5/1/2011 V R I. = ds. by definition is the ratio of potential difference of the wire ends to the total current flowing through it.

Transcription

1 Session : Fundamentals by definition is the ratio of potential difference of the wire ends to the total current flowing through it. V R I E. dl L = σ E. ds A R = L σwt W H T

2 At high frequencies, current tends to distribute near the surface of a conductor δ π f µσ 3 Dimension s nm pitch skin depth δ 3 Fclk λ Cu Technology node Scale of GSI interconnections is continually shrinking toward dimensions comparable with the mean free path of the electrons. At the same time, interconnects operate at higher frequencies such that skin depth becomes in the same order of mfp of electrons. 4

3 In D-L-S model the metal is divided into different subsystems Nucleus + Core electrons Valance electrons The kinetic theory of gas is applied to the metal gas v v v = eeτ m τ 4τ ne τ J = nev = E = σ E * m λ = v F τ v t 5 λ λ J D J D λ << D λ D λ ρ ρ b ρ ρ b ( + ( p) ) D p = specularity parameter (the fraction of electrons that have elastic collisions at the wire surfaces) (0<p<) diffuse scattering specular scattering p = 0 p = 6 3

4 δ Skin effect λ δ Anomalous skin effect λ D D D > δ >> λ δ = ρ πµ f b R f D > δ λ δ = ρ πµ f λ ρ = ρb ( + ) δ R f 3 7 w λ Anomalous Skin Effects Skin Effects room temperature: Size Effects λ 39nm λ nm Cu δcu 600nm 0GHz Al Low Temp δ λ 4

5 o A capacitor is a passive electronic component that stores energy in the form of an electrostatic field. Q = D ds S C V Eid L l In its simplest form, a capacitor consists of two conducting plates separated by an insulating material called the dielectric. The capacitance is directly proportional to the surface areas of the plates, and is inversely proportional to the separation between the plates. 9 Q C 0 + C + C3 C C3 V Q C C 0 C C3 C 3 V = + + Q 3 C3 C3 C 30 C3 C V 3 0 5

6 Volume-based method: (finite-element, finite-difference) B.C. C m. v = 0. dv dn = 0 C C. cր, c ց ;. cց, c ր + : accuracy, any complex structure - : time consuming Software: Maxwell, HFSS, Raphael m m Surface-based method: (integral equation) Green s function Integral equation (panel method, method of moments) G( r, r ) = 4π r r = V G( r, r ) σ ( r ) da surface Q = surface σ ( r ) da Software: Boundary Elements Method, FASTHENRY 6

7 Random-walk method: (stochastic) N V = V ( x ) center square i N i = Software: Random Logic Corp, QuicCap Random walk: best for self cap for complicated net Surface based: best for small coupling capacitance Volume based: best for dealing with multiple dielectrics 3 T W H S C m C C 4 7

8 W T C = ε Cm = ε H S 5 C W T = ε H H 0. C m T W S = ε H T H

9 o An inductor is a passive electronic component that stores energy in the form of a magnetic field. In its simplest form, an inductor consists of a wire loop or coil. The inductance is directly proportional to the number of turns in the coil. Inductance also depends on the radius of the coil and on the type of material around which the coil is wound 7 t RC v ( t) = V ( e ) c dd t0.5 = 0.693RC t 0.9 =.3 RC Using parallel plate approximation ρε L t0.5 = TH 8 9

10 V ( L, s) = V ( s) CL Rs ( c + r ) + ( + s L ) in src sinh src R C s cosh src 9 δt δt ( ) V ( t, L) = V + K e + K e + dd RT K =.0 R δ σ = = T + CT + + C + T π 4.04 RC RT CT + RT + CT + ( π ) tυ = RC( RT CT + RT + CT + ( π ) ) ln( ) + 0.RC υ t =.3( R C + R C + RC ) + RC 0.9 t = 0.69( R C + R C + RC ) RC 0.5 s L s L s L s L 0 0

11 Two coupled lumped RC lines Two coupled distributed RC lines Combining these equations V ( x, t) ( c c ) V ( x, t) c V ( x, t) r x t t = + m m = + m m V ( x, t) ( c c ) V ( x, t) c V ( x, t) r x t t Assuming r = r = r and c = c = c simplifies to: x x ( V + V ) = rc ( V + V ) t ( V V ) = r( c + cm ) ( V V ) t

