The Wires

1. EE4271VLSI Design Dr. Shiyan Hu Office: EERC 518 The Wires Adapted and modified from Digital Integrated Circuits: A Design Perspective by Jan M. Rabaey, Anantha Chandrakasan, and Borivoje Nikolic.

2. Modern Interconnect

3. Modern Interconnect - II

4. Interconnect Delay Dominates 300 250 Interconnect delay 200 150 Delay (psec) 100 Transistor/Gate delay 50 0 0.25 0.8 0.5 0.35 0.25 0.18 0.15 Technology generation (m) Source: Gordon Moore, Chairman Emeritus, Intel Corp.

5. Wire Model

6. Capacitor • A capacitor is a device that can store an electric charge by applying a voltage • The capacitance is measured by the ratio of the charge stored to the applied voltage • Capacitance is measured in Farads

7. 3D Parasitic Capacitance • Given a set of conductors, compute the capacitance between all pairs of conductors. 1V + - - + + + - C=Q/V - + - - -

8. Simplified Model • Area capacitance (Parallel plate): area overlap between adjacent layers/substrate • Fringing/coupling capacitance: • between side-walls on the same layer • between side-wall and adjacent layers/substrate m3 m2 m2 m2 m1

9. The Parallel Plate Model (Area Capacitance) Capacitance is proportional to the overlap between the conductors and inversely proportional to their separation

10. Wire Capacitance • More difficult due to multiple layers, different dielectric =8.0 m3 multiple dielectric =4.0 m2 m2 m2 =3.9 =4.1 m1

11. Simple Estimation Methods - I • C = Ca*(overlap area) +Cc*(length of parallel run) +Cf*(perimeter) • Coefficients Ca, Cc and Cf are given by the fab • Cadence Dracula • Fast but inaccurate

12. Simple Estimation Methods - II • Consider interaction between layer i and layers i+1, i+2, i–1 and i–2 • Cadence Silicon Ensemble • Accuracy 50%

13. Library Based Methods • Build a library of tens of thousands of patterns and compute capacitance for each pattern • Partition layout into blocks, and match with the library • Accuracy 20%

14. Accurate Methods In Industry • Finite difference/finite element method • Most accurate, slowest • Raphael • Boundary element method • FastCap, Hicap

15. Fringing versus Parallel Plate Coupling capacitance dominates.

16. Wire Resistance • Basic formula R=(/h)(l/w) •  : resistivity • h: thickness, fixed for a given technology and layer number • l: conductor length • w: conductor width l h w

17. Sheet Resistance • Simply R=(/h)(l/w)=Rs(l/w) • Rs: sheet resistance Ohms/square, where h is the metal thickness for that metal layer. Given a technology, h is fixed at each layer. • l: conductor length • w: conductor width l w

18. Typical Rs (Ohm/sq)

19. Contact and Via • Contact: • link metal with diffusion (active) • Link metal with gate poly • Via: • Link wire with wire

20. Interconnect Delay

21. Analysis of Simple RC Circuit i(t) R v(t) vT(t) C ± state variable Input waveform

22. v0u(t) v0 v0(1-e-t/RC)u(t) Analysis of Simple RC Circuit Step-input response: match initial state: output response for step-input:

23. 0.69RC • v(t) = v0(1 - e-t/RC) -- waveform under step input v0u(t) • v(t)=0.5v0  t = 0.69RC • i.e., delay = 0.69RC (50% delay) v(t)=0.1v0 t = 0.1RC v(t)=0.9v0 t = 2.3RC • i.e., rise time = 2.2RC (if defined as time from 10% to 90% of Vdd) • For simplicity, industry uses TD = RC (= Elmore delay) • We use both RC and 0.69RC in this course. In textbook, it always uses 0.69RC.

