Efficient Numerical Modeling of Random Rough Surface Effects

in Interconnect Internal Impedance Extraction

CHEN CHEN QuanQuan & WONG & WONG NgaiNgai

Department of Electrical & Electronic EngineeringDepartment of Electrical & Electronic Engineering

The University of Hong KongThe University of Hong Kong

2

OutlineOutlineBackground

Modeling (Effective Parameters)

Computation (Modified SIE Method)

Results

Conclusion

3

Surface Roughness in InterconnectsSurface Roughness in Interconnects

4

Sources of Surface RoughnessSources of Surface RoughnessUnintentional sourcesUnintentional sources–– Technology limitationsTechnology limitations–– Process variationsProcess variations

Intentional sourcesIntentional sources–– Electronic deposition Electronic deposition –– Chemical etchingChemical etching–– AnnealingAnnealing

Enhance the cohesion between metal and Enhance the cohesion between metal and dielectricdielectric

5

Impact of Surface Roughness on Internal Impedance

Impact of Surface Roughness on Internal Impedance

Current under smooth surface Current under rough surface

More resistive lossLonger

current pathLargercurrent loop

Higher internal inductance

Higher resistance

Interaction between rough surface and currentInteraction between rough surface and current

6

High Frequency EffectsHigh Frequency Effects•• Rough surface effects is insignificant in low Rough surface effects is insignificant in low frequencies frequencies (large skin depth, small roughness)(large skin depth, small roughness)

•• It becomes significant in high frequencies It becomes significant in high frequencies (comparable skin depth and roughness)(comparable skin depth and roughness)

Skin depth = 6.5microns when f = 0.1 GHz (From Intel)

7

ModelingModeling

Model the impact of random rough surface on interconnect internal

impedance

8

Effective Resistivity Effective Resistivity ρρee & Effective Permeability & Effective Permeability μμee

Capture the increase of resistance and internal Capture the increase of resistance and internal inductance caused by surface roughnessinductance caused by surface roughness

Current Extraction Tools

Layout Information

Rough Surface Module

Effective Resistivity & PermeabilityRough Surface

RLC Circuit

Effective ParametersEffective Parameters

9

Analytical FormulationAnalytical FormulationFor effective resistivity

–– h h –– RMS height; RMS height; δδ-- skin depthskin depth

–– Widely used in practical design, BUTWidely used in practical design, BUT

–– Inaccurate Inaccurate (only h is considered)(only h is considered)

For effective permeability–Unavailable

2121 tan 1 .4e

hρ ρπ δ

−⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

10

Numerical Formulation of ρeNumerical Formulation of ρe

Power loss equivalence Ps = Pr

{ }0

20

Re d ( )2 rSe zU z

HP

H l lρδδρ == ∫

Smooth surface power loss2

0

2s

H lP

ρδ

=

Rough surface power loss

( ){ }*0 Re d

2r S

HP zU zρ= ∫

V

11

Numerical Formulation of μeNumerical Formulation of μe

{ }2 *

0 Im d ( )2r S

HW zU zμδ= ∫

{ }200

Im d ( )2 rSe

WH

zUlH

zl

μδμδ

== ∫

20

2s

H lW

μδ=

Smooth surface magnetic energy

Rough surface magnetic energy

Magnetic energy equivalence Ws = Wr

V

12

Governing EquationGoverning Equation2

0 01 ( ) ( , ')d ( , ') ( ) d 12S cp

y z G z zzG z z U z H H zz n

∂ ∂⎛ ⎞= + + ⎜ ⎟∂ ∂⎝ ⎠∫ ∫

2( ) ( )( ) 1ˆ

y z H zU zz n

∂ ∂⎛ ⎞= + ⎜ ⎟∂ ∂⎝ ⎠

0( )H r H r S= ∈Boundary condition

(1)0 1( , ') ( ' )G z z H k z z= −

Surface unknown

Green’s function

13

Modeling of Random Rough SurfaceModeling of Random Rough Surface

Described by Described by stationary stationary stochastic processstochastic process with with Probability Density Probability Density FunctionFunction and and Correlation FunctionCorrelation Function

