Ecient Three-Dimensional Extraction Based on
Static and Full-Wave Layered Green's Functions
Jinsong Zhao
Wayne W.M. Dai
Department of Computer Engineering
UC Santa Cruz
Santa Cruz, CA 95064
Abstract
Integral equation approaches based on layered media Green's
functions are often used to extract models of integrated circuit structures. The primary advantage of these approaches
over equivalent-source based schemes is the dramatic reduction in problem size. When combined with an SVDaccelerated scheme for the solution of the associated dense
linear system, this leads to a substantial speedup.
In this paper we derive and solve for these multilayered
3D Green's functions using a transmission line circuit analog. A generalized image method for an arbitrary number
of layers is presented. This method is rapidly convergent
for near-eld interactions. For the far eld, a Chebyshev
interpolation approach is adopted, where a database is precomputed (using a Fast Hankel Transform) and stored. The
combination of these two approaches leads to an extremely
ecient scheme for the generation of Green's functions.
We combine the SVD-accelerated integral equation solver
IES3 with the multilayered Green's function approach, apply
it to the extraction of IC parasitics and passive components,
and we demonstrate its speed, accuracy and versatility via
a number of examples.
1 Introduction
Extraction and modeling in multilayered circuit environments play a signicant role in the design of modern ICs,
packages and printed circuit boards. Shrinking circuits and
increasing operating frequencies have made it critical that
three dimensional eects be modeled accurately. General
purpose eld solvers based on nite dierence or nite element schemes (e.g., Raphael, Ansoft) can be used. However,
they require volume discretization, and it is dicult to enforce radiating boundary conditions for open regions. This
results in large computation times and memory use. Integral
equation schemes (such as FastCap [10]) based on equivalent sources can also be used. They work by introducing
additional equations to enforce boundary conditions at region interfaces; this can result in a prohibitive increase in
problem size. In this paper, we present a general-purpose
Sharad Kapur
David E. Long
Bell Laboratories, Lucent Technologies
600 Mountain Avenue
Murray Hill, New Jersey 07974
method based on layered Green's functions which exploits
the radial invariance of the media. The principal advantage
of this scheme is that the unknowns are restricted to only the
source region, i.e., no additional equations need to be introduced to enforce boundary conditions. This leads to a substantial reduction in problem size compared to equivalentsource based schemes. The layered Green's functions are
combined
with the SVD-accelerated integral equation solver
IES3 [6, 7] (which is not restricted to any specic kernel),
giving a dramatic speedup over all competing approaches.
Layered Green's functions decouple the process structure
from the circuit geometry and can be precomputed, compressed and stored. Once the process variables are xed,
the Green's functions remain unchanged for each new extraction. Layered Green's functions have traditionally been
used in a 2.5D simulation context where the sources are
conned to innitely thin sheets. This approach has been
popular in the microwave and antenna communities. For
these communities, 2.5D modeling of the structures is adequate because conductor thickness is much smaller than the
width. However, in IC and packaging contexts this assumption is no longer valid. The thickness of the conductors is
often on the same order as the widths.
In the spectral domain it can be shown that the full
wave Green's function can be decoupled into three sets of
one-dimensional wave equations [8]. Poisson's equation for
the static Green's function leads to a one-dimensional wave
equation [3]. These one-dimensional wave equations have a
transmission line circuit analog. The boundary conditions
to be enforced are obtained by solving the equivalent circuit.
In this paper, we derive and solve for these multilayered 3D
Green's functions. A generalized image method for an arbitrary number of layers is presented. The implementation is
based on a heap data structure which dynamically sorts the
images for the rapid computation of the near eld. While
the image method is always numerically stable, it is ecient
only in the near eld; in the far eld slow convergence makes
it impractical. For the far eld, a database is eciently constructed with a Fast Hankel Transform and compressed with
an adaptive Chebyshev interpolation procedure. The singularity in the Green's function in the near eld obviates the
use of any 3D interpolation procedure. The combination of
the two approaches, i.e., an image scheme for the near-eld
and a lookup scheme for the far eld, leads to an extremely
ecient implementation. The typical cost of building the
database for a given process takes less than one hour for 3D
problems and a few minutes for 2.5D. The cost of evaluating
a single interaction between a source and observation pair
is on the same order of that of the free-space Green's func-
tion evaluation.
