2 Electro-thermal stresses

The temperature distribution defined by (4.6)-(4.8) is used to set up the mechanical problem and the required equation for stress development due to thermal expansion is,

$$\sigma_{ij}=B\alpha(T-T_0)\delta_{ij}+\lambda\varepsilon_{kk}\delta_{ij}+2\mu\varepsilon_{ij}.$$ (219)

$B$ is the bulk modulus, $\mu$ and $\lambda$ are the Lame' constants, $\alpha$ is the thermal expansion coefficient, and $\varepsilon_{ij}$ are the components of the strain tensor.

The strain tensor is connected to the local displacements $u_i(x_1,x_2,x_3)$ through relations,

$$\varepsilon_{ij}(u)=\frac{1}{2}\Bigl(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\Bigr),\quad i,j=1,2,3,$$ (220)

and the mechanical stress equilibrium condition is given by,

$$\frac{\partial \sigma_{11}}{\partial x_1}+\frac{\partial \sigma_{21}}{\partial x_2}+\frac{\partial \sigma_{31}}{\partial x_3}=0,$$

$$\frac{\partial \sigma_{12}}{\partial x_1}+\frac{\partial \sigma_{22}}{\partial x_2}+\frac{\partial \sigma_{32}}{\partial x_3}=0,$$ (221)

$$\frac{\partial \sigma_{13}}{\partial x_1}+\frac{\partial \sigma_{23}}{\partial x_2}+\frac{\partial \sigma_{33}}{\partial x_3}=0.$$

Replacing (4.9) and (4.10) into (4.11) one obtains an equation system for unknown local displacement functions $u_i(x_1,x_2,x_3)$.

Equations (4.6)-(4.8) and (4.9)-(4.11) model time dependently the evolution of stress in the interconnect structures. These equations are solved by means of the finite element method [76,11]. After determining local displacement functions $u_i(x_1,x_2,x_3)$, the stress tensor is calculated using relations (4.9) and (4.10).

Depending on the interconnect layout, electro-thermal stresses can have tensile or compressive nature. High tensile stresses can cause break-up of the material and delopement of voids, on the other hand compressive stresses induce generation of extrusions.