raphael reference manualjmbussat/physics290e/fall-2006/tcad_documentation/raphael_ref.pdfraphaeltm...

RaphaelTM

Interconnect Analysis Program

Reference ManualVersion Y-2006.03, March 2006

ii

Copyright Notice and Proprietary InformationCopyright © 2006 Synopsys, Inc. All rights reserved. This software and documentation contain confidential and proprietary information that is the property of Synopsys, Inc. The software and documentation are furnished under a license agreement and may be used or copied only in accordance with the terms of the license agreement. No part of the software and documentation may be reproduced, transmitted, or translated, in any form or by any means, electronic, mechanical, manual, optical, or otherwise, without prior written permission of Synopsys, Inc., or as expressly provided by the license agreement.

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Table of Contents

CONTENTS

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About This Manual xi

Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiRelated Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiConventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiCustomer Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Accessing SolvNet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiContacting the Synopsys Technical Support Center . . . . . . . . . . . . . xiii

Chapter 1 Using Raphael 1-1Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

2D and 3D Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1GDS II Stream Format Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

Raphael Graphical User Interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3Field Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

RC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3RC2-BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4RC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4RC3-BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4RI3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4

RIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4GDS II Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5DPLOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5Taurus Visual Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6Command Description Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6Command Editor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7Naming Convention for Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7Raphael Flow to Extract Parasitics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8

Flow Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9

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Table of Contents Raphael Tutorial

Chapter 2 RC2: 2D Resistance, Capacitance, and Inductance 2-1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1RC2 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3

PARAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4BOX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5CIRC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6CIRC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8POLY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10COPY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11WINDOW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12MERGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13POTENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14CAPACITANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14CURRENT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15RESISTANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15INDUCTANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16Z0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16SPICE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16EXTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18OPTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

Theory of Floating Conductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20Selection of Linear Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21Examples Using RC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22

Example 1: Inductance Simulation of a Line Above Ground Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22

Example 2: Three Lines Above a Plane. . . . . . . . . . . . . . . . . . . . . . 2-24Example 3: Current Density and Resistance Analysis. . . . . . . . . . . 2-29Example 4: SPICE Model Extraction . . . . . . . . . . . . . . . . . . . . . . . 2-32Example 5: Floating Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-34Example 6: Anisotropic Dielectric Materials. . . . . . . . . . . . . . . . . . 2-36

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-38

Chapter 3 RC2-BEM: 2D Field Solver by Boundary Element Method 3-1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2FD, FEM, BEM Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

RC2-BEM Command Line Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3Notes on the Current Version of RC2-BEM. . . . . . . . . . . . . . . . . . . . . . 3-4Comparison of RC2-BEM and RC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

Example 1: Microstrip Lines Above a Ground Plane . . . . . . . . . . . . 3-6Example 2: Inhomogeneous Dielectric Layers . . . . . . . . . . . . . . . . . 3-9

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Example 3: Modeling of the Power-Plane Resistance . . . . . . . . . . . 3-13References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-15

Chapter 4 RC3: 3D Resistance, Capacitance, and Thermal Resistance 4-1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1RC3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3

PARAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4BLOCK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5CYLINDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9SPHERE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11POLY3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12COPY3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15MERGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16WINDOW3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17POTENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19CAPACITANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19CURRENT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19RESISTANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19TEMPERATURE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19THERMORES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-20EXTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-20OPTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-20

Theory of Floating Conductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21Selecting Linear Solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21Creating Graphics Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-22Interface to Taurus Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-23Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24

Example 1: Current Density and Resistance of a Cylindrical Via . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25

Example 2: Capacitance and Potential Analysis of a Cross-Over Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-29

Example 3: Floating-Gate Transistor . . . . . . . . . . . . . . . . . . . . . . . . 4-35Example 4: Anisotropic Dielectric Materials. . . . . . . . . . . . . . . . . . 4-37

Chapter 5 RC3-BEM: 3D Field Solver by Boundary Element Method 5-1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1RC3-BEM Command-Line Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2Notes on the Current Version of RC3-BEM. . . . . . . . . . . . . . . . . . . . . . 5-3Comparison of RC3-BEM and RC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

Example 1: Single Plate Above a Plane. . . . . . . . . . . . . . . . . . . . . . . 5-4Example 2: Crossover Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8Example 3: A Trapezoidal Conductor Between Two

