Numerical ElectroMagnetics & Semiconductor Industrial Applications

Numerical ElectroMagnetics&

Semiconductor Industrial Applications

Ke-Ying Su Ph.D.

National Central University

Department of Mathematics

10 Summary: RC extractor & ElectroMagnetic (EM) field solver

P. 2 Ke-YingSu Ph.D.

Contents

(1) Design flow & EDA tools• Methods in Raphael 2D & 3D, QRC, PeakView, Momentum, & EMX.

(2) Quasi-static-analyses (C extraction)• corss-section profile vs Green's function• process variation vs method of moment• 2D & 3D models in a RC techfile

(3) PEEC (RLK extraction)• Partial-Element-Equivalent-Circuit (PEEC)• RLK relations in spiral inductors and interconnects

(4) Full-wave analyses (S-parameter extraction)• Maxwell's equations• S-parameters from current waves

(5) Double Patterning Technology & Solution


Schematic

Pre-Layout Simulation

Place & Route Layout

Design Rule Check DRC

Layout vs Schematic LVS

RC ExtractionRC

Post-Layout Simulation

Spec.

Tape out

Yes

No

Design House

Foundry

foun

dry

supp

ort

AMD, nVidia, Qualcomm, Broadcom, MTK, etc.

TSMC, UMC, etc.

EDA

Synopsys, Cadence, Mentor, Magma, etc.

I. Design Flow:


RCLK extraction:Semiconductor industry: parasitic Capacitance (C), Resistance (R), Inductance (L) extraction.


EDA tools

C model RLCK model

Numerical

Analytical

Quasi-static analysis Full-wave analysis

2D & 3DRaphael

2D engine

2.5D RC extractor

3D QuickCap

3D EMX

3D HFSS

3D Momentum

2.5D RC extractorRL extractor

3DHelic

3D Lorentz


• integral equation in frequency domain

• Galerkin’s procedure

3D EMX

2D & 3DRaphael

• Boundary element method (BEM)

• Finite difference Method (FD)

3D QuickCap

• Laplace’s equation

• Floating random walk method

3D Lorentz

• Mixed potential integral equation

• Partial Element Equivalent Circuit (PEEC)

3D Momentum

• Method of moment

• microwave full wave mode

• faster RF quasi-static mode

QRC RC extractorRL extractor

• Partial Element Equivalent Circuit (PEEC) for RLCK extraction


2. Spectral potential function of a charge

1. Laplace’s equation

3. Matrix pencil method

4. Spectral potential function of a charge

II. Quasi-static analyses (2D & 3D)

“Complex images for electrostatic field computation in multilayered media,”Y.L. Chow, J.J.Yang, G.E.Howard, IEEE MTT vol.39, no.7, July 1991, pp.1120-1125.

“A multipipe model of general strip transmission lines for rapid convergence of integral equation singularities,”G.E.Howard, J.J.Yang, Y.L. Chow, IEEE MTT vol.40, no.4, April 1992, pp.628-636.

Cross-section of a dielectric layer

Cross-section of multi-dielectric layers

A given cross-section profile is related to a Green’s function.


6. Method of moment (Galerkin’s procedure)

)(')'()',( rVdrrfrrc ii

i

drrfrVdrdrrfrfrrc jjii

i )()(')()'()',(

for all j

-w/2+ w/2-

f1 fn

…

-w/2+ w/2-

c1f1 cnfn

…

11 jxixijxi bcA

bAci1

V

cCap i

2D model:

Capacitance per unite length (fF/um)

i

ii rfcr )()(

i

ii xfcx )()(

fi is the basis function.

charge distribution

Infinite long transmission

line

Approximated charge

distribution

Integral basis functions with above equation

become a matrix:

solve the unknown ci :

Final capacitance from charges:

)(')'()',( rVdrrrr

5. Spectral potential function

let

then

is the process variation.


3D model: capacitance (fF)

Open-end Gap discontinuity

Cross-together

))((lim 2DtotalL

oc LCLCC

“Static analysis of microstrip discontinuities using the excess charge density in the spectral domain,”J. Martel, R.R. Boix and M. Horno, IEEE MTT vol.39, no.9, Sep. 1991, pp.1625-1631.

“Microstrip discontinuity capacitances for right-angle bends, T junctions and Crossings,”P.Silvester and P. Benedek, IEEE MTT vol.21, no.5, April 1973, pp.341-347.


Models in 2.5D RC technology files


III. Partial Element Equivalent Circuit (PEEC)

1972, Albert E. Ruehli (IBM)

to solve interconnect problems on packages.

IEEE MTT, vol.42, no.9, Sep. 1994, pp.1750-1758

Project: IBM & MIT

Assume

Integral equation from Maxwell’s equations

Let

whereIi is the current inside filament i.Ii is a unit vector along the length of a filamentwi(r) is the basis function of filament i.

Filaments in a conductor for skin and proximity effects.


Define

then

where

Ex: 2 conductors

Then

l1

l2


Ex: Spiral inductor or interconnect

Self inductance

2|| aaa lll

Mutual inductance

|| dada llll

Laa > 0

Lad > 0

0ba ll Lab = 0

|| caca llll Lac < 0

Same current directions have a positive mutual inductance.

Orthogonal current directions have no mutual inductance.

Oppositive current directions have a negative mutual inductance.

