Wire Sizing with Scattering Effect for Nanoscale Interconnection

Transcription

1 Wire Sizing with Scattering Effect for Nanoscale Interconnection Sean X. Shi and David Z. Pan Dept. of Electrical & Computer Engineering University of Texas at Austin

2 Outline Introduction Model of Scattering Effects Wire Delay & Sizing with Scattering Effects Conclusion

3 Interconnect Delay Dominates 3 (source: ITRS 003)

4 What is Wire Sizing? WS is an effective way to reduce distributed RC delay Continuous wire shaping [Fishburn+, TCAD 96, DATE97; Chen+, DAC 96, DAC 97] Discrete wire sizing [Cong-Leung, ICCAD 93] Rd C L C L -width sizing (-WS) [Cong-Pan, DAC 99] w opt C L -width sizing (-WS) [Cong-Pan, DAC 99] 4 w w l l C L

5 What is Scattering Effect? Mesoscopic scale effect for interconnect Example: surface roughness 5 Dunn and Kaloyeros, (000)

6 Two Kinds of Scattering Effects Surface Roughness Effect Wetting and nucleation of Cu FS model: [Fuchs, 938] and [Sondheimer, 95] Grain Boundaries Effect Polycrystalline structure of Cu MS model: Mayadas and Shatzkes [970] Electrical impacts Electron movement will be bumpy Higher resistivity than bulk metal Complicated quantum mechanical effect to model them Dunn and Kaloyeros, (000) 6

7 Introduction Outline A Simple Scattering Effect Model Wire Delay & Sizing with Scattering Effects Conclusion 7

8 Model of Scattering Effect Modeling can be very complicated λ: bulk mean free path (39nm) l ρ( ) ρ α( ) β(, ) ρ( ) ρ w = 0 + w + h w R = w 0 w t w t w exp t w arctan( wt) π 3 λcos( θ) cos( ϕ) = dy dx dϕ sin ( θ) cos ( θ) ( p ) dθ β( tw, ) 4πtw t w π + arctan( w t) 0 w p exp λcos( θ) cos( ϕ) arctan( wt) t w π 3 + dy dx dϕ sin θ cos θ p 4π tw t w arctan( w t) 0 ( ) ( ) ( ) 8 l w exp λcos( θ) cos( ϕ) dθ w p exp λcos( θ) cos( ϕ ) [Durkan and Welland 000]

9 A Simple Model for Scattering Effect Based on the published measurement data from various sources, we obtain the following empirical resistivity model with scattering effect ρ w = + ( ) ρb Resistivity is wire-width dependent Kρ is an empirically fitting coefficient K ρ w 9

10 Simple but Capture the Essence 5 Experimental Data Fitting Results Experimental Data Fitting Line Resistivity [μω cm] 4 3 ρ B =.0 (μω cm) K ρ =.03 (kω nm ) Resistivity [μω cm] 8 4 ρ=.5967; k= Wire Width [nm] Thickness [nm] Infineon [Steinhoegl+ 00] IBM [Rossnagel+ 004] Comparison with measurement data 0

11 Confirmation from ITRS 05 Our simple model matches the newest ITRS 05 (just released at public.itrs.net)

12 Outline Introduction Model of Scattering Effect Wire Delay & Sizing with Scattering Effects Conclusion

13 Wire Delay with Scattering Delay of minimum-width wire Real delay (with scattering) may be up to 3x of that without scattering T WS = R c l w+ c l+ C d a f L T Wmin /T Wmin,NoS ca l w cf l Kρ CL ρb + w l wt Wire Length [mm] nm 3nm 45nm 65nm 90nm 3

14 Wire Sizing is More Effective under Scattering Wire sizing sensitivity (effectiveness) -6x more effective T NonScattering w (, ) ( w, l) l l = [ Rcl] + c + C t t w d a f ρ Lρ T w l w = [ Rcl] + c + C t t w 4 (dt scattering /dw) / (dt nonscattering /dw) w / w min nm 3nm 45nm 65nm l l l l l d a f ρb LρB ck a ρ c f + CL Kρ 3 tw t t w

15 New Wire Sizing Function Delay of a single-width wire sizing Optimal width with width-dependent resistivity ca l w cf l Kρ l T WS = Rd ca l w+ cf l+ C L CL ρb + w w t T w= 0 WS a θ woptimal = w = cos w 3 3 min ckl cρ l C ρ K ( c l C ) 54R c t where a + + +, cos R c t ( ckl ) a ρ + cfρbl+ CLρB a ρ f B L B ρ f L d a = θ = 3 d a 5

16 Optimal Wire Sizing with and without Scattering Previous works with fixed resistivity Either too small (bulk) Or too big (conservative) An example of fixed (not width-dependent) resistivity The optimal wire size will be overestimated (by 0 x min width) This will cause area waste and routability problem Can also be undersized (w NoS -w OPT )/w MIN wire length [mm] nm 3nm 45nm 65nm 90nm Comparison of optimal wire sizing (normalized by min width) 6

17 Delay Reduction with WS The optimal wire sizing delay reduction increases as technology scales (more scattering effects) 7

18 New Model for Wire shaping Rd Revisit the wire shaping problem [Fishburn 97, Chen 97] L L x ρ K B ρ dx T = Rd ca dx f ( x) + cf dx+ C L + CL + ( ca f ( l) + cf ) dl + f ( x) f ( x) t d L L 0 (,, ) = R C + F x u u dx Euler s Differential Equation = x ( ( ) ( )) if: I F x, u x, u x dx is minimized, x0 d then: Fu xu, x, u x F xu, x, u x dx ( ( ) ( )) = u ( ) ( ) 8 ( ) C L

19 Polynomial Expansions ( ) ( ) f x = u x = na x = a + a x+ 3a x + 4a x + L a a a a min 3 4 = w n= n n ( ρ 3 ρ) ( ρ ρ) 4 CL ( ρb a + 3Kρ ) ca a B a+ K + cf a B a+ K = ( ρ ) a c a a c K C ρ a = a B f ρ L B 3 CL ( 3Kρ + ρb a ) ( 6 3 ) 9 CL ( 36Kρ + ρb a ) ( ) ( ρ ρ ) ca ρb a a + a a K a a = 3 ρ 3 cf ρb a + 3 a a3 + 5 Kρ a 3 CL 8 B a a3 C 36K + a C 36K + a L 9 [ ρ ] ( ρ ρ ) L B L B

20 Wire Shape f(x) Expansion ( ) g x = a ; ( ) = + ( + 3 ) 3 4 ( ) = ( ) g x a a x a x ; g3 x a a x a x a x a x ; L gn x a x a x a x n 3 i i 3 i ( ) = i + ( i + i ) i= Wire Width [nm] g g g3 g4 g Wire Length [μm] Fig. 6 Wire shaping with different order polynomial approximation g 3 (x) is good enough g 3 (x) is different from [Fishburn 97] Can be used for delay estimation modeling 0 Delay [ps] Polynomial Order

21 Conclusion Scattering effect should be considered for interconnect of nanoscale/mesoscale IC design The most recently released ITRS 05 has a lot of coverage on scattering effect A semi-empirical model is developed It fits the measurement data well Matches ITRS 05 Suitable for interconnection optimization Wire delay & sizing impacts are studied Future work: Variational modeling (surface roughness )