Coupled ana lytical and boundary elem en t m ethod for 3-D in terconnecting res istance ca lcula tion s FEM BEM. FDM FEM,

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1 ISSN CN gN ( ) J T singhua U niv (Sci & Tech), , V o l. 44, N o. 9 32g ,, (, ) :,,,,,, R aphael, 2 3, 0. 1% 1%, R aphael, : ; ; ; ; : TN 47 : (2004) : A Coupled ana lytical and boundary elem en t m ethod for 3-D in terconnecting res istance ca lcula tion s WANG Xiren, YU W enjia n, WANG Zeyi (D epartmen t of Computer Sc ience and Technology, Tsinghua Un iversity, Be ij ing , China) Abstract: D eep subm icron p rocessing techno logy is w idely used, but the interconnecting structures are becom ing very comp lex so resistance calculations based on two2dim ensional models can no longer p rovide sufficiently accurate results. This paper p resents a three2dim ensional resistance calculation m ethod, called the com bined analytical and boundary elem ent m ethod. The m ethod cuts selected interconnecting lines, then calculates the resistances of straigh t sections using analytical fo rm ula and the resistances of mo re comp lex sections w ith comp lex interconnections using the imp roved BEM. T he resistances of the separate subregions are com bined to calculate the resistance of the entire region. Experim ents on actual layouts show that compared w ith the comm ercial Raphael algorithm, the p ropo sed m ethod is 2 3 o rders of m agnitude faster, uses m uch less m emo ry (about 0. 1% to 1% ), and is more accurate than Raphael w ith default m esh partitions. The results illustrate that the coup ling m ethod used in the algo rithm is efficient and accurate. Key words: very large scale integration; interconnecting resistance; 32D extraction; analytical and boundary elem ent m ethods; coup ling,, (VL S I) [1 ],,, [2 ], [1 ], [3, 4 ], [5 ], 5 GH z, ; [6 ], ( FDM ) [7 ] ( FEM ) [8 ] (BEM ) [2, 6 ] L ap lace FDM FEM, [2, 5, 6 ],,,,, [ 9 ], FDM [ 10 ], FEM,,,,, ;, BEM ;, : : (2002AA 1Z1460SOC) : (19782), ( ),, :,, E2m ail: w angzy@ tsinghua. edu. cn

2 1278 ( ) 2004, 44 (9), BEM, ( ), 1,, (via), 1 IBM [1 ], 8,, [ 2, 6, 9, 10 ] 1 IBM , R = Θlgs, (1) : Θ, l, S, 2, j ( ) V j= 1, V k= 0 (k j ), j k R jk = V jk = V j - V k = 1, (2) I k I k I k I k k, I k = Ρ 9u 9n # k d#, (3) : # k k, Ρ, u, n [6 ], 2. 3,, L ap lace [6 ] 9 2 u 9x u 9y u 9z 2 = 0, (4), u L ap lace Green (4) [6 ] csu s + # q3 ud# = # u 3 qd#, (5) : q 9ug9n, u 3 L ap lace, q 3 9u 3 g9n, #,, A x = f, (6) : x q u (6), (3) (2) 3,,,, B EM, 3. 1 [ 11 ],,, 2, E1 E2, FUL L,, 2. 2,,,, 3, d,, d = 0. 5w (w, 3 P 1

3 , : 1279 ),, ;,,,, d, 2d;,,, 3a, P 1, P 2 P 8 P 1 P 2 P 3 P 4 P 1, P 5 P 1, P 6 P 7, P 8 P 1 d 0. 5 P 1 (), L 1< 2d, L 2 2d, FULL 3b P 2 P 3 P 4 2d, P 1, N 1; P 5 P 6 P 7 N 3 P 8 P 1, d, N 5P 1 N 1 N 3 N 5, N 2 N , N 2 N 4, N 1 P 2 P 3 P 4, N 3 P 5 P 6 P 7, N 5 P 8,,,, 4,,, , (1),, BEM, BEM, 1),,,,, (4 ),,,, BEM,, 2),, ;,, u q, (), 5, 1 2 B,, 5B