24. Elmore Delay • 50%-50% point delay • Delay=RC • (Precisely, 0.69RC) Delay

25. Elmore Delay - III What is the delay of a wire?

26. Elmore Delay – IV Assume: Wire modeled by N equal-length segments For large values of N: Precisely, should be 0.69RC/2

27. Elmore Delay - V n2 n1 n1 n2 C/2 C/2 R R=unit wire resistance*length C=unit wire capacitance*length

28. RC Tree Delay 4 4 2 2 7 2 7 24+4*2=32 3.5 1 2 1 3.5 Unit wire cap=1, unit wire res=1 2*(1+3.5+3.5+2+2)=24 24+7*3.5=48.5 Precisely, 0.69*48.5 RC Tree Delay=max{32,48.5}=48.5

29. More Accurate RLC Delay Model I=V/R at t=0 is not right since you assume that you can see R with 0 time At time t=0, switch is on. This effect is not felt everywhere instantaneously. Rather, the effect is propagated with a speed u. Denote by c0 the speed of light, epsilon the permittivity and mu the permeability of the dielectric of the medium which the wire is in, L and C the unit wire inductance and capacitance, respectively. According to Maxwell’s law,

30. RLC Delay - II Voltage and Current at time t1 and t2 R0 is the resistance you can really see at t1. R cannot be seen yet.

31. RLC Delay - III • What is R0? • The front of the voltage travels from 0 to l. Suppose that the distance it moves is dx, the capacitance to be charged is Cdx. The charge is thus dQ=CdxV. • Current I=dQ/dt=CVdx/dt=CVu • where is called characteristic impedance.

32. RLC Delay - IV • R0 is a function of the medium • For Printed Circuit Board (PCB), it is about 50-75 ohm • For any x between 0 and l, we always have Ix=Vx/R0,Il=Vl/R0 when x=l • Note that there is a resistor R. We should have Il=Vl/R • What happens if R!= R0?

33. RLC Delay - V • At load, the wave will be reflected back to the source. • The amplitude and polarity of this reflected wave are such that the total voltage, the sum of incident voltage and reflected voltage, satisfies Il=Vl/R • If the incident voltage is V, denote by pV the reflected voltage, where p is called the reflection coefficient. • If incident current is V/R0, then reflected current is –pV/R0 • Thus, (V+pV)/(V/R0–pV/R0)=R. • p=(R/R0-1)/(R/R0+1) • R=R0, p=0, no reflection • R=infty, p=1, wire is unterminated • R=0, p=-1, wire is short-circuited • There can be multiple rounds of reflections.

34. RLC Delay Example • Consider a wire of length l, R0=100 ohm, R=900 ohm driven by the source resistance (transistor equivalent resistance) Rs= 14 ohm. Source voltage is 12V as a step input at time t=0. We want to compute the waveform at the end of l. • Reflection coefficient • At t=0, V1=12*R0/(R0+Rs)=10V since it cannot see R yet • At t=td=l/u, wave V1 arrives at the end and is reflected as V2=pRV1=8V. The total voltage at the end is V1+V2 =18V • At time t=2td, wave V2 arrives at the source and reflected as V3=pSV2=-0.75*8=-6V • At time t=3td, wave V3 arrives at the end and is reflected as V4=PRV3=-4.8V, so the total voltage at the end is V1+V2+V3+V4=7.2V • Continues this process. Next total voltage at the end is 13.7V. • The total voltage at l will converge to 12*R/(R+Rs)=11.7V Rs=14

35. RLC Delay Example - II Voltage at the end of l

36. v0u(t) v0 v0(1-e-t/RC)u(t) When To Use RLC Model RC Model, V0=12 • The voltages at first few td have large magnitudes and are quite different from RC model. This is because Rs<R0. • When Rs>>R0, V1 is small and is the reflected voltage V2. • The total voltage at the end of the wire will gradually increase to 11.7V, which is the same as predicted by RC model. • Thus, RLC model should only be used when Rs is small (see also Figure 4-21 in the textbook) since RLC model is expensive to compute. • RLC model can be used when the switching is fast enough since signal transition time is proportional to Rs.

37. Summary • Wire capacitance • Fringing/coupling capacitance dominates area capacitance • Wire resistance • RC Elmore delay model for wire • For single wire, 0.69RC/2 • RC tree • RLC model for wire • Reflection • When to use