CAD-ORIENTED SEMI ANALYTIC APPROACH FOR CAPACITANCE MATRIX COMPUTATION OF MULTILAYER VLSI
INTERCONNECTS
H. Ymeri\ B. Nauwelaers\ K. Maex^'^
^Katholieke Universiteit Leuven, Department of Electrical Engineering ( ESAT), Division ESAT-TELEMIC, Leuven-Heverlee, Belgium ^IMEC, Leuven, Belgium
Key words; semiconductors, electronics, microelectronics, IC, Integrated Circuits, lossy Interconnections, VLSI multilayer circuits, Very Large Scale of integration multilayer circuits, capacitance matrices, interconnect conductors, Fourier projection mettiod, mutual capacitance, Mei method. Green functions, CAD, Computer Aided Design
Abstract : In ttiis paper, we present a new approacli for capacitance matrix calculation of lossy multilayer VLSI interconnects based on quasi-static analysis and Fourier projection technique. The formulation Is independent from the position of the interconnect conductors and number of layers In the structure, and is especially adequate to model 2-D and 3-D layered structures with planar boundaries. Thanks to the quasi-static algorithms considered for the capacitance analysis and the expansions in terms of convergent Fourier series the tool is reliable and very efficient; results can be obtained with relatively little programming effort. The validity of the technique is verified by comparing its results with on-surface MEI method, moment method for total charges in the structure, and CAD-oriented equivalent-circuit methodology, respectively.
Semi-analitični pristop izračuna kapacitivnostne matrike večslojnih povezav v VLSI vezjih primeren za računalniško
podprto načrtovanje
Ključne besede: polprevodniki, elektronika, mikroelektronika, IC vezja integrirana, povezave vmesne Izgubne, VLSI vezja integracije zelo visoke stopnje večplastna, matrike kapacitivne, vodniki povezav medsebojnih, Fourier metoda projekcije, kapacitivnost medsebojna, Mei metoda, Green funkcije, GAD snovanje računalniško podprto
Izvleček: V prispevku predstavljamo nov pristop k izračunu kapacitivnostne matrike izgubnih večslojnih povezav v VLSI vezjih. Le-ta sloni na kvazi-statlčni analizi in Fourier metodi projekcije. Oblika je neodvisna od števila in položaja prevodnih povezav v sloju in je posebno primerna za modeliranje 2-D in 3-D struktur s planarnimi mejami. Vsied uporabe kvazi-statičnih algoritmov za analizo kapacitivnosti in razvoja v konvergentne Fourier vrste, je omenjeno orodje zanesljivo in izredno učinkovito; rezultate lahko dobimo z relativno malo truda pri programiranju. Veljavnost tehnike smo preverili s primerjavo rezultatov dobljenih z dvema drugima pristopoma.
1.	Introduction
Calculation of the capacitance matrix in multilayer IC interconnects is a well-known problem that can be solved by many analytical and numerical techniques /1-9/. Often these procedures were based on the integral equation formulation, differential equation formulation, or have been the results of extensive numerical simulations using adequate empirical corrections.
This letter proposes a new and more general formulation for computation of capacitance matrix of the most common 2-D interconnect structures using quasi-static analysis and Fourier projection approach.
2.	Background of the method
In the formulation, 2-D L-layered interconnect structures with planar boundaries are considered. Each layer is linear, homogeneous, and isotropic, and has permittivity e®
and conductivity a®, where I = 1.....L. For lossy medium
the complex permittivity ise® = 8®-jo/co. The point charge source is located along y = 0, x = Xs and z = Zs, respectively(see Fig. 1).
Fig. 1. Geometry of a layered structure for multilayer Green's function determination.
Inside each layer I and excluding the source point layer, the potential function (p® satisfies
vV"'=o	(1)
and the induction vector D® is obtained from
(2)
Here, the problem is solved by developing each potential cp® as a Fourier series. In the source layer, the general solution to (1) can be written as
(x^.,yf,Zf) = (ppiXj,yf,Zf) + (pH(Xj,yf,Zf) (3) where cpp is the source term given by

(Xf-xj'+yjHZf-z^^Y
1/2
(4)
and
^H i^f')'/' ) = Z	Zf) +
n,m>0

(5)
where kn = nn/a, km = m7t/b, Knm = (kn^ + km^)^''^, (xf,yf,Zf) are field point coordinates, and a and b are dimensions of the structures in x and y direction.
Considering (2), D'®' is given by
D'-" (x^., yj, Zf) = Bp (Xf, yf ,Zf) + D^ {x^, y^, Zj )(6) with
T>p{Xj,y^.,Zf) =
An:

iXj-xS-+y]HZf-zS-
3/2
(7)
and
D, {X f, y ,. ,z,) = e'"' ^^ K fc:,' exp(i^,„„ ) +
ii,m>0