Interconnect: even

la

ld

Interconnect: odd

la

lc

la

lb

lcld le

lf

Spiral inductor


Layers : M3-M2 (0.5GHz) | (K=L12/sqrt(L11*L22) ) Width Space | R L K Ctotal Cc (um) (um) | (Ohm/um) (nH/um) (fF/um) (fF/um) ---------------------+------------------------------------------------------------------- 0.08 0.08 | 3.9e+00 7.5e-04 0.69 2.1e-01 4.9e-02 0.08 0.24 | 2.0e+00 7.0e-04 0.62 2.2e-01 4.2e-02

Example: RLCK from Fast-Henry (RLK) & Raphael (C)

Layers : M3-M2 (5GHz) | (K=L12/sqrt(L11*L22) ) Width Space | R L K Ctotal Cc (um) (um) | (Ohm/um) (nH/um) (fF/um) (fF/um)---------------------+-------------------------------------------------------------------- 0.08 0.08 | 4.0e+00 7.4e-04 0.68 2.1e-01 4.9e-02 0.08 0.24 | 2.1e+00 6.9e-04 0.61 2.2e-01 4.2e-02

Layers : M3-M2 (10GHz) | (K=L12/sqrt(L11*L22) ) Width Space | R L K Ctotal Cc (um) (um) | (Ohm/um) (nH/um) (fF/um) (fF/um)---------------------+-------------------------------------------------------------------- 0.08 0.08 | 4.2e+00 7.2e-04 0.66 2.1e-01 4.9e-02 0.08 0.24 | 2.2e+00 6.7e-04 0.60 2.2e-01 4.2e-02

low frequency: uniform

Current density in a conductor cross-section

high frequency: skin effect

Frequency (GHz)

RL

RL relations vs frequency


IV. Full wave analyses (Electromagnetic field theory)

D

B

JDjH

BjE

0

x

y

Js(x,y)

JL(x,y) JL(x,y)

Jx(x,y)Jy(x,y)

Jy(x,y)

1. Spectral domain Maxwell’s equations

“Application of two-dimensional nonuniform fast Fourier transform (2-D NUFFT) technique to analysis of shielded microstrip circuits,” K.Y. Su and J.T.Kuo, IEEE MTT vol.53, no.3, March. 2005, pp.993-999.


2. Method of moment

yi

m

xi

n mny

mnx

mnyymnyx

mnxymnxx

y

x mn eeJ

J

GG

GG

yxE

yxE

),(~

),(~

),(~

),(~

),(~

),(~

),(

),(

2D-NUFFT

g1

L4 1L

L3

L2

g2 w1

w2

w2

w2

x

y

GPOF GPOF

a

b

yc

t

z

r

L5 L5

(a) |Jx(x,y)| @2.47GHz (b) |Jy(x,y)| @2.47GHz

calculate Jx & Jy


3. Calculate S parameters from currents

Let Iim be the current on the ith (i=1, 2) transmission line at the mth excitation (m=1, 2), in the regions far from the circuit and generators.

where i is the phase constant of the ith transmission line, z01 and z02 are reference planes,Iim+ and Iim- are incident and reflect current waves.

where Z01 and Z02 are characteristic impedance of the ith transmission line.


Ex. Passive devices and components

inductor capacitor RF MOS parasitic effects

“Scalable small-signal modeling of RF CMOS FET based on 3-D EM-based extraction of parasitic effects and its application to millimeter-wave amplifier design,”W.Choi, G.Jung, J.Kim, and Y.Kwon, IEEE MTT vol.57, no.12, Dec. 2009, pp.3345-3353.From Google search.From Google search.


1

23

4

5

6

7

1

2 3

4

5

67

1

2 3

4

5

67

Even circle: 2 colors

V. Double Patterning Technology & Solution

Problem

Design

Mask

2 colors decomposition

2 masks variations

Designer

Foundry

uncertain

random

Can not estimate margin

Solution

Design

RLCK networkwith

overlaySensitivity

Post-layout simulation

Monte Carlo simulation:for all possible

decomposition & variation

Designer: the worst margin to protect circuit

Foundry: the best decomposition to gain yield


Backup


Appendix

(1)

(2)

(3)

(4) Determine “M” for accuracy and efficiency.

It was developed to solve signal processing problems, but is applied to solve IC problems.

IEEE Antenna Pro. Mag, vol.37, no.1, Feb. 1995, pp.48-56


Appendix

IEEE Microwave and Guided wave, vol.8, no.1, Jan. 1998, pp.18-20

f and are finite sequences of complex numbers.Tj=2j/N, j=-N/2,…,N/2-1. wk are non-uniform.

Sjsj Sj+1Sj-1 Sj+q/2Sj-q/2

1 2 3 4 5 6 7 8 9-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

l

rl(

s j)

sj = 0.5207

q = 8

IEEE MTT vol.53, no.3, March. 2005, pp.993-999.

The (q+1)2 nonzero coefficients.

The square 2D-NUFFT

Some of these 2D coefficients approach to zero rapidly.

NUFFT : 1D 2D

Xt-1

Ys+q/2

Ys-q/2

Ys

Ys+1

Ys-1

Xt-q/2 Xt+q/2Xt+1Xt

12/

2/

12/

2/

M

Mm

N

Nn

inyimxmnst

steeGD

Numerical ElectroMagnetics & Semiconductor Industrial Applications

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