2

1 2

1P ( ) exp( )22zyhhπ

= −

η --- correlation length

21 2

1 2 2( , ) exp( )g

z zC z z

η−

= −

Most rough surfaces in reality are randomMost rough surfaces in reality are random

14

Conventional Statistical Solver-- Monte-Carlo Method

Conventional Statistical Solver-- Monte-Carlo Method

Input Layout information

Random Rough Surface Generator

Direct Integral Equation Solver

Computation Output Mean of the solution

S1 S2 S3 Sn-1 SnA large number of

surface realizations

R1 R2 R3 Rn-1 Rn

Corresponding solutions

15

Limitations of Monte-Carlo methodLimitations of Monte-Carlo methodProbabilistic natureProbabilistic nature–– Mean value is also a random variableMean value is also a random variable

Slow convergenceSlow convergence–– More than 2500 More than 2500

runs to converge runs to converge within 1%within 1%

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Number of Monte−Carlo runs

Co

nve

rgen

ce o

f m

ean

of ρ

e (%

)

16

ComputationComputationEfficient Stochastic Integral Equation

(SIE) method

17

Stochastic Integral EquationStochastic Integral Equation (SIE)(SIE) methodmethod–– Use mean value as unknownUse mean value as unknown–– OneOne--pass solution pass solution –– Deterministic natureDeterministic nature

Two stepsTwo steps– Zeroth-order approximation

(Uncorrelatedness assumption )– Second-order correction

18

Zeroth-Order ApproximationZeroth-Order ApproximationDirectly applying ensemble average on both sides

2

0 01 ( ) ( , ')d ( , ') ( ) d 12S cp

y z G z zz G z z U z H H zz n

∂ ∂⎛ ⎞= + + ⎜ ⎟∂ ∂⎝ ⎠∫ ∫

1( ) ( ( )) ( )f x P f x f x+∞

−∞= ∫Ensemble average

2

0 01 ( ) ( , ')d ( , ') ( ) d 12S cp

y z G z zz G z z U z H H zz n

∂ ∂⎛ ⎞= + + ⎜ ⎟∂ ∂⎝ ⎠∫ ∫

Assuming the Green’s function G and the surface unknown U are statistically independent (Uncorrelatedness Assumption)

New unknown

19

Inaccurate zeroth-order approximation

Zeroth-order ApproximationZeroth-order ApproximationMatrix equation format

AU L=

(0) 1U A L−=Cause: uncorrelatedness assumption

( , ') ( ) ( , ') ( )G z z U z G z z U z≠

(0) (0)TV L U=

2( , ) d d ( , ) ( , ; , )ik i k i k i k i k i kA G z z y y P y y G y y z z= = ∫∫

Deterministic quantities

( , ')G z z

( )U z

20

Primitive Second-Order CorrectionPrimitive Second-Order CorrectionImproveImprove the accuracy of the mean Vthe accuracy of the mean V

(0) (2)V V V= +Second-order

correction term(2) ( )TV trace A D−=

(0) (0)( ) ( ) ( ) ( )Tvec D A A A A U U= − ⊗ − ⊗

Zeroth-order approximation term

Limit: Also timeLimit: Also time--consumingconsuming

2 2N NF

×

21

Computational BottleneckComputational BottleneckHigh dimensional infinite integration in High dimensional infinite integration in FF

iknm ik mn ik mnF A A A A= −

4d d d d ( , , , ) ( , ) ( , )ik mn i k m n i k m n i k m nA A y y y y P y y y y G y y G y y= ∫∫ ∫∫Standard technique: Gauss Hermite quadrature– Complexity grows exponentially with the integral

dimension (Curse of dimensionality)

