Slide 1 SLIP 2004

Payman Zarkesh-Ha, Ken Doniger, William Loh, and Peter Bendix

LSI Logic CorporationInterconnect Modeling Group

February 14, 2004

Prediction of Interconnect Adjacency Distribution:Derivation, Validation, and Applications

Slide 2 SLIP 2004

Outline

­Motivation

­ Interconnect Adjacency Model­Derivation

­Validation

­Applications

­Conclusion

Slide 3 SLIP 2004

Effect of Geometry on Capacitance

Coupling Cap.Gnd Cap.

0.20

0.15

0.10

0.05

0.00

Line

Cap

acita

nce

[fF/µ

m]

Ref.

-26%

-51%

+80%

+42%

Gnd Cap.

Coupling Cap.

Slide 4 SLIP 2004

­ How to choose the geometry for system level modeling of interconnect capacitance:

­ A real system is a mix of many cases.

­ A statistical approach is required to predict the capacitance distribution more realistically.

Motivation

­ We will derive the interconnect geometry distribution that produces circuit variations.

Slide 5 SLIP 2004

Outline

­Motivation

­ Interconnect Adjacency Model­Derivation

­Validation

­Applications

­Conclusion

Slide 6 SLIP 2004

­ Wire placement in a random logic network is complex and irregular enough to be modeled by a probability distribution involving “coin flipping”.

­ Average channel utilization, p, is known.

Assumptions

Slide 7 SLIP 2004

What is Adjacency?

Interconnect Length

InterconnectPitch

Neighbor 2Neighbor 1

Neighbor 3

Adjacent length 1 Adjacent length 2

Adjacent length 3

Interconnect adjacency is the fractional length of an interconnect that possesses neighbors at minimum spacing.

Slide 8 SLIP 2004

Routing Channel

Definitions

Grid

Empty spaceInterconnectWindow

Victim Line

Slide 9 SLIP 2004

Simple Case: Unit Length System

Window of size n=9

BernoulliDistribution

45 )p1(p!4!5

!9)5(f −

×=

n = total number of gridsk = number of filled gridsp = probability of filling

knk )p1(p)!kn(!k

!n)k(f −−

−

=

Slide 10 SLIP 2004

Realistic System

­ Wires are random length (not all unit length).

­ Window size, n, is not constant anymore.

­ Wires may cross window boundaries.

Derivation details available in the paper.

Slide 11 SLIP 2004

­ Average channel utilization, p.

­ Minimum and maximum segment length.

­ Grid size.

­ Segment length distribution.

� Can be derived from Rent’s rule by partitioning.

� Can be fit empirically.

Model Input

Slide 12 SLIP 2004

M2 Segment Length Distribution

1 10 100

100000

10000

1000

100

10

1

Wire Length [µm]

Nu

mb

er o

f se

gm

ents

Metal 2 Distribution:( ) 1.2

int−∝ llN

Technology = 130nmMetal layer = M2Min wire length = 1.0 µmMax wire length = 100 µm

Slide 13 SLIP 2004

Other Segment Length Distribution

1 10 100

10000

1000

100

10

1

Wire Length [µm]

Nu

mb

er o

f se

gm

ents

Metal 3 Distribution:( ) 3.1

int−∝ llN

1 10 100

10000

1000

100

10

1

Wire Length [µm]N

um

ber

of

seg

men

ts

Metal 4 Distribution:( ) 2.1

int−∝ llN

Slide 14 SLIP 2004

Outline

­Motivation

­ Interconnect Adjacency Model­Derivation

­Validation

­Applications

­Conclusion

Slide 15 SLIP 2004

0% 50% 100% 150% 200%

0.20

0.15

0.10

0.05

0.000% 50% 100% 150% 200%

0.20

0.15

0.10

0.05

0.00

Real Data Model

Comparison with Actual Data PDF

Channel utilization = 40%Min wire length = 1.0 µmMax wire length = 100 µmGrid size = 0.01 µm

