UCLA Modeling and Optimization for VLSI Layout Professor Lei He

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 )

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

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

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

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

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.,

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,...

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

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

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

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

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.

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 (28.4%)

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 (28.4%) (16.3%)

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 (28.4%) (16.3%) + neighbors (1.8%)

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.

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 C i,i C i,i-2 --2x4x8x12x

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.

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 C i,i C i,i C i,i-1

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 C i,i C i,i C i,i-1

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.

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

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

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

S2   S1S2 2 Independence of Neighbors C i,i differs less than 1.0%. (S1,S2)(1,2)(1,3)(1,4)(1,  ) lhs rhs victim S1

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.

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

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

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

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

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

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

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

Illustration of Capacitance Computation victim Compute the lumped cap for victim

victim  Find Nearest Neighbors on Same Layer

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

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

victim  Find All Crossovers and Crossunders

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

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

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

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

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.

Experimental Results 2 1/2-D3-DError net pF6.5713pF-0.54% net pF pF-3.33% Good match in terms of lumped capacitance!

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

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

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

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

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

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/  ) W=1.05um W=0.7um

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