findings collectively provide quantitative evidence that both numerical convergence and material-specific bonding characteristics strongly affect defect energetics. Such findings further highlight the exponential dependence of equilibrium vacancy concentration (C₀) on its formation energy, which signifies that even minor deviations in computed values can lead to substantial errors in the predicted defect densities. Consequently, an accurate prediction of vacancy formation energy of any metal at 0 K is crucial for the reliability prediction of a metallic system.