Electromigration wire width calculator

How it works

Electromigration slowly moves metal atoms along a wire that carries current, until a void opens or a hillock shorts to a neighbour. A wire must be wide enough that its current density stays under the limit for the life and temperature it will see. This tool finds the narrowest width that meets the average, RMS and peak current limits, and the vias the net needs.

1. Reduce the waveform to three currents. Electromigration follows the average current, self-heating follows the RMS current and the peak current limits stress on the metal. A direct current gives all three the same value. A unidirectional pulse of peak Ip and duty D has an average of Ip x D and an RMS of Ip x sqrt(D). A bidirectional wire carries current both ways and part of the damage heals: each direction is credited with the other direction's average times a recovery factor R, and the larger remainder is the effective average.pulse: I avg = Ip x D; I rms = Ip x sqrt(D); bidirectional: I eff = max(0, I fwd − R x I rev, I rev − R x I fwd) with I fwd = Ip x D x forward share
2. Scale the limit with Black's equation. The foundry's DC limit is quoted at a reference temperature and lifetime. Black's equation says the median life falls with the current density to the power n and rises with exp(Ea / kT). Solving it for the current density that gives a target life lets the limit move to your own temperature and life. A hotter wire or a longer life lowers the limit.MTTF = A x J^(−n) x exp(Ea / kT); J use / J ref = (L ref / L use)^(1/n) x exp((Ea / (n k)) x (1 / T use − 1 / T ref))
3. Find the width for each limit. Every limit is a current per micrometre of width, so each width is the current divided by its limit. Electromigration uses the effective average and the scaled DC limit. Self-heating uses the RMS current and its limit. The peak limit uses the peak current. A minimum width rule can also set the answer. The width that meets everything is the largest of the four.w EM = I avg / (limit DC x scale); w RMS = I rms / limit RMS; w peak = I peak / limit peak; w = max(w EM, w RMS, w peak, w min)
4. Convert current density limits. Some design manuals give the limit as a current density in MA/cm2 instead of mA per micrometre. The two are tied by the metal thickness t: one MA/cm2 is ten mA per square micrometre. The tool accepts either and shows the density the finished wire runs at.limit [mA/um] = J limit [MA/cm2] x 10 x t [um]
5. Allow for the Blech effect (optional). A short segment can be immune to electromigration when the product of current density and length stays below a critical value, because stress builds up and pushes back. If you enter that product and the segment length, the tool also finds the narrowest width that stays immune and takes the smaller of the two electromigration widths. The RMS and peak widths still apply.immune if J x L <= (J L)c; w Blech = 10 x I avg x L / (t x (J L)c) with I in mA, L, t and w in um and (J L)c in A/cm
6. Count the vias. A via has its own current limit. It is scaled with the same temperature and life factor when you choose so, derated by the current sharing of the array, and the number of vias is the current divided by what each carries, rounded up. A peak limit per via can set the count instead.vias = ceiling(max(I avg / (via limit x scale x array efficiency), I peak / (via peak limit x array efficiency)))

Sources

The methods and default figures rest on these public sources. The wording here is our own; read the source for its full text.

• Electromigration: A brief survey and some recent results (opens in a new tab), IEEE Transactions on Electron Devices, volume 16, issue 4, pages 338 to 347 (J. R. Black), April 1969. Black's equation for the median life of a metal line under current.
• JESD63: Standard Method for Calculating the Electromigration Model Parameters for Current Density and Temperature, JEDEC Solid State Technology Association, Reaffirmed April 2023. Defines the exponent n and the activation energy as the model parameters, and how to estimate them from test data.
• JEP122H: Failure Mechanisms and Models for Semiconductor Devices (opens in a new tab), JEDEC Solid State Technology Association, September 2016. Lists failure mechanisms with activation energies and the current density sensitivity used for life estimates.
• Electromigration and its impact on physical design in future technologies (opens in a new tab), Proceedings of the ACM International Symposium on Physical Design (J. Lienig), 24 March 2013. Layout measures that raise current density limits, including short-length (Blech) and reservoir effects.