Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Electromigration Blech length
Electromigration AimsBefore you startIntroductionThe theory of electromigrationElectromigration damageFlux divergence (I)Flux divergence (II)Avoiding electromigration problemsAlternatives to aluminium metallizationSummaryQuestionsGoing further
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Blech length

The likelihood of electromigration damage depends not only on current density, but also on the absolute length of conducting segments. As shown by Blech, prevention of electromigration damage is possible when the product of the current density, j, and line length, l, is below a critical value (as shown below):

$${(jl)_c} = {{\Delta \sigma {\rm{ }}\Omega } \over {{Z^ * }e\rho }} $$

where Ω is the activation volume (m3) and σ is the hydrostatic component of the mechanical stress (kg m-1 s-2).

This relationship can be derived simply, by looking at the different contributions to flux within the metallization line. Electromigration of atoms, often results in a build up of stress within the metallization line. Regions of compressive and tensile stresses develop, resulting in a variation of stress (stress gradient) along the line. This stress gradient contributes to the gradient in the chemical potential, and thereby effects the mass flux, J:

$$J = {{DC} \over {RT}}\left( {{Z^ * }e\rho j - \Omega {{\Delta \sigma } \over {\Delta x}}} \right) $$


The animation below depicts the development of stress gradient during a Blech length experiment.


If the stress gradient is permitted to grow without the onset of damage (i.e. the stresses within the metallization line does not exceed the critical absolute stress levels for onset of damage), eventually its contribution to the chemical potential gradient is equal and opposite to that of the electric field. At that point, there is no gradient in chemical potential and no net flux of mass – thus the stress gradient does not continue to grow as it has achieved steady-state.

$$\eqalign{ & {Z^ * }e\rho j = \Omega {{\Delta \sigma } \over {\Delta x}} \cr & {{\Delta \sigma } \over {\Delta x}} = {{{Z^ * }e\rho j} \over \Omega } \cr} $$

Therefore, can approximate to:

$$\eqalign{ & {{\Delta \sigma } \over l} = {{{Z^ * }e\rho j} \over \Omega } \cr & {(jl)_c} = {{\Delta \sigma {\rm{ }}\Omega } \over {{Z^ * }e\rho }} \cr} $$

There is no electromigration damage below (jlc). An estimation can therefore be made for the longest conductor, which would have “infinite” life.

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