prediction of interconnect adjacency distribution ...€¦ · slide 7 slip 2004 what is adjacency?...
TRANSCRIPT
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