Electrical Modeling of Wirebonds in Stacked ICs Using 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.

{kjhan, madhavan, engin}@ece.gatech.edu

Abstract This paper discusses an efficient method for simulation

and modeling of three-dimensional (3-D) interconnections such as wire bonds and through-hole vias. The proposed method is based on the volume electric field integral equation (EFIE), and approximates current density distribution with cylindrical conduction mode basis functions (CMBF). The geometrical fitness and orthogonal property of the basis functions are found useful in capturing skin effect and current crowding in cylindrical conductors. The Analyses of wire bonds using the proposed method demonstrate various 3-D coupling issues including wire bending effect, vertical coupling, and ground wiring.

Introduction The continuing trend of integration technology in modern

electronics has reached the system integration level of System on Package (SoP) and System in Package (SiP). As the role of packaging is critical in the performance of these modern microsystems, the design and modeling of three-dimensional (3-D) interconnections such as bonding wires and through-hole vias became especially important.

Among various choices of the 3-D interconnections, bonding wires have been a preferable technology since late 90’s [1]. The popularity of wire bonding comes from the benefits of high flexibility, high reliability, and low defect rates [2]. However, long interconnection length limits the use of bonding wires in high-speed applications. In addition, complicated 3-D configuration of about a thousand wires is exposed to various couplings as shown in Figure 1, which make the design of 3-D bonding wires more challenging.

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

For modeling of the large 3-D bonding wire structure, the previous modeling methods for single-chip or radio-frequency (RF) applications could be useful, but they have limitations in practice as well. Measurement data may be a direct source for modeling of bonding wires, as extensively discussed in [3], but the modeling procedure based on full measurement requires increased design cycles and cost, especially for high-density 3-D interconnections. Analytic methods using partial inductances [4] or transmission line model [5] can be used for modeling of large 3-D interconnections efficiently since numerical computations are rarely required. However, analytic formulas of mutual inductance are found to be inaccurate especially when two conductors are close with each other [6]. In addition, the simple expression does not include high-frequency skin and proximity effects.

In order to capture the high-frequency effects accurately, various methods of computational electromagnetics (EM) can be a general choice. Commercial or public EM simulation tools have been applied for modeling of bonding wires (e. g. [7]). In addition, finite difference time domain (FDTD) [8] and method of momentum (MoM) [9] were shown to be efficient for accurate modeling. However, the performance of CPU and memory is still under capability for such simulators to solve the full coupling model of the large 3-D interconnection problem. Therefore, requirements for simulation tools to model modern 3-D integrations are the capabilities 1) to address the problem involving large number of conductors and 2) to characterize frequency dependent behavior of bonding wires.

To meet the requirements for 3-D interconnection modeling, this paper proposes the analysis method that combines the electric field integral equation (EFIE) with modal basis functions [10-11]. The proposed method uses the same governing integral equation in the classical partial element equivalent circuit (PEEC) method [12], and applies global basis functions [13] to describe current density distribution. A main improvement of the proposed method comes from using cylindrical conduction mode basis functions (CMBF) to describe skin and proximity effects on the circular cross section of bonding wires or through-hole vias. With the combination of multifunction method to the original approach, the efficient simulation of large interconnection structure is possible.

This paper focuses on using the cylindrical CMBF-based method for the modeling of bonding wires. Some examples of simple and complicated bonding wire configurations are simulated. Then, various coupling issues in the 3-D integrations are discussed from the perspectives of the interconnection design.

978-1-4244-2231-9/08/$25.00 ©2008 IEEE 1225 2008 Electronic Components and Technology Conference

Overview of Analysis Procedure This section discusses the basic analysis procedure used in

this paper. The overall procedure is composed of defining the governing integral equation, approximating current density distribution with basis functions, constructing equivalent circuit formula, and computing partial impedances. In addition to the basic procedure, using multi function method (MFM) accelerates the process of matrix formation.

A. EFIE By using the Green’s function expression of scalar and

vector potentials, the governing Maxwell’s equation on a point in a conductor is converted to the following volume EFIE.