f ! f^x

n,m>0
z r)jcos(/:„X.) }'/■) fl ^
«,m>0
In the other layers, the solutions are
n,m>0
Dl!,l exp(-/r,„„ zf)Jcos(/:„.x^) cos(^„, y)
(9)
and
D"^ (x, ,yj-,Zj) = £''\Z K k'l exp(7^„„, ) +
n,m>0
Zf)Jsin(A;„ Xj) cos(^„,}'/)[!,
.....Z,) +
n,m>0
exp(-i^„„, Zf )\cos{k,Xf) sin(/:„, yf)
/i.w>0
exp(-i^„„, z f)Jcos(^„ X^) cos(/c,„ ) ^ ^^
The potential function distribution (p® and the normal component of electric induction vector D® are expressed by series expansions in terms of solutions of the Laplace equation (1). One such expansion is written down for each homogeneous region of the layered structure in Fig. 1. The expansion coefficients Cnm® and Dnm® of the different series are related to each other and to the charge density distribution on the interconnect conductors via boundary conditions. Then, coefficients Cnm® and Dnm® are determined recursively. In this way we have found the multilayer Green's function G(rf; rs) of the probelm. By deriving the Green's function over a multilayer dielectric region and al-
lowing evaluation of potential distribution in any layer, we can place interconnect conductors anywhere in the multilayer structure, and therefore solve for the capacitance per unit length matrix for an arbitrary number of conductors.
3. Capacitance matrix calculation
In the following the complex capacitance calculation procedure will be treated in more detail. In an equivalent circuit, the value of a capacitance is the ratio of the free charge associated with a voltage difference between two interconnect conductors or between an interconnect conductor and the reference (e.g. the ground plane or the point at infinity), and that voltage difference. The values of these capacitances are known as network capacitances.
According to the equivalent source principle for the electromagnetic field, we can replace the rectangular conductor (c)(see Fig. 2a) with a piece of surface charge density distribution ac(rs) around the surface Sc, as shown in Fig.
2b. Using a Green's function of the medium	that
incorporates all boundary conditions in the structure in Fig.
2b (see Sect. 2), the voltage at any point r^ is generated
by the charge density Oc(rs) on all conductors (c = 1,...,N)

(11)
C=1
Element Cq of the capacitance matrix [C] may be calculated as the charge Qo per unit length on conductor (c) when the voltage on conductor (j) is 1 and 0 V on all other conductors. The charge per unit length on conductor (c) is the integral of the surface charge density ac(rs) over the circumference of conductor (c): Qc = , CTc (r. . The
charge distribution on every conductor (c) may be approximated by a number Nb of well-chosen basis functions ac,r=i.....Nb(rs) along the contour of the conductor:
^r(rv) = ^^^■■'■^'■.'■(''v)- The problem has been reduced
b)
Fig. 2. Geometry of a layered structure with (a)
embedded conductors, and (b) charge density distribution on the discretized surface of the conductors.
to the computation of the discrete charge constants {Wc=i...N, r=i...Nb}. As the result we obtain a series of simultaneous equations and represent them as follows:
N Ni,
l^J^KrPjl^.rl,	(12)
where Vj=i...N is the voltage on any conductor i}), with
Pj\c.r
J(j)Jic)

U)

"(13)
as potential coefficients of the Galerkin matrix. Solving the matrix equation (12) on a computer, we can determine the constants {Wc,r} and then the capacitance per unit length Ccj can be obtained in the form:

