Wideband Electrical Modeling of Large Three-DimensionalInterconnects using Accelerated Generation of Partial Impedances with

Cylindrical Conduction Mode Basis Functions

Ki Jin Han, Madhavan Swaminathan, and Ege Engin

School of Electrical and Computer Engineering, Georgia Institute of technology,266 Ferst Drive, Atlanta, Georgia 30332, U.S.A.

Abstract - For wideband modeling of large and complicatedthree-dimensional interconnects, this paper proposes anefficiency improvement in solving electric field integral equationwith cylindrical conduction mode basis functions. Based on themultifunction method, the improved method reducescomputational cost by using smaller number of higher-orderbasis functions for computing mutual inductances between far­separated conductors. From the modeling examples of through­hole vias and bonding wires in stacked Ie's, the proposed methodis verified for application to real three-dimensional interconnects.

Index Terms - 3-D interconnect, cylindrical conduction modebasis function, electric field integral equation, multifunctionmethod, proximity effect, skin effect, through hole vias, bondingwires.

I. INTRODUCTION

The trend of integrating subsystems in a three-dimensional(3-D) packaging such as system-in-package (SiP) requiresaccurate and efficient electrical design of the 3-Dinterconnects in the packaging. A technical challenge ofelectrical design arises in the generation of the entire couplingmodel of over a thousand interconnects such as bond wiresand through-hole vias (THV) since they are complicatedlycoupled in vertical as well as in parallel, as shown in Figure 1.The other design challenge is in the construction of widebandinterconnect model for the design of system packaging, whichincludes analog, radio-frequency (RF), and digital sub systems.

For these modeling issues, a trade off between modelingtime and accuracy should be considered in existing methods.

Figure 1. An example of 3-D bonding wire integration. (Photocourtesy of Amkor Technology, Inc.)

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For example, equation-based methods provide parasitic valuesof interconnects with little computational effort [1-2], but theyare not available for modeling of high-frequency effects suchas skin and proximity effects. In contrast, various numericalmethods can capture the high-frequency effects, but theirapplication to the large and complicated 3-D interconnectproblem requires much simulation time and memory.

To improve speed with maintaining accuracy in the 3-Dinterconnect modeling, we have proposed using the electricfield integral equation (EFIE) combined with globally definedcylindrical conduction mode basis functions (CMBF) in [3-4].The proposed method was found to be efficient for capturinghigh-frequency effects, compared with the conventionalpartial element equivalent circuit (PEEC) method [5] and theEFIE with CMBF's on the rectangular coordinate [6].

In addition to the previous research, this paper presents theacceleration of generating partial impedances by using themultifunction method (MFM), which uses variousapproximations in computing integrals for obtaining mutualinductances. MFM reduces computational burden for solvinga system of large number of conductors because of thedecrease in time for computing matrix elements and the sparseproperty of the higher-order mode sub matrix. For applicationexamples, we simulated more practical structures such asTHV array and vertically- and horizontally-coupled bondingWIres.

II. OVERVIEW OF EFIE WITH CYLINDRICAL CMBF

The electrical modeling of interconnects is to find anequivalent circuit model that accurately describes frequency­dependent conduction losses and electromagnetic couplings.For this purpose, the classical PEEC method [5] and itsvariations are preferred since they extract the equivalentcircuit model directly from the solution of the followingvolume EFIE.

lei,m) + jm£ fG(r,r')l(r',m)dV'= -V<I>(r,m) , (1)u 47£ v'

where (j is conductivity, OJ is angular frequency, p ispermittivity, and G(r,r') = I/lr - r'l, which is Green's functionwith the retardation term neglected. For the approximation of

1297

III. MULTIFUNCTION METHOD

difference of the conductor segment since the surface integralwith the basis function becomes unity. If the basis is in thehigher-order skin effect modes, the integral becomes zero dueto the harmonic function in the higher-order functions.Therefore, the equivalent circuit is composed as shown inFigure 2, where the components from the skin effect mode areconnected to the physical nodes, and the other componentsfrom the proximity effect modes form closed loops.