),('),'()',(1

),('),'()',(4

),(

'0

'

ωωε

ωωπµω

σω

rdVrqrrG

rdVrJrrGjrJ

V

V

rrrr

rrrrrrr

∫

∫

Φ=

Φ−∇=+, (1)

where σ is conductivity, ε0 is electric permittivity, ω is angular frequency, µ is magnetic permeability, ),( ωrJ rr is current density, ),( ωrq r is electric charge, and

'/1)',( rrrrG rrrr−= , which is Green’s function. Since the main

contribution to wire parasitic elements is from inductances, this paper does not consider the second integral of electric charges. In other words, displacement current is neglected in the interconnection analysis, so the resonance at high frequencies is not captured. B. Cylindrical CMBF

For numerical computation of (1), several approximate methods have been proposed. One popular way is the PEEC method [12], which uses the staircase approximation to describe the current density distribution. This approximation is equivalent to using simple piecewise constant basis functions, so requires to divide a conductor into many filaments. In the process of discretizing large number of interconnections, the PEEC method results in a large dense matrix. Another approximate way assumes the current density distribution with combination of a few global basis functions [13]. The global bases are obtained from the following diffusion equation of current density in a conductor.

01 2

=⎟⎠⎞

⎜⎝⎛ +

+×∇×∇ JjJrr

δ, (2)

where µσπδ f/1= is the skin depth. The main benefit of using the global CMBF is that descretization of conductor is not necessary, and that the resulting system matrix becomes smaller. In addition, simplified equivalent network is directly obtained.

A main difference of the proposed method from the original work [13] is in the type of CMBF. Since the cross section of bonding wires and THV is circular rather than rectangular, the solutions of (2) should be obtained from cylindrical coordinates. The angular symmetry of cylindrical CMBF results in the following set of global bases [10].

Skin-effect mode ( 0=n ):

⎩⎨⎧ ∈⋅−

=elsewhere

VrrrJw iiiAz

ii

i

0)ˆ)((0

ˆ

00

rrrr ρα , (3)

Proximity-effect, direct mode ( 0>n ):

⎩⎨⎧ ∈⋅−

=elsewhere

VrnrrJw iiiinAz

indin

i

0)cos()ˆ)((ˆ rrr

r ϕρα , (4)

Proximity-effect, quadratic mode ( 0>n ):

⎩⎨⎧ ∈⋅−

=elsewhere

VrnrrJw iiiinAz

inqin

i

0)sin()ˆ)((ˆ rrr

r ϕρα , (5)

where Ain is an effective area to normalize the integral of the basis functions. The skin-effect mode basis (3) is the fundamental mode required for all conductor segments to describe the skin effect, which exists regardless of vicinity of other conductors. On the other hand, orthogonal pairs of two proximity-effect bases are higher order functions to describe current crowding or proximity effect. The number of higher-order bases depends on the geometrical configuration of conductors such as distance and conductor diameter. For each higher order, two orthogonal bases are required to capture current crowding in arbitrary direction. Figure 2 summarizes patterns of various cylindrical CMBF, weighted summation of which approximates the current density in a conductor. C. Equivalent Circuit

Equivalent circuit equation of a conductor segment is obtained by substituting the approximate current density into (1) and applying the inner product based on Galerkin’s method. In this process, each term of the left side of (1) is

(a) Skin-effect mode

(b) 1st proximity-effect (direct) mode

(c) 2nd proximity-effect (direct) mode

Figure 2. Absolute values of cylindrical CMBF for different modes and frequencies. Conductor material is copper and radius is 25 um. (left: 100 MHz, center: 1 GHz, right: 10 GHz)

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converted to voltage drop due to resistance and inductance, and the right side of (1) is converted to the voltage difference between two nodes. That is,

∑∑ =+qn

imdjnqimdpjnqqn

jnqimdpjnq VLIjRI,

,,,

,, ω , (6)

where

∫ ⋅=iV

iijnqiimdjnqimdp dVrwrwR ),(),(1 *,, ωω

σrrrr ,

∫ ∫ ⋅=i jV

ijjiV

jjnqiimdjnqimdp dVdVrrGrwrwL ),(),(),(4

*,,

rrrrrr ωωπµ , and

∫Φ−=iS

iiimdiimd SdrwrVrrrr ),()( * ω .

