0. Numerical ElectroMagnetics & Semiconductor Industrial Applications
National Central University
Department of Mathematics
Numerical ElectroMagnetics & Semiconductor Industrial Applications
10 Summary:
RC extractor & ElectroMagnetic (EM) field solver
Ke-Ying Su Ph.D.

1. 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

2. Layout vs Schematic LVS Pre-Layout Simulation Post-Layout Simulation
I. Design Flow:
Design House
AMD, nVidia, Qualcomm, Broadcom, MTK, etc.
Design Rule Check DRC
Spec.
Layout vs Schematic LVS
foundry support
EDA
Schematic
Synopsys, Cadence, Mentor, Magma, etc.
Pre-Layout Simulation
RC Extraction RC
Post-Layout Simulation
Place & Route Layout No
Yes
Foundry
Tape out
TSMC, UMC, etc. 2

3. RCLK extraction:
Semiconductor industry: parasitic Capacitance (C), Resistance (R), Inductance (L) extraction. 3

4. Quasi-static analysis
EDA tools
C model
RLCK model
Quasi-static analysis
Full-wave analysis
Analytical
2D & 3D
Raphael 3D
EMX 2D
engine
2.5D
RC extractor
2.5D
RC extractor
RL extractor 3D
Momentum 3D
Lorentz 3D
Helic 3D
QuickCap 3D
HFSS
Numerical

5. integral equation in frequency domain Galerkin’s procedure
Momentum 3D
EMX
integral equation in frequency domain
Galerkin’s procedure
Method of moment
microwave full wave mode
faster RF quasi-static mode
QRC
RC extractor
RL extractor 3D
Lorentz
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)
Laplace’s equation
Floating random walk method

6. II. Quasi-static analyses (2D & 3D)
Cross-section of a dielectric layer
1. Laplace’s 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
“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
 A given cross-section profile is related to a Green’s function.

7. 2D model: Capacitance per unite length (fF/um)
5. Spectral potential function
Infinite long transmission line
let
then
charge distribution
6. Method of moment (Galerkin’s procedure)
Integral basis functions with above equation
for all j
-w/2+d
w/2-d f1 fn …
d is the process variation.
become a matrix:
fi is the basis function.
solve the unknown ci :
-w/2+d
w/2-d
c1f1
cnfn …
Approximated charge distribution
Final capacitance from charges:

8. 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
“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

9. Models in 2.5D RC technology files

10. 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
Project: IBM & MIT
Integral equation from Maxwell’s 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.

11. Then
Define
then
where l1 l2
Ex: 2 conductors

12. Ex: Spiral inductor or interconnect
Interconnect: even la ld
Ex: Spiral inductor or interconnect la lb lc ld le lf
Spiral inductor
Interconnect: odd la lc
Self inductance 
Laa > 0
Mutual inductance 
Lad > 0
Same current directions have a positive mutual inductance. 
Lab = 0
Orthogonal current directions have no mutual inductance. 
Lac < 0
Oppositive current directions have a negative mutual inductance.

13. Example: RLCK from Fast-Henry (RLK) & Raphael (C)
low frequency: uniform
Current density in a
conductor cross-section
high frequency: skin effect
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)
| e e e e-02
| e e e e-02
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)
| e e e e-02
| e e e e-02
Frequency (GHz) R L
RL relations vs frequency
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)
| e e e e-02
| e e e e-02

14. IV. Full wave analyses (Electromagnetic field theory)
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

15. 2. Method of moment 2D-NUFFT  calculate Jx & Jy
(a)
(b)

16. 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 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.

17. 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
From Google search.
From Google search.

18. V. Double Patterning Technology & Solution1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7
Even circle: 2 colors
Problem
Solution
Can not estimate margin
Designer
Design
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

19. Backup 19

20. Determine “M” for accuracy and efficiency.
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.

21. Some of these 2D coefficients approach to zero rapidly.
Appendix
IEEE Microwave and Guided wave, vol.8, no.1, Jan. 1998, pp.18-20
IEEE MTT vol.53, no.3, March. 2005, pp
NUFFT : 1D  2D
The square 2D-NUFFT
Some of these 2D coefficients approach to zero rapidly.
f and a are finite sequences of complex numbers.
Tj=2pj/N, j=-N/2,…,N/2-1. wk are non-uniform.
The (q+1)2 nonzero coefficients.