Energy variation in diffusive void nucleation induced by electromigration
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RESEARCH PAPER
Energy variation in diffusive void nucleation induced by electromigration Yuexing Wang1 · Yao Yao2,3 · Zhang Long1 · Leon Keer4 Received: 14 December 2019 / Revised: 21 April 2020 / Accepted: 9 May 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract An energy approach is proposed to describe electromigration induced void nucleation based on phase transformation theory. The chemical potential for an individual migrated atom is predicted by diffusion induced back stress equivalent principle. After determining the chemical potential for the diffusing atoms, the Gibbs free energy controlling the void nucleation can be determined and the mass diffusion process is considered. The critical void radius and nucleation time are determined analytically when the Gibbs free energy approaches the extreme value. The theoretical predictions are compared with the experimental results from literatures and show good accuracy. The proposed model can also be applied to other diffusion induced damage processes such as thermomigration and stress migration. Keywords Energy · Electromigration · Void nucleation · Diffusion Abbreviations a The radius of a sphere void a0 The critical radius of electromigration void nucleation A The cross section area of the flux atoms across a void B The applicable modulus C0 The atom concentration e The unit electric charge D Diffusion coefficient D0 The diffusion pre-exponential factor Ds0 The surface diffusion pre-exponential factor Db0 The lattice diffusion pre-exponential factor Dg0 The grain boundary diffusion pre-exponential factor E Electric field intensity
* Yao Yao [email protected]; [email protected] 1
Institute of Electronic Engineering, China Academy of Engineering Physics, Mianyang 621999, China
2
School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China
3
School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, China
4
Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208, USA
Ee Young’s module G A parameter combined Z*, e, E and Ω written as G = Z*eE/Ω Gb The Gibbs free energy j The electric current density J The diffusion flux kb The Boltzmann’s constant K Thermodynamic temperature unit for Kelvins L A virtual length to consider the stress evolution in space N The total depleted atoms in a unit cubic volume p A parameter combined D, B, Ω, kb and T written as DBΩ/(kbT) Q The activate energy Qs The surface diffusion activate energy Qb The lattice diffusion activate energy Qgb The grain boundary diffusion activate energy R Universal gas constant S The surface area t The time T Temperature V0 The critical void nucleation volume W The elastic energy W1 The released stored elasticity energy W2 The energy stored in the defects such as dislocations or grain boundaries Z* The effective charge number σ Electromigratio
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