When combined with the integral equation
solver IES3 , this leads to a substantial speed up over competing approaches (such as those implemented in FastCap
and Raphael).
2 Green's Functions for Layered Media
The following discussion on layered Green's functions is presented in the context of static capacitance extraction. The
full-wave extension of the techniques developed in this paper
is straightforward with only a few changes.
(a) For electric or magnetic walls use a short or open
circuit;
(b) For an open layer, use a terminating resistor with
resistance given by the equivalent characteristic
impedance;
(c) For an imperfect layer, use a resistor with resistance given by equivalent surface impedance.
2. For the intermediate layers, use a transmission line
with the equivalent characteristic impedance, propagation constant, and length.
2.1 Wave Equation and Transmission Line Equivalence
It is well known that the basic equations in electromagnetics
are the wave propagation equations. A complete solution in
3D is computationally infeasible. However, in layered media,
by taking advantage of radial invariance, Maxwell's equations can be reduced to three independent one-dimensional
wave equations via TE and TM decomposition [8]. Matching
boundary conditions for a one-dimensional problem can be
simplied by solving an analogous transmission line circuit.
A uniform transmission line can be characterized by three
parameters: the propagation constant , the characteristic
impedance Zc , and the line length l. Voltage and current
propagate along the line based on the Telegrapher's equations:
2.2 Green's Functions
A Green's function is the response of a physical system due
to a point exciting source. More specically, the Green's
function is the underlying solution of the associated partial
dierential equation. For capacitance extraction problems
in layered media the associated equation to be solved is Poisson's equation, i.e., the Green's function G satises
1 (z0 )(0)
(10)
r2 G(z; ) = ? 2
where (z; ) and (z0 ; 0 ) are the observation and source points
in cylindrical coordinates. Fourier transforming in the radial
direction yields the spectral domain Green's function G~
dv = ?Z i
c
dz
di
= ?Ycv + is
dz
(1)
(2)
where is is the current source density, and Yc = 1=Zc is
the characteristic admittance. The boundary conditions for
the transmission line circuit at the source-free interfaces are
that voltage and current be continuous, i.e., v+ = v? and
i+ = i? . These conditions are automatically satised by
enforcing KCL and KVL.
In layered media, the potential satises the propagation equation
@2
@z 2
? 2 = ?E
(3)
where E is the source of excitation. satises the following
boundary conditions at the media interface:
+ = ?
(4)
_
_
(5)
+ + = ? ?
where the dot denotes the derivative with respect to z. This
is equivalent to solving a transmission line with the same
propagation constant. We reduce the problem of solving for
to the problem of solving for the voltage on a line with
the following transformation.
v
(6)
_
i ? (7)
Yc (8)
is E
(9)
The equivalent transmission line circuit is constructed as
follows:
1. For terminal layers such as ground, open layers, etc:
@2
@z 2
1 (z0 ):
? 2 G~ (z) = ? 2
(11)
For a xed spectral number , this equation can be mapped
to an equivalent transmission line equation as described in
the previous subsection.
One set of transmission line equations is sucient to describe the static Green's functions because of the one dimensional source charge. On the other hand, for full-wave problems, the three-dimensional source current results in three
dierent sets of transmission line equations, each with its
own characteristic impedance. Furthermore, because of retardation, the singularities and discontinuities in the spectral domain Green's functions involve additional complications that will not be discussed in detail here but that have
been successfully tackled [8].
3 Computations of 3D Green's functions
The Green's functions can be easily derived and solved in
the spectral domain. However, since the integral equation
solution is usually obtained in the space domain, a transformation from the spectral domain to the space domain is
required. This involves the use of a two-dimensional Fourier
transform, which, due to radial symmetry, reduces to a one
dimensional Hankel Transform.
