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UCLA Modeling and Optimization for VLSI Layout Professor Lei He lhe@ee.ucla.edu http://eda.ee.ucla.edu/

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n Programming homework n Last lecture: Placement n Today: Wrap up placement Interconnect modeling n Student presentation: April 29 th, Thermal modeling (by Mehul Shah) May 2 nd, Dynamic and leakage power modeling (Phoebe and Qun) n Read: Three papers on interconnect modeling Especially [Xu-He’01] (checked on May 2 nd )

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Chapter 5 Interconnect RLC Modeling n Table and formula based capacitance extraction n Table and formula based inductance extraction n RC or RLC circuit model generation n Numeric based interconnect modeling

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Capacitance Extraction n Introduction n Table lookup method n Formula-based method

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What’s Capacitance? n Simplest model: parallel-plate capacitor It has two parallel plates and homogeneous dielectric between them The capacitance is permittivity of dielectric A area of plate ddistance between plates The capacitance is the capacity to store charge charge at each plate is one is positive, the other is negative ++ - - +Q -Q

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General Picture n For multiple conductors of any shapes and materials, and in any dielectric, there is a capacitance between any two conductors m1m1 m3m3 m2m2 c 23 c 13 c 12 n Mutual capacitance between m1 and m2 is C 12 = q 1 /v 2 q 1 is the charge of m1 v 1 =0 and v 3 = 0

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Capacitance Matrix n Capacitance is often written as a symmetric matrix m1m1 m3m3 m2m2 c 23 c 13 c 12 C = -c 21 c 22 -c 23 -c 31 -c 32 c 33 c 11 -c 12 -c 13 n is the self-capacitance for a conductor e.g., c 11 =c 12 +c 13 n The charge is given by e.g.,

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Application in VLSI Circuits n Total cap for a wire delay, power n Mutual cap between wires signal integrity n Conductors: metal wire, via, polysilicon, substrate n Dielectrics: SiO 2,...

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Characteristics of Coupling Capacitance n Coupling capacitance virtually exists only between adjacent wires or crossing wires Cx Cx Cx Capacitance can be pre-computed for a set of (localized) interconnect structures

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2.5D Capacitance Extraction [Cong-He-Kahng-et al, DAC’97] Propose and validate five foundations to simplify capacitance extraction Develop a simple yet accurate 2.5D capacitance extraction

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Verification of Foundations n Method: 3D analysis by FastCap [Nabors-White, TCAD’91] n Geometrical parameters: 0.18 process [NTRS’94] 3.0 2.5 1.0

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Key Factor to Enable Foundations n Minimum metal density requirement Metals occupy > 30% area on anywhere on routing layer Foundry may introduce dummy metals for metal sparse areas dummy metal

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Foundation I Effect of Ground and Neighbors Both ground, and neighboring wires on the same layer, have significant shielding effects. Thus, both must be considered for accurate modeling.

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Shielding Effect of Ground and Neighbors layer i layer i-2 lumped capacitance for victim on layer i coupling between victim and aggressor on layer i-2 C i,i C i,i-2 C i,i C i,i-2 no GND458.4130.1(28.4%)

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Shielding Effect of Ground and Neighbors layer i layer i-2 lumped capacitance for victim on layer i coupling between victim and aggressor on layer i-2 C i,i C i,i-2 C i,i C i,i-2 no GND + GND 458.4130.1(28.4%) 486.679.49(16.3%)

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Shielding Effect of Ground and Neighbors layer i layer i-2 lumped capacitance for victim on layer i coupling between victim and aggressor on layer i-2 C i,i C i,i-2 C i,i C i,i-2 no GND + GND 458.4130.1(28.4%) 486.679.49(16.3%) + neighbors142824.77(1.8%)

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Foundation II Coupling between Layers i and i-2 Coupling between wires on layer i and wires on layers i-2 is negligible when the metal density on layer i exceeds a certain threshold.

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Coupling between Layers i and i-2 layer i layer i-1 layer i-2 lumped capacitance for victim on layer i coupling between victim and aggressor on layer i-2 C i,i C i,i-2 486.6534.5581.3622.2635.9 79.4948.4521.993.472.47 C i,i C i,i-2 --2x4x8x12x

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Foundation III Coupling Effect of Layers i+2 and i-2 During capacitance extraction for wires on layer i, layers i+2 and i-2 can be treated as ground planes with negligible error. There is no need to look beyond layers i+2 and i-2.

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Coupling Effect of Layers i+2 and i-2 i i-1 i-2 i+1 i+2 layer lumped capacitance for victim on layer i coupling between victim and central crossover on layer i+1 C i,i C i,i+1 coupling between victim and central crossunder on layer i-1 C i,i-1 418.9 52.35 C i,i C i,i+1 52.26 C i,i-1

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Coupling Effect of Layers i+2 and i-2 i i-1 i-2 i+1 i+2 layer lumped capacitance for victim on layer i coupling between victim and central crossover on layer i+1 C i,i C i,i+1 coupling between victim and central crossunder on layer i-1 C i,i-1 418.9 52.3552.59 C i,i C i,i+1 52.2652.53 C i,i-1

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Foundation IV Coupling Effect of Neighbors Coupling analysis to wires on the same layer need only consider nearest neighbors independently, with the widths of same- layer neighbor wires having negligible effect on the coupling.