The calculation of each resistance and inductance in (6) is a major source of computational cost in the proposed method. Some of analytic formula and other techniques for the multiple integrals are discussed in the previous work [10]. The modal voltage difference term (Vimd) is the actual voltage difference between two conductor nodes for equations from the fundamental (skin-effect) mode basis, but becomes zero for equations from the higher-order basis functions. In the representation of equivalent network, therefore, the circuitries from the proximity-effect modes form closed loops, and the branches of circuitries from the skin-effect modes are connected to each other. The matrix expression of the conductor currents and voltages are as follows.

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡0V

II

ZZZZ i

p

s

ppps

spss , (7)

where Zss, Zsp, Zps, and Zpp are sub matrices of impedances from skin-effect and proximity-effect CMBF, Is and Ip are current sub vectors, and Vi is the node voltage difference vector in a conductor. An example of the entire equivalent network model of two wires is illustrated in Figure 3. D. Efficiency Improvement with Multi Functions

The procedure discussed in the previous subsections has a benefit that the system matrix of the equivalent network (7) is

“Loops” fromproximity-effect modes

“Branches”from skin-effect modes

Conductor segment approximation

of a bonding wire

Figure 3. Equivalent network example of two wires.

smaller than the matrix of the classical PEEC method. However, calculation of multiple integrals (6) for each frequency step is more complicated than the PEEC method. In addition, for modeling of real 3-D interconnections in SiPs, more reduction of computational cost may be necessary.

One idea of reducing computational cost is to determine the number of higher-order bases differently for each neighboring conductor [11]. For example, higher-order basis functions are not necessary when the distance between two conductors is sufficiently large, or when the coupling coefficient between two conductors is small enough. Thus, we can capture current crowding effectively by merely calculating interaction between close conductors. The reduction of calculation clearly shortens simulation time, and makes Zpp sparse. The criteria of determining the number of basis functions can be constructed based on the physical distance or the coupling coefficient between two conductors. In case of using the physical distance criteria, analytic method of finding the minimum distance [14] is useful.

In addition, if the two conductors are even further separated, the frequency-dependent integrals are not necessary. In this case, the variations of current density in conductors are negligible, so the following thin-filament approximation [15] can be used instead.

∫ ∫=i jz

ijz

jijip dzdzrrGL ),(4,,

rr

πµ (8)

If more loss of accuracy is acceptable, the following center-to-center approximation [16] is also available.

ij

jijip R

llL

πµ

4,, = (9)

where li, lj, and Rij are the length of the ith and the jth conductor, and the distance between center of two conductors, respectively. The threshold distance of using the frequency-independent approximation is usually higher than that of omitting higher-order bases. The approximations reduce the number of matrix elements calculated for each frequency step.

Bonding wire Simulation Examples In this section, we apply the previously proposed method

to bonding wire simulation. With a number of different interconnection structures, various effects of bonding wires geometry and 3-D couplings are investigated. All wires in this section have conductivity of gold (4.1×107 S/m) and diameter of 25 um. A. Geometric model of a bonding wire

For the accurate and efficient simulation of bonding wires, proper geometric model of bonding wires should be constructed. In the case of bonding wires in traditional packages, three-segment model was favorable choice [4]. However, in the modern 3-D integrations, realistic curvatures should be described to capture coupling effects more accurately. For example, exact geometries are obtained by using the image processing of bonding wires [7]. In this paper, we used a simpler arc model [17] that is close to the real 3-D bonding wire model and easy to shape with simple geometric parameters. As shown in Figure 4, a wire shape is formed by defining three points on a wire, which is usually a

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1P

3P

2P

1h

2h

mh

1d 2d0d

Figure 4. Arc model of a bonding wire

starting point, a mid point, and an ending point. B. A single bonding wire – effect of bending

A basic question about bonding wires characteristics is what becomes different when a straight conductor bends to form a bonding wire. The proposed method is flexible to observe these geometric variations. Figure 5 (a) shows a single bonding wire, which has the same length of 1 mm with different span angles from 0 to 180 degrees.

