Energy variation in diffusive void nucleation induced by electromigration

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.

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Abbreviations

a :

The radius of a sphere void

a 0 :

The critical radius of electromigration void nucleation

A :

The cross section area of the flux atoms across a void

B :

The applicable modulus

C 0 :

The atom concentration

e :

The unit electric charge

D :

Diffusion coefficient

D 0 :

The diffusion pre-exponential factor

D s0 :

The surface diffusion pre-exponential factor

D b0 :

The lattice diffusion pre-exponential factor

D g0 :

The grain boundary diffusion pre-exponential factor

E :

Electric field intensity

E e :

Young’s module

G :

A parameter combined Z*, e, E and Ω written as G = Z*eE/Ω

G b :

The Gibbs free energy

j :

The electric current density

J :

The diffusion flux

k b :

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

Q s :

The surface diffusion activate energy

Q b :

The lattice diffusion activate energy

Q gb :

The grain boundary diffusion activate energy

R :

Universal gas constant

S :

The surface area

t :

The time

T :

Temperature

V 0 :

The critical void nucleation volume

W :

The elastic energy

W 1 :

The released stored elasticity energy

W 2 :

The energy stored in the defects such as dislocations or grain boundaries

Z * :

The effective charge number

σ :

Electromigration induced back stress

τ :

The characteristic length of the maximum elasticity zone

μ :

The chemical potential of the diffusion atoms

ρ :

Electrical resistivity

γs :

The surface energy

χ :

The ratio between the applicable modulus and the Young’s modulus (B/E0)

λ :

The eigenvalue of corresponding voids shape (λ = Z*eEr2/(γsΩ))

Ω :

The atom volume

Φ :

The electric potential whose gradient represents electric field intensity

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Acknowledgements

The authors would like to acknowledge the financial support by the National Natural Science Foundation of China (Grant 11772257), Natural Science Foundation of Shaanxi Providence (Grant 2020JM-103) and Fundamental Research Funds for the Central Universities (Grant G2019KY05212).

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Correspondence to Yao Yao.

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Wang, Y., Yao, Y., Long, Z. et al. Energy variation in diffusive void nucleation induced by electromigration. Acta Mech. Sin. (2020). https://doi.org/10.1007/s10409-020-00963-2

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Keywords

  • Energy
  • Electromigration
  • Void nucleation
  • Diffusion