More specically, for a xed z and z0 , if the impedance
transfer function of the equivalent transmission line from the
source to the observation point is Z , then the space domain
Green's function is:
Z1
G() = 1
(12)
20 0 Z ( )J0 ( )d:
This transform is computationally expensive. Although ecient codes such as the Fast Hankel Transform alleviate this
problem, it is still prohibitive to perform these computations
on-line.
Fortunately, the Green's functions in the far eld are
smoothly varying as a function of distance and can be precomputed and compressed (via interpolation) o-line. An
adaptive Chebyshev interpolation procedure is used to exploit the particular nature of the 1=r-like dependence in the
Green's functions. We can tabulate and store the Chebyshev
coecients in a relatively small database.
The 1=r singularity in the Green's function in the near
eld obviates the use of any 3D interpolation procedure (due
to excessive storage costs). Instead, it is more ecient to use
a method based on images and to perform this computation
on-line. The method of images is based on an innite series expansion which is equal to the Hankel Transform integral. The image method [14] is generalized for an arbitrary
number of layers and is briey described in the following
subsection.
3.1 Generalized image method
The image method is one of the most popular approaches
for capacitance extraction in multilayered media [14]. But
its use is usually limited to media with only a small number
of layers for the following reasons:
1. For more than two layers, the innite image series expression is relatively hard to derive and implement;
2. There is no robust convergence testing procedure for
the image series. Moreover, in the far eld an enormous number of images are required to converge to a
reasonable tolerance.
For multilayered media in modern printed or lithographied
circuits, the traditional image method is usually abandoned
because of the above diculties [15, 10]. Instead, an equivalent-source formulation is employed where the dielectric
interfaces are discretized.
Wave-tracing is a generalized image method to obtain all
the possible images by inspecting the physical meaning of
the spectral domain Green's function [16]. Recall the mechanism in the transient response of transmission line circuits
in time domain. At the initial time, the current source generates starting voltage waves (up and down) at the excitation point. The two waves can independently propagate.
If the wave meets an impedance discontinuity, it splits into
a reected wave and a transmitted wave according to the
boundary condition. The reected and transmitted waves
then independently propagate. Each wave is attenuated by
the splitting process and the distance it travels. Waves passing the observation point are called images.
We developed a Prioritized Wave-Tracing method using
a priority queue to store and retrieve the wave information.
Rather than a naive implementation which recursively traces
waves, each wave is instead assigned a key value which is
calculated from the wave's amplitude and distance. Waves
are dynamically sorted in a Priority Queue (PQ) with the
strongest ones at the head. The pseudo-code is as follows:
Prioritized Wave-Tracing
insert initial waves in the PQ;
while the last image contribution is not small enough f
Extract the wave at the head of the PQ;
if the wave passes the observation point then
accumulate Green's function;
record image information;
split the wave into reected and transmitted;
assign the key to the two new waves;
insert them into the PQ;
g
The key advantage of this implementation is that only
the dominant waves are traced, and that a good stopping
criterion is available.
The image method is not feasible for the far eld since the
number of images grows exponentially. For a typical MCM
structure at distance 10m the number of images required
to achieve accuracies of 10?4 and 10?3 is shown in Figure 1.
100000
Number of images with eps=1e-4
Number of images with eps=1e-3
10000
1000
100
10
1
1e-09
1e-08
1e-07
1e-06
1e-05
Horizontal distance(m)
0.0001
0.001
Figure 1: The number of images grows exponentially in the
far eld. This example is for a typical MCM-D technology
with 5 dielectric layers and a ground plane.
3.2 The Fast Hankel Transform
In this subsection we discuss the method used to compute
the space domain Green's function for the far eld. Traditionally one of the following three techniques has been used.
Discrete complex image [2, 16]. This method approximates the smooth part of spectral domain Green's
function by a sum of complex exponentials. However,
serious accuracy and stability problems make it impractical for more than two layers.