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Ground Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-10Example 4: Substrate Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 5-12

Chapter 6 RI3: 3D Resistance and Inductance with Skin Effect 6-1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1Program RI3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4RI3 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

PARAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6NODE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6PLANE_NODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7SINGLE_BAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9MULTI_BAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10PLANE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12OPTIONS3I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13MATRIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13MERGE3I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14EXT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14REF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15FREQUENCY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15ADMITTANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15SPARAMETER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16OUTPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16

Equivalent Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16Selection of Extraction Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17

Example 1(raexi31): Inductance of Two Parallel Microstrips. . . . . 6-17Example 2 (raexi32): Skin Effect Simulation . . . . . . . . . . . . . . . . . 6-20Example 3 (raexi33): Circular Ring Above Ground Plane . . . . . . . 6-23Example 4 (raexi34): Inductance of Four Bond Wires . . . . . . . . . . 6-27

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-31

Chapter 7 Raphael Interconnect Library 7-1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1Raphael Interconnect Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1Running RIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3RIL Interactive Commands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

ADD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5CHECK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5GENERATE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5INPUT CHECK / PLOTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6LIST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6PRINT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

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NEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6SAVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7VISUALIZE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7TABLE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7QUIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

Directory Structure and Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8Customization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8

start_up File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8RIL Environment Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9

SPICE Netlist Generation and Naming Convention. . . . . . . . . . . . . . . 7-10RIL Example Sessions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12

Session 1: 3D Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-12Session 2: 2D Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16Session 3: RIL and STUDIO Visualize . . . . . . . . . . . . . . . . . . . . . . . . . . 7-21

Chapter 8 Advanced Parser of Interconnect Structures 8-1Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1 Running APIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3Using the APIS Program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5

APIS Input and Output Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5Assigning Metal and Text Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6Boolean and Size Operations at Layout Layers . . . . . . . . . . . . . . . . . 8-6Specifying 3D Arrangement of Metal Layers . . . . . . . . . . . . . . . . . . 8-8Net Constructing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9Specifying Dielectric Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-10Specifying Conformal Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-11Specifying Net Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-12Defining Pins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13Generating RI3 Bars and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14

APIS Program Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18HEADER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-18ASSIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-20BOOLEAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-21SIZE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-22GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-23DIELECTRIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-24CONFORMAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-26RI3_OPTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-28WINDOW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-29TEXT_OPTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-31VOLTAGE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-32CONNECTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33

Labeling File Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-34

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RIL Templates A-1

3D Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-12D Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

Boundary Conditions for the BEM Solvers B-1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1Open-Space Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1Neumann Boundary Condition (or, Magnetic Ground Plane) . . . . . . . . B-2Dirichlet Boundary Condition (or Electric Ground Plane) . . . . . . . . . . . B-3Two Magnetic Planes Facing Each Other. . . . . . . . . . . . . . . . . . . . . . . . B-3

Capacitance Theory C-1

Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-2

Ground Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3Inhomogeneous Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-42D vs. 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4Capacitance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4Two-Terminal Capacitances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-5Method of Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-6Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-6Thomson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-7Boundary-Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-7Finite-Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-8Finite-Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-8Relationship Between [L] and [C] . . . . . . . . . . . . . . . . . . . . . . . . . . . C-8Resistance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-9

Analytic Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-10

Inductance Theory D-1

Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-1Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2Magnetic Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2Effective Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-3Internal and External Inductances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-4Current Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-4Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-5Equivalent Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-5

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Taurus Topography Interface E-1

Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2Taurus Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2Taurus Topography 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3Raphael RC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3

Command-Line Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-3Input File Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-4

REGION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-5OPTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-6

Example Using the Taurus Topography Interface . . . . . . . . . . . . . . . . . E-7Step 1: Creating a Mask Layout with Taurus Layout . . . . . . . . . . . . E-7Step 2: Creating an Input File for Taurus Topography . . . . . . . . . . . E-9Step 3: Computing Capacitance, Resistance with Raphael RC3 . . . E-11

Capacitance Post-Processing F-1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1Cpost SPICE Input Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-2Cpost Command Input Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-2

INPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-3OUTPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-3FLOAT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-3GROUND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4DEFAULT GROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4DEFAULT FLOAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4SIGNAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4INDUCTANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-4UNIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-5END . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-6QUIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-6