Figure 5 (b) shows the variation of self inductance with different span angles. As the span angle increases, dc and ac inductances decrease because the mutual coupling between conductor segments reduces the original self inductance of the straight conductor. For values of resistance, no variation is found with different span angles. C. Two wires coupled with a ground wire – effect of 3-D coupling

Vertical coupling is a new issue in the design of 3-D interconnections. Basic coupling mechanism of the vertical coupling is the same as that of the horizontal coupling. However, the vertical coupling occurs under asymmetric geometry in the 3-D structure, and sometimes results in unbalanced signaling and grounding in spite of the balance in the schematic design.

In order to demonstrate the effect of the 3-D vertical coupling, a test bonding wire structure in Figure 6 is simulated with different tilt angles of the lower ground wire. Two upper wires have the same shape, and are assumed to be used for transmission of the same signals.

Figure 7 shows the variations of resistances and inductances, where the difference of self inductances between two wires becomes significant as the tilt angle of the ground

span angle

arc length = 1 mm

104 106 108 10100.65

0.7

0.75

0.8

0.85

0.9

indu

ctan

ce (n

H)

frequency (Hz)

0306090120150180

(a) a wire model with bending (b) inductances of a wire

Figure 5. Inductances of a 1-mm single wire with different span angles.

Wire increases. We found that the variation of loop inductances can be avoided when two upper wires are used for differential signaling, but the effect of asymmetric vertical coupling does not completely vanish. This simple example shows that the design of 3-D interconnections should be more complicated to maintain signal balances. D. Six wires with the ideal ground plane

Figure 8 shows the configuration of the last example of six bonding wires. Three short wires (1 to 3) and the other three long wires (4 to 6) are connected to the lower and the upper ICs, respectively. To include the effect of the infinite ground

300 um

37 um 50 um100 um

100 um

50 um

50 um

220 um

140 umzy

(a) y-z plane view

tilt angle

wire 1

wire 2

ground wire

xy

(b) x-y plane view

Figure 6. Configuration of two signal wires (on the upper chip) with a ground wire (on the lower chip). Three cases are tested with different tilt angles (0, 5, and 10 deg.) of the lower wire. Pitch between two wires is 50 um.

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10-1

100

resi

stan

ce (o

hm)

frequency (Hz)

self resistances

mutual resistances

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0.15

0.2

0.25

0.3

0.35

0.4

L 12 (n

H)

frequency (Hz)

0 deg.5 deg.10 deg.

(a) Resistance (angle = 10 deg.) (b) Mutual inductance

104 106 108 10100.1

0.15

0.2

0.25

0.3

0.35

0.4

L 11 (n

H)

frequency (Hz)

0 deg.5 deg.10 deg.

104 106 108 10100.1

0.15

0.2

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0.3

0.35

0.4

L 22 (n

H)

frequency (Hz)

0 deg.5 deg.10 deg.

(c) Self inductance of wire 1 (d) Self inductance of wire 2

Figure 7. Resistances and inductances of two wires for different tilt angles of the ground conductor.

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plane on Z = 0, six symmetric image wires are added in the simulation setup.

The sparsity pattern of the partial impedance matrix is shown in Figure 9, which is subdivided according to the interaction between skin-effect (fundamental) and proximity-effect (higher-order) modes. Using different number of higher-order bases reduced number of nonzeros in Zpp, so the inversion of the matrix becomes efficient. Although Zss is a dense matrix, some of the matrix elements are obtained from the frequency-independent thin-filament approximation. Thus, frequency-dependent part of Zss is actually a banded matrix.

A focus of the bonding wire simulation is to observe variations of equivalent components of wires with different locations of ground wires. As shown in Figure 10 to 12, four cases are tested with wires 1, 2, 4, and 5 grounded, respectively. From the geometry in Figure 8, we can estimate that using 1 or 4 as ground brings imbalance in impedance values. For example, as shown in Table 1, difference of impedances between two upper wires in the case of wire 4 grounded is about 3.4 ohms. Each ground wire also effects slightly on wires on the other chip, and the effect will be more significant when the vertical coupling is stronger.

-6

-4

-2

0

2

x 10-4

-10

12

34

5

x 10-4

0

2

4

6

x 10-4

xY

Z

123

456

-1 0 1 2 3 4 5

x 10-4

-1

0

1

2

3

4

5

6

7

x 10-4

Y

Z

(a) 3-D plot (b) y-z plane view Figure 8. Configuration of six bonding wire model. (Six image wires are not shown.) Pitch among wires is 50 um.