4-D infinite integration

( )1 dimension O N− →( )44 dimension O N− →

22

5 10 15 20 25 307

7.5

8

8.5

9

9.5

10

10.5

11

11.5x 10

−5

Number of quadrature points

|<A

ijAm

n>|

Gauss HermiteAccurate Value

More than 30quadrature points for 1D integration

(81000 points for 4D)

No. of Points for Gauss Hermite QuadratureNo. of Points for Gauss Hermite Quadrature

23

Improved Formulation of SIE (2D case)Improved Formulation of SIE (2D case)Translation invariance of GreenTranslation invariance of Green’’s functions function

2( , ) d d P ( , ) ( , )) (2Dik i k i k i k i kA G y y y y y y G y y= = ∫∫

( , ) ( , ) (0, )i k i i d dG y y G y y y G y= + =

1dˆd P ( ) ( ) (1( ) )ˆ Dd d dik d y y G yA G y= = ∫

ˆ( , ) ( )i j dG y y G y=

PP1d 1d –– Probability density function of yProbability density function of ydd

ThusThus

24

Partial Probability Density FunctionPartial Probability Density Function

2 2

2 2 22 2

2 ( ) ( )1P ( , ) exp2 (1 )2 1

i i i d i di d

y cy y y y yy yh ch cπ

⎛ ⎞− + + += −⎜ ⎟−− ⎝ ⎠

2 2

2 2 22 2

21( , ) exp2 (1 )2 1

i i k ki k

y cy y yP y yh ch cπ

⎛ ⎞− += −⎜ ⎟−− ⎝ ⎠

( ) ( )

1d 2

2

2

P ( ) d P ( , )

1 exp4 12 1

d i i i d

d

y y y y y

yh ch cπ

= +

⎛ ⎞−= ⎜ ⎟⎜ ⎟−⎝ ⎠

∫

－

25

Improved Formulation of SIE (4D case)Improved Formulation of SIE (4D case)

4d d d d ( , , , ) ( , ) ( , )ik mn i k m n i k m n i k m nA A y y y y P y y y y G y y G y y= ∫∫ ∫∫

1 1 1 2 1 22dˆ ˆd d P ( , ) ( ) ( )d d d d d dik mn y y y yA A G y G y= ∫∫

1 2d k i d n my y y y y y= − = −LetLet

1 2 1 2 1 22d 4P ( , ) d d P ( , , , )d d d d i i d m m dy y y y y y y y y y= + +∫WhereWhere

26

ResultsResults

27

Numerical VS Analytical Formulation(Different Correlation Length)

Numerical VS Analytical Formulation(Different Correlation Length)

28

SIE VS MIE (Mean ρe Ratio)SIE VS MIE (Mean ρe Ratio)

Gaussian surface (σ= 1μm)

29

SIE VS MIE (Mean μe Ratio)SIE VS MIE (Mean μe Ratio)

Gaussian surface (σ= 1μm)

30

CPU Time Comparison (Unit: second)

CPU Time Comparison (Unit: second)

MethodMethodh /h /δδ= 1= 1 h /h /δδ= 2= 2

MIEMIE

(1500run)(1500run)7160.37160.3 14484.714484.7

SIESIE

(Original)(Original)6367.36367.3 6544.76544.7

SIESIE

(Modified)(Modified)198.7198.7 200.3200.3

(Gaussian rough surface (Gaussian rough surface ------ σσ=1=1μμm, m, ηη=1 =1 μμm)m)

32X

31

Conclusion

An efficient numerical approach for modeling the impact of surface

roughness on interconnect internal impedance

Conclusion

An efficient numerical approach for modeling the impact of surface

roughness on interconnect internal impedance

32

Numerical effective parameters–Model the rough surface effects on internal

impedance –Take all statistical information into account

Modified SIE method–One-pass solution for mean values–Halve infinite integral dimension by partial

PDF formulation

ConclusionConclusion

33