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

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P. 1 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

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

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P. 3 Ke-YingSu Ph.D. RCLK extraction: Semiconductor industry: parasitic Capacitance (C), Resistance (R), Inductance (L) extraction.

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P. 4 Ke-YingSu Ph.D. EDA tools C modelRLCK model Numerical Analytical Quasi-static analysisFull-wave analysis 2D & 3D Raphael 2D & 3D Raphael 2D engine 2D engine 2.5D RC extractor 2.5D RC extractor 3D QuickCap 3D QuickCap 3D EMX 3D EMX 3D HFSS 3D HFSS 3D Momentum 3D Momentum 2.5D RC extractor RL extractor 2.5D RC extractor RL extractor 3D Helic 3D Helic 3D Lorentz 3D Lorentz

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P. 5 Ke-YingSu Ph.D. integral equation in frequency domain Galerkin’s procedure 3D EMX 3D EMX 2D & 3D Raphael 2D & 3D Raphael Boundary element method (BEM) Finite difference Method (FD) 3D QuickCap 3D QuickCap Laplace’s equation Floating random walk method 3D Lorentz 3D Lorentz Mixed potential integral equation Partial Element Equivalent Circuit (PEEC) 3D Momentum 3D Momentum Method of moment microwave full wave mode faster RF quasi-static mode QRC RC extractor RL extractor QRC RC extractor RL extractor Partial Element Equivalent Circuit (PEEC) for RLCK extraction

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P. 6 Ke-YingSu Ph.D. 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 “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 Cross-section of a dielectric layer Cross-section of multi-dielectric layers A given cross-section profile is related to a Green’s function.

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P. 7 Ke-YingSu Ph.D. 6. Method of moment (Galerkin’s procedure) for all j -w/2+ w/2- f1f1 fnfn … -w/2+ w/2- c1f1c1f1 cnfncnfn … 2D model : Capacitance per unite length (fF/um) f i 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 c i : Final capacitance from charges: 5. Spectral potential function let then is the process variation.

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P. 8 Ke-YingSu Ph.D. 3D model : capacitance (fF) Open-endGap 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

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P. 9 Ke-YingSu Ph.D. Models in 2.5D RC technology files

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P. 10 Ke-YingSu Ph.D. 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 Assume Integral equation from Maxwell’s equations Let where I i is the current inside filament i. I i is a unit vector along the length of a filament w i (r) is the basis function of filament i. Filaments in a conductor for skin and proximity effects.

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P. 11 Ke-YingSu Ph.D. Define then where Ex: 2 conductors Then l1l1 l2l2

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P. 12 Ke-YingSu Ph.D. Ex: Spiral inductor or interconnect Self inductance Mutual inductance L aa > 0 L ad > 0 L ab = 0 L ac < 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 lala ldld Interconnect: odd lala lclc lala lblb lclc ldld lele lflf Spiral inductor

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P. 13 Ke-YingSu Ph.D. 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) | 3.9e e e e | 2.0e e e e-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) | 4.0e e e e | 2.1e e e e-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) | 4.2e e e e | 2.2e e e e-02 low frequency: uniformCurrent density in a conductor cross-section high frequency: skin effect Frequency (GHz) R L RL relations vs frequency

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P. 14 Ke-YingSu Ph.D. 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

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P. 15 Ke-YingSu Ph.D. 2. Method of moment 2D-NUFFT (a) |J x (b) |J y calculate J x & J y

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P. 16 Ke-YingSu Ph.D. 3. Calculate S parameters from currents Let I i m 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, z 01 and z 02 are reference planes, I im + and I im - are incident and reflect current waves. where Z 01 and Z 02 are characteristic impedance of the ith transmission line.

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P. 17 Ke-YingSu Ph.D. Ex. Passive devices and components inductorcapacitorRF 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.

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P. 18 Ke-YingSu Ph.D 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 network with overlay Sensitivity 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

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P. 19 Ke-YingSu Ph.D. Backup

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P. 20 Ke-YingSu Ph.D. 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

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P. 21 Ke-YingSu Ph.D. Appendix IEEE Microwave and Guided wave, vol.8, no.1, Jan. 1998, pp f and are finite sequences of complex numbers. T j =2 j/N, j=-N/2,…,N/2-1. w k are non-uniform. IEEE MTT vol.53, no.3, March. 2005, pp The (q+1) 2 nonzero coefficients. The square 2D-NUFFT Some of these 2D coefficients approach to zero rapidly. NUFFT : 1D 2D

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