Electric Field Integral Equation Electric Field Integral Equation Combined with Cylindrical Conduction Combined with Cylindrical Conduction

Mode Basis Functions for Electrical Mode Basis Functions for Electrical Modeling of Three-Dimensional Modeling of Three-Dimensional

InterconnectsInterconnects

June 11, 2008June 11, 2008

Ki Jin Han*, Madhavan Swaminathan, and Ege EnginKi Jin Han*, Madhavan Swaminathan, and Ege EnginSchool of Electrical and Computer Engineering,School of Electrical and Computer Engineering,

Georgia Institute of TechnologyGeorgia Institute of Technology{kjhan, madhavan, engin}@ece.gatech.edu{kjhan, madhavan, engin}@ece.gatech.edu

45th Design Automation ConferenceAnaheim, California

22

ContentsContents IntroductionIntroduction Cylindrical conduction mode basis Cylindrical conduction mode basis

functions (CMBF’s)functions (CMBF’s) Electric field integral equation (EFIE) Electric field integral equation (EFIE)

formulation with CMBF’sformulation with CMBF’s Simulation examplesSimulation examples ConclusionsConclusions

33

BackgroundBackground Interconnections in 3-D Interconnections in 3-D

integrationintegration– Role of interconnections Role of interconnections

is critical for signal is critical for signal transmission.transmission.

– Design and modeling of Design and modeling of interconnections are interconnections are important. important.

Challenges in electrical Challenges in electrical design of SiPdesign of SiP– Modeling of the entire Modeling of the entire

coupling among coupling among interconnects in an SiP.interconnects in an SiP.

– High-frequency High-frequency modeling, including skin modeling, including skin and proximity effects.and proximity effects.

Introduction (1)

Bonding wires in a 3-D integration. (Photo courtesy of Amkor Technology,

Inc.)

* R. Chatterjee, R. Tummala, “The 3DASSM Consortium: An Industry/Academia Collaboration,” online article in http://ap.pennnet.com

Through-silicon via (TSV) interconnections in a Si substrate*

Wideband modeling of the entire coupling (between a thousand wires) is required for multi-functional and high-

density integrations.

44

Approach in this PaperApproach in this Paper State of the artState of the art

– Existing modeling tools: Have issues in accuracy or Existing modeling tools: Have issues in accuracy or speed for modeling of large number of 3-D interconnects.speed for modeling of large number of 3-D interconnects.

– Electric field integral equation (EFIE) combined with Electric field integral equation (EFIE) combined with conduction mode basis function (CMBF)*conduction mode basis function (CMBF)*

AdvantagesAdvantages– Describes current density distribution with a few CMBF’s.Describes current density distribution with a few CMBF’s.– Provides simplified equivalent circuit model.Provides simplified equivalent circuit model.

Issues for 3-D interconnect modelingIssues for 3-D interconnect modeling– Not geometrically suitable for 3-D cylindrical structures.Not geometrically suitable for 3-D cylindrical structures.– Allocating basis functions is difficult to describe various Allocating basis functions is difficult to describe various

proximity effects.proximity effects. Approach in this paper: EFIE combined with Approach in this paper: EFIE combined with

cylindrical CMBFcylindrical CMBF– Maintains advantages of the original CMBF-based Maintains advantages of the original CMBF-based

approach.approach.– Geometrically suitable for 3-D interconnections (bonding Geometrically suitable for 3-D interconnections (bonding

wires, through hole vias).wires, through hole vias).– Automatically captures skin and proximity effects. Automatically captures skin and proximity effects.

Introduction (2)

* L. Daniel, J. White, and A. Sangiovanni-Vincentelli, “Interconnect Electromagnetic Modeling using Conduction Modes as Global Basis Functions,” Topical Meeting on Electrical Performance of Electronic Packages, EPEP 2000, Scottsdale, Arizona, Oct. 2000.

55

Cylindrical CMBFCylindrical CMBF Derived from solutions of Derived from solutions of

the current density the current density diffusion equation in a diffusion equation in a conductor.conductor.

The fundamental mode The fundamental mode (skin-effect mode) basis (skin-effect mode) basis captures skin effect.captures skin effect.

