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- National Central University
- Department of Mathematics

Numerical ElectroMagnetics Semiconductor

Industrial Applications

10 Summary RC extractor ElectroMagnetic (EM)

field solver

Ke-Ying Su 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

I. Design Flow

Design House

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

Design Rule Check DRC

Spec.

Layout vs Schematic LVS

foundry support

Schematic

EDA

Synopsys, Cadence, Mentor, Magma, etc.

Pre-Layout Simulation

RC Extraction RC

Post-Layout Simulation

Place Route Layout

No

Yes

Tape out

Foundry

TSMC, UMC, etc.

RCLK extraction

Semiconductor industry parasitic Capacitance

(C), Resistance (R), Inductance (L) extraction.

EDA tools

C model

RLCK model

Quasi-static analysis

Full-wave analysis

Analytical

2D 3D Raphael

3D EMX

2D engine

2.5D RC extractor

3D Momentum

2.5D RC extractor RL extractor

3D Lorentz

3D Helic

3D QuickCap

3D HFSS

Numerical

3D Momentum

3D EMX

- integral equation in frequency domain
- Galerkins procedure

- Method of moment
- microwave full wave mode
- faster RF quasi-static mode

3D Lorentz

QRC RC extractor RL extractor

- Mixed potential integral equation
- Partial Element Equivalent Circuit (PEEC)

- Partial Element Equivalent Circuit (PEEC) for

RLCK extraction

2D 3D Raphael

3D QuickCap

- Boundary element method (BEM)
- Finite difference Method (FD)

- Laplaces equation
- Floating random walk method

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

Cross-section of a dielectric layer

1. Laplaces equation

2. Spectral potential function of a charge

Cross-section of multi-dielectric layers

3. Matrix pencil method

4. Spectral potential function of a charge

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.

? A given cross-section profile is related to a

Greens function.

2D model Capacitance per unite length (fF/um)

5. Spectral potential function

Infinite long transmission line

let

then

charge distribution

6. Method of moment (Galerkins procedure)

Integral basis functions with above equation

for all j

d is the process variation.

become a matrix

fi is the basis function.

solve the unknown ci

Approximated charge distribution

Final capacitance from charges

3D model capacitance (fF)

Open-end

Gap

discontinuity

Cross-together

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. Micro

strip 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

Integral equation from Maxwells equations

Assume

Let

where Ii is the current inside filament i. Ii

is a unit vector along the length of a

filament wi(r) is the basis function of filament

i.

Filaments in a conductor for skin and proximity

effects.

Then

Define

then

where

Ex 2 conductors

Ex Spiral inductor or interconnect

Self inductance

?

Laa gt 0

Mutual inductance

?

Lad gt 0

Same current directions have a positive mutual

inductance.

?

Lab 0

Orthogonal current directions have no mutual

inductance.

?

Lac lt 0

Oppositive current directions have a negative

mutual inductance.

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

Layers M3-M2 (0.5GHz)

(KL12/sqrt(L11L22) ) Width Space

R L K

Ctotal Cc (um) (um) (Ohm/um)

(nH/um) (fF/um) (fF/um)

------------------------------------------------

---------------------------------------- 0.08

0.08 3.9e00 7.5e-04 0.69

2.1e-01 4.9e-02 0.08 0.24

2.0e00 7.0e-04 0.62 2.2e-01 4.2e-02

Layers M3-M2 (5GHz)

(KL12/sqrt(L11L22) ) Width Space

R L K Ctotal

Cc (um) (um) (Ohm/um) (nH/um)

(fF/um) (fF/um) ---------------

-------------------------------------------------

------------------------- 0.08 0.08

4.0e00 7.4e-04 0.68 2.1e-01

4.9e-02 0.08 0.24 2.1e00

6.9e-04 0.61 2.2e-01 4.2e-02

Layers M3-M2 (10GHz)

(KL12/sqrt(L11L22) ) Width Space

R L K

Ctotal Cc (um) (um) (Ohm/um)

(nH/um) (fF/um)

(fF/um) -----------------------------------------

------------------------------------------------

0.08 0.08 4.2e00 7.2e-04

0.66 2.1e-01 4.9e-02 0.08 0.24

2.2e00 6.7e-04 0.60 2.2e-01

4.2e-02

IV. Full wave analyses (Electromagnetic field

theory)

1. Spectral domain Maxwells 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

2D-NUFFT

? calculate Jx Jy

(a) Jx(x,y) _at_2.47GHz

(b) Jy(x,y) _at_2.47GHz

3. Calculate S parameters from currents

Let Iim be the current on the ith (i1, 2)

transmission line at the mth excitation (m1, 2),

in the regions far from the circuit and

generators.

where bi 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.

V. Double Patterning Technology Solution

Even circle 2 colors

Problem

Solution

Can not estimate margin

Design

Designer

Design

Designer the worst margin to protect circuit

Mask

Foundry

RLCK network with overlay Sensitivity

Foundry the best decomposition to gain yield

2 colors decomposition

uncertain

Post-layout simulation

Monte Carlo simulation for all possible

decomposition variation

2 masks variations

random

Backup

Appendix

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

(1)

(2)

(3)

(4)

Determine M for accuracy and efficiency.

Appendix

IEEE Microwave and Guided wave, vol.8, no.1, Jan.

1998, pp.18-20

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

The square 2D-NUFFT

NUFFT 1D ? 2D

Some of these 2D coefficients approach to zero

rapidly.

f and a are finite sequences of complex

numbers. Tj2pj/N, j-N/2,,N/2-1. wk are

non-uniform.

The (q1)2 nonzero coefficients.