Black's equation

Blacksche the equation is the average failure time (MTTF: Mean Time To Failure ) of a conductive path due to damage caused by electro-migration in response to the temperature and the electrical current density. It can be used lifetimes of interconnects in integrated circuits (ICs ), which have been tested under the influence of increased temperature and increased current density, extrapolate lifetimes under realistic conditions. The equation was erected in the 1960s by James R. Black and is used with some adjustments made to date.

Background

Integrated circuits have to be able to operate reliably for years or even decades. Computer as they are commonly available in the market, should, if possible 40,000-60,000 hours and in the military field work even over one hundred thousand hours without failure. It is therefore not useful to test these under operating conditions and to wait for a statistically significant portion of the sample fails. In order to be able to work with the lowest possible cost to gain useful information about the area bounded by electromigration lifetime, the experiments are carried out under accelerated conditions. This is achieved by increasing the current density and the ambient temperature. Typical values ​​for the temperature of 175 to 300 ° C and the current density j = 106 A/cm2 - 107 A/cm2. This stress combination reduces the downtime of years to weeks or even days. The lifetime under normal operating conditions can then be extrapolated to the Black 's equation.

The determination of the failure criterion represents a further difficulty dar. You can select the total failure of this line, a certain change in resistance or short circuit of two adjacent lines.

Mean Time Between Failures

Are each performed test having an N Leiterbahnart at different stress temperatures and / or current densities. The downtime of several tests under the same conditions are normally distributed approximately logarithmically. Therefore, the mean time between failures of the individual failure times tf ( i)

Be calculated. Here, i runs through all of the samples from 1 to N.

Blacksche equation

For many metals are found empirically following relationship:

A, n and EA are adaptation parameters, k is the Boltzmann constant and T is the absolute temperature.

EA can be interpreted as the activation energy of the rate-determining step for the development of the lesions.

The exponent n is dependent on the current density used, the strip temperature (mainly the temperature gradient ), the geometry of the conductor track and the conductor material. In the literature, values ​​1 to 7 In the original published by Black form of the equation is n = 2 This value is expected if the nucleation of damage is rate-determining. If, however, the growth of the damage is rate-determining, one rather expects n = 1