TRANSMISSION LINE MODEL PARAMETERS FOR VERY HIGH SPEED VLSI INTERCONNECTS IN MCMs USING FEM

WITH SPECIAL ELEMENTS

S .Y. Kulkarni K . D. Pat il K . V . V. Mur t hy Department of Electrical Engineering

Indian Institute of Technology Bombay - 400 076(INDIA)

Abstract In this paper Finite element method(FEM) with

higher order isoparametric elements have been employed to compute electrical representative pa- rameters (like R,L and C, measured in PUL) of interconnects used in high-speed VLSl .sysre~r,.s, es- pecially, in multichip modules(AfCMs). Varaou.3 2-0 VLSI interconnect/dielectric packaging struc- tures involve infinite domain for analysis. Special quadrilateral infinite elements are being used to obtain parameters accurately. The singular points introduced by the sharp corners of signal conduc- to r boundaries are specially treated to drastically reduce the number of degrees of freedom and com- putation time in implementing FEM b y using spe- cial singular elements.

1 Introduction Package design should be such that increased per-

formance and improved reliability should be achieved a t reduced cost. A design approach to achieve these objectives is to reduce the number of interconnec- tion and reduce the total wire length. MCMs can of- fer these advantages since the chips are spaced more closely and signal lines are fabricated in multilayers

The VLSI interconnects operating at very high speeds need to be modelled as distributed parame- ter transmission line model. The central +,: k m in computation of these parameters is the soluLion of Laplace equation in two dimensions subject to apprc- priate boundary conditions. The most common a p proach for this problem is through conformal m a g pings constructed from Schwartz - Christoffel trans- formation. However, there are many other approches to solve Laplace equation, like Transmission line ma- trix method, Method of Moments(MOM), Variational Method. Finite difference Method and Finite element Method(FEh1). All these well-known methods have been discuused in detail in many standard books and papers.

Modern design techniques require a rapid and ac- curate method for determining these parameters for wide range of geomhtries. The finite element method is best suited for such applications. Finite element formulation to solve Laplace equation and techniques

to compute the transmission line parameters, usually measured in per-unit-length(PUL), will be briefly re- viewed in section (2), which is required to formulate the problem discussed in the present paper.

The signal lines will have sharp corners, which form singular points with field singularity of the order O(r+),popularly called as inverse square root sin- gularity. Special singular elements are used to account this singularity. For interconnect structure, whose do- main of analysis is large, finite elements alone cannot model field distribution at far field region properly. Hence, infinite elements are used at the boundaries of the mesh. Formulation of singular elements and in- finte elements and their shape functions are given in section(3).

Section 4) lists the computed values of the ele- ments of [A and [C] matrices for various configura- tions. The complex structure that is being analysed has 5 layers of dielectrics(ceramic) and 6 conductors of rectangular cross-section.

2 Finit e Element Formulation For electrostatic problems that are to be solved

in connec lion with the evaluation of capacitance ma- trix, the general algorithm starts with the solution of Laplace equation. The two dimensional potential dis- tribution is given by

with associated Dirichlet and Nuemann boundary con- ditions[l]. The well known principle of minimum po- tential energy is employed to solve for Laplace equa- tion. The energy functional corresponding to Laplace equation, is given by

and the boundary condition q5 = 4 0 on S+ The domain D is divided into finite number of ele- ments. The potential in each element is given by

n

i=l

260 1063-9667/95 $4.00 0 1995 IEEE 8th International Conference on VLSI Design -January 1995

Where N,(z , y) are called as shape functions, which are represented in terms of the nodal values through suitable interpolation functions, and 4, is the potential at node i . Substituting for 4 from eqn.(3), we get

(4) w(e) = .+'"'TK'")&) 1 - 4'"'T4(e) 2

Where 4'") is the vertex values of potential and I@ is called as element Flux-potential matrix, given bY

lde) = JJD(onr,(2, Y> v N&l y))detJWv (5)

The total energy of the problem domain is given by

rU=C&) (6) all'd

w = ;MI*[K1[41 - 14" (7)

Applying minimum energy principle, we get

[Kl[41 = PI (8)

Solving the above set of linear equations by introduc- ing necessary boundary conditions. we get

~W'~41 = [Bl' (9)

Where [q* is a square, symmetric, nonsingular and banded matrix of order N x N. [B]' is an array of known nodal potentials obtained by inserting Dirichlet boundary conditions. This equation set is solved by using Banded Gaussian elimination technique.