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Effect of Non-immediate Neighbors victim C i,i :lumped capacitance for victim. layer i ClCl CrCr C i,i 1436 C l 616.6 C r 616.5

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Effect of Non-immediate Neighbors victim C i,i :lumped capacitance for victim. layer i ClCl CrCr ClCl CrCr C i,i 14361436(0%) C l 616.6639.8(+3%) C r 616.5639.5(+3%)

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Effect of Neighbor Widths layer i victim w w C i,i 764.5765.2764.9764.4 W1234W1234 C i,i varies less that 0.3% for different neighbor widths.

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S2 S1S2 2 Independence of Neighbors C i,i differs less than 1.0%. (S1,S2)(1,2)(1,3)(1,4)(1, ) lhs639.2600.0582.5559.7 rhs638.0597.1578.9553.1 victim S1

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Foundation V Interaction between Layers i-1 and i+1 The joint interaction of layers i-1 and i+1 on layer i is negligible; therefore, corrections for orthogonal crossovers and crossunders can be performed independently.

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i i-1 i+1 layer i+2 couplings between victim and crossunders No crossover 36.0137.2036.9937.58 Independence of Crossovers and Crossunders

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i i-1 i+1 layer i+2 couplings between victim and crossunders No crossover 36.0137.2036.9937.58 Full of crossovers Independence of Crossovers and Crossunders

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i i-1 i+1 layer i+2 couplings between victim and crossunders No crossover 36.0137.2036.9937.58 Full of crossovers35.6636.8736.7037.35

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Table (Cap coefficients) generation One-time use of 3-D method Capacitance computation table lookup with linear interpolation and extrapolation Table-Based 2.5D Capacitance Extraction

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Table Generation for Lateral, Area and Fringe Capacitances layer i w ss Functions of (w,s) Pre-computed for per-side per unit-length

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Table Generation for Crossing Capacitances layer i Function of (w,s,w c,s c ) w s scsc wcwc scsc

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C i,i Per-corner C over (w,s,w c,s c ) = 4 s scsc wcwc s scsc wcwc scsc w w Table Generation for Crossing Capacitances

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Illustration of Capacitance Computation victim Compute the lumped cap for victim

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victim Find Nearest Neighbors on Same Layer

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victim w S1 L1 Add in Per-Side Area, Fringe and Lateral Capacitances Per-side area capacitance = C A (w,s 1 ) * L1 Per-side fringe capacitance = C F (w,s 1 ) * L1 Per-side lateral capacitance = C L (w,s 1 ) * L1

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victim Add in Per-Side Area, Fringe and Lateral Capacitances

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victim Find All Crossovers and Crossunders

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wcwc scsc S1 victim w Add in Crossing Capacitances Corner-by-Corner One-corner crossover correction = C over (w, S1,w c,s c )

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wcwc S1 victim w Add in Crossing Capacitances Corner-by-Corner One-corner crossover correction = C over (w, S1,w c, )

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wcwc S1 victim w Add in Crossing Capacitances Corner-by-Corner One-corner crossover correction = C over (w, ,w c, )

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wcwc scsc victim w Add in Crossing Capacitances Corner-by-Corner One-corner crossover correction = C over (w, ,w c,s c )

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Summary of Capacitance Computation Find nearest neighbors on the same layer Add in per-side lateral, area and fringe capacitances w.r.t. each neighbor Find all crossovers and crossunders Add in crossing capacitances corner-by-corner w.r.t. each crossover and crossunder Sum of capacitance components in above steps is the lumped capacitance of the victim.

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Experimental Results 2 1/2-D3-DError net16.53552pF6.5713pF-0.54% net23152.42pF3261.17pF-3.33% Good match in terms of lumped capacitance!

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Formula based on horizontal and vertical parameters n [Sakurai-Tamaru,ED’83][Wu-Wong-et al, ISCAS’96] single line parallel lines …...

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Single Line [Sakurai-Tamaru,ED’83] w FpFp FfFf FfFf t h n Unit-length cap n Error less than 6% when

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Single Line of Length L [Sakurai-Tamaru,ED’83] w t h n Line of length L

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Parallel Lines on Same Layer [Sakurai-Tamaru,ED’83] w w s t h n Unit-length cap n Error less than 10% when

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Parallel Lines on Same Layer [Wu-Wong-et al, ISCAS96] w w s t h n Unit-length cap n Recall [Sakurai-Tamaru,ED’83] w s

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Comparison n n [Wu-Wong-et al] is better in smaller width and spacing numerical Wu-Wong-et al Sakurai Noramlzied space (s/h) Normalized cap (C/ ) 3 4 5 6 1 2 3 4 15 W=1.05um W=0.7um

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Parallel Lines within Two Grounds [Wu-Wong-et al, ISCAS96] w w s t h1h1 n One ground w s h1h1 n Two grounds where

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