Minimum Void Size and 3-Parameter Lognormal Distribution for EM Failures in Cu Interconnects

Abstract

z = HLog@tD − Log@t 50 DL ê σ F@tD= p1 * F1@t;t 501 ,σ1D +H1−p1L * F2@t;t 502 , σ2D F@tD=F1@t;t 501 ,σ1D+F2@t;t 502 ,σ2D−F1@t;t 50 1 ,σ1D * F2@t;t 502 ,σ2D f@t;t 50 , σD = 1 t è!!!!!!!!!! 2 πσ 2 e −JLogB t t 50 FN 2 í2 σ 2F@t; t 50 , σD = ‡ 0 t f@x; t 50 , σD x ABSTRACT Broad failure time distributions were observed for line depletion electromigration in Cu interconnects for various structures without sufficient liner contact and via redundancy. The root cause for this behavior was identified as the sensitivity of failure times to the void size, shape and location. Application of the traditional 2-parameter lognormal distribution model to corresponding stress data often results in very pessimistic EM lifetime projections. A 3-parameter lognormal distribution was found not only to fit the experimental data better, especially for the early portion of the failure time distributions, but also to generate more accurate lifetime projections for void-size-limited EM. Given the nature of EM wear-out, deeper consideration indicates that a 3-parameter lognormal distribution has a sounder physical basis than a 2-parameter lognormal distribution. The new parameter introduced in the model, the minimum failure time (X 0 ), scales with via size over several technology generations, further validating the minimum void size explanation. [KEY

Application of three-parameter lognormal distribution in EM data analysis

“…Traditional EM analysis with two-parameter lognormal fitting leads to unacceptably short lifetime projections for such failure distributions. As discussed by the current authors [5], a more appropriate three-parameter lognormal distribution model can better fit the data and produce more reasonable projections. In this paper, we discuss the application of a three-parameter lognormal model in EM data analysis.…”

“…In what follows we will denote by T the random variable representing the failure time distributed in accordance with (1). The major visual difference between a two-parameter and a three-parameter lognormal distribution is a straight line vs. a downward bending curve at the early fail portion (for a positive X 0 ) in the normal probability plot for logarithms of failure times [5,9]. The difference between these two distributions depends heavily on the shape factor, r, and the difference between t 50 and X 0 .…”

“…Therefore, unlike defect-caused failures, there is no instant EM induced failure; a finite minimum failure time is needed for an EM failure. For distributions with a minimum threshold failure time, X 0 , a three-parameter lognormal fit has been found to be more appropriate [5]. The probability density function (PDF) f(t) of the three-parameter lognormal distribution is given by…”

“…When the circuit designers follow this J use guideline in their product design, the product is protected from premature EM wear out failures. The projection of J use from stress data is given by a version of Black's equation that specifies the acceleration mechanism with respect to stress conditions [5,6] …”

“…This via size shrinking results in a smaller critical void size, and consequently a shorter time to cause an EM failure. Furthermore, for interconnect structures with poor redundancy (both via and liner redundancy), EM failure times are very sensitive to void location and shape, and show broader distributions [1,5]. Traditional EM analysis with two-parameter lognormal fitting leads to unacceptably short lifetime projections for such failure distributions.…”