Two orthogonal higher-Two orthogonal higher-order mode (proximity-order mode (proximity-effect mode) bases effect mode) bases captures proximity effects captures proximity effects in arbitrary orientations.in arbitrary orientations.

elsewhere0

ˆ)(ˆ

)( 000,

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i

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zrw

elsewhere0

))(cos(ˆ)(ˆ

)(,iiniiin

in

i

ni

VrnrrJA

zrw

-1.5 -1 -0.5 0 0.5 1 1.5

x 10-5

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-5 Order 0, Frequency 1.0e+008 Hz

X

Y

-1.5 -1 -0.5 0 0.5 1 1.5

x 10-5

-1.5

-1

-0.5

0

0.5

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1.5x 10

-5 Order 0, Frequency 1.0e+010 Hz

X

Y

-1.5 -1 -0.5 0 0.5 1 1.5

x 10-5

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-5 Order 1, Frequency 1.0e+008 Hz

X

Y

-1.5 -1 -0.5 0 0.5 1 1.5

x 10-5

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-5 Order 2, Frequency 1.0e+008 Hz

X

Y

-1.5 -1 -0.5 0 0.5 1 1.5

x 10-5

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-5 Order 1, Frequency 1.0e+010 Hz

X

Y

-1.5 -1 -0.5 0 0.5 1 1.5

x 10-5

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-5 Order 2, Frequency 1.0e+010 Hz

X

Y

CMBF (1)

66

Basic FormulationBasic Formulation

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

'

rdVrJrrGjrJ

V

),(),(

,

rwIrJ jnqqn

jnqj

),(),(1

4),(

1

,,

rdVrw

rrIjrwI

jVjjjnq

jdnjnq

dnjnqjnq

i

i j

i

S iiimdi

V iV j

ji

jjnqiimdqn

jnq

qniijnqV iimdjnq

Sdrwr

dVdVrr

rwrwIj

dVrwrwI

)()(

1)()(

4

)()(1

*

*

,

,

*

j : conductor index, n: order indexq: orientation index

V imd

imd

dVxrw

xrw

),(

),,(

*

(3) Applying innerproduct

(Galerkin’s method)

(2) Approximationof current density

with CMBF’s

(1) EFIEConstruction of partial impedance matrix

jnqimdpR ,,

jnqimdpL ,,

imdV

dVrwrw ijnqV iimd )()(1 *

V V jnqimd dVdVrr

rwrw'

* ''

1)'()(

4

S imdS imd SdrwrSdrwr

)()()()( **

imdqn

jnqimdpjnqqn

jnqimdpjnq VLIjRI ,

,,,

,,

ijijij VIj

)( LR

jnqimdpij R ,,R

jnqimdpij L ,,L

The global partial impedance matrix equation (N: number of conductors)

N

i

N

j

NNN

ij

N

N

j

NNN

ij

N

V

V

V

I

I

I

j

I

I

I

11

1

1111

1

111

LL

L

LL

RR

R

RR

ijR

ijL

EFIE (1)

77

Calculation of Partial Calculation of Partial ImpedancesImpedances

Partial resistancesPartial resistances– The resistance matrix is purely diagonal.The resistance matrix is purely diagonal.

Partial self inductancesPartial self inductances– Coordinates transform for angular coordinates enables Coordinates transform for angular coordinates enables

an analytic integral, and address singular points.an analytic integral, and address singular points. Partial mutual inductancesPartial mutual inductances

– Pre-computation of frequency independent integrals Pre-computation of frequency independent integrals improves efficiency of frequency-sweep simulation.improves efficiency of frequency-sweep simulation.

EFIE (2)

ij ij j i

j i

jijij

z

i

z

ijiimijjnj

V V

minjjnim

dddddzdzww

L

, ,

,*

,

,*

,,

'

1)ˆˆ(),(),(

4

''

1)'()(

4

rrzz

drdrrr

rwrw

Frequency dependentpart

Frequency independentpart

88

Equivalent CircuitEquivalent Circuit Interconnects are Interconnects are

approximated into approximated into cylindrical cylindrical conductor segment conductor segment model.model.

Cylindrical CMBF Cylindrical CMBF are used as global are used as global basis functions.basis functions.

Partial resistances Partial resistances and inductances of and inductances of are computed.are computed.

Combined Combined equivalent circuit is equivalent circuit is constructed.constructed.

