Civil Engineering Reference
In-Depth Information
The global component‚ is location-dependent. For example‚ across
the die‚ it can be modeled by a slanted plane and expressed as a simple function
of the die location:
where is its die location‚ and are the location-dependent gradients
of parameter indicating the spatial variations of parameter along the
and
direction respectively.
The local component‚ is proximity-dependent and layout-specific. The
random residue‚ stands for the random intra-chip variation and is modeled
as a random variable with a multivariate normal distribution to account for
the spatial correlation of the intra-chip variation:
where is the covariance matrix of the distribution. When the parameter
variations are assumed to be uncorrelated‚ is a diagonal matrix; spatial
correlations are captured by the off-diagonal cross-covariance terms in a general
using methods such as those described in Section 6.2.2. A fundamental
property of covariance matrices says that
must be symmetric and positive
semidefinite.
If the impact of only the global and random components are considered‚
then under the model of Equation (6.2)‚ at a given location‚ the true value of
a parameter
at location
can be modeled as:
where
is the nominal design parameter value at die location (0‚0).
6.2.2
Spatial correlations
To model the intra-die spatial correlations of parameters‚ the die region may
be partitioned into Since devices or wires close to
each other are more likely to have similar characteristics than those placed
far away‚ it is reasonable to assume perfect correlations among the devices
[wires] in the same grid‚ high correlations among those in close grids and low
or zero correlations in far-away grids. For example‚ in Figure 6.3‚ gates
and (whose sizes are shown to be exaggeratedly large) are located in the
same grid square‚ and it is assumed that their parameter variations (such as
the variations of their gate length)‚ are always identical. Gates and lie
in neighboring grids‚ and their parameter variations are not identical but are
highly correlated due to their spatial proximity. For example‚ when gate has
a larger than nominal gate length‚ it is highly probable that gate will have a
larger than nominal gate length‚ and less probable that it will have a smaller
than nominal gate length. On the other hand‚ gates and are far away from
each other‚ their parameters are uncorrelated; for example‚ when gate has a
