Modeling and Optimization for VLSI Layout

1
Modeling and Optimization for VLSI Layout

• Professor Lei He
• lhe_at_ee.ucla.edu
• http//eda.ee.ucla.edu/

2

• Programming homework
• Last lecture Placement
• Today
• Wrap up placement
• Interconnect modeling
• Student presentation
• April 29th, Thermal modeling (by Mehul Shah)
• May 2nd, Dynamic and leakage power modeling
  (Phoebe and Qun)
• Read
• Three papers on interconnect modeling
• Especially Xu-He01 (checked on May 2nd)

3
Chapter 5 Interconnect RLC Modeling

• Table and formula based capacitance extraction
• Table and formula based inductance extraction
• RC or RLC circuit model generation
• Numeric based interconnect modeling

4
Capacitance Extraction

• Introduction
• Table lookup method
• Formula-based method

5
Whats Capacitance?
Q
-Q

• Simplest model parallel-plate capacitor
• It has two parallel plates and homogeneous
  dielectric between them
• The capacitance is
• ? permittivity of dielectric
• A area of plate
• d distance between plates
• The capacitance is the capacity to store charge
• charge at each plate is
• one is positive, the other is negative
  

6
General Picture

• For multiple conductors of any shapes and
  materials, and in any dielectric, there is a
  capacitance between any two conductors

• Mutual capacitance between m1 and m2 is C12
  q1/v2
• q1 is the charge of m1
• v1 0 and v3 0

7
Capacitance Matrix

• Capacitance is often written as a symmetric
  matrix

8
Application in VLSI Circuits

• Conductors metal wire, via, polysilicon,
  substrate
• Dielectrics SiO2 ,...

• Total cap for a wire
• delay, power
• Mutual cap between wires
• signal integrity

9
Characteristics of Coupling Capacitance

• Coupling capacitance virtually exists only
  between adjacent wires or crossing wires

• Capacitance can be pre-computed for a set of
  (localized) interconnect structures

10
2.5D Capacitance ExtractionCong-He-Kahng-et al,
DAC97

• Propose and validate five foundations to simplify
  capacitance extraction
• Develop a simple yet accurate 2.5D capacitance
  extraction

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Verification of Foundations

• Method 3D analysis by FastCap Nabors-White,
  TCAD91
• Geometrical parameters 0.18 process NTRS94
  

12
Key Factor to Enable Foundations

• Minimum metal density requirement
• Metals occupy gt 30 area on anywhere on routing
  layer
• Foundry may introduce dummy metals for metal
  sparse areas

13
Foundation IEffect of Ground and Neighbors
Both ground, and neighboring wires on the same
layer, have significant shielding effects. Thus,
both must be considered for accurate modeling.
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Shielding Effect of Ground and Neighbors
Ci,i
Ci,i-2
layer i
no GND
458.4
130.1(28.4)
layer i-2
Ci,i
lumped capacitance for victim on layer i
Ci,i-2
coupling between victim and aggressor on layer i-2
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Shielding Effect of Ground and Neighbors
Ci,i
Ci,i-2
layer i
no GND
458.4
130.1(28.4)
GND
486.6
79.49(16.3)
layer i-2
Ci,i
lumped capacitance for victim on layer i
Ci,i-2
coupling between victim and aggressor on layer i-2
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Shielding Effect of Ground and Neighbors
Ci,i
Ci,i-2
layer i
no GND
458.4
130.1(28.4)
GND
486.6
79.49(16.3)
neighbors
1428
24.77(1.8)
layer i-2
Ci,i
lumped capacitance for victim on layer i
Ci,i-2
coupling between victim and aggressor on layer i-2
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Foundation IICoupling between Layers i and i-2
Coupling between wires on layer i and wires on
layers i-2 is negligible when the metal density
on layer i exceeds a certain threshold.
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Coupling between Layers i and i-2
layer i
layer i-1
layer i-2
-- 2x 4x 8x 12x
Ci,i
486.6 534.5 581.3 622.2 635.9
Ci,i-2
79.49 48.45 21.99 3.47 2.47
Ci,i
lumped capacitance for victim on layer i
Ci,i-2
coupling between victim and aggressor on layer i-2
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Foundation IIICoupling Effect of Layers i2 and
i-2
During capacitance extraction for wires on layer
i, layers i2 and i-2 can be treated as ground
planes with negligible error. There is no need to
look beyond layers i2 and i-2.
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Coupling Effect of Layers i2 and i-2
layer
i2
i1
i
i-1
i-2
418.9
Ci,i
Ci,i1
52.35
Ci,i-1
52.26
Ci,i
lumped capacitance for victim on layer i
Ci,i1
coupling between victim and central crossover on
layer i1
Ci,i-1
coupling between victim and central crossunder on
layer i-1
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Coupling Effect of Layers i2 and i-2
layer
i2
i1
i
i-1
i-2
418.9 418.9
Ci,i
Ci,i1
52.35 52.59
Ci,i-1
52.26 52.53
Ci,i
lumped capacitance for victim on layer i
Ci,i1
coupling between victim and central crossover on
layer i1
Ci,i-1
coupling between victim and central crossunder on
layer i-1
22
Foundation IVCoupling Effect of Neighbors
Coupling analysis to wires on the same layer need
only consider nearest neighbors independently,
with the widths of same-layer neighbor wires
having negligible effect on the coupling.
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Effect of Non-immediate Neighbors
victim
layer i
Ci,i 1436 C l 616.6 Cr 616.5
Ci,i lumped capacitance for victim.
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Effect of Non-immediate Neighbors
victim
victim
layer i
Ci,i 1436 1436(0) C l 616.6 639.8(3) Cr 616.5 6
39.5(3)
Ci,i lumped capacitance for victim.
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Effect of Neighbor Widths
W 1 2 3 4
Ci,i 764.5 765.2 764.9 764.4
Ci,i varies less that 0.3 for different neighbor
widths.
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Independence of Neighbors
S2
S2
S1
S2