FFT schemes [5, 4]. These methods enforce Dirichlet or Neumann boundary conditions on a bounding
box, and use 2D FFTs to perform the transformation.
They require extremely ne sampling to cover sucient spectral content to achieve reasonable accuracy.
Numerical integration [9]. As mentioned, the two dimensional Fourier transform of the Green's function in
the spectral domain is reduced to a one-dimensional
Hankel transform. Numeric evaluation of this transform, although robust, is relatively time consuming.
However, the use of the Fast Hankel Transform (as
discussed below) reduces the cost signicantly.
The spectral domain Green's function can be split into
two parts: A region near the origin which may contain poles
and branch points, and a smooth tail. For the rst part,
adaptive Gaussian integration along a contour excluding the
poles is used. The tail is smoothly decaying and oscillating when convolved with the Bessel function kernel. The
Fast Hankel Transform [1] provides an ecient procedure for
computing the Hankel Transform from 0 to 1 of a smooth
function f . In order to apply this technique, we extend the
tail back to the origin by matching the function and its rst
few derivatives at the start of the tail with a polynomial.
We then correct for the contributions of the polynomial in
the initial region.
The Fast Hankel Transform is a digital lter for evaluating the following integral:
f () =
Z
0
1
F (k )Jk (k )dk :
(13)
By a change of variables, = ex; k = e?y , the integral
reduces to a convolution, which is then computed using
g (x) =
N
X
i=1
wi G(x ? ai )
(14)
where g(x) = exf (ex ), G(x) = F (e?x), and where wi and
ai are sets of weights and abscissa respectively.
3.3 3D Green's function database
A space domain Green's function with radial invariance has
three variables: z, z0 , and . For a 3D problem every
required value cannot be stored because of memory constraints. Instead we use an interpolation strategy that takes
advantage of the fact that the majority of source and observation points lay on a xed number of sheets rather than on
the side walls of the conductor. The database stores the interaction between every pair of sheets. We use the database
with a mixed-dimension interpolation scheme. Adaptive
Chebyshev interpolation is applied in the radial direction
and Lagrange interpolation is used for the other two. The
accuracy of the interpolation is controlled. More importantly, the cost of evaluating a Green's function is on the
same order as that of the free-space Green's function.
Figure 2 shows the various interpolation schemes.
1. For sheet to sheet interactions such as between points
A and D, we use 1D interpolation on .
2. When only one point is on a sheet (e.g., points A and
C) we use 2D interpolation.
3. When neither point is on a sheet (e.g., points B and
C) we use 3D interpolation.
4.1 Comparison to other tools
We begin by comparing IES3 to the FMM-based FastCap
program (version 2) [10] for two examples: a 1x1 bus crossing and a 3x3 bus crossing in a layered medium. A comparison to Raphael [13] is also made for the 1x1 bus structure.
The criteria for the discretization of the conductors (and
for FastCap, the ground plane and dielectric interfaces) was
to ensure convergence to within 1% accuracy. For FastCap
it was sucient to make the ground plane twice the extent
of the conductors. Both examples were run with the dielectric layer information shown in Figure 4. The sizes of
the layers are typical for a CMOS process but the dielectric
constants were arbitrarily chosen. The ratios are small to
ensure that FastCap does not exhibit numerical ill conditioning [12]. FastCap was optimized at the same level of
our program and was run in its default mode. The relative tolerance for IES3 was set to give the same accuracy as
FastCap's default two-term multipole expansions. Table 1
summarizes the results of the experiments.
For each structure we show the additional panels required by FastCap for the discretization of the ground plane
and dielectric interfaces. It was interesting to see that the
two approaches for modeling the dielectric interfaces give
good agreement.
Since the number of unknowns for running a problem
in FastCap is dependent on the size and discretization of
the ground and dielectric interfaces, it is dicult to make
an accurate timing comparison. Typically, our algorithm
is three to six times faster. More importantly, once the
layer structure is xed the accuracy of the solution is only
dependent on discretization of the conductors surfaces.