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-7Example 1: Batch Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-7Example 2: Multiple Input Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-8Example 3: Interactive Mode and Inductance . . . . . . . . . . . . . . . . . . F-9

Frequently Asked Questions (FAQs) G-1

RC2 and RC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-1RI3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-2RPD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-2

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Glossary Glossary-1

Index Index-1

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ABOUT THIS GUIDE

About This Manual1

This guide covers all aspects of using Raphael and using the Raphael graphical user interface (GUI) and its four models. The Raphael GUI helps to systematically perform the large number of simulations necessary to completely characterize the interconnect parasitics for different process technologies. The program transfers the three-dimensional (3d) field solver accuracy to the Layout Parameter Extrac-tor (LPE) tools.

This manual contains the following topics:

• A general purpose 2D program for solving Poisson’s equation.• RC2-BEM, a 2D field solver.• RC3, a 3D program for solving Poisson’s equation.• RC3-BEM, a viable 3D field solver to RC3 for 3D static field analysis.• RI3, a computer program for conducting 3D inductance.• Raphael Interconnect Library, a database program that generates and stores

electrical model parameters for interconnect elements.• Advanced Parser of Interconnect Structures (APIS) utility for building 3D

interconnect structures using mask layout files, written in the industry-standard GDS II format.

AudienceThis manual is for anyone wanting to use Raphael.

Related PublicationsFor additional information about Raphael, see:

• Synopsys Online Documentation (SOLD), which is included with the software for CD users or is available to download through the Synopsys Electronic Software Transfer (EST) system

• Documentation on the Web, which is available through SolvNet at http://solvnet.synopsys.com

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• The Synopsys MediaDocs Shop, from which you can order printed copies of Synopsys documents, at http://mediadocs.synopsys.com

• You might also want to refer to the documentation for the following related Synopsys products:- For information on Raphael installation procedures, see the TCAD

Products and Utilities Installation Manual.- For information on the use of STUDIO Visualize tool, see the STUDIO

Visualize User Manual.- For information on the Taurus-Layout tools, see the Taurus Layout User

Manual.

ConventionsThe following conventions are used in Synopsys documentation.

Convention Description

Courier Indicates command syntax.

Italic Indicates a user-defined value, such as object_name.

Bold Indicates user input—text you type verbatim—in syntax and examples.

[ ] Denotes optional parameters, such as write_file [-f filename]

... Indicates that a parameter can be repeated as many times as nec-essary:pin1 [pin2 ... pinN]

| Indicates a choice among alternatives, such aslow | medium | high

\ Indicates a continuation of a command line.

/ Indicates levels of directory structure.

Edit > Copy Indicates a path to a menu command, such as opening the Edit menu and choosing Copy.

Control-c Indicates a keyboard combination, such as holding down the Control key and pressing c.

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Customer SupportCustomer support is available through SolvNet online customer support and through contacting the Synopsys Technical Support Center.

Accessing SolvNetSolvNet includes an electronic knowledge base of technical articles and answers to frequently asked questions about Synopsys tools. SolvNet also gives you access to a wide range of Synopsys online services including software downloads, Documentation on the Web, and entering a call to the Support Center.

To access SolvNet:

1. Go to the SolvNet Web page at http://solvnet.synopsys.com.

2. If prompted, enter your user name and password. (If you do not have a Synop-sys user name and password, follow the instructions to register with SolvNet.)

If you need help using SolvNet, click SolvNet Help in the Support Resources section.

Contacting the Synopsys Technical Support CenterIf you have problems, questions, or suggestions, you can contact the Synopsys Technical Support Center in the following ways:

• Open a call to your local support center from the Web by going to http://solvnet.synopsys.com (Synopsys user name and password required), then clicking “Enter a Call to the Support Center.”

• Send an e-mail message to your local support center.- E-mail [email protected] from within North America. - Find other local support center e-mail addresses at

http://www.synopsys.com/support/support_ctr.• Telephone your local support center.

- Call (800) 245-8005 from within the continental United States.- Call (650) 584-4200 from Canada.- Find other local support center telephone numbers at

http://www.synopsys.com/support/support_ctr.

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http://solvnet.synopsys.com

http://solvnet.synopsys.com

http://www.synopsys.com/support/support_ctr

http://www.synopsys.com/support/support_ctr

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CHAPTER 1

R

Using Raphael1

OverviewRaphael is the result of a collaboration with Hewlett-Packard (HP) to develop a simulator for parasitic modeling in the semiconductor industry.