Once the potential distribution is known, then one can find the charge enclosed in a conductor by apply- ing Gauss's law.

Where 2 represents a normal component of 4 at dl. then

Q i

vj c,ij = - 1 (VI, v,, ."...) y-1,5+1, ... V,) = 0 (11)

for i = j , C,, is the self capacitance between the ith conductor and ground and for i # j, Czj is the mutual capacitance between ith and j th conductor. Then per unit length capacitance matrix for (n t 1) conductors is

The capxitance matrix [C] is always symmetric and positive definite. Calculation of per-unit-length(PUL) inductance matrix is carried out from the capacitance matrix [CO], obtained by replacing the dielectric with air. As per the quasi-TEM approximation, we have

1 [L] = -[co]-l 72 (13)

Where U is the velocity of light. Resistance R of the conductor is obtained in terms of the geometrical de- tails and specific resistivity(p,) [2]

3 Special Elements 3.1 Singular Elements

The finite element formulation involving singulari- ties can ble described either by using (a) Direct method or (b) Singular element formulation method. The di- rect method requires a large number of degrees of free- dom to obtain the results with reasonable accuaracy, whereas, the use of special singular elements with sin- gular functions is one of the most appealing formula- tions. Though there are numerous techniques to de- velop singular elements, most of these methods involve defining interpolation functions that result in singular- ity. The number of degrees of freedom can be reduced by using relatively large number of singular elements adjacent to the singularity and standard elements in the regions far from singularity domain[3].

Usuall:y, in the isoparametric formulation, both potential(4) and co-ordinate(x) are represented by the same interpolation function.

4 = a0 + ala + a 2 2 + 1 . - + unan 3: =: bo + b i a -t ha2 + * * * + bnan

(14) (15)

By taking derivative of 4 w.r.t x and substituting for a and its derivative, for a particualr positioning of nodes will give[dI]

I 2 I-" x 2-n _ - "---[U~(-)" + 2 a 2 ( - ) 7 + + . . + n % ] (16) dx nb, b, bn

As x -+ 01, the leading potential term is of the order O ( X e ) . For parabolic interpolation(n = a), the order of singulai~ity will be O ( X 3 ) as X -+ 0. This singular- ity is also popularly called as "Inverse square root singularity". Figl(a) shows the singular elenient[51 and Figl(b) is its parent 8-noded isoparanietric ele- ment. Cosidering node(1) as the singular point, we have

h 4

21 = 27 = xg = o ,x2 = 26 = -, 2 3 = X q = 2.5 = h (17)

y1 = y4 = y7 = ys = 0, y3 = -&j = -1, -1

9 2 = -y6 = - 4

261

5 I , X-cmlt

Fig( 1) 8-noded Singular Element.

Where i = 1,2,3, ...., 8 and Ci,ql = fl for corner nodes and zero for midside nodes. 3.2 Infinite Elements

If the domain of analysis is large, the E’EM with general elements cannot model field distribution accu- rately, alone. Usually, an artificial boundary will be created of larger dimension to account this infinite do- main. However, this increases the number of elements and hence, the number of nodes. This obviously in- creases the computation time. Another way of dealing with this type of problem is to formulate special types of elements, called as infinite elements, and to use them a t the boundaries. These special infinite ele- ment is shown in figs.Z(a). Fig2(b) shows its parent 4-noded element. Following are the shape functions of

00

I

I Fig(2) Infinite elements.

these elements.

10 + CCl>(l + 77771) (19) -(CCt + 1) N = exP[

Where i = 1,2,3 and 4, Ct = vi = f l for corner nodes and L is decay length. It can be noted from the shape functions that these infinite elemnts use exponential decay principle. Sirniiar eiernents are aiso reported in ref[6,7].

4 A computer program VIAFEM has been developed

locally using FEM, mainly to analyse VLSI intercon- nects. Program is supported with a library of elements which includes 3-noded CST to 1Znoded isoparamet- ric elements, singular element and infinite elements. A rigorous excercise of comparision of the results ob- tained using VIAFEM with the results obtained using other techniques like conformal mapping, TLM,MOM etc is reported in ref[8]. It is observed from this that the results of VIAFEM match very well with the re- sults obtained by other techniques listed above. Fol- lowing few examples list the parameter values for var- ious structures. Many more such illustrative problems have been worked out seperately and corresponding results are with authors.

