ÁùºÏ²ÊÖ±²¥¿ª½±

The existing solution to avoid electromigration is to ensure that wires with potential large current densities have proper widths to hold them. Due to chemical-mechanical polishing (CMP) effects which reduces the thickness of wires, a thinner wire may be able to hold a larger current density than a wider one. So there could be more than one ranges of EM compliant widths for a wire that has a large current density. Current density tables are typically used to calculate EM compliant widths.

In a design, net routing always needs to meet various constraints, such as EM and resistance constraints. An automatic methodology of checking EM and resistance constraints is needed to improve their designer productivity. With such capability, users then can check their routing results at any stage of the routing flow and make adjustments accordingly to achieve EM and resistance constraints compliance. 

The EM constraint tells us how much current a connection object can sustain continuously to achieve a predefined mean time to failure (MTTF). There are 4 types of EM constraints for the 4 kinds of equivalent currents below:

1. The absolute current
2. The average current
3. The peak current
4. The root-mean-square (RMS) current 

To do an EM constraint check for a net, designers define the equivalent current of this type for each analog or digital cell pin, according to which the EM/R checking engine calculates the equivalent current of this type for each connection object of the net and then performs the following checks: 

• Use a current density table for the corresponding EM type supplied by the user to determine the proper width ranges to carry the current for each connection object of a net. If the actual width isn¡¯t within any EM compliant width range, a flag is sent to the user or routing results should be automatically adjusted. 
• Use via current tolerance constraints for the corresponding EM type supplied by the user to determine the min number of cut to carry the current for each via of a net. If the actual value is smaller than the EM compliant min number of cuts, a flag is sent to the user or routing results should be automatically adjusted.

Factors affecting electromigration:

• Wire Material. It is known that pure copper used for Cu-metallization is more electromigration-robust than aluminum. Copper wires can withstand approximately five times more current density than aluminum wires while assuming similar reliability requirements

• Wire Temperature. In Black¡¯s equation, which is used to compute the mean time to failure of metal lines, the temperature of the conductor appears in the exponent, i.e. it strongly affects the MTTF of the interconnect. The temperature of the interconnect is mainly a result of the chip environment temperature, self-heating effect of the current flow, heat of the neighboring interconnects or transistors, and thermal conductivity of the surrounding materials.

Black¡¯s Equation: MTTF= CJ^-ne(Ea/kT) where

?      C= a constant based on metal line properties;

?      J = the current density;

?      n = integer constant from 1 to 2

?      T = temperature in deg K;

?      k = the Boltzmann constant; and

?      Ea = Activation Energy 

• Wire Size. As Black¡¯s equation shows, apart from the temperature, it is the current density that constitutes the main parameter affecting the MTTF of a wire. Since the current density is obtained as the ratio of current I and cross-sectional area A, and since most process technologies assume a constant thickness of the printed interconnects, it is the wire width that exerts a direct influence on current density: The wider the wire, the smaller the current density and the greater the resistance to electromigration.

If the frequency is less than a critical value f₀ = ?(MTF_dc), the interconnect will follow a DC electromigration behavior. System fails even before the onset of reverse current. A gradual improvement in MTF happens as frequency is increased above f₀. This is due to increased effectiveness of damage healing during reverse period.

At the beginning of positive and negative pulses, atoms and vacancies start to migrate along grain boundaries and interfaces. This migration can recover with opposing stress. A shorter stress period means a relatively small displacement of atoms and vacancies, which is easy to be healed.

Within a very high frequency range, the damage healing process can overcome all defects which are brought during the other half period. However, an interconnect is never immortal. It can fail because of temperature gradients only. In this case Joule Heating sets the lifetime based on RMS current density. Waveform of a stress also has an impact on failure rates. For example, local melting will happen and cause failures for very large peak current densities even if the RMS current density is not very high.