# Modification of BEM for Precise Capacitance Extraction

1. Introduction

It is well-known that parasitic effects are no longer negligible for modern integrated circuit technology. Among the most important parasitic effects limiting the chip performance are interconnect related problems like extensive signal delays and crosstalk. The main numeric problem is extraction of parasitic capacitance, because it is a more complicated task than resistance extraction. The latest codes use directly 3D electrostatic solvers for this purpose. But the disadvantage of such approach is prohibitively long solution time for the problem. So the simplest model of 3D structures, when layout is splitted to 2D small pieces, can be used when it is necessary to handle complete IC's in a reasonable amount of time.

Usually 2D pieces are approximated with empirical formulas [1,2], which are valid only for the simplest shapes of conductors, usually rectangular (Figure 1). For more complicated shapes or for more accurate results, numerical methods are used. A standard approach is using Finite Element Method (FEM) for solving electrostatic problems. However for the 2D case, the Boundary Element Method (BEM) has a number of advantages. The first advantage is that a 2D problem reduces to a 1D problem in BEM, and it is not necessary to use the 2D mesh. The second advantage is that it calculates the surface charge directly, and hence the accuracy of output results is higher than for FEM.

Figure 1. Capacitance C1-C5 can be approximated by empirical formulas for C1-C5. Formulas are parameterized only by geometrical constants like w1, w2, w3 and h1, h2.

2. Capacitance Problem and BEM

The Capacitance problem for N conductors is defined
as calculation of capacitance matrix C_{ij} for these conductors.
Symmetrical capacitance matrix C_{ij} is defined by the
following simultaneous equations (Figure 2):

where Q_{i} is the electric charge on i-th
conductor, U_{j} is the potential on the j-th conductor.

Figure 2. An example of a system
that consists of three conductors.

Capacitors C_{12}, C_{13}, C_{23} are elements
of the capacitance matrix.

So, in order to find C_{ij}
we must know Qi for
a fixed distribution of U_{j}. Numerically,
each coefficient C_{ij} is
equal to Qi for U_{j}=1
and U_{k}=0, for all k j.
Qi is calculated as .
*dS*, where
is the surface charge density,
is the conductor surface.

The BEM is convenient for this purpose because the unknown function for BEM is s and it is defined along the conductor contour only (problem reduction: 2D to 1D). It is in contrast with FEM, where the electrostatic problem is solved for potentials at first, then potentials must be numerically differentiated to obtain the electric field (the accuracy is reduced at this step) and after that the surface charge density is calculated from the electric field.

3. Improvements of BEM for the Precise Capacitance Extraction

The BEM uses the integral analog of the Laplas equation to solve the electrostatic problem.

If we write this equation for the conductor surface, we obtain the integral equation for surface electric charge density (2D case):

where =
(x,y) and =
(x_{0},y_{0})
are observation and integration points , L,
L is the contour of conductors, G(,)
is the kernel of the integral equation.

3.1. Spline interpolation
of surface charge density

In BEM, this equation is written at a finite number of nodes
on the conductor surface and the values of electric charge density
are found at the same nodes. One way to increase the accuracy of
the solution is using a high order approximation of the charge density.
The third order spline interpolation gives a good smooth solution
[3] (Figure 3) :

where l is contour parametric length, l_{i-1}
l
l_{i},
h_{i} = l_{i}
- l_{i-1},
i=2,…,N, N is number of nodes,

is the second moment of the spline, fki are the weight functions of the spline:

At the endpoint of an interpolation segment the following condition of free bounds is used:

M_{1} =
M_{2}, M_{N-1}
= M_{N}.

Using spline features we can obtain the linear equation system for the second moment of the spline: AM=B, where

The solution of equation system is:

Here a_{ik} are
the coefficients of A^{-1}.
Using this expression, we obtain the following equation for surface
charge density:

Using this equation and the definite integral

we obtain the final system of linear equations:

Solving this system, we obtain a solution
with interpolation accuracy o(h^{4}).

Figure 3. Discretization of segment for spline interpolation of
surface charge.

3.2. Extraction of a Solution
Singularity at Sharp Edges

Another difficulty in BEM is that the solution has singularities
at harp edges of conductors. The singularity of a solution in general
case has the following dependence [4]:

^{t-1},
where r is the distance from the edge, and t>0 is the degree
of the singularity. For the case t<1 the charge density ca not
be described by cubic spline interpolation with acceptable accuracy.
The following method is used to avoid this problem.

The conductor contour is divided into smooth continuous
parts. The junction points of the parts belong to the edges of conductors.
So the whole contour consists of the sum of these parts L=L_{k}.
On each part L_{k} the charge density can be rewritten as:

where *
is a smooth function and can be fitted by spline with high accuracy,
t_{1k} and t_{2k}
are the degrees of singularity at the beginning
and at the end of part L_{k}.
The singularity degree for a sharp conductor edge with angle
is equal to t = /(2-)
(Figure 4).

Figure 4. The singularity
degree of the solution at the

end of the segment L_{k} is
defined by the angle .

For this case we can rewrite integral (8) as:

and solve the system of linear equation (9) for *.

Integrals (11) are computed numerically using Gauss quadratures. Since the integral kernel contains singularities, for accurate calculation the method of integral kernel singularity extraction is used. The idea of the method is as follows. The integral kernel F(x) is split into the product of two factors S(x) and R(x), and the integral can be rewritten as the sum of two integrals:

R(x) is a function without singularities. S(x) has an integrable singularity at point a, and the integral in the second term (12) can be calculated analytically. The first integral has smooth kernel and can be calculated numerically without problems.

4. Conclusion

The spline approximation of surface charge and extraction of solution singularity in BEM permits to calculate capacitance with high accuracy. Implementation of BEM with described improvements in extraction code can provide accurate estimation of parasitic capacitances.

**Reference:**

- U. Choudry and A. Sangiovanni-Vincentelli, "Automatic generation of analytical models for interconnect capacitances," IEEE Trans. Computer-Aided Design, vol. 14, pp. 470-480, 1995.
- N. Arora, K. Raol, R. Schumann and L. Richardson, "Modeling and Extraction of Interconnect Capacitances for Multilayer VLSI Circuits," IEEE Trans. Computer-Aided Design, vol. 15, pp. 58-67, 1996.
- Ivanov V.Y. "Automation of computer design of electronic devices". Preprint CS SBAS USSR, 1977 N-40. (In Russian).
- R. Mittra , S.W. Lee " Analytical Technique in the Theory of Guided Waves", The Macmillian Company, New York, 1971.