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

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

• Coordinates transform for angular coordinates enables an analytic integral, and address singular points.

• 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:
• Wire bonds, connecter pin array, and through hole via 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