Numerical ElectroMagnetics

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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.
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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

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

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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.
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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
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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.
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Models in 2.5D RC technology files
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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.
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Then
Define
then
where
Ex 2 conductors
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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.
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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
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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.
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2. Method of moment
2D-NUFFT
? calculate Jx Jy
(a) Jx(x,y) _at_2.47GHz
(b) Jy(x,y) _at_2.47GHz
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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.
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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.
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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
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Backup
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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.
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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.