Transcript I. Overview - Texas A&M University

II.Parasitic Extraction
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Importance
Capacitance extraction
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2D/2.5D algorithms
Finite difference method
Boundary element methods
Monte Carlo method
Inductance extraction
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Importance
Interconnect delay dominates gate delay
45
40
35
30
25
20
15
10
5
0
Gate
Interconnect
(Al+SiO2)
Interconnect
(Cu+lowk)
Sum (Al+SiO2)
Sum
(Cu+lowk)
85
0
50
0
35
0
25
0
18
0
13
0
10
0
delay (ps)
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technology
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Math and Physics Review
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A vector field: F: RnRn that assigns
each x a vector f(x)
A scalar field: f: RnR that assigns
each x a scalar f(x)
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Example Vector Fields
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Example Scalar Field
G( x, y ) =
1
2
2
x +y
G=1/7
G=1/6
G=1/5
G=1/4
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Gradient
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The gradient of a scalar field f, denoted grad(f), is
where  is called del or nabla
The gradient of a scalar field f is a vector field
F=(Fx, Fy, Fz)
The direction of grad(f) is the orientation in which
the directional derivative has the largest value and
|grad(f)| is the value.
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Divergence
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The divergence of a vector field F, denoted
div(F), is
The divergence of a vector field is a scalar
field
It gives the rate at which "density" exits a
given region of space
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Electric Field
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For a point charge q at position r, the
electric field at r’
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MKS Measurement
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Measures
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Permittivity 0 and permeability 0
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Length in meters
Mass in kilograms
Time in seconds
0=1/(c20)
For vacuum, 40=111.27 pF/m
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Potential field
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For static electric field E, the potential
(or voltage) (x,y,z) is defined as
E=–, or
q
where  E  ds =  E x dx + E y dy + E z dz
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Capacitor
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A capacitor is a device that can store an
electric charge by applying a voltage
The capacitance is measured by the
ratio of the charge stored to the applied
voltage
Capacitance is measured in Farads
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Parasitic Capacitance
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Given a set of conductors, compute the
capacitance between all pairs of conductors.
-
+
+
+
-
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1V
+
+
-
- -
C=Q/V
-
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2D Methods
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Area capacitance:
area overlap between
adjacent layers
Coupling capacitance:
between side-walls on
the same layer
Fringing capacitance:
between side-wall and
adjacent layers
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m2
m2
m2
m1
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2D Method
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C = Ca*(overlap area)
+Cc*(length of parallel run)
+Cf*(perimeter)
Coefficients Ca, Cc and Cf are given by
the fab
Cadence Dracula
Fast but inaccurate (100%)
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2.5D Method
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Consider interaction between layer i
and layers i+1, i+2, i–1 and i–2
Consider distance between conductors
on the same layer
Cadence Silicon Ensemble
Accuracy 50%
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Library Based Methods
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Build a library of tens of thousands of
patterns and compute capacitance for
each pattern
Partition layout into blocks, and match
with the library
Accuracy 20%
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3D Methods
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Finite difference/finite element method
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Most accurate, slowest
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Raphael
Boundary element method
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FastCap, Hicap
Monte Carlo random walk
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QuickCap
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Finite Difference Method
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Gauss’ Law
Let Vi be potential at each grid point
Approximate derivative by finite difference
Add boundary conditions
Solve a linear system
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FDM (cont’d)
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For conductor exterior, solve Laplace
differential equation
l
In 2D:
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k
i
m
j
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FDM (cont’d)
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Boundary conditions
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Potential on conductors are given
To avoid solving an infinite problem, set
boundary condition on enclosing box:
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Closed box: V=0 on the box
Open box: dV/dn=0 on the box
box
conductors
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FDM (cont’d)
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Once we have E, use Gauss law to compute
charge
where S is any enclosed surface and q is the
amount of charge in S
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Boundary Element Method
Laplace integral equation, where
 (x) is known surface potential,
 (x’) is charge density,
da’ is incremental conductor surface area,
x’ is on da’, and 1/|x-x’| is kernel.
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BEM (cont’d)
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Partition conductor surfaces into n small
panels A1,…, An.
Assume charge qi on each panel Ai
We have linear system Pq=v, where
q=(q1,…,qn) is the vector of unknown
charges, v=(v1,…,vn) is the vector of
known panel potential.
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BEM (cont’d)
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Each entry Pij of potential coefficient
matrix P represents the potential at
panel Ai due to unit charge on panel Aj
Solution q of linear system Pq=v gives
capacitance
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BEM Example
1
Conductor 1
2
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Conductor 2
1
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Step 1. Discretization
Conductor 1
A1
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Conductor 2
A2
A3
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Step 2. Compute P
1
A1
A2
2
A3
1
A4
2
P12 = P34 =1/(40)*(1/1), P13 = P24 =1 /(40)*(1/3)
P23 = 1/(40)*(1/2), P14 = 1/(40)*(1/4)
P11 = P22 = P33 = P44 =?
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Step 3. Approximate Pii
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Compute potential at the center of a
disk due to uniform charge distribution
1/2
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Step 4. Linear System
Solve it using Matlab:
q1=3.0966, q2=3.1664, q3=–0.7201, q4=–0.3532
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Step 5. Compute Capacitance
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Compute capacitance
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C11=q1+q2=3.0966+3.1664=6.263,
C12=q3+q4=–0.7201–0.3532= –1.0733
Repeat for other conductors
Final capacitance matrix:
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Computational Complexity
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Straightforward method
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Discretization: O(N)
Compute P: O(N2)
Solve Pq=v: O(N3)
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Assignments #2 (due 1/28)
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2.1 Use finite difference method to compute the
capacitance of three conductor surfaces. Consider
the surfaces have 0 thickness.
- 1  x  1, - 1  y  1, z = - 10
- 1  x  1, - 1  y  1, z = 0
- 1  x  1, - 1  y  1, z = 10
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2.2 Use boundary element method to solve the
same problem. Compare the time and results.
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2.3 Use FastCap on eesun1 or download it from
http://rleweb.mit.edu/vlsi/codes.htm to compute the
problem again
2.4 Prove the capacitance matrix must be symmetric,
even if the conductors are of different sizes and
shapes
Research Problem (not required to turn in): How to
use the symmetry property of the capacitance matrix
to save computation time?
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