Example(1)In this example, 8-noded singular el- ements are used to analyse a simple single conduc- tor structure. The region around singular point is meshed using these special singular elements and 8- noded isoparametric elements are used in the remain- ing region, as shown in fig(3). This is also called as hybrid finite element method.It is really interesting to note that it is possible to get more accurate results with fewer degrees of freedom and elements(99 nodes and 32 elements), exploring the symmetry of structure if, singular elements are used. However, this accuracy is observed for 189 nodes with 80 six-noded isopara- metric elements and 213 nodes with 60 eight-noded elements, for the same structure. Table I lists these values. Computation time decreases drastically(31.2 %) if this hybrid mesh used.

Implement at ion and Results

1.5 &-- 1.5 4 5

6

7 t_l; 3

Fig(3)Mesh of Singular & 8-noded elements.

Table I : Capacitance values for singular element mesh

I I FEM I c11 I FEM I cii I TLM n Singular Elts. I pf/cm I 8-noded e h . I pf/cm I C11 nn=75,ne=24 I 1.7822 I nn=231,ne=60 I 1.7697 I 1.7664

1 1 nn=99,ne=32 I 1.7667 1 nn=245,ne=70 I 1.7671 I - 1 1 Example(2) First part of this example mainly de- scribes the impact of using higher order elements on accuracy of the results. Fig(4) shows a simple struc- Lure having two finite thickness conductors spaced equally i r i a iiiicrostrip coriiiguration. It can be noted

262

that results obtained using 8-noded iso. elements match very well with TLM and MOM results. These results are listed in Table 11.

Second part of the example shows the impact of using infinite elements at the boundaries to account unbounded region problems. In the Fig (4), infi- nite elements are used a t the boundaries and 8-noded iso.elements in the remaining area. Table I11 lists the self and mutual capacitance values, computed by cre- ating an artificial boundary of conducting walls. Re- sults are noted for different enclosure heights. It can be noted that accurcy of the capacitance values in- creases with increase in enclosure height(h). However, same accuracy is observed using infinite elements with very nominal enclosure dimensions. This also removes the ambiguity regarding the selection of enclosure di- mensions and offers accurate results at reduced com- putation time( 39.8 % less than conventional FEM).

Unbounded region &J

I I I

h

I t

d I

d

Fig(4)Simple 2-cond.pstrip struct.

Table I1 : Capacitance values f o r differnt element meshes .

TLM Method I 1.638 I MOMMethod I 1.637 1

Table 111 : Capacitance values(pf/cm) for different enclosure dimensions

Note : IE - Infinite Elements.

Example(3) In this example a case of multconduc- tor and multilayer(MCM) structure is being analysed wing speck! elcrrxnts formdated above and 8-noded higher order isoparemetric elements, as shown in Fi 5. Following matrices give the entries of [L] and [Cq values

Capacitance Matrix(C in pf/cm) MCML configuration

2.506 -0.926 -0.014 -0.442 -0.121 -0.0080 -0.926 3.096 -0,928 -0.116 -0.337 -0.1218 -0.014 -0.928 2.586 -0.007 -0.118 -0.4830 -0.442 -0.116 -0.007 4.686 -0.462 -0.0153 -0.121 -0.337 -0.118 -0.462 S.836 -1.464 -0.Do8 -0.121 -0.483 --0.015 -1.464 4.686

Inductance Matrix L in nH/cm) for MCML con L guration

3.256 1.236 0.507 0.609 0.347 0.221 1.236 3.031 1.147 0.374 0.292 0.361 0.507 1.147 3.256 0.225 0.333 0.579 0.609 0.374 0.226 2.626 0.748 0.259 0.347 0.292 0.333 0.748 2.438 0.714 0.221 0.361 0.579 0.259 0.714 2.626

E , = 6.5

E , = 4.3 p- 4.0

E , = 9.5 k 4 . 0 2.0+ k 12.9 X -P

Fig(5) Multilayer Multiconductor struct.

5 Coinclusion The electrical representative parameters of very

high speed VLSI interconnects can be obtained accu- rately by accounting singularities in interconnects and modeling far field distribution appropriately by using singular and infinite elements, respectively. This also reduces computation time drastically.

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3. Dennis M. Tracey and Thonias S.Cook."Analysis of power type singularities using finite elements." IJNME, Vol.11, 1977, pp 1225-1233.

Kluwer Academic Pub. 1990. 4. Arnold V.1, Singularities of caustics and wavefronts"

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pp 16 13-1625,1980. 8. S.YXulkarni and K.V.V.Murthy "Computation of very high

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v01.1 i , i g u , p p 8698.

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