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3 Models

In this section the models used for the analysis of the failure mechanisms are described.
In Section 3.1 the electro-thermal model is explained followed by the different EM models regarding bulk metals, interfaces between conducting materials, and the mechanical model for elastic materials influenced by plastic deformation due to EM induced material transport. This section is followed by the elucidation of void nucleation. The last section of this chapter deals with the evolution model which gives the ability to track the cross section reduction and thereby the resistance development in interconnect structures.

3.1  Electro-Thermal Model

For an ohmic material the current density and the electrical field is linearly related by the conductivity \( \sigma _{E}. \)

(3.1) \{begin}{gather} \mathrm {\b {J}}=\sigma _{\mathrm {E}}\mathrm {\b {E}} \mathref {(3.1)} \{end}{gather}

For a conservative electrical field the field can be expressed as the gradient of an electrical potential \( \phi _{\mathrm {E}}. \)

(3.2) \{begin}{gather} \mathrm {\b {E}}=-\nabla \phi _{\mathrm {E}} \mathref {(3.2)} \{end}{gather}

Inserting (3.2) into (3.1) and applying the divergence results in [138]

(3.3) \{begin}{gather} \nabla \cdot \mathrm {\b {J}}=-\nabla \ (\sigma _{\mathrm {E}}\nabla \phi _{\mathrm {E}}) . \mathref {(3.3)} \{end}{gather}

The left side of this equation must be \( 0 \) in a charge free region due to the conservation of charge. Under the assumption of a constant conductivity the equation can be simplified to Laplace’s equation [66].

(3.4) \{begin}{gather} \nabla ^{2}\phi _{\mathrm {E}}=0 \mathref {(3.4)} \{end}{gather}

For interfaces with normal vector \( \mathrm {\b {n}} \) between a conducting material and a non-conducting material the current density has to vanish and obey therefore

(3.5) \{begin}{gather} \mathrm {\b {J}}\cdot \mathrm {\b {n}}=0. \mathref {(3.5)} \{end}{gather}

Applying (3.4) results in

(3.6) \{begin}{gather} \displaystyle \frac {\partial \phi _{\mathrm {E}}}{\partial n}=0. \mathref {(3.6)} \{end}{gather}

The heat flow in materials can be written in the form [121]

(3.7) \{begin}{gather} \displaystyle \rho _{\mathrm {M}}c_{\mathrm {T}}\frac {\partial T}{\partial t}=\nabla \ (\lambda _{\mathrm {T}}\nabla T)+\dot {q}, \mathref {(3.7)} \{end}{gather}

where \( \rho _{\mathrm {M}} \) is the material density, \( c_{\mathrm {T}} \) is the specific heat capacity, \( \lambda _{\mathrm {T}} \) is the thermal conductivity, and \( \dot {q} \) is the heat generation per time. The heat generated in interconnect systems is the Joule heat given by

(3.8) \{begin}{gather} \dot {q}=\mathrm {\b {J}}\cdot \mathrm {\b {E}}=\sigma _{\mathrm {E}}|\mathrm {\b {E}}|^{2}=\sigma _{\mathrm {E}}|\nabla \phi _{\mathrm {E}}|^{2}. \mathref {(3.8)} \{end}{gather}

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