Raphael simulates the electrical and thermal analysis of two- and 3D (2D and 3D) structures, such as on-chip interconnect structures, packaging, multi-chip mod-ules, and printed circuit boards. Analysis is made in the following electrical areas:

• Resistance

• Capacitance

• Inductance (including skin effects in 3D structures)

• Current flow densities

• Thermal and potential distribution

2D and 3D Solvers

Within the Raphaell program are five interconnect solvers, or analyzers, designed for different simulation domains. The 2D electrical parameters are best analyzed with RC2 and RC2-BEM, as described in Chapter 2 and Chapter 3. RC3 and RC3-BEM (see Chapter 4 and Chapter 5) solve 3D thermal and electrical problems. The calculation of inductances in 3D structures formed by bars and planes can be analyzed by RI3, which is described in Chapter 6.

Different areas of parasitic simulation analysis present special design challenges. To facilitate the generation of 3D structures, Raphael allows for the input of two- and 3D structures from different sources. Because the input languages of the solv-ers allow for parametric definition of variables, a design of a 3D structure can be developed from these sources. However, this approach is problematical when the number or complexity of the input structures is above a certain limit.

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Overview Raphael Reference Manual

Figure 1-1 diagrams the Raphael data flow, which begins with input from three different sources with different structures and ends with Raphael output.

Raphael offers a graphical user interface (GUI) that is specially designed for the creation and interactive use of interconnect databooks used to characterize differ-ent technologies (see the Raphael Tutorial). The GUI offers four modes of opera-tion:

• Parasitics Database and LPE Tools Interface

• Field Solvers

• Interconnect Library

• Taurus Layout and Net Extraction System (NES)

Parasitics Database is of special interest to silicon foundries in the characteriza-tion of the interconnects of different technologies, such as CMOS 0.35 or BICMOS 0.80 micron technology. This mode of operation requires a Raphael solver license. The database can then be distributed to IC designers for use in interconnect parasitics extraction. With the LPE Tools interface, IC designers can create the rule files for such LPE tools as Cadence’s Dracula, Diva, and Vampire and Mentor Graphics’ xCalibre and ICextract. The field solvers, RC2, RC2-BEM, RC3, RC3-BEM, and RI3, can be accessed through the GUI. The Raphael Inter-connect Library (RIL), described in Chapter 7, consists of a set of parametrically defined structures that ease Raphael simulations.

Figure 1-1 Raphael data flow

Manual Input

GDS II

Raphael

Electrical Lumped Values

Distributions

PotentialCurrentTemperature

SPICE Netlists

RIL

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Raphael Reference Manual Raphael Graphical User Interface

GDS II Stream Format Interface

One of the most important features of the input structure generation for Raphael is the GDS II Stream format interface, which is implemented in Taurus Layout (See the Taurus Layout Tutorial). To generate the initial structure, a key step in a Raphael simulation, Raphael can use predefined models from RIL or read GDS II layout information. The structures, which can also be generated manually, are pre-sented as lumped parameters, distributions, or SPICE net lists.

Raphael Graphical User InterfaceThe Raphael GUI facilitates characterization of interconnect technologies, such as CMOS 0.35 μm or BICMOS 0.80 μm, and interactively visualizes and models the interconnect characteristics. A companion to this manual, the Raphael Tutorial, describes this interface in greater detail.

To run the interface, on the UNIX command line, type:

raphael or raphael -g

The Raphael main window opens to show four modules:

• Parasitics Database

• LPE Tools Interface

• Field Solvers

• Interconnect Library

This manual discusses the contents of the Field Solvers modules and the Interconnect Library. This material is to be used in conjunction with the Raphael Tutorial, which also describes the Parasitics Database and the LPE ToolsInterface.

Field SolversThe Raphael program provides five interconnect analyzers, RC2, RC2-BEM, RC3, RC3-BEM, and RI3.

RC2

The RC2 solver may be used for the entire range of 2D problems. RC2 solves the Laplace and Poisson equations using the finite difference method, with automatic gridding and regridding, as discussed in detail in Chapter 2.

RC2 interprets all structures as infinitely long in the direction normal to the simu-lation domain. Resistance, capacitance and inductance are presented as the corre-sponding electrical units per unit length. For example, capacitance is given as Farad/meter.

