Figure 2.3: Measured process variation in a wafer [30]. Friedberg et al. used Electrical
Linewidth Metrology (ELM) to measure the process variation in all the dies
of the wafer. Inter-die and intra-die variations can be clearly observed.
proposed a model for each parameter variations. The results show that having
a statistical characterization of variations can reduce IC power prediction error
from 30% to 7%. Their work signaled benefit of variations modeling. However,
their analysis used a test array circuit and it can not be extended for modeling
legacy ICs that are not equipped with the sensors.
Liu [47] proposed a new modeling approach that described systematic vari-
ations as an affine function of the device’s geometric coordinates. To model
random variations, he recommended three spatial correlation functions: expo-
nential, Gaussian, and linear. Using generalized least square fitting, he chose a
13
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