Phase field simulation of the inclusion instability and splitting processes in interconnects due to interface diffusion induced by electromigration

Zhou, Linyong; Huang, Peizhen; Zhang, Jiaming

doi:10.2140/jomms.2023.18.39

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Phase field simulation of the inclusion instability and splitting processes in interconnects due to interface diffusion induced by electromigration

Linyong Zhou, Peizhen Huang and Jiaming Zhang

Vol. 18 (2023), No. 1, 39–58

DOI: 10.2140/jomms.2023.18.39

Abstract

Based on the bulk free energy density and the degenerate mobility constructed by the quartic double-well potential function, a phase field model is established to simulate the evolution of inclusions in interconnects due to interface diffusion in an electric field. The corresponding phase field governing equations are derived and the reliability of the program is proved by the agreement between the numerical simulation results and the theoretical analysis of the inclusion. The evolution of elliptical inclusions under different electric fields $χ$ , different aspect ratios $β$ , and different electric conductivity ratios $λ$ is calculated using the mesh adaptation finite element method. The results show that the drift velocity of the circular inclusion is proportional to the electric field and inversely proportional to the electric conductivity ratio. There exist critical values of the electric field $χ_{c}$ , the aspect ratio $β_{c}$ and the electric conductivity ratio $λ_{c}$ . When $λ \leq λ_{c}$ , $χ \geq χ_{c}$ or $β \geq β_{c}$ , the elliptical inclusions will split into several small inclusions. When $λ > λ_{c}$ , $χ < χ_{c}$ or $β < β_{c}$ , the elliptical inclusions will drift along the direction of the electric field as a relatively stable shape. The smaller the electric conductivity ratio of the inclusions $λ$ , the greater the electric field $χ$ or the aspect ratio $β$ , the easier it is for the elliptical inclusions to split. Moreover, the time required for splitting increases with increasing the electric conductivity ratio or the aspect ratio, and decreases with increasing the electric field. In addition, the interconnect line with two inclusions is more complex, and the inclusions will split into more small ones under the high electric field strength, and the phenomenon of multiple merging and multiple splitting will occur.

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Keywords

phase field method, electromigration, interface diffusion, finite element method, inclusion evolution

Milestones

Received: 20 November 2021

Revised: 5 June 2022

Accepted: 10 August 2022

Published: 24 March 2023

Authors

Linyong Zhou
The State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing China

Peizhen Huang
The State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing China

Jiaming Zhang
The State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing China

Vol. 18, No. 1, 2023