June 11, 2008 Ki Jin Han*, Madhavan Swaminathan, and Ege Engin



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Electric Field Integral Equation Combined with Cylindrical Conduction Mode Basis Functions for Electrical Modeling of Three-Dimensional Interconnects

  • June 11, 2008

  • Ki Jin Han*, Madhavan Swaminathan, and Ege Engin

  • School of Electrical and Computer Engineering,

  • Georgia Institute of Technology

  • {kjhan, madhavan, engin}@ece.gatech.edu


Contents

  • Introduction

  • Cylindrical conduction mode basis functions (CMBF’s)

  • Electric field integral equation (EFIE) formulation with CMBF’s

  • Simulation examples

  • Conclusions



Background

  • Interconnections in 3-D integration

    • Role of interconnections is critical for signal transmission.
    • Design and modeling of interconnections are important.
  • Challenges in electrical design of SiP

    • Modeling of the entire coupling among interconnects in an SiP.
    • High-frequency modeling, including skin and proximity effects.


Approach in this Paper

  • State of the art

    • Existing modeling tools: Have issues in accuracy or speed for modeling of large number of 3-D interconnects.
    • Electric field integral equation (EFIE) combined with conduction mode basis function (CMBF)*
      • Advantages
        • Describes current density distribution with a few CMBF’s.
        • Provides simplified equivalent circuit model.
      • Issues for 3-D interconnect modeling
        • Not geometrically suitable for 3-D cylindrical structures.
        • Allocating basis functions is difficult to describe various proximity effects.
  • Approach in this paper: EFIE combined with cylindrical CMBF

    • Maintains advantages of the original CMBF-based approach.
    • Geometrically suitable for 3-D interconnections (bonding wires, through hole vias).
    • Automatically captures skin and proximity effects.


Cylindrical CMBF

  • Derived from solutions of the current density diffusion equation in a conductor.

  • The fundamental mode (skin-effect mode) basis captures skin effect.

  • Two orthogonal higher-order mode (proximity-effect mode) bases captures proximity effects in arbitrary orientations.



Basic Formulation



Calculation of Partial Impedances

  • Partial resistances

    • The resistance matrix is purely diagonal.
  • Partial self inductances

  • Partial mutual inductances

    • Pre-computation of frequency independent integrals improves efficiency of frequency-sweep simulation.


Equivalent Circuit

  • Interconnects are approximated into cylindrical conductor segment model.

  • Cylindrical CMBF are used as global basis functions.

  • Partial resistances and inductances of are computed.

  • Combined equivalent circuit is constructed.



5 by 5 Via Array (1)

  • Geometry



5 by 5 Via Array (2)

  • Self & mutual loop inductances



10 by 10 Via Array (1)

  • Geometry



10 by 10 Via Array (2)



10 by 10 Via Array (3)



102 Bonding Wires on 3 Stacked Chips (1)

  • Geometry



102 Bonding Wires on 3 Stacked Chips (2)

  • Impedances at 10 GHz

    • A wire in class 2 is grounded.


Conclusions

  • EFIE combined with the cylindrical CMBF’s

    • Captures skin and proximity effects with a small number of basis functions.
    • Geometrically suits various 3-D interconnects:
  • Improvements in computation of partial impedances

    • Using analytic expressions, pre-computation for frequency-independent parts, and efficiency enhancement schemes.
    • Large 3-D interconnect structures can be modeled with the proposed method.
  • Future Work

    • Inclusion of capacitive coupling and finite ground effect.
    • Extension of the proposed method for modeling of TSV.


Acknowledgements

  • Mixed Signal Design Tools Consortium (MSDT) at the Packaging Research Center, Georgia Institute of Technology

  • Matsushita

  • EPCOS

  • Infineon

  • Sameer

  • NXP



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