Task Supervisor:Prof' Y'L' Le Coz

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Random-Walk Algorithms for Electromagnetic
Analysis of Massively Coupled IC-Interconnect
Structures
(IFC Task IV Circuit System Design
Modeling)

• Task Supervisor Prof. Y.L. Le Coz
• Students G. Hariharan (MS '02), J.
  Kalyanasundharam (PhD '05),
• D. Krishna (PhD '04)
• Research Associate Dr. R.B. Iverson
• Industry Affiliates Dr. P. Bendix, Dr.
  W. Loh, Dr. D. Petranovic (LSI Logic)

Center for Integrated Electronics Rensselaer
Polytechnic Institute
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Sponsors Acknowledgements

• SRC MARCO Interconnect Focus Center (IFC)
• SRC Custom-Funding Program (CADTS, D. Edwards, W.
  Joyner)
• LSI Logic Corporation (R.D. Schinella)
• DARPA (IFC)
• NYSTAR (IFC, NY State)

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Background Anticipated Result

• Understanding and predicting the electrical and
  optical behavior of on-chip IC interconnects is
  critical to meeting ITRS objectives for the
  present decade.
• Fundamentally one must solve Maxwells equations
  in one of two ways
• One-Step direct solution of the underlying
  field equation (wave-equation PDE)
• Two-Step lumped-element parasitic extraction,
  and solution of the resulting circuit equations
  (PEEC, RLCM coupled ODEs).
• Anticipated Result Creation of efficient, fully
  parallel stochastic (random-walk) algorithms for
  massively coupled, One-Step and Two-Step Maxwell
  solution.
• Philosophy Evaluate IC-interconnect delay and
  cross-talk metrics to directly support high-level
  timing and verification CAD tools encountered in
  the typical industrial IC-design flow.

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Electrical IC-Interconnect RLC IR Extraction

• Approach
• Stochastic evaluation of a Feynman diagrammatic
  expansion.

• Objectives
• Create a novel random-walk (RW) algorithm for
  RLC impulse-response (IR) extraction within
  electrical IC-interconnect networks.
• Maintain full parallelism for future hardware
  speed up, and validate the extraction algorithm
  by comparison with exact, analytical solutions.
• CAD support for other IFC Electrical and
  Optical activities, once this Task has
  sufficiently matured.

See Poster!
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A Key Accomplishment for This Year (Electrical)

• Generalized a new RW algorithm to extract IR
  moments for IC-interconnect networks
• discovered a Sum-over-Paths Postulate (v)
• achieved computational efficiency (v)
• inherently parallel, with little inter-processor
  communication. (v)
• Successfully validated our extraction algorithm,
  by comparison with exact, analytical solutions
• uncoupled RLC lines (v)
• coupled RLC lines. (v)
• The algorithm is clearly valid for arbitrary
  RLC-interconnect networks!

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s-Domain Expansion

• Transfer function

• Taylor-series expansion about s 0

• IR moments

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Feynman Sum-over-Paths Postulate
The order-sn term in an RLC-circuit
transfer-function expansion is a sum of
weight-factor products over all possible unique
paths within the associated network transition
diagram. The product of weight-factors,
constructed over each path, is similarly of order
sn.
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Two-Stage-Line Transition Diagram, Full Circuit
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Two-Stage-Line Transition Diagram, m0(I)
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m0 Moment for a Two-Stage Line, Diagram Sum
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Two-Stage-Line Transition Diagram, m1(I)
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m1 Moment for a Two-Stage Line, Diagram Sum
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Two-Stage-Line Transition Diagram, m2(I)
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m2 Moment for a Two-Stage Line, Intermediate
Diagram Sum
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Two-Stage-Line Transition Diagram, Over-Counting
Correction
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m2 Moment for a Two-Stage Line, Corrected Diagram
Sum
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Uncoupled Coupled RLC Lines, Computational Data
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m1 m2 Moment vs. Number of Samples, Uncoupled
Lines
5 m1 accuracy after 100 samples! 10 m2
accuracy after 100 samples!
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m1 m2 Moment vs. Number of Samples, Coupled
Lines
5 m1 accuracy after 100 samples! 10 m2
accuracy after 100 samples!
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Future Directions

• Extension to RLCM Interconnect Networks (next
  year)
• RW approach is well suited to massively coupled
  systems
• Characteristic of long-range mutual-inductance.
• Application to large, realistic benchmark
  problems (year after next)
• Include, perhaps, as many as 1M1G interactions!
• Bus power-distribution grid networks.
• Extension to m3 (beyond current program scope)
• Higher-order IR moments are important when
  evaluating inductive overshoot.
• Parallel-Hardware Implementation (beyond current
  program scope)
• Expect speed-up factors as large a 1000, with
  nearly 100 processor efficiency.

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3D Maxwell Solution for Optical IC Interconnects

• Approach
• Stochastic evaluation of a Feynman diagrammatic
  expansion.

• Objectives
• Attempt to create a novel random-walk (RW)
  algorithm for IR extraction within optical
  IC-interconnect networks.
• If possible, validate for both materially
  homogeneous heterogeneous domains.
• CAD support for other IFC Electrical and
  Optical activities, once this Task has
  sufficiently matured.

See Poster!
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Tech Transfer, Industrial Interactions,
Publications

• Interactions and Technology Transfer
• IFC MARCO Program Annual Review (Oct. 2002,
  Atlanta, GA).
• Y.L. Le Coz, A Sum-over-Paths Impulse-Response
  Moment-Extraction Algorithm for RLC
  IC-Interconnect Networks (presentation, LSI
  Logic, Santa Clara, CA P. Bendix, W. Loh, P.
  Zarkesh-Ha, et. al, June 2003).
• IFC MARCO Program Quarterly Review (SUNY-Albany,
  Albany, NY, June 2003).
• SRC CADTS Program Annual Review (Oct. 2002, Ann
  Arbor, MI).
• Recent Publications
• Y.L. Le Coz, D. Krishna, G. Hariharan, D.M.
  Petranovic, W.M. Loh, P. Bendix, A
  Sum-over-Paths Impulse-Response Moment-Extraction
  Algorithm for RLC IC-Interconnect Networks (in
  preparation, Solid-State Electronics).
• Y.L. Le Coz, D. Krishna, D.M. Petranovic, W.M.
  Loh, P. Bendix, A Sum-over-Paths
  Impulse-Response Moment-Extraction Algorithm for
  RC IC-Interconnect Networks (to appear,
  Solid-State Electronics).
• Y.L. Le Coz, D. Krishna, D.M. Petranovic, W.M.
  Loh, P. Bendix, A Sum-over-Paths
  Impulse-Response Moment-Extraction Algorithm for
  IC-Interconnect Networks Verification, Coupled
  RC Lines (to appear, ICCAD-2003, Nov. 2003).
• Y.L. Le Coz, J. Kalyanasundharam, D. Krishna,
  R.B. Iverson, A Stochastic Algorithm for 3D
  Maxwell Solution within Optical On Chip IC
  Interconnects Materially Homogeneous, Scalar
  Wave-Equation Benchmarks (VMIC-03, Sept. 2003).
• Y.L. Le Coz, D. Krishna, D.M. Petranovic, W.M.
  Loh, P. Bendix, A Sum-over-Paths
  Impulse-Response Moment-Extraction Algorithm for
  IC-Interconnect Networks Verification, Uncoupled
  RC lines (VMIC-02, Nov. 2002).