Interconnect Modeling

Transcription

1 Interconnect Modelng Modelng of Interconnects Interconnect R, C and computaton Interconnect models umped RC model Dstrbuted crcut models Hgher-order waveform n dstrbuted RC trees Accuracy and fdelty Prepared by CK 1

2 Interconnect Resstance Computaton Resstance R R Mawell s Equaton of Dfferental Form Mawell s equaton for statc electrc feld: ( ε E) = ρ In Cartesan coordnate system, gradent operator s = v v n y ny z v n ε: permttvty of the feld regon E: strength of electrc feld ρ: charge densty of the feld regon z Prepared by CK

3 Posson s Equaton ( ε E) = ρ Base on E =, the Possson s equaton s ( ε ) = ρ ε: permttvty of the feld regon E: strength of electrc feld ρ: charge densty of the feld regon aplace s Equaton ( ε ) = ρ If assume ρ = 0 and homogenous delectrc, potental satsfes aplace s equaton ( ε ) = ε = 0 = s aplacan operator In Cartesan coordnate system, aplace s equaton s y = 0 z = 0 Prepared by CK 3

4 Eact Etracton of Resstance Solvng aplace s equaton, Ψ = 0 Relaaton method: Potental at every pont s the average potental of ts neghbors Consder square grds [ ( 1, y) ( 1, y) (, y 1) (, 1) ] / 4 (, y) = y Pont on an edge, mrror the potental about the edge, e.g., rght vertcal edge [ ( 1, y) (, y 1) (, 1) ] / 4 (, y) = y Repeatedly replace the potental of all ponts wth the average of ther neghbors Equpotentals and Current Flow Current flow potental Prepared by CK 4

5 Break nes for Varous Shapes Potental Dstrbuton for Parallel and Dagonal Contacts Prepared by CK 5

6 Equvalent Perpendcular Contacts l p = mn(, l ) = l p / r c l p = mn(, l ) = l p / 4 r c Resstance Computaton for Non-rectangular Shapes [Horowtz-Dutton, T-CAD July 83] R = R = R = 4 1 R = 1 All lnes are rectlnear or dagonal (45º) Prepared by CK 6

7 Calculatng the Resstance of a Conve Regon Resstance Computaton for Non-rectangular Shapes (cont d) Rato = Rato = 1 A 1 C 1 1 Rato = E Rato = B D 1 Rato = 1 Shape Rato Resstance A 1 1 A 5 5 B 1.5 B B.55 B 3.83 C C.3 C 3.66 C 4.97 D 1.5 D D.43 D 3.74 E E 1.8 E 3.33 E 4.71 Prepared by CK 7

8 Accuracy of Resstance Etracton Shape Rato Etracted Eact %Error Resstance Resstance A A B B B.55.6 B C C.3.5 C C D 1.5. D D D E E E E Eact resstances are obtaned by solvng aplace s equaton Current Spreadng from a Small Contact Prepared by CK 8

9 Typcal sheet resstance for conductors Sheet Resstance Ω/SQ. Materal Mn. Typcal Ma. Metal[Al] Slcdes Dffuson (n and p) Polyslcon hat s Capactance? Q -Q Smplest model: parallel-plate capactor It has two parallel plates and homogeneous delectrc between them The capactance s C = ε ε: permttvty of delectrc A: area of plate d: dstance between plates The capactance s the capacty to store charge charge at each plate s Q = CV one s postve, the other s negatve A d Prepared by CK 9

10 General Pcture m1 c1 c13 m m3 c3 For multple conductors of any shapes and materals, and n any delectrc, there s a capactance between any two conductors Mutual capactance between m1 and m s C 1 = q 1 /v q 1 s the charge of m1 v 1 = 0 and v 3 = 0 Capactance Matr Capactance s often wrtten as a symmetrc matr m1 c1 c13 m m3 c3 c11 -c1 -c13 C = -c1 c -c3 -c31 -c3 c33 c c ( j ) s the self-capactance for a conductor, m = j= 1 j e.g., c 11 = c 1 c 13 The charge s gven by e.g., q = c v c v c v 1 = c ( v1 v) c13( v1 v3) m mm m T Q = C ( V ) 3 Prepared by CK 10

11 Physcal Desgn Doman Conductors: metal wre, va, polyslcon, substrate Delectrcs: SO,... vctm vctm cross-secton Total cap for a wre delay, power Mutual cap between wres sgnal ntegrty top-vew 3D Model 3D model res wth fnte wdth, thckness and length Compute self-cap and mutual cap 3D model D model D model res wth fnte wdth and thckness, but nfnte length along the 3rd dmenson Compute the unt-length cap (also called cap coeffcent) Prepared by CK 11

12 Appromaton of 3D Structure by a D Model Pck a 3rd dmenson Chop 3D structure nto sectons wth dstngush profles 1 3 3rd dmenson 3rd dmenson Appromaton of 3D structure va D Model Solve each dstngush profle va D model two types of profles wth unt-length cap c 1 and c Type I: 1 Type II: 3 3rd dmenson Total cap s c 1 ( 1 3 )c Prepared by CK 1