Metal layer = M2Technology = 130 nm

Adjacency

Prob

abili

ty D

ensi

ty F

unct

ion

AdjacencyPr

obab

ility

Den

sity

Fun

ctio

nPeak @ 0% Peak @ 100%

Peak @ 200%

Peak @ 0% Peak @ 100%

Peak @ 200%

Slide 16 SLIP 2004

0% 50% 100% 150% 200%

1.00

0.75

0.50

0.25

0.00

Adjacency

Cu

mu

lati

ve P

rob

abili

ty D

ensi

ty

DataModelDataModelDataModel

Layer: M2

Comparison with Actual Data CDF

Slide 17 SLIP 2004

Comparison with Actual Data CDF

0% 50% 100% 150% 200%

1.00

0.75

0.50

0.25

0.00

Adjacency

Cu

mu

lati

ve P

rob

abili

ty D

ensi

ty

DataModelDataModelDataModel

Layer: M3

0% 50% 100% 150% 200%

1.00

0.75

0.50

0.25

0.00

Adjacency

Cu

mu

lati

ve P

rob

abili

ty D

ensi

ty

DataModelDataModel

Layer: M4

Metal layer = M4Channel utilization = 42%Gate pitch = 1.0 µmMax wire length = 150 µmGrid size = 0.01 µm

Metal layer = M3Channel utilization = 50%Gate pitch = 1.0 µmMax wire length = 150 µmGrid size = 0.01 µm

Slide 18 SLIP 2004

Sensitivity to Length Distribution

0% 50% 100% 150% 200%

1.00

0.75

0.50

0.25

0.00

Adjacency

Cu

mu

lati

ve P

rob

abili

ty D

ensi

ty

α = -2.31

α = -1.89

Layer: M2

0% 50% 100% 150% 200%

1.00

0.75

0.50

0.25

0.00

AdjacencyC

um

ula

tive

Pro

bab

ility

Den

sity

lmax = 100µm

Layer: M2

lmax = 1µm

lmax = 10µm

Change in α Change in lmax

Slide 19 SLIP 2004

0% 50% 100% 150% 200%Adjacency

Cu

mu

lati

ve P

rob

abili

ty D

ensi

ty 1.00

0.75

0.50

0.25

0.00

p=80%

p=40%

p=10%

0% 50% 100% 150% 200%Adjacency

Cu

mu

lati

ve P

rob

abili

ty D

ensi

ty 1.00

0.75

0.50

0.25

0.00

p=80%

p=40%

p=10%

Adjacency Distribution Change with p

Slide 20 SLIP 2004

Outline

­Motivation

­ Interconnect Adjacency Model­Derivation

­Validation

­Applications

­Conclusion

Slide 21 SLIP 2004

Statistical Crosstalk Analysis

0% 50% 100% 150% 200%

1.00

0.75

0.50

0.25

0.00

Adjacency

Cu

mu

lati

ve P

rob

abili

ty D

ensi

ty%Nets at risk

(∼16%)

Range subjectto crosstalk

Adj > 150%

Slide 22 SLIP 2004

Sensitivity Analysis

0.25 0.27 0.29 0.31 0.33 0.35

16%

12%

8%

4%

0%

Dielectric Thickness [µm]

Cap

acit

ance

Ch

ang

e [%

]

Isolated line

Dense line

Reference

Adjacency = 0%

Adjacency = 200%

Slide 23 SLIP 2004

Interconnect Statistical Reference

10%

8%

6%

4%

2%

0%

Pro

bab

ility

Den

sity

Model

Data

0 20% 40% 60% 80% 100%

Metal Density

0% 50% 100% 150% 200%

1.00

0.75

0.50

0.25

0.00

Adjacency

Cu

mu

lativ

e P

rob

abili

ty D

ensi

ty

DataModelDataModel

Pro

bab

ility

Den

sity

Line Capacitance [aF/µm]

5%

4%

3%

2%

1%

0%80 100 120 140 160 180 200

0 0.2 0.4 0.6 0.8 1

10%

8%

6%

4%

2%

0%

Pro

bab

ility

Den

sity

Model

Data

0 20% 40% 60% 80% 100%

Metal Density

M3

M4

M2

Slide 24 SLIP 2004

Outline

­Motivation

­ Interconnect Adjacency Model­Derivation

­Validation

­Applications

­Conclusion

Slide 25 SLIP 2004

Conclusions

í Compact model for Interconnect Adjacency Distribution for random logic networks is derived.

í The model uses only system level generic parameters such as segment length distribution and channel utilization. This enables us to predict future system level performance.

í Comparison to product data confirms the accuracy of the model.

í Some possible applications of the adjacency are proposed.

Slide 26 SLIP 2004

We acknowledge Ralph Iverson from Magma Design Automation for his generous support in the extraction of the real data.

Acknowledgement