5 integral equation in frequency domain Galerkin’s procedure Momentum3DEMXintegral equation in frequency domainGalerkin’s procedureMethod of momentmicrowave full wave modefaster RF quasi-static modeQRCRC extractorRL extractor3DLorentzMixed potential integral equationPartial Element Equivalent Circuit (PEEC)Partial Element Equivalent Circuit (PEEC) for RLCK extraction2D & 3DRaphael3DQuickCapBoundary element method (BEM)Finite difference Method (FD)Laplace’s equationFloating random walk method
6 II. Quasi-static analyses (2D & 3D) Cross-section of a dielectric layer1. Laplace’s equation2. Spectral potential function of a chargeCross-section of multi-dielectric layers3. Matrix pencil method4. 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 functionInfinite long transmission lineletthencharge distribution6. Method of moment (Galerkin’s procedure)Integral basis functions with above equationfor all j-w/2+dw/2-df1fn…d is the process variation.become a matrix:fi is the basis function.solve the unknown ci :-w/2+dw/2-dc1f1cnfn…Approximated charge distributionFinal capacitance from charges:
8 3D model: capacitance (fF) Open-endGapdiscontinuityCross-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
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, ppProject: IBM & MITIntegral equation from Maxwell’s equationsAssumeLetwhereIi is the current inside filament i.Ii is a unit vector along the length of a filamentwi(r) is the basis function of filament i.Filaments in a conductor for skin and proximity effects.
12 Ex: Spiral inductor or interconnect Interconnect: evenlaldEx: Spiral inductor or interconnectlalblcldlelfSpiral inductorInterconnect: oddlalcSelf inductanceLaa > 0Mutual inductanceLad > 0Same current directions have a positive mutual inductance.Lab = 0Orthogonal current directions have no mutual inductance.Lac < 0Oppositive current directions have a negative mutual inductance.
13 Example: RLCK from Fast-Henry (RLK) & Raphael (C) low frequency: uniformCurrent density in aconductor cross-sectionhigh frequency: skin effectLayers : 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-02Layers : 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-02Frequency (GHz)RLRL relations vs frequencyLayers : 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 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, ppFrom Google search.From Google search.
18 V. Double Patterning Technology & Solution 123456712345671234567Even circle: 2 colorsProblemSolutionCan not estimate marginDesignerDesignDesignDesigner:the worst margin to protect circuitMaskFoundryRLCK networkwithoverlaySensitivityFoundry:the best decomposition to gain yield2 colorsdecompositionuncertainPost-layoutsimulationMonte Carlo simulation:for all possible decomposition & variation2 masksvariationsrandom
20 Determine “M” for accuracy and efficiency. AppendixIt 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. AppendixIEEE Microwave and Guided wave, vol.8, no.1, Jan. 1998, pp.18-20IEEE MTT vol.53, no.3, March. 2005, ppNUFFT : 1D 2DThe square 2D-NUFFTSome 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.