Efficient Numerical Modeling of Random Rough Surface Effects in Interconnect Internal Impedance Extraction

Transcription

1 Efficient Numerical Modeling of Random Rough Surface Effects in Interconnect Internal Impedance Extraction CHEN Quan & WONG Ngai Department of Electrical & Electronic Engineering The University of Hong Kong

2 Outline Background Modeling (Effective Parameters) Computation (Modified SIE Method) Results Conclusion

3 Surface Roughness in Interconnects 3

4 Sources of Surface Roughness Unintentional sources Technology limitations Process variations Intentional sources Electronic deposition Chemical etching Annealing Enhance the cohesion between metal and dielectric 4

5 Impact of Surface Roughness on Internal Impedance Interaction between rough surface and current Current under smooth surface Longer current path More resistive loss Larger current loop Current under rough surface Higher resistance Higher internal inductance 5

6 High Frequency Effects Rough surface effects is insignificant in low frequencies (large skin depth, small roughness) It becomes significant in high frequencies (comparable skin depth and roughness) Skin depth = 6.5microns when f = 0.1 GHz (From Intel) 6

7 Modeling Model the impact of random rough surface on interconnect internal impedance 7

8 Effective Parameters Effective Resistivity ρ e & Effective Permeability μ e Capture the increase of resistance and internal inductance caused by surface roughness Layout Information Current Extraction Tools Rough Surface Module Rough Surface RLC Circuit Effective Resistivity & Permeability 8

9 Analytical Formulation For effective resistivity ρ h RMS height; δ- skin depth Widely used in practical design, BUT Inaccurate (only h is considered) For effective permeability Unavailable e = ρ + π 1 1 tan 1.4 h δ 9

10 Numerical Formulation of ρ e Smooth surface power loss Power loss equivalence P s = P r P s = ρ H 0 δ Rough surface power loss P r * ρh = 0 Re d S l { zu ( z) } δ Pr ρδ ρ e = = Re d ( ) H S H l l 0 0 { zu z } V 10

11 Numerical Formulation of μ e Smooth surface magnetic energy W r W s μδ H Rough surface magnetic energy * μδ H = 0 Im d ( ) S { zu z } Magnetic energy equivalence W s = W r = 0 l μ e W r μδ = = Im d ( ) H l S δ H l 0 0 { zu z } V 11

12 Governing Equation S 1 y( z) G( z, z') d zg( z, z ') U ( z) = H0 + H 0 dz 1+ cp z n Green s function Gzz (, ') = H ( k z z') (1) 0 1 Surface unknown Boundary condition yz ( ) Hz ( ) U( z) = 1+ z nˆ H () r = H r S 0 1

13 Modeling of Random Rough Surface Most rough surfaces in reality are random Described by stationary stochastic process with Probability Density Function and Correlation Function 1 z P 1( y) = exp( ) π h h z1 z C ( z, z ) = exp( ) g 1 η η --- correlation length 13

14 Conventional Statistical Solver -- Monte-Carlo Method Input Layout information Random Rough Surface Generator S 1 S S 3 S n-1 S n A large number of surface realizations Direct Integral Equation Solver R 1 R R 3 R n-1 R n solutions Corresponding Computation Output Mean of the solution 14

15 Limitations of Monte-Carlo method Probabilistic nature Mean value is also a random variable Slow convergence More than 500 runs to converge within 1% Convergence of mean of ρ e (%) Number of Monte Carlo runs 15

16 Computation Efficient Stochastic Integral Equation (SIE) method 16

17 Stochastic Integral Equation (SIE) method Use mean value as unknown One-pass solution Deterministic nature Two steps Two steps Zeroth-order approximation (Uncorrelatedness assumption ) Second-order correction 17

18 S Zeroth-Order Approximation Directly applying ensemble average on both sides S 1 yz ( ) Gzz (, ') d z G( z, z') U( z) = H0 + H 0 dz 1+ cp z n Ensemble average + f( x) = P1 ( f( x)) f( x) Assuming the Green s function G and the surface unknown U are statistically independent (Uncorrelatedness Assumption) 1 yz ( ) Gzz (, ') d z G( z, z') U( z) = H0 + H 0 dz 1+ cp z n New unknown 18

19 Zeroth-order Approximation Matrix equation format Gzz (, ') AU Inaccurate zeroth-order approximation U = A L (0) 1 Cause: uncorrelatedness assumption = A Gz (, z) dyd yp( y, y) Gy (, y; z, z) = = ik i k i k i k i k i k GzzUz (, ') ( ) Gzz (, ') Uz ( ) L V U( z) = L U (0) T (0) Deterministic quantities 19

20 Primitive Second-Order Correction Improve the accuracy of the mean V Zeroth-order approximation term V = V (0) + V () () T V = trace A D ( ) Second-order correction term vec D A A A A U U (0) (0) ( ) = ( ) ( ) T ( ) F N N Limit: Also time-consuming 0

21 Computational Bottleneck High dimensional infinite integration in F 4-D infinite integration F = A A A A ik nm ik mn ik mn A A dydy dy d y P ( y, y, y, y ) G( y, y ) G( y, y ) = ik mn i k m n 4 i k m n i k m n Standard technique: Gauss Hermite quadrature Complexity grows exponentially with the integral dimension (Curse of dimensionality) 1 dimension O( N) 4 4 dimension O N ( ) 1

22 No. of Points for Gauss Hermite Quadrature 11.5 x 10 5 <A ij A mn > Gauss Hermite Accurate Value More than 30 quadrature points for 1D integration (81000 points for 4D) Number of quadrature points

23 Improved Formulation of SIE (D case) Translation invariance of Green s s function Thus G( y, y ) = G( y, y + y ) = G(0, y ) i k i i d d A G( y, y ) dydy P ( y, y ) G( y, y ) (D) = = ik i k i k i k i k A ˆ d P ˆ ik G( yd ) yd 1d ( yd) G( yd) (1D ) = = G( y, y ) = Gˆ ( y ) i j d P 1d Probability density function of y d 3

24 Partial Probability Density Function 1 y cy y + y π h 1 c h (1 c ) i i k k ( i, k) = exp P y y 1 ( ) ( ) P( yi cyi yi + yd + yi + y d yi, yd) = exp π h 1 c h (1 c ) P ( y ) = dyp ( y, y + y ) 1d d i i i d 1 y d = exp ( ) 4 h π 1- c h ( 1 c) 4

25 Improved Formulation of SIE (4D case) A A dydy dy d y P ( y, y, y, y ) G( y, y ) G( y, y ) = ik mn i k m n 4 i k m n i k m n Let y = y y y = y y d k i d n m 1 AA d d P ˆ ˆ ik mn yd yd d ( yd, yd ) Gy ( d ) Gy ( d ) Where = P d ( yd, y 1 d ) = dy d P d y 1 d 4( yi, yi + yd, y, ) 1 m ym + yd 5

26 Results 6

27 Numerical VS Analytical Formulation (Different Correlation Length) 7

28 SIE VS MIE (Mean ρ e Ratio) Gaussian surface (σ= 1μm) 8

29 SIE VS MIE (Mean μ e Ratio) Gaussian surface (σ= 1μm) 9

30 CPU Time Comparison (Unit: second) Method MIE (1500run) SIE (Original) SIE (Modified) h /δ=/ = 1 h /δ=/ = X (Gaussian rough surface --- σ=1μm, η=1 μm) 30

31 Conclusion An efficient numerical approach for modeling the impact of surface roughness on interconnect internal impedance 31

32 Conclusion 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 3

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