Method of Modeling a Translational-Invariant and Continuous Spatial Correlations (16-Nov-2009)
Disclosed is a method of modeling the spatial correlation or the across-chip variation of a process/device/circuit parameter. The resulting spatial correlation is (1) both translational invariant and (2) continuous over whole region.
Method of Modeling a Translational-Invariant and Continuous Spatial Correlations
Problem solved. Circuit design and circuit yield can be improved by using better and more accurate circuit and device models in circuit design and circuit simulations. Including spatial correlations in circuit and device models can improve the quality and accuracy of those models. This disclosure provides a method of constructing a spatially correlated statistical model. The model describes the correlation among a set of device instances (or device parameter instances, or logic circuit instances), and the degree of correlation between any two instances of device decreases with increasing distance between the two instances.
Importantly
the modeled spatial correlation is both translational invariant and always continuous.
Prelude. Consider spatial correlation among a set of semiconductor device parameters. Examples include FET channel length, FET channel width, OP resistor's resistance value, MIMCAP's capacitance value, VNCAP (BEOL metal wires formed capacitor) capacitance values, ring oscillator's period/speed, other logic circuits (
NAND,
,
NO
R
, etc)
period. In each of
the above examples, each instance has the same mean value and the same standard deviation, but the correlation between any two instances vary (say, decrease) with the separation/distance between them. We consider a rectangular region for a part of chip region or for the whole chip region. In the following, we denote the mean,
p
0, and standard deviation, σ, for a
stochastic/random variables
p
(
x
) located at different position
x
in a chip/die (or located on
different chips on a wafer) as
≡( ) ( ) ,
)
1
/
2
1
/
2
σ =
−
=
−
= p
p
p
p
p
p x
x
x
x
x
(1)
(
)
2
0 σ
,
(
)
(
)
(
)
(
2
0
) denotes a two-dimensional location within a chip. We denote the correlation among them as
where x = (
x
,
y
C −
≡ x
( )( ) ( )
( )
p
(
x
1
)
−
p
(
x
1
)
p
(
x
2
)
−
p
(
x
2
)
(
p
(
x
)
p
( 2
0
0
p
(
p
x
1
,
x
)
=
1−)( ) .
)
2
2
p
(
x
1
)
−
p
(
x
)
2
1
p
(
x
)
−
p
(
x
)
2
σ
2
2
(2)
It is easy to see that
.
x
x C
C = (3)
The absolute value of any correlation coefficient is upper-ward bounded,
,
1
)
,
( 2
x1 ≤
( 1
2
2
1 x
,
)
(
x
,
)
C (4)
and the unity value is achieved at zero separation,
.
1
)
,
( 1
x1 =
x
C (5)
Homogeneity: If a spatial correlation depends only on the separation of two points but does not depend on the absolute coordinates of any point,
,
)
,
(
)
,
( 2
1
2
1
2
1 y
y
x
x (6)
then the spatial correlation is said to be translational invariant. Characterizing a spatial correlation as translational invariant reduces the characterization task from a two-dimensional task to a one-dimensional task. Experimentally, there is no data suggesting that the spatial
f
C −
−
x
=
x
x
1
correlation in semiconductor process/device/circuit is translati...
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