Numerical ElectroMagnetics & Semiconductor Industrial Applications

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

2. 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

3. 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 Foundry Tape out TSMC, UMC, etc.

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

5. 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

6. 3D 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

7. 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.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 Green’s function.

8. 2D model: Capacitance per unite length (fF/um) 5. Spectral potential function Infinite long transmission line let then f1 fn … charge distribution 6. Method of moment (Galerkin’s procedure) Integral basis functions with above equation -w/2+d w/2-d c1f1 cnfn for all j d is the process variation. … become a matrix: fi is the basis function. -w/2+d w/2-d solve the unknown ci : Approximated charge distribution Final capacitance from charges:

9. 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. “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.

10. Models in 2.5D RC technology files

11. 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 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.

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

13. la Ex: Spiral inductor or interconnect lb lf ld la Interconnect: even ld lc le la Self inductance  Laa > 0 Spiral inductor lc Interconnect: odd 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.

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

15. 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.993-999.

16. 2. Method of moment 2D-NUFFT  calculate Jx & Jy (a) |Jx(x,y)| @2.47GHz (b) |Jy(x,y)| @2.47GHz

17. 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.

18. 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.

19. 1 1 5 1 5 5 2 2 3 3 3 6 6 6 2 7 4 7 4 7 4 V. Double Patterning Technology & Solution 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

20. Backup

21. 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.

22. 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. NUFFT : 1D  2D The square 2D-NUFFT Some of these 2D coefficients approach to zero rapidly. f and aare finite sequences of complex numbers. Tj=2pj/N, j=-N/2,…,N/2-1. wk are non-uniform. The (q+1)2 nonzero coefficients.