victim
(S1,S2) (1,2) (1,3) (1,4) (1,?) lhs 639.2 600.0
582.5 559.7 rhs 638.0 597.1 578.9 553.1
Ci,i differs less than 1.0.
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Foundation VInteraction between Layers i-1 and
i1
The joint interaction of layers i-1 and i1 on
layer i is negligible therefore, corrections for
orthogonal crossovers and crossunders can be
performed independently.
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Independence of Crossovers and Crossunders
layer
i2
i1
i
i-1
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Independence of Crossovers and Crossunders
layer
i2
i1
i
i-1
30
Independence of Crossovers and Crossunders
layer
i2
i1
i
i-1
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Table-Based 2.5D Capacitance Extraction

• Table (Cap coefficients) generation
• One-time use of 3-D method
• Capacitance computation
• table lookup with linear interpolation and
  extrapolation

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Table Generation for Lateral, Area and Fringe
Capacitances
w
s
s

• Functions of (w,s)
• Pre-computed for per-side per unit-length

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Table Generation for Crossing Capacitances

• Function of (w,s,wc,sc)

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Table Generation for Crossing Capacitances
s
s
wc
sc
wc
sc
sc
w
w
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Illustration of Capacitance Computation
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• Find Nearest Neighbors on Same Layer

victim
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• Add in Per-Side Area, Fringe and Lateral
  Capacitances

victim
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• Add in Per-Side Area, Fringe and Lateral
  Capacitances

victim
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• Find All Crossovers and Crossunders

victim
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• Add in Crossing Capacitances Corner-by-Corner

victim
S1
wc
sc
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• Add in Crossing Capacitances Corner-by-Corner

victim
S1
wc
42

• Add in Crossing Capacitances Corner-by-Corner

victim
S1
wc
43

• Add in Crossing Capacitances Corner-by-Corner

victim
wc
sc
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Summary of Capacitance Computation

• Find nearest neighbors on the same layer
• Add in per-side lateral, area and fringe
  capacitances w.r.t. each neighbor
• Find all crossovers and crossunders
• Add in crossing capacitances corner-by-corner
  w.r.t. each crossover and crossunder

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Experimental Results
2 1/2-D
3-D
Error
net1
6.53552pF
6.5713pF
-0.54
net2
3152.42pF
3261.17pF
-3.33
Good match in terms of lumped capacitance!
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Formula based on horizontal and vertical
parameters

• Sakurai-Tamaru,ED83Wu-Wong-et al, ISCAS96
• single line
• parallel lines
• ...

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Single Line Sakurai-Tamaru,ED83
w
Ff
t
Ff
h
Fp

• Unit-length cap
• Error less than 6 when

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Single Line of Length L Sakurai-Tamaru,ED83
w
t
h

• Line of length L

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Parallel Lines on Same Layer Sakurai-Tamaru,ED8
3
w
s
w
t
h

• Unit-length cap

• Error less than 10 when

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Parallel Lines on Same Layer Wu-Wong-et al,
ISCAS96
w
s
s
w
w
t
h

• Unit-length cap

• Recall Sakurai-Tamaru,ED83

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Comparison
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numerical
Wu-Wong-et al
5
Sakurai
Normalized cap (C/?)
W1.05um
4
W0.7um
3
3
4
15
1
2
Noramlzied space (s/h)

• Wu-Wong-et al is better in smaller width and
  spacing

52
Parallel Lines within Two Grounds Wu-Wong-et
al, ISCAS96
w
s
s
w
w
h1
t
h1

• Two grounds where

• One ground