In addition, the 1x1 bus crossing example was run in
Raphael, which is based on nite dierences. This tool enforces the Neumann boundary condition on a bounding box.
In this case the box was made to be four times the size
of
the structure. Raphael converged to within 2% of IES3 and
FastCap answers.
−6
x 10
4
3.5
3
D
C
}
2.5
Chebyshev nodes
2
1.5
5
0
Chebyshev nodes
4
1
{
−6
x 10
A
Figure 2: Various interpolation schemes for multilayer structure.
4 Experimental Results
In this section, we present experimental results for IES3 using the layered
static and full-wave Green's functions. All
tests for IES3 and FastCap were run on an SGI machine
(200 MHz R10000 CPU).
3
2
B
−6
x 10
2
3
1
4
5
0
Figure 3: Discretization of and charge distribution on the
3x3 bus. All wires are 1m 5:2m the layer thickness
and are separated by 1:1m.
4.2 Full-wave simulation of MCM inductors
In this subsection we present results of a 2.5D full-wave simulator [7] using the full-wave layered Green's functions and
IES3
FastCap
Example
Conductor Time Ground Dielec. Total Time Dierence
panels
panels panels panels
1x1 bus crossing
1172
31
637 2548 4357 191
0:19%
3x3 bus crossing
4384
177
637 2548 7569 511
0:38%
Table 1: Comparison of IES3 to FastCap (time in seconds)
0.595m
r = 1
0.65
0.595
r = 2 : 2
0.65
0.635
r = 3 : 9
0.65
0.28
0.361
r = 7
r = 3 : 9
Figure 5: Layout of the two metal layer MCM inductor.
Figure 4: The layer structure for the 1x1 and 3x3 bus crossing examples
IES3 . The simulator has been run on a number of examples, including capacitors, inductors, and lters on CMOS
and MCM substrates. In this section, we report simulation
results for a inductor on a seven layer MCM substrate. The
layout is shown in Figure 5. The inductor is constructed
with the top two metal layers of the process. The inductor was discretized into 1962 panels and the impedance was
calculated over a range of frequencies. The simulation of
a single frequency point takes about 100 seconds. Current density over the structure near resonance are shown
in Figure 6. Comparisons between the simulation results
and measurement are shown in Figure 7. The graphs show
the magnitude and phase of the complex impedance. The
self resonant frequency is accurately captured.
5 Conclusion and future work
In this paper we used a transmission line analog to derive the
spectral domain Green's function for multilayered media.
To eciently evaluate the corresponding three-dimensional
space domain Green's function, we proposed a combination
of two techniques. For near-eld interactions, we use a generalized image method. In the far eld, we construct a
database using a Fast Hankel Transform and adaptive Gaussian quadrature. The database is eciently represented with
adaptive Chebyshev interpolation. The time to evaluate the
layered Green's function is usually about twice the time required to evaluate the free-space Green's function. The cost
of precomputing the database for a typical process is less
Figure 6: Density of current ow in the inductor near resonance.
than 20 minutes.
We combined the SVD-accelerated solver IES3 with the
multilayered Green's function and applied it to extraction
problems on CMOS, MCM and
GaAs substrates. For static
extraction,3 we compared IES3 to FastCap and Raphael and
found IES to be substantially more ecient. The Green's
functions
were also used in a 2.5D full-wave solver based
on IES3 to simulate an MCM inductor. The results were
compared to measurement.
Future work can be summarized as follows:
1. Combining the layered Green's functions with an equivalent-source formulation to handle conformal dielectrics and other irregular boundaries.
2. Sensitivity analysis of process variations via the layered Green's functions.
Impedance
5
10
[6]
4
10
[7]
3
Ohms
10
2
10
[8]
1
10
0
10 8
10
9
10
10
10
11
10
Frequency(Hz)
[9]
Phase
2
[10]
1.5
1
Radians
0.5
[11]
0
−0.5
−1
[12]
−1.5
−2 8
10
9
10
10
10
11
10
Frequency
Figure 7: Comparison of simulation (circles) to measurement (lines)
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