13 Appromaton va Quas-3D Model D model effects of sdewalls along the 3rd dmenson a correcton capactance for each sdewall Classfcaton (orthogonal to 3D/D) Numercal method accurate any geometrc structures etremely epensve Formula-based method effcent lmted accuracy and geometrc structures nsght nto dependency on desgn parameters Table lookup accuracy -> numercal method effcency -> analytcal method More fleble handlng of geometrc structures than analytcal method Prepared by CK 13

14 Framework for Numercal Method ❶ Assume voltage [0, 0,, 1,, 0],.e., only conductor has unt voltage ➋ Compute charge q j for every conductor j ➌ Obtan mutual cap c j =q j, and self-cap (sum of mutual cap) ➍ Iterate through steps 1-3 usng dfferent voltage assgnments V m = vctm m1 c1 m c4 c3 m4 m3 C mm = 0 -c c1 c -c3 -c4 0 -c c4 0 0 Solutons to Conductor Charge Dfferental-based use Mawell s equaton n dfferental form Integral-equaton based use Mawell s equaton n ntegral form Prepared by CK 14

15 Capactance Etracton - Electrostatc Analyss For m conductors solve m potental problems for the conductor surface charges Each problem has one conductor at 1V, the rest at zero potental Two Vews of Capactance Can derve C 1 from C 1, C 1, and C Set up a system of charge neutralty equatons C 1 C C 1 1 V V Solve for V and q, then C 1 = q 1 C C 1 C 1 q = 0 q Prepared by CK 15

16 Prepared by CK 16 Volume Methods - Fnte-Dfferences/Fnte-Elements Solve aplace s equaton n the conductor eteror Appromate dervatves by fnte-dfferences Conductors Provde potental boundary condtons Voltage one on conductor 1, zero on conductor -D eample Conductor 1 Conductor m k l j 0 )) ( ) 0.5(( )) ( ) 0.5(( = j k j j k k l m l l m m Volume Methods Generate Sparse Matrces One equaton for each grd node In 3-D, each equaton nvolves at least 7 varables Solve by matr soluton methods Sparse Gaussan Elmnaton Incomplete Cholesky Conjugate-Gradent Method (ICCG) Multgrd methods

17 Two Dfferent Boundary Condtons Closed bo: Overestmates self C, underestmates couplng C Open Bo: Underestmates self C, overestmates couplng C Components of Interconnect Capactance Classfcaton based on profles of nteractng nterconnects Area component Due to overlappng area of two nterconnects on dfferent layers Frngng component Due to sde-walls of an nterconnect and the surface of an nterconnect on a dfferent layer ateral component Due to sde-walls of two nterconnects on the same layer Prepared by CK 17

18 Capactance: The Parallel Plate Model H Electrcal-feld lnes SO t o Substrate C nt = ε o t o S S S S C, wre = = S Parallel-Plate Capactor d (0) = V ( d) = 0 aplace s equaton s = y z = 0 ( ) = k k 1 (0) = k 1 0 k = V d) = k d k 0 ( 1 = ( ) = V d V Prepared by CK 18

19 Parallel-Plate Capactor ( ) = V d V unt-area charge = unt-area cap = d ε parallel-plate cap = ε ( ) = ε V ε A d d Typcal Capactance/Unt Area for 1µm CMOS Interconnect ayer Polyslcon to Substrate Metal1 to Substrate Metal to Substrate Metal3 to Substrate N Dffuson to Substrate (@ 0 Volt) P Dffuson to Substrate (@ 0 Volt) ff/µm ± ± ± ± ± ± 0.06 Prepared by CK 19

20 Frngng Capactance H H - H/ Frngng Component Area Component Frngng and Area Capactances H π C = εo t o to t t o o ln 1 H H H C = ε o to to 0.5 H 1.06 to 0.5 Prepared by CK 0

21 Typcal Frngng Capactance/Unt ength for 1µm CMOS Interconnect ayer Polyslcon to Substrate Metal1 to Substrate Metal to Substrate Metal3 to Substrate ff/µm ± ± ± ± Frngng-Feld Effect Prepared by CK 1

22 Interwre Capactance evel evel 1 SO Substrate Creates crosstalk Couplng Capactances Area component Frngng component C X frngng = c f 1 / 0 / 0 ( e e ) 1 0 measures how frnge capactance vares for ncremental length of the frngng surface ateral component C d 1 X lateral Prepared by CK

23 Interwre Capactance for 1µm CMOS Metal1 to Polyslcon Metal to Polyslcon Metal to Metal1 Area ff/µm Frngng ff/µm Impact of Interwre Capactance Prepared by CK 3

24 Sgnfcance of Couplng Capactance C/Ctotal Mn. Spacng Mn. Spacng Technology (u m) Capactance Crosstalk Assume C X = 5 ff For 55mm overlap of X and Y C C C XY, area XY, frnge XY V X = ff = 1.9 ff = ff C X = C C X = 0.35V XY 5V Prepared by CK 4

25 ma (mm) Couplng Nose 10% Vdd mn. Spacng % Vdd mn. Spacng Technology (u m) Prepared by CK 5