(14)
r=l
The lossy semiconducting substrate is taken into account by the complex permittivity
^cv = - J-
o
CO
(15)
where Es is the permittivity and a conductivity of the semiconducting substrate (silicon).
Due to the quasi-TEM character of the electromagnetic fields in the examined structure the frequency dependent distributed admittance per unit length Y can be calculated as
Y = G + jcoC = j CO
AV
(16)
where Q is the total charge per unit length, AV denote the voltage difference between the conductors, G is the conductance per unit length (losses) and C is the capacitance per unit length.
4. Discussion of the results
In this section we apply the new procedure to calculate some examples. In these examples we use multilayer IC interconnects whose strip conductors are infinitely thin (zero-thickness) or of rectangular cross-section and very thick (as usually in on-chip interconnets).
Example 1
Let us consider the system of four strip conductors embedded in a two-layered dielectric region with structure as shown in Fig. 3, where the conductors are numbered from left to right and upperto lower as 1, 2, 3 and 4, respectively. Numerical values for the capacitance matrix elements, generated by the proposed approach (has been used the moment method) and by a on-surface MEI procedure/1/ and moment method with total charge in structure /5,8/, respectively, are given in Table I. Note that the discrepancies between the values generated by our approach and one by/5,8/ are practically smaller than 0.2% over a wide range of physical dimensions and material parameters (all treated cases are not reported in this letter).
- Magnetic wall
Example 2
In order to prove the validity of the given approach seifand mutual per unit length shunt admittance (capacitance and conductance per unit length) calculated using our procedure are compared with the results of the full-wave analysis (spectral domain approach) in conjunction with equivalent circuit modeling technique /9/. In Fig. 4, an asymmetric coupled interconnect structure is depicted with the following electrical and geometrical parameters:
tsi = 500 jim, tox = 2 |j,m, wi = 4 |j,m, W2 = 1 j^m, Ti = Ta = 1 fxm, 8si = 11.8, psi = 0.01 £2cm, Box = 3.9 and s = 4 |im.
-X-
T,
T,
SiO,
Silicon
Fig. 4.
Asymmetric coupled interconnects on lossy silicon substrate.
Fig. 5a shows the variation in the distributed self and mutual capacitance per unit length Cii(co), C22(co), and Ci2(w), as a function of the frequency. Similarly, Fig. 5b shows the variation of the distributed self and mutual conductance per unit length Gii((o), 612(0)), and 622(0)) as a function of frequency. The solid lines are computed using the new multilayer Green's function procedure and the dashed lines are the results from the equivalent-circuit model approach /9/. It is observed that the values of the self and mutual
Fig. 3. Geometry of the structure from example with four strips (W/H1= S/H1- H2/H1= 1/3, H3/ H1-'2/3, e ri= 5 and e 1)
Capacitance (pF/m)	MoM [5,7]	MEI [1]	This letter
c„	70.158	69.514	70.158
C,2	-12.842	-12.832	-12.839
C,3	-12.960	-13.110	-12.967
C,4	-22.240	-23.014	-22.230
C22	87.327	87.028	87.227
C 23	-54.195	-55.462	-54.234
C 24	-4.052	-3.988	-4.049
C33	133.935	128.86	128.50
C34	-14.16	-14.93	-14.21
C44	135.70	141.31	135.94
Table I. Capacitance matrix of the structure of Fig. 3.
1,5
Q. I
8
2. O
-0.5
	.......... Our model Circuit model [9]
C11	
	
C22	
C12	
!	
4	6
Frequency (GHz)
10
Fig. 5a Self and mutual capacitance per unit length of asymmetric coupled interconnects on lossy silicon substrate.
0,15r
0.11-
0,05
O
-0.05
-0.1
Our model Circuit model [9]
Gil
' G22
G12
4	6
Frequency (GHz)
10
Fig. 5b Self and mutual conductance per unit length of asymmetric coupled Interconnects on lossy silicon substrate.
capacitance and conductance per unit length, respectively, are in good agreement with those of /9/. As expected, the lossy silicon semiconducting substrate has significant impact on the frequency-dependence of the capacitance and conductance per unit length as compared to the lossless or low loss dielectric substrate.
ical calculations. This method results in a very simple formulation of the problem that is well suited for computer solutions with relatively little programming effort.
References
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VLSIJ., 16(1993) 33 /4/ F. Stellari and A. L. Lacaita, IEEE Trans. Elec. Dev., 47 (2000) 222
/5/ C. Wei, R. F. Harrington, J. R. Mautz, and T. K. Sarkar, IEEE
Trans. Microwave Theory Tech. MTT-32 (1984) 439 /6/ T. Sakurai, IEEE Trans. Elec. Dev., 40 (1993) 118 /7/ 0. P. Yan and T. N. Trick, IEEE Trans. Elec. Dev Lett, 3 (1982) 391
/8/ A. R. Djordjevio, M. B. Bazdar, T. K. Sarkar, and R. R Harrington, LINPAR: matrix parameters for multiconductor transmission lines, Artech House, inc.. New York, 1999
/9/ J. Ziieng, Y.-C. Nahm, A. Weissiiaar, and V. K. Tripatini, Proo. of 1999 International Microwave Symposium, Paper M01D-3, Analneim, CA, June 1999
5. Conclusion
In this paper, we have discussed a technique for capacitance matrix extraction over a multilayer Si substrate. We derived the appropriate Green's function using quasi-stat-ic analysis and Fourier projection method. The potential function and electric induction vector components are defined as a series expansions in terms of the Laplace equation which are periodic in the direction parallel to the plane of interconnect conductors. The proposed semi-analytical procedure allows us: first, to assess in an analytical and simple way the integral equations of the problem, and second, to obtain a fast convergence of the numerical results due to the averaging technique used in the Galer-kin approach which leads to better accuracy in the numer-
/-/. Ymeri, B. Nauwelaers, Katholieke Universiteit Leuven, Department of Electrical Engineering (ESAT), Division ESAT-TELEMIC, Kasteelpark Arenberg 10, B-3001 Leuven-Heverlee, Belgium
K. Maex
iMEC, Kapeldreef 75, B-3001 Leuven, Belgium
Prispelo (Arrived): 18.11.2001
Sprejeto (Accepted): 25.1.2002