Conductor #2

Conductor #2

Conductor #1

For solving the problem of large number of interconnects,additional improvement of computation speed is useful. Oneidea of reducing the computational effort is applying themultifunction method (MFM) [7], which uses simpleapproximate formula or lower-order integrations forcalculation of mutual inductances between sufficientlyseparated conductors. For example, thin-wire approximateformula, which is independent of the local coordinates of eachconductor segment, can reduce computation time considerably.

A multifunction approach used for the CMBF-based methodis to control the number of higher-order modes for twoconductor segments. Although the required number of basesfor a conductor segment is determined by the minimumdistance from nearby conductors, all the bases need not beused for calculation of mutual inductances with far-separatedconductors. That is, using smaller number of bases forconductors with large separation is sufficient for maintainingaccuracy. An apparent benefit of the variable number of basesis the reduction of the computation time for calculating partialmutual inductances. In addition, the higher-order sub matrixof the partial impedance matrix becomes sparse, so we cansave memory for storing non-zero elements.

Figure 3 shows the relative error of the couplingcoefficients with adding higher-order proximity effect modesin two-parallel-conductor problem. As the spacing betweenconductor segments increases, the required number of higher­order bases becomes smaller. With a permissible relative error

,'-------------, ,'--------- ,~ , ~I I II I II I II I II I II I II I vL , II I I

\ ~ \ ~,-------------, ,-------------,VL,imp = LjOJLimp,jn/jnq '-y---J '- ./

j,n,q "Branches" "Loops"jnpcoeimq from skin-effect modes from proximity-effect modes

Figure 2. Equivalent circuit example for two conductor segments.

where

Rp,imd,jnq =~ f W;md(~,£i))· Wjnq(~,£i))dV; ,Vi

- Jl ff -* - ). - - )G - - )dV dV andLp,imd,jnq - 4Jr Wimd 1j OJ W jnq rj OJ 1j rj j i '

Vi Vj

V:md =-fet> ~)W~d ~ OJ)dSi ·

Si

Calculating multiple integrals in the partial impedances canbe a source of computational cost in the proposed method.However, the cost can be minimized by using analyticexpressions and other techniques. At first, partial resistancesmatrix is diagonal because of the orthogonal property of thebasis functions, and all the elements can be found fromanalytic expressions. In the case of calculating partial selfinductances, a coordinate transform reduces a numericalintegration, and efficiently controls singular points in Green'sfunction. Computational cost in calculating partial mutualinductances is reduced by using the analytic expression for theintegral over the axial variables. In addition, the multifunctionmethod discussed in the next section finds smaller number ofrequired higher order elements, so accelerates the formation ofthe global partial impedance matrix.

The modal voltage V;md in (3) is simplified for differentconduction modes. If the basis function is in the skin effectmode, the modal voltage is the same as the nodal voltage

the current density in (1), the CMBF-based methods use thefollowing combination of global basis functions [4, 6].

Jj(r,OJ) ~ Lljnq wjnq(r,{o) , (2)

where

_ _ {~j I n a r- ~ .Pj cos n qJj - qJjq rE VjW jnq r OJ = .In

o elsewhere

is the (n, q) mode cylindrical CMBF in the jth conductor, Ajn isthe effective area for normalization, I n is the nth order Besselfunction, and a 2 = - j OJJ.lU •

The cylindrical CMBF's are obtained from the solutions ofdiffusion equation of the current density on the circular crosssection. Among the possible solutions of the equation, thefundamental-order (n=O) basis, which is independent ofangular variable, is essential to describe skin effect of eachconductor. If a conductor is significantly influenced by nearbyconductors, the higher-order (n>O) bases are necessary tocapture various proximity effects. In case of using theproximity effect basis of a certain order, two orthogonal (d­and q-) modes are sufficient to describe current crowding inarbitrary orientations [4].

By substituting (2) into (1) and applying the inner productbased on the Galerkin's method, we can obtain the followingequivalent circuit equation, which is composed of partialimpedances and modal voltages.

LljnqRp,imd,jnq + jmLljnqLp,imd,jnq =~md ' (3)n,q n,q

n,q

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IV. EXAMPLES

.035

--S9'- #1 -- #2

-e- #8--#9

12

~"~ 10

~] 9

] 8

e

~ ~ ~ .-_...I'IZ- 34180

0.036

0.042

0.043,.-------::-.....,.........------,---------,

0.035,...........~_~1_eoI_4...._ .........

0.041

~ 0.044) ---9-- Conductor #1g 0.039~ --e-- Conductor #9

] 0.0384-<

] 0.037

1.6

1.8

0.2

0.4

I 1.4

g 1.2S.~ I

~ 0.8

~ 0.6

0.03

104 106 108

frequency (Hz)

Figure 6. Self resistances and inductances of diagonal conductors inthe THV array.

104

106

108

10 10 ~04 106 108 10 10

frequency (Hz) frequency (Hz)

Figure 7. Mutual resistances and inductances between diagonal

conductors that are separated to 50J2 microns.

Figure 5. The sparsity pattern of global partial impedance of the ten­by-ten THV array model.

Figure 6 shows self resistances and inductances of thediagonal conductors. The parasitic values become diversingdue to various proximity effects in high frequencies. That is,inductances of conductors near to the ground conductor(conductor # 9 for example) decrease more than otherconductors far from the ground. Figure 7 also showsvariations of mutual impedances in high frequencies. Theunbalances in self and mutual parasitic components affecthigh frequency performance of 3-D packages.

(10-3 for example), we can determine thresholds of conductordistances (about 0.1, 0.5, and 0.7 for example) that findsufficient number of bases, as illustrated in Figure 3.

o10

A. Through Hole Via Array

Through hole via (THV) is an emerging interconnectstructure in the 3-D packaging technology. A test example inthis paper is shown in Figure 4, where ten-by-ten identicalcopper conductors are distributed with the same spacing. Ninediagonal conductors were selected to observe their frequency­dependent resistances and inductances, which are differentdue to the grounded conductor at an edge.

By applying MFM, required numbers of higher-ordermodes among conductor segments were determined based onthe thresholds in Figure 3, resulting in the sparse higher-order­element part (Zpp) in the global matrix, as shown in Figure 5.

Figure 4. Geometry and port number indication of a ten-by-ten (ahundred) copper THV array.

5'u _IS 10

G)

ou00

~ 10-20.::sou.5 -3~ 10 ~~ ~_~!",,,,,,,,,-=,,,"I,,,,,= _l::,...= - - __

E ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - ~ ~~ =~

.~ = j::: = '- L- =J =IK3 - K 21

~ ==~~~~=~=~~~ =c~~=~ ~~ -~-r - -r- --r-~--r---­

-~- -- -r ~--r-~--r----

~_-----.1~ ~~.~~1--l-1__ .--.-J

o 0.2 0.4 0.6 0.8Diameter / Distance

Figure 3. Relative errors in coupling coefficient at 10 GHz withadding higher-order modes in two-parallel-conductor configuration.

The diameter of conductors varies from 20 to 40 microns.

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-e-Ld,l_4

... Ld,lO_13

0.67

0.69 _.~----~--------,

0.62

0.63

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v Ld,7-10

0.62

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.5 0.64

REFERENCES

[1] H. Patterson, "Analysis of Ground Bond Wire Arrays forRFICs," Proc. Radio Frequency Integrated Circuits Symposium,pp. 237-240, June 1997.

[2] S. Ray, A. K. Gogoi, and R. K. Mallik, "Closed FormExpression for Mutual Partial Inductance between Two SkewedLinear Current Segments," Proc. Antennas and PropagationSociety International Symposium, pp. 1278-1281, July, 1999.

[3] K. 1. Han, E. Engin, and M. Swaminathan, "CylindricalConduction Mode Basis Functions for Modeling of InductiveCouplings in System-in-Package (SiP)," Proc. Topical Meetingon Electrical Performance of Electronic Packaging, pp. 361 ­364, October 2007.

[4] K. J. Han, M. Swaminathan, and E. Engin, "Electric FieldIntegral Equation with Cylindrical Conduction Mode BasisFunctions for Electrical Modeling of Three-dimensionalInterconnects," accepted for presentation in 45th DesignAutomation Conference, June 2008.

[5] A. E. Ruehli, "Equivalent Circuit Models for Three­Dimensional Multiconductor Systems," IEEE Trans. MicrowaveTheory and Techniques, vol. 22, no. 3, pp. 216-221, March 1974.

[6] L. Daniel, J. White, and A. Sangiovanni-Vincentelli,"Interconnect Electromagnetic Modeling using ConductionModes as Global Basis Functions," Proc. Topical Meeting onElectrical Performance of Electronic Packages, pp. 84-89,October 2000.

[7] G. Antonini and A. E. Ruehli, "Fast Multipole andMultifunction PEEC Methods," IEEE Trans. Mobile Computing,vol. 2, no. 4, pp. 288-298, October 2003.

ACKNOWLEDGEMENT

This work was supported by the Mixed Signal Design ToolsConsortium (MSDT) of the Packaging Research Canter,Georgia Tech, under project number 2126QOR.

0.6~04 106 10K 1010 0.6~04 106 10K 10 10

frequency (Hz) frequen:y (Hz)

(a) (b)Figure 9. Differential inductances of wires for different groundwiring conditions. (a) Side wires are grounded. (b) Mid wires aregrounded.

z

x

.t~

.4~

r ·7~ Gbip~:)OO~1 10 ~12 Wnpitcbltthebottom:SOum

~Wire diameter:2Sum~15 . .• Approximatewirelqths: 696.84um

13 ~ SlO.47um~ 323.99um

Figure 8. Geometry and port number indication of a 3-D fifteen-wire­bond example. (left upper: side view, left lower: top view, right: 3-Dview.)

B. Horizontally and Vertically Coupled Bonding Wires

Bonding wires used in 3-D integrations connect ports onstacked IC's to printed traces on substrates. Wiring in high­density integration results in complicated configuration oflarge number of bonding wires, and finding coupling model ofarbitrary oriented wires is a difficult task. To verify thecapability of the proposed method in the 3-D bonding wireproblem, we generated a fifteen bonding wire model on athree stacked IC's, as shown in Figure 8. Similar to theprevious THV array example, frequency-dependent variationsof resistances and inductances are obtained from the bondingwire example.

An interesting finding from the simulation is shown inFigure 9, where pairs of two wires form differential ports andthree conductors are grounded. As known in design practice,the unbalanced ground as in Figure 9 (a) causes differences ofinductances in high frequencies. Furthermore, we also observea little difference of inductances in Figure 9 (b), where threecenter wires are grounded to maintain the balance. Thisimperfect balance comes from the geometric asymmetry of thethree grounded conductors (#7 - #9), which provides differentgrounding conditions for two conductor pairs. This exampleshows that more deliberate design strategy is necessary for theelectrical design of interconnects in the 3-D structure.

For modeling of large 3-D interconnects, this paperproposed an improvement in EFIE with cylindrical CMBF's.By using smaller number of higher-order basis forcomputation of mutual inductance, we could reduce thesimulation time and memory for matrix storage. Theapplication to the THV array and the 3-D bonding wiresexamples validates the efficiency of the proposed method forthe wideband modeling of large interconnects in practice.

V. CONCLUSION

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