12 Boundary conditions Transformation Solution: V u( t) I (0, t) R = V (0, t) ; I (0, t) R = V (0, t) dd s s I( L, s) = CL V ( L, s) ; I( L, s) = CL V ( L, s) t t V = ( V V ) ; V = ( V + V ) + V + = rc V+ x t Vdd u ( t ) I + (0, t ) R s = V + (0, t ) I+ ( L, t) = CL V+ ( L, t) t V ( ) = r c + cm V x t Vdd u ( t ) I (0, t ) R s = V (0, t ) I ( L, t) = CL V ( L, t) t 3 Sakurai single line solution RT + CT + K =.0 R + C + σ = T.04 π 4 RT CT + RT + CT + ( π ) T V t x l V K σt (, = ) ( + exp( )) dd RC Plus solution + V R dd T + CT +.04t V+ ( t, x = L).0 exp( ) + RC + + π RT CT + RT + CT + ( π ) RT + CT + 4 Minus solution V R dd T + CT +.04t V ( t, x = L).0 exp( ( ) ) π R C+ C m RT CT + RT + CT + ( π ) RT + CT + 4 4

13 Active line V + V V = Quiet line = + V V V + V P t P 5 Solving for t : t P + K σ C σ σ = ln + + K σ C C + m R( C + Cm) RC Assuming CL << C simplifies to: V P C C m V R + C C C =.0 dd T m m π RT + 4 C + Cm C + Cm C + Cm VP Cm V C + C dd m parallel plate approximation T S W H + T S 6 3

14 Finite rise time?! 7 Solution: Transient voltage parameters 8 4

15 Time delay expressions: As T rise 0 converges to Sakurai 9 Generalized delay formula for RC>T rise Coupled line solutions: 30 5

16 Hspice comparison 3 Peak crosstalk expression 3 6

17 Hspice comparison 33 Scaling independent: VP Cm V C + C dd m parallel plate approximation T S W H + T S Length dependence 34 7

18 Scaling dependence 35 Material dependence 36 8

19 Driver resistance dependence 37 o An inductor is a passive electronic component that stores energy in the form of a magnetic field. In its simplest form, an inductor consists of a wire loop or coil. The inductance is directly proportional to the number of turns in the coil. Inductance also depends on the radius of the coil and on the type of material around which the coil is wound 38 9

20 Inductance of rectangular wires with return path at infinity 39 Inductance of rectangular wires with return path in perfect ground plane 40 0

21 Loop inductance for coplanar ground lines 4 VS RS V ( x, t) inf x In its simplest 4

22 Z 0 σ t Vinf ( x, t) = VS e [ I0( σ t ( x lc) ) + Z0 + RS k 0 t x lc k Ik ( σ t ( x lc) ) 4 ( ) Γ + Γ ] k = t x lc Γ + u ( t x lc) where RS Z σ = r l ; Γ = R + Z S 0 0 Note that: 0 rx Z0 Vinf ( x, x lc) = VS e Z0 + RS Z 43 VS RS l V (, ) fin l t Delay model: 44

23 Capacitive load Delay model: td max tf, 0.37rcL RScL C rl R Z L ( ) S 0 R tr x=0 x=l + - C L 45 Transmission line effects should be considered when the rise or fall time of the input signal (t r,t f ) is smaller than the time-of-flight of the transmission line (t flight ). t r (t f ) <<.5 t flight Transmission line effects should only be considered when the total resistance of the wire is limited: R < 5 Z 0 The transmission line is considered lossless when the total resistance is substantially smaller than the characteristic impedance R < Z 0 / 46 3

24 n= n=0 n=50 n=

25 49 coupled distributed RLC interconnects A (active) and Q (quiet). 3 coupled distributed RLC interconnects 50 5

26 n= n=0 n=50 n=

27 line RC: VP Cm V C + C dd m line RLC: VP π Cm V 4 C + C dd m 3 line RLC: VP π Cm V 3 C + C dd 4 3 m 53 7