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RIL Raphael Reference Manual

RC2-BEM

RC2-BEM is an alternative field solver for RC2 based on the boundary element method. Although the current version of RC2-BEM has more limited applications than RC2, for those application where RC3-BEM can be applied, it provides a fast alternative for RC2. Differences between the RC2 and RC2-BEM solvers are dis-cussed in Chapter 3. Several examples in that chapter benchmark the RC2 and RC2-BEM solvers.

RC3

The RC3 solver, described in Chapter 4, may be used for the calculation of resis-tances, capacitances, current density distribution, and thermal and potential distri-bution in three dimensions. As with RC2, RC3 uses the finite difference method for discretization of equations with automatic gridding and regridding. Basic structures are specified with commands that define basic built-in modules such as sphere, cylinder and extruded polygons.

RC3-BEM

Similar to RC2-BEM, RC3-BEM is an alternative field solver for RC3 based on the boundary element method. Unlike the 2D solver, the increase in speed of RC3-BEM over RC3 is marginal or worse for complex geometries. Differences between the RC3 and RC3-BEM solvers are presented in Chapter 5 along with benchmarking examples.

RI3

The RI3 solver may be used for 3D inductance and resistance simulations, includ-ing the variation of these parameters with frequency due to the skin effect. All other electrical and thermal characteristics should be calculated using RC3.

RI3 is more limited than RC3 in the scope of the geometrical structures it can cal-culate. The RI3 solver is oriented to square section wires and RC3 to more general shapes. Because the structures that can be analyzed by RI3 can also be analyzed by RC3, the input files for these structures may be used by both solvers. However, because some RC3 structures are too complex to be analyzed with RI3, the input files are not consistently interchangeable. The methods used by RI3 are described in Chapter 6.

RILThe Raphael Interconnect Library (RIL) performs complex simulations on pre-defined structures with an easily accessible user interface. See Chapter 7 for a description of this library.

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Raphael Reference Manual GDS II Interface

When a new structure is calculated, RIL defines the parameters. The RIL interface also generates the input files and executes the corresponding solvers for each new structure. Currently, RIL does not use BEM solvers (RC2-BEM and RC3-BEM). Once a database of parasitic values is generated, the dependence of each value on one of the library element parameters can be inspected with STUDIO Visualize. For more information on the STUDIO Visualize one-dimensional plotting tool, refer to the corresponding chapter in the STUDIO man-ual, which lists the complete set of RIL library elements.

GDS II InterfaceThe GDS II Stream format is the most widely used data format for layout repre-sentation. A GDS II file stores layout information as polygons, lines, or labels in up to 1,024 different layers. The structures drawn in the file are usually intended as mask information for IC processing or for the manufacture of package lead-frames. A GDS II file cannot store information related to material properties, or process information related to the intended use of a particular structure, or thick-ness of a given layer. Information must be stored as flat or hierarchical. For exam-ple, for flat information, every polygon is drawn in its intended position. For information referred to as hierarchical, a given structure is drawn only once, with pointers to places where the structure should be reproduced.

The GDS II interface to Raphael is implemented in Synopsys TCAD’s IC Layout Interface program, Taurus Layout. With this interface it is possible to load a GDS II Stream format file that is either flat or hierarchical. The interface also defines the process information that identifies ordering of layers and properties of materi-als. This information is used when generating Raphael input files for parasitics extraction simulation. (Refer to the Taurus Layout Tutorial.)

DPLOTDPLOT is Raphael’s 2D and 3D plotting package used for visualization of struc-ture and for graphical presentation of results, such as potential, temperature and current density distributions. The full DPLOT manual is provided in the Supple-mental Information section at the end of this manual.

To visualize results of Raphael simulations, type:

raphael dplot [FILE]

The syntax of the input file is explained in the DPLOT manual. If <file> is omit-ted, DPLOT takes the input from the standard input. DPLOT reads <file>, it plots according to the input, and writes the results to the output file <file>.out. Usually, DPLOT reads the structure and the results to be plotted from a file of type.pot that it is created after an RC2, RC3 or RI3 simulation. For more information on this subject see the examples for simulation and plotting in Chapter 2, Chapter 4, and Chapter 6.

