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Worst-case Delay Analysis Considering the Variability of Transistors and Interconnects. Takayuki Fukuoka, Tsuchiya Akira and Hidetoshi Onodera Kyoto University. Outline. Motivation Worst-delay Analysis Classification of the Worst-delay Direction Conclusions. Motivation. Technology scaling

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worst case delay analysis considering the variability of transistors and interconnects

Worst-case Delay Analysis Considering the Variability of Transistors and Interconnects

Takayuki Fukuoka, Tsuchiya Akira and Hidetoshi Onodera

Kyoto University

outline
Outline
  • Motivation
  • Worst-delay Analysis
  • Classification of the Worst-delay Direction
  • Conclusions
motivation
Motivation
  • Technology scaling
    • Increasing significance of variation
  • Transistor and Interconnect variations affect delay variation
    • Gate length, width etc.
    • Metal width, ILD (inter layer dielectric) etc.
  • Worst-delay corner depends on many parameters (Drive strength, Interconnect length etc.)

Where is the worst-delay corner?

How the corner changes?

outline1
Outline
  • Motivation
  • Worst-delay Analysis
    • Interconnect Model
    • Delay Model
    • Worst-delay Corner
    • Case Study
  • Classification of the Worst-delay Direction
  • Conclusions
interconnect model
Interconnect Model
  • Interconnect structure variation (W, T and H)
    • Pitch (S+W) is constant
  • R, C and RC variations
  • Not statistically independent of R and C variations
    • As R increases, C decreases

Cross section model for interconnects

r and c variations
R max

RC max

R max

RC max

R variation[%]

R variation[%]

C max

C max

C variation[%]

C variation[%]

R and C variations

ITRS2005 80nm Intermediate

W, T, H 3σ=20%

  • C max
    • Interconnect becomes thick
    • (W+, T+, H-)
  • R and RC max
    • Interconnect becomes thin
    • (W-, T-, H-)

S=W

S=3W

  • Wider Spacing
    • C variation decreases
    • R variation does not change

Opposite direction

delay model
Delay Model

Delay formula of a RC distributed line

[S.Sakurai, IEEE trans. ’93]

  • Transistor variation Rtr variation
  • Interconnect variation W, T, H variations

Every part of the interconnect is uniformly fluctuated

delay variation model
Delay Variation Model
  • Delay is linear combination of parameters
  • W, T, H and Rtr are normally distributed
  • Delay is normally distributed
    • Statistical worst-delay is

: nominal value : standard deviation

worse delay corner
Worse-delay Corner

Normalization

relative values of sensitivity coefficients

Worst-delay corner

( )

thickness

width

(W-14%, T-14%)

case study
Case Study
  • ITRS2005 80nm
    • High performance model:
    • Intermediate Interconnect:
  • W, T, H and Rtr variations:
  • Realistic Drive strength and Interconnect length
    • Optimally-buffered interconnect length: 94um
    • Optimal drive strength: 32X

buffer

Optimum length

experimental results drive strength
Experimental Results(drive strength)

thickness

Drive strength: 1X

Drive strength: 32X

7%

-2%

10%

width

Opposite direction

-8%

experimental results spacing
Experimental Results (Spacing)

Spacing: S=W

Drive strength: 1X

Spacing: S=3W

Drive strength: 1X

Wider spacing

W and T effects (C effect) become small

Worst-delay corner depends on many parameters (drive strength, spacing, etc.)

outline2
Outline
  • Motivation
  • Worst-delay Analysis
  • Classification of the Worst-delay Direction
  • Conclusions
idea of classification
Idea of Classification
  • Worst-delay Direction
    • Interconnect becomes thick (W+, T+) or thin (W-, T-)
  • Dominant factor
    • C: interconnect thick delay increases
    • R, RC: interconnect thin delay increases

We compare the proportion of each term

The largest term is the dominant factor

example of c dominant case
1

1

2

2

3

3

4

4

Example of C-dominant case

RC dominant

Drive strength:1X Spacing: S=W

C dominant

  • C dominant
    • Second term (RtrC) > Forth term (RCL)
    • Drive strength is small.
    • As interconnect becomes thick (C increases), delay increases.
    • Rtr also affects delay
  • RC dominant
    • Long interconnect
    • As Interconnect becomes thin (R increases), delay increases.
example of r and rc dominant case
1

1

2

2

3

3

4

4

Example of R-and RC-dominant case

Drive strength:32X Spacing: S=W

RC dominant

R dominant

large drive strength

Rtr decreases and CL increases

(RtrC decreases and RCL increases)

As interconnect becomes thin, delay increases.

Optimum drive strength

intermediate vs global
Intermediate vs. Global

Intermediate (thin)

Global (thick)

The boundary of each dominant region changes

depending on layer

R and RC dominant regions become smaller.

Global: R is small

minimum spacing vs wider spacing
Minimum Spacing vs. Wider Spacing

Spacing: S=W

Spacing: S=3W

C variation becomes smaller.

C dominant regions become smaller.

Wider spacing

conclusions
Conclusions
  • We propose a criterion for classifying the worst-delay direction
  • Worst-delay corner is context-dependent
    • Small drive strength:

Thicker interconnect worst-delay

    • Large drive strength or long interconnect:

Thinner interconnect worst-delay

  • This criterion is used as a guideline for the selection of interconnect parasitic values used for the worst-delay calculation.
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