Capacitance Etaction n Intoduction n Table lookup metod n Fomula-based metod n Numeical metod Classification otogonal to D/D n Numeical metod accuate an geometic stuctues etemel epensive n Fomula-based metod efficient limited geometic stuctues insigt into dependenc on design paametes n Table lookup accuac -> numeical metod efficienc -> analtical metod geometic stuctues moe feasible tan analtical metod
Famewok fo Numeical Metod Assume voltage [,,,,, ], i.e., onl conducto i as unit voltage Compute cage q j fo eve conducto j Obtain mutual cap c ij q j, and self-cap sum of mutual cap Iteate toug steps - using diffeent voltage assignments victim m m c c c m m V m -c C mm -c c -c -c -c -c Solutions to Conducto Cage n Diffeential-based use Mawell s equation in diffeential fom n Integal-equation based use Mawell s equation in integal fom
Oveview of Diffeential-Based Metod Potential is known fo conductos n Solve potential voltage j in te wole field egion l based on Mawell s equation in diffeent fom l using finite element metod FEM o finite diffeent metod FDM n Compute cages using gadient of potential j Field Teo Concepts b Adolf J. Scwab, Spinge publises, 988 Mawell s Equation of Diffeential Fom n Mawell s equation fo static electic field: ε E ρ In Catesian coodinate sstem, gadient opeato is v v v n n n z z ε: E: ρ: pemittivit of te field egion stengt of electic field net cage densit of te field egion
Poisson s Equation ε E ρ n Base on ε: E: ρ: E, we obtain te Poisson's equation ε ρ pemittivit of te field egion stengt of electic field net cage densit of te field egion Laplace s Equation ε ρ n If assume and omogenous dielectic, potential j satisfies Laplace s equation ε ε is Laplacian opeato In Catesian coodinate sstem, Laplace s equation is z
FDM to Solve Potential Discetization of te field egion into individual elements Appoimation of potential j witin an element Assembl of te sstem equation Solution of potential j b appling bounda conditions to sstem equations Discetization n Discetization in D model fo a conducto ove a gound Gid lines cove te full field Te conducto bounda must fall at a gid line 5
6 Appoimation n Appoimating potential b Talo seies,, Appoimation n Wen, adding j, j, j and j ields n In geneal
Sstem Solution n Tee ae n linea equations fo n gid points Sstem mati is spase n Sstem mati is solved using bounda conditions. on te suface of te metal. on te gound Summa on FDM fo Capacitance Etaction n Solve potential voltage j in te wole field egion l based on Mawell s equation in diffeent fom ε ρ l sstem mati is lage and spase n Fo eve conducto j, compute its cage ε nds q j n Obtain mutual cap and self cap 7
Oveview of Integal-Based Metod Assume voltage [,,,,, ], i.e., onl conducto i as unit voltage Fo an conducto j, compute its cage q j l based on Mawell s equation of integal fom l use bounda element metod BEM Obtain mutual cap c ij q j, and self-cap sum of mutual cap Iteate toug steps - using diffeent potential assignments Mawell s Equation in Integal Fom n Mawell s equation fo static electic field: s ε E v nds Q ε: E: Q: pemittivit of te field egion stengt of electic field net cage witin s 8
Simple Case Q s ε E v nds Q ε E E π n Q Q ε π n v d Q E nd πε πε n We obtain P Q P is potential coefficient πε Geneal Case ' Q σ ' n n G, ' σ ' ds ' allsufaces G, ' is te geen function in case of omogeneous dielectic G, ' σ ' Q πε πε ' is cage densit 9
BEM fo Statified Stuctue e e e e n e is a constant witin eac lae fo a statified stuctue Cage is esticted to conducto sufaces and dielectic intefaces Geen function is weigted sum of tose unde te omogeneous case n It can be solved b bounda element metod BEM onl boundaies between diffeent mateials ae discetized Suface Cage via BEM Discetization of te bounda into individual elements Appoimation of potential j fo an element Assembl of te sstem equation Solution of cage densit
Discetization n Divide sufaces of m conductos into n panels elements simplest model: unifom cage densit witin eac panel n Cage densit is ig nea conducto edge Leads to lage numbe of panels if constant densit is assumed witin a panel n Smalle panels nea edges FastCap Appoimation j k ' l k cage panel evaluation panel q l n Te potential j k at cente of panel k is te potential fo te panel n j k is caused b cage on all n panels k n ql al l panell G, ' ds ' a l and q l ae aea and cage of panel l Integation along ove panel l '
Assembl k n ql al l panell G, ' ds ' n Cage q n fo n panels is given b nn n n [ p ] q kl te following ae known: Potential coefficient: p kl al panell G, ' ds' Potential fo n panels --j n Solution to Cage Cage [ p nn n kl] q n Te mati in BEM is dense but smalle tan tose fo FEM and FDM n Solved b Gaussian elimination in On time n: numbe of panels n n Can be solved in Onm time fo m conductos FastCap Genealized conjugate esidual metod GCR, faste tan Gaussian elimination multipole epansion local epansion
Monopole Epansion R k R k q l n Wen >> R, cages q l can be eplaced b te sum of q l at cente of inne cicle fo evaluation point aving distance lage tan Fist-Ode Local Epansion R l l cage point q l k evaluation points n Wen >> R, points of inne cicle ave potential same as te on in te cente of te inne cicle
Summa on Numeical Metod n diffeential-equation based Compute conducto suface cage via te gadient of field potential, wic is solved b finite element metod FEM o finite diffeent metod FDM Te wole field egion is discetized, and sstem mati is lage and spase n integal-equation based Compute conducto suface cage based on Mawell s equation of integal fom via bounda element metod BEM Onl boundaies of diffeent mateials bot conductos and dielectics ae discetized, and sstem mati is small and dense Moe efficient tan diffeential-equation based metod, but applicable onl to statified stuctues Homewok n Suve of tee BEM papes Summa fo eac metod Compaison between tem Rougl ~ pages n Refeences: Z.Q. Ning and P. Dewilde, TCAD, Dec., 988 SPIDER: capacitance modeling fo VLSI Inteconnections K. Nabos and J. Wite, TCAD, Nov., 99 FastCap: A multipole acceleated -D capacitance etaction pogam W. Si, J. Liu, N. Kakani, and T. Yu, DAC 98 best pape awad A Fast Hieacical Algoitm fo D Capacitance Etaction