4 1280 ( ) 2004, 44 (9) 3. 4 G i, ( ), R j, G i R j G i G ϖ ( 6), G 6( 4) 1), 1V,, 0V ; 2) N s, G ϖ, N s ; 3) N c i G i, n i, n λ i, G i n λ i (6 ), n i (n i- 1) g2 G i,, n λ i+ n i (n i- 1) g2; 4) G ϖ N s, N s ( ), 1) ; 5) G ϖ G i,, K irchhoff ;, n λ i+ n i (n i- 1) g2, 3) ; 6) 4) 5), ; 7),, I k, (2) 8) 1),, 2) 7), G BEM, ) FUL L,, 3) B EM 4), , Λm Λm Λm Λm Λm Λm, 2 3, ( B EM ) R ap hael ( , FDM 2) 1 ) S un U ltra E n terp rise 450, 248M H z, 1 2, ; R ap hael, (R ap hael 2 ) ; 11 M B s % BEM R aphael M B s % BEM R aphael ,,, R aphael ;, BEM,, 6% 0. 8%, R aphael, 2

5 , : 5,,,, (References) [ 1 ] B row n A S. Fast film s [J ]. IE E E S p ectrum, 2003, 40 (2) : [ 2 ] SUN W eikai, HON G W ei, DA I W W ei2m ing. R esistance extraction using superconvergence accelerated boundary elem ent m ethod [A ]. A sia Pacific M icrow ave Conf P roc [C ]. Piscataw ay, N J: IEEE, [ 3 ] KAO W H, LO Chi2Yuan, Basel M. Parasitic extraction: current state of the art and future trends [A ]. IEEE Int Sym on C ircuits and System s [C ]. P iscataw ay, N J: IEEE, [ 4 ] YU W enjian, WAN G Zeyi, GU J iangchun. Fast capacitance extraction of actual 32D VL S I interconnects using quasi2m ultip le m edium accelerated BEM [J ]. IE E E T rans on M icrow ave T heory and T ech, 2003, 51 (11) : [ 5 ],. [J ]. ( ), 2002, 42 (01) : YU W enjian, WAN G Zeyi. Efficient equation o rganization fo r 32D m ulti2dielectric parasitic capacitance extraction [J ]. J T singhua U niv (S ci and T ech), 2002, 42 (01) : ( in Chinese) [ 6 ] WAN G Zeyi, WU Q im ing. A two2dim ensional resistance sim ulator using the boundary elem ent m ethod [J ]. T rans on Com p u ter2a ided D esign of S y stem s, 1992, 11 (4) : IE E E Integ rated C ircu its and [ 7 ] Gleg H M, D rake J M. Calculation of m ultiterm inal resistance in integrated circuits [J ]. and S y stem, 1986, 33 (4) : IE E E T rans on C ircuits [ 8 ] M itsuhashi T, Yoshida K. A resistance calculation algorithm and its app lication to circuit extraction [J ]. IE E E T rans on Com p uter2a ided D esign of Integ rated C ircuits and S y stem s, 1987, 6 (3) : [ 9 ] M cco rm ick S P. EXCL : A circuit extracto r fo r IC designs [A ]. ACM IEEE 21st D esign A utom ation Conference [C ]. N ew Yo rk: IEEE, 1984, [ 10 ] WAN G Yucheng, O verhauser D, Basel M. A ccurate parasitic resistance extraction fo r interconnection analysis [A ]. Custom Integrated C ircuits Conference [C ]. P iscataw ay, N J: IEEE, [ 11 ] W alton A J, Ho lw ill R J, Robertson J M. N um erical sim ulation of resistive interconnects for integrated circuits [J ]. IE E E J S olid2s tate C ircuits, 1985, 20 (6) : (1276 ) (References) [ 1 ] H eflin J, H endler J. A po rtrait of the sem antic w eb in A ction [J ]. IE E E Intelligent S y stem, 2001, 16 (2) : [ 2 ] Sure Y, Erdm ann M, A ngele J, et al. O ntoedit: co llabo rative onto logy developm ent fo r the sem antic w eb [A ]. P roceedings of the F irst International Sem antic W eb Conference 2002 [C ]. Berlin, Germ any: Sp ringer2v erlag H eidelberg, [ 3 ],,,. W eb [J ]., 2002, 29 (12) : XU J ianjun, L IAN G Bangyong, L I Juanzi, et al. The intelligent w eb services based on the O nto logy [J ]. Chinese J Com p uter S cience, 2002, 29 (12) : ( in Chinses) [ 4 ] H andschuh S, Staab S, C iravegna F. S2CREAM : sem i2autom atic creation of m etadata [J ]. E x p ert U p d ate, 2002, 5 (3) : [ 5 ] Caroll J J. U nparsing RD FgXM L [R ]. HPL , HP lab: H P, 2001, http: ggwww. hp l. hp. com gtechrepo rtsg2001ghpl pdf, [ 6 ] M ladenic D. Text2learning and related intelligent agents: A survey [J ]. IE E E Intelligent S y stem s, 1999, 14 (4) :