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Note:Synopsys will continue to include DPLOT as part of the Raphael release; however, DPLOT is no longer being developed. Please use Taurus Visual to visualize output from Raphael. Taurus Visual is easier to use, has higher capacity, and better quality graphics.

Taurus Visual SupportIn addition to the original .pot file for DPLOT, Raphael supports Taurus Visual, Synopsys TCAD’s advanced graphic package. Taurus Visual uses the Technology Data Format (TDF), developed by Synopsys TCAD internally, based on the Hier-archical Data Format (HDF) from the University of Illinois at Urbana-Champaign. Raphael uses the .tdf extension for the TDF graphic file.

Command Description Format The commands to input information and run the individual solvers are described in the input sections of the corresponding chapters. This section describes the con-ventions used in these input sections.

Several special characters are used in the formatted parameter list that appears at the beginning of each command description: square brackets [ ], braces {}, and parentheses ( ). These characters are part of the description, not part of the com-mand.

The special character semicolon (;) can be used as the delimiter of a parameter definition.

The character plus (+), when placed as the first character in a command line, means that this line is a continuation of the previous line. When the character plus (+) is placed at the end of a line, it means that the following line is a continuation of the current line.

The special characters star (*) and dollar sign ($) can be used indicate comment lines when either one of them is placed as the first character of a line.

The input is in free format, and upper and lowercase characters are interpreted dif-ferently, i.e., the input is case sensitive, meaning any instruction is recognized when written in all lowercase or all uppercase letters.

In the following command, the term group refers to a parameter by itself or a set of parameters enclosed in a matched pair of square brackets, braces, or parenthe-ses. For example,

{(PARM1; [PARM2; [PARM3;]] PARM4;) PARM5;}

constitutes a valid group, composed of the subgroups: (PARM1; [PARM2; [PARM3;]] PARM4;) and PARM5. The first subgroup may be further subdi-

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Raphael Reference Manual Command Editor

vided into the subgroups: PARM1, [PARM2; [PARM3;]] and PARM4, and so on.

Square brackets enclose groups that are optional. For example,

NEWCOMMAND [PARM1;] [PARM2; PARM3;] [PARM4; [PARM5;]]

indicates that in the NEWCOMMAND command, the parameter PARM1 is optional. The group [PARM2; PARM3;] is optional, but if PARM2 is specified, then PARM3 must also be specified. The group [PARM4; [PARM5;]] is optional, but PARM5 may be specified only if PARM4 is specified.

When one of a list of groups must be selected, the groups are enclosed in braces and separated by the special word, OR, that does not form part of the command. For example,

NEWCOMMAND {PARM1; OR PARM2; OR (PARM3; PARM4;)}

indicates that the NEWCOMMAND command requires that one and only one of the three groups PARM1, PARM2, or (PARM3; PARM4;) be specified.

Parentheses enclose groups that are to be considered as single items in higher level groupings. For example, in the above NEWCOMMAND command, the group (PARM3; PARM4;) constitutes one of three possible choices and is, therefore, enclosed in parentheses. With the exception of the semicolon, the special charac-ters are used only for documenting the input syntax and do not form part of the actual input to Raphael.

Command EditorAn important feature of Synopsys TCAD’s STUDIO environment is the STUDIO Command Editor. The command editor introduces the syntax of Synopsys TCAD software tools. To learn more about the Command Editor, refer to the STUDIO User’s Manual.

Naming Convention for ExamplesAll [email protected] examples presented in this manual are named as follows:

• For RC2 and RC2-BEM: raexc2#

• For RC3 and RC3-BEM: raexc3#

• For RI3: raexi3#

where # is the number for the particular example.

Input file names do not need extension. The extensions added to the input file names to generate the names of the output files are discussed in the chapters that describe the solvers.

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Raphael Flow to Extract Parasitics Raphael Reference Manual

The corresponding DPLOT input files for plotting the results from each of the three solvers use the same name as the solver example with the prefix dp. For example, the DPLOT input file for plotting the results of the first RC2 analysis is dpraexc21.

Raphael Flow to Extract Parasitics The following sequence of steps defines the Raphael work flow, from generating a parasitics database to obtaining the final SPICE models of the interconnects that need to be extracted. Refer to Figure 1-2 for a graphical representation of the flow.

• Input technology characteristics to Raphael.

• Make field solver simulations on the structures in the layout to generate the parasitics database.

• