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3.2  Electromigration in Bulk Metals

The mass transport due to electromigration is modeled as the flow of vacancies. First the equations for EM in bulk were written in this form in [116] and further developed in [34, 136]. The governing equation of the vacancy concentration is given by the conservation law

(3.9) \{begin}{gather} \displaystyle \frac {\partial C_{\mathrm {v}}}{\partial t}=-\nabla \cdot \mathrm {\b {J}}_{\mathrm {v}}+G, \mathref {(3.9)} \{end}{gather}

with a flow term and a generation/annihilation term. The flow is the sum of the four flows driven by diffusion, by EM, a gradient of the pressure, and a gradient of the temperature distribution.

(3.10) \{begin}{gather} \mathrm {\b {J}}_{\mathrm {V}}=\mathrm {\b {J}}_{\mathrm {D}}+\mathrm {\b {J}}_{\mathrm {E}\mathrm {M}}+\mathrm {\b {J}}_{\mathrm {S}}+\mathrm {\b {J}}_{\mathrm {T}} \mathref {(3.10)} \{end}{gather}

The first flow contribution is a diffusion induced flow expressed by

(3.11) \{begin}{gather} \mathrm {\b {J}}_{\mathrm {D}}=-\mathrm {\b {D}}_{\mathrm {v}}\ \nabla C_{\mathrm {v}}, \mathref {(3.11)} \{end}{gather}

where \( \mathrm {\b {D}}_{\mathrm {v}} \) is the diffusion coefficient tensor. This coefficient is in general a tensor of second order as shown in Section 2.2.

The second component is the electromigration induced flow given by

(3.12) \{begin}{gather} \mathrm {\b {J}}_{\mathrm {E}\mathrm {M}}=\mathrm {\b {D}}_{\mathrm {v}}\frac {|Z^{*}|e}{k_{\mathrm {B}}T}{C_{\mathrm {v}}\mathrm {\b {E}}}, \mathref {(3.12)} \{end}{gather}

where \( Z^{*} \) is the effective valence, \( e \) is the elementary charge, \( k_{\mathrm {B}} \) the Boltzmann constant, \( T \) the temperature, and \( \mathrm {\b {E}} \) the electrical field.
Due to a gradient in the hydraulic pressure \( \sigma \) a flux is driven which is modeled by the third term as

(3.13) \{begin}{gather} \displaystyle \mathrm {\b {J}}_{\mathrm {S}}=-\mathrm {\b {D}}_{\mathrm {v}}\ \frac {f\Omega }{k_{\mathrm {B}}T}C_{\mathrm {v}}\nabla \sigma \mathref {(3.13)} \{end}{gather}

with \( f \) being the relation between the volume of a vacancy and the volume of an atom \( \Omega \). Therefore, \( f \) is in the range between zero and one. For the crystal lattice of metals the ratio is in the range of \( f=0.2-0.4 \) [39, 41].

The fourth term is the flow driven by temperature gradient.

(3.14) \{begin}{gather} \displaystyle \mathrm {\b {J}}_{\mathrm {T}}=\mathrm {\b {D}}_{\mathrm {v}}\ \frac {Q^{*}}{k_{\mathrm {B}}T^{2}}C_{\mathrm {v}}\nabla T \mathref {(3.14)} \{end}{gather}

For vacancies there is an equilibrium concentration to which the concentration converges in the absence of any outer perturbation. This phenomenon is modeled by the so called Rosenberg-Ohring term [21, 110].

(3.15) \{begin}{gather} G=\displaystyle \frac {C_{\mathrm {v},\mathrm {e}\mathrm {q}}-C_{\mathrm {v}}}{\tau } \mathref {(3.15)} \{end}{gather}

\( C_{\mathrm {v},\mathrm {e}\mathrm {q}} \) is the equilibrium concentration and \( \tau \) is the characteristic relaxation time of the vacancy concentration. The equilibrium concentration depends on the temperature according to an Arrhenius law [102]

(3.16) \{begin}{gather} C_{\mathrm {v},\mathrm {e}\mathrm {q}}=C_{\mathrm {v},0}\mathrm {e}^{-\frac {E_{\mathrm {a}}}{k_{\mathrm {B}}T}} . \mathref {(3.16)} \{end}{gather}

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