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 C₁₂, C₁₃, C₂₃ 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 = (x₀,y₀) 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:

M₁ = M₂, 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(h⁴).

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.