Editing Electromigration (section)

=== Balance of atom concentration ===
A governing equation which describes the atom concentration evolution throughout some interconnect segment, is the conventional mass balance (continuity) equation

:<math>\frac{\partial N}{\partial t} + \nabla\cdot\vec J = 0</math>

where <math>N(\vec x, t)</math> is the atom concentration at the point with a coordinates <math>\vec x=(x, y, z)</math> at the moment of time <math>t</math>, and <math>J</math> is the total atomic flux at this location. The total atomic flux <math>J</math> is a combination of the fluxes caused by the different atom migration forces. The major forces are induced by the [[electric current]], and by the gradients of temperature, [[stress (physics)|mechanical stress]] and concentration. <math>\vec J = \vec J_c + \vec J_T + \vec J_\sigma + \vec J_N</math>.

To define the fluxes mentioned above:

: <math>\vec J_c = \frac{NeZD\rho}{kT}\,\vec j</math>. 
Here <math>e</math> is the [[electron]] charge, <math>eZ</math> is the effective charge of the migrating atom, <math>\rho</math> the [[resistivity]] of the conductor where atom migration takes place, <math>\vec j</math> is the local current density, <math>k</math> is the [[Boltzmann constant]], <math>T</math> is the [[absolute temperature]]. <math>D(\vec x, t)</math> is the time and position dependent atom diffusivity.

: <math>\vec J_T = -\frac{NDQ}{kT^2}\nabla T</math>. &nbsp;We use <math>Q</math> the heat of thermal diffusion.

: <math>\vec J_\sigma = \frac{ND\Omega}{kT}\nabla\! H</math>, 
here <math>\Omega=1/N_0</math> is the atomic volume and <math>N_0</math> is initial atomic [[concentration]], <math>H=(\sigma_{11}+\sigma_{22}+\sigma_{33})/3</math> is the [[hydrostatic stress]] and <math>\sigma_{11},\sigma_{22},\sigma_{33}</math> are the components of principal stress.

: <math>\vec J_N = -D\,\nabla\! N</math>.

Assuming a vacancy mechanism for atom [[diffusion]] we can express <math>
D</math> as a function of the hydrostatic stress <math>D = D_0\exp\left(\tfrac{\Omega H - E_A}{kT}\right)</math> where <math>E_A</math> is the effective [[activation energy]] of the thermal diffusion of metal atoms. The vacancy concentration represents availability of empty lattice sites, which might be occupied by a migrating atom.