Wire Modeling

João Canas Ferreira

Universidade do PortoFaculdade de Engenharia

April 2016

Topics

1 Interconnect Structure

2 Capacitance

3 Resistance

4 Propagation delay

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On-chip interconnects

Schematic Physical

I Real interconnects have a 3D structure

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Interconnect usage

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Example of 3D interconnect structure

Source: [unknown]

I Metal 1 - metal 4I Vias

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Global vs. local interconnect

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Topics

1 Interconnect Structure

2 Capacitance

3 Resistance

4 Propagation delay

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Interconnect capacitance

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Parallel plates

Cint =εditdi

WL

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Relative permittivity of typical dielectrics

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Fringe capacitance

Capacitance per length unit (approximation):

Cwire = Cpp + Cfringe =εdiwtdi

+2πεdi

log(2 tdi/H + 1)w = W –

H2

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Influence of fringe capacitance

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Capacitance between wires

Source: [Rabaey03]

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Impact of interwire capacitance

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Interwire capacitance for a 0.25 µm CMOS process

Source: [Rabaey03]

Areas in aF/µm2, lengths in aF/µm (fringe capacitance, in gray).

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Topics

1 Interconnect Structure

2 Capacitance

3 Resistance

4 Propagation delay

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Resistance of a single wire

Source: [Rabaey03]

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Resistivity of different materials

Source: [Rabaey03]

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Polycide gate

Reducing polysilicon resistivity by a factor of 8–10:

Silicides: WSi2, TiSi2, PtSi2 and TaSi

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Typical sheet resistance

Source: [Rabaey03]

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Example: Intel 0.25 µm process

Source: [Rabaey03]

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Topics

1 Interconnect Structure

2 Capacitance

3 Resistance

4 Propagation delay

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Lumped parameter model

Source: [Rabaey03]

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Elmore delay

Source: [Rabaey03]

à There is just one resistive path between r and any node i (Rii).à Shared path resistance between the paths from r to nodes i and j (Rij):

Rik =∑

Rj where Rj ∈ [path(r→ i) ∩ path(r→ k)]

à Equivalent time constant (dominant pole):

τri =N∑

k=1

CkRik

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Special case: the RC chain

τ =N∑

i=1

Ci

i∑j=1

Rj =N∑

i=1

CiRii

In this case, the path resistance and the shared path resistance are the same.

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Interconnect model by segments

à Consider a line of length L composed of N equal segments

τ =(

LN

)2(rc + 2rc + · · · + Nrc) =

(rcL2

) N(N + 1)

2N2 = RCN + 12N

with

R = rL C = cL

à For large N:

τ =RC2

=rcL2

2

à This is the same result that would be obtained by treating the wire as atransmission line.

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Response of RC wire to a voltage step

Source: [Rabaey03]

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Intermediate RC models

Source: [Rabaey03]

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Distributed RC line

τ = RSCW +RWCW

2= RSCW + 0.5rwcwL2

tp = 0.69RSCW + 0.38RWCW

Practical rule: use a distributed line model only if tpRC ≥ tgate.

Criterion for critical wire length:

Lcrit =

√tpgate

0.38rc

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References

à Some of the figures used come from:

Rabaey03 J. M. Rabaey et al, Digital Integrated Circuits, 2nd edition,Prentice Hall, 2003.http://bwrc.eecs.berkeley.edu/icbook/

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