Random variations result from arbitrary fluctuations in the manufacturing
process. These variations can be decomposed into inter-die ≠inter and intra-die
variations ≠intra. Inter-die variations represent the differences among the dies for
the same wafer. Inter-die variation is a random variable equaling some constant
value for each chip. Intra-die variations represent the differences among the
devices on one chip. Thus, the total random variation for gate и is
c = ≠inter+ctra∙
Finally, total variation can be written as
≠u = C+≠u
= α0 + a1xu + a2yu + ≠inter + Ctra
= Fj,∕5 + Cra (2-1)
Where Fu = [l,a⅛, yu]τ and β = [ɑɑ + Ctθr> αι> α2]τ∙ Note that Fu contains the
gates location information. The term Fu β for a specific gate is constant. Ctra
is a Gaussian random vector with zero mean and correlation matrix Σ [47]
∑u,υ = p{Fu - Fv).
p is the correlation function and can have three forms [47]: p(∙) = exp(-α2∣∣ ∙ ∣∣)
(exponential), p(∙) = exp(-α2∣∣ ∙ ∣∣2) (Gaussian), or /?(•) = max{0,1 — α2∣∣ ∙ ∣∣}
(linear).
19
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