“Loops” from

proximity-effect m

odes

“Branches” from skin-effect modes

EFIE (7)

99

GeometryGeometry

X

Y

10 11

5 6 7

1 2

12

Ground

1 Testing ports

m100

m30

m60

5 by 5 Via Array5 by 5 Via Array (1) (1)Examples (1)

1010104

106

108

1010

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

self

loop

indu

ctan

ce (

nH)

frequency (Hz)

L1,1

L2,2

L5,5

L6,6

L7,7

L10,10

L11,11

L12,12

104

106

108

1010

2

4

6

8

10

12

14

16x 10

-3

mut

ual l

oop

indu

ctan

ce (

nH)

frequency (Hz)

5 by 5 Via Array 5 by 5 Via Array (2)(2)

Self & mutual loop inductancesSelf & mutual loop inductances

Group 1

Group 2

Group 3

Group 4

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

G

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

1 2 G

10 11 12

5 6 7

- At low frequencies, all inductances are mainly functions of distance between lines.- At high frequencies, inductances decrease variously according different proximity effects.

Examples (2)

1111

GeometryGeometry

10 by 10 Via Array 10 by 10 Via Array (1)(1)

X

Y

m100

m25

m30

Examples (3)

1212

10 by 10 Via Array 10 by 10 Via Array (2)(2)

107 HzAll

conductors are uniformly

excited.108 Hz109 Hz

Examples (4)

1313

10 by 10 Via Array 10 by 10 Via Array (3)(3)

107 Hz108 Hz109 Hz+1 volt

-1 volt

Examples (5)

1414

GeometryGeometry

yx

Class 1

Class 2Class 3

z

y

z

102 Bonding Wires on 3 Stacked Chips 102 Bonding Wires on 3 Stacked Chips (1)(1)

Bonding wires- Material: gold- Diameter: 25 um- Pitch: 60 um

Pack

ag

e h

eig

ht

~ 1

.4 m

m

- Length Class 1: 1.26 mm Class 2: 0.82 mm Class 3: 0.48 mm

Y. Fukui et al., “Triple-Chip Stacked CSP,” Proc. IEEE ECTC, pp. 385-389, 2000.

Examples (6)

1515

Impedances at 10 GHzImpedances at 10 GHz– A wire in class 2 is grounded.A wire in class 2 is grounded.

5 10 15 20 25 300.1

0.2

0.3

0.4

0.5

0.6

0.7

resi

stan

ce (

ohm

)

location index5 10 15 20 25 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

indu

ctan

ce (

nH)

location index

102 Bonding Wires on 3 Stacked Chips 102 Bonding Wires on 3 Stacked Chips (2)(2)

Class 1

Class 2

Class 3

groundedconductor

Examples (7)

Elapsed time for 19 frequency points 81,749 secs (22.7 hrs)

(CPU: Intel Xeon 2.8 GHz, RAM: 2 GB)

Full modeling of 102 bonding wires is not available with existing tools!

1616

ConclusionsConclusions EFIE combined with the cylindrical CMBF’sEFIE combined with the cylindrical CMBF’s

– Captures skin and proximity effects with a small number Captures skin and proximity effects with a small number of basis functions.of basis functions.

– Geometrically suits various 3-D interconnects:Geometrically suits various 3-D interconnects: Wire bonds, connecter pin array, and through hole via Wire bonds, connecter pin array, and through hole via

interconnectsinterconnects Improvements in computation of partial Improvements in computation of partial

impedancesimpedances– Using analytic expressions, pre-computation for Using analytic expressions, pre-computation for

frequency-independent parts, and efficiency frequency-independent parts, and efficiency enhancement schemes.enhancement schemes.

– Large 3-D interconnect structures can be modeled with Large 3-D interconnect structures can be modeled with the proposed method.the proposed method.

Future WorkFuture Work– Inclusion of capacitive coupling and finite ground effect.Inclusion of capacitive coupling and finite ground effect.– Extension of the proposed method for modeling of TSV. Extension of the proposed method for modeling of TSV.

Conclusions (1)

1717

AcknowledgementsAcknowledgementsMixed Signal Design Tools ConsortiumMixed Signal Design Tools Consortium

(MSDT) (MSDT) at the Packaging Research Centerat the Packaging Research Center,,

Georgia Georgia Institute of Institute of TechTechnologynology

MatsushitaMatsushitaEPCOSEPCOS

InfineonInfineonSameerSameer

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