0 100 200 300 400 500

0

50

100

150

200

250

300

350

400

450

500

ZssZsp

Zps

Zpp

Figure 9. The sparsity pattern of global partial impedance of the six bonding wire model with image conductors. (Number of nonzeros is 49296.)

Frequency dependencies of mutual couplings in Figure 12 are also different for balanced grounding (wire 2 or 5) and unbalanced grounding (wire 1 or 4). Current crowding to the

104 106 108 101010-2

10-1

100

frequency (Hz)

resi

stanc

e (o

hm)

#1 grounded

23456

104 106 108 101010-2

10-1

100

frequency (Hz)

resi

stanc

e (o

hm)

#2 grounded

13456

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10-1

100

frequency (Hz)

resi

stanc

e (o

hm)

#4 grounded

12356

104 106 108 101010-2

10-1

100

frequency (Hz)

resi

stanc

e (o

hm)

#5 grounded

12346

Figure 10. Wire resistances with different ground wires

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0.15

0.2

0.25

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0.4

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self

indu

ctan

ce (n

H)

#1 grounded

23456

104 106 108 10100.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

frequency (Hz)

self

indu

ctan

ce (n

H)

#2 grounded

13456

104 106 108 10100.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

frequency (Hz)

self

indu

ctan

ce (n

H)

#4 grounded

12356

104 106 108 10100.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

frequency (Hz)

self

indu

ctan

ce (n

H)

#5 grounded

12346

Figure 11. Self inductances of wires for different ground wires

Table 1. Absolute values of wire impedances at 10 GHz with

different ground wires (ohms) wire # gnd

1 2 3 4 5 6

1 N/A 5.5659 6.8789 17.988 17.824 18.115 2 5.6367 N/A 5.6370 18.015 17.781 18.015 4 7.4776 7.3946 7.5300 N/A 11.495 14.919 5 7.4894 7.3772 7.4892 11.619 N/A 11.619

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0.1

0.2

0.3

0.4

0.5

0.6

0.7co

uplin

g co

effic

ient

frequency (Hz)

#1 grounded

K23

coupling bet. upper wires

coupling bet. upper & lower wires

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0.2

0.3

0.4

0.5

0.6

0.7

coup

ling

coef

ficie

ntfrequency (Hz)

#2 grounded

K13

coupling bet. upper wires

coupling bet. upper & lower wires

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0.2

0.3

0.4

0.5

0.6

0.7

coup

ling

coef

ficie

nt

frequency (Hz)

#4 grounded

K56

coupling bet. lower wires

coupling bet. upper & lower wires

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0.2

0.3

0.4

0.5

0.6

0.7

coup

ling

coef

ficie

nt

frequency (Hz)

#5 grounded

K46

coupling bet. lower wires

coupling bet. upper & lower wires

Figure 12. Coupling coefficients between wires with different ground wires.

ground conductor at high frequencies reduces the mutual inductance between signal lines, so the variation of mutual coupling over the frequency sweep is more significant for the cases of using balanced ground wires.

Conclusion For efficient simulation and modeling of complicated 3-D

couplings of interconnections in SiP, this paper presented an efficient method to solve the volume EFIE using cylindrical CMBF. The geometrical fitness and orthogonal property of the basis functions are found to be useful to capture skin effect and current crowding in cylindrical conductors. Application of the proposed method to bonding wire simulation shows various phenomena in 3-D bonding wires such as bending, vertical coupling, and ground wiring. These 3-D coupling issues need to be considered in the design of 3-D interconnections.

Future work of including capacitive coupling and finite ground effect should be completed for the modeling of bonding wires in practice.

Acknowledgments This work was supported by the Mixed Signal Design

Tools Consortium (MSDT) of the Packaging Research Canter, Georgia Tech, under project number 2126Q0R.

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using Conduction Modes as Global Basis Functions,” Proc. 9th Topical Meeting Elect. Performance Electron. Packag., Scottsdale, AZ, Oct. 2000, pp. 84-89.

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