Editing Electromigration (section)

== Failure mechanisms ==

=== Diffusion mechanisms ===
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In a homogeneous crystalline structure, because of the uniform lattice structure of the metal ions, there is hardly any momentum transfer between the conduction electrons and the metal ions. However, this symmetry does not exist at the grain boundaries and material interfaces, and so here momentum is transferred much more vigorously. Since the metal ions in these regions are bonded more weakly than in a regular crystal lattice, once the electron wind has reached a certain strength, atoms become separated from the grain boundaries and are transported in the direction of the current. This direction is also influenced by the grain boundary itself, because atoms tend to move along grain boundaries.

Diffusion processes caused by electromigration can be divided into grain boundary diffusion, bulk diffusion and surface diffusion. In general, grain boundary diffusion is the major electromigration process in aluminum wires, whereas surface diffusion is dominant in copper interconnects.

=== Thermal effects ===
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In an ideal conductor, where atoms are arranged in a perfect [[crystal structure|lattice]] structure, the electrons moving through it would experience no collisions and electromigration would not occur.  In real conductors, defects in the lattice structure and the random thermal vibration of the atoms about their positions causes electrons to collide with the atoms and [[scattering|scatter]], which is the source of electrical resistance (at least in metals; see [[electrical conduction]]).  Normally, the amount of momentum imparted by the relatively low-[[mass]] electrons is not enough to permanently displace the atoms.  However, in high-power situations (such as with the increasing current draw and decreasing wire sizes in modern [[VLSI]] [[microprocessor]]s), if many electrons bombard the atoms with enough force to become significant, this will accelerate the process of electromigration by causing the atoms of the conductor to vibrate further from their ideal lattice positions, increasing the amount of electron [[scattering]].  High [[Current (electricity)|current density]] increases the number of electrons scattering against the atoms of the conductor, and hence the rate at which those atoms are displaced.

In integrated circuits, electromigration does not occur in [[semiconductor]]s directly, but in the metal interconnects deposited onto them (see [[Fabrication (semiconductor)|semiconductor device fabrication]]).

Electromigration is exacerbated by high current densities and the [[Joule heating]] of the conductor (see [[electrical resistance]]), and can lead to eventual failure of electrical components. Localized increase of current density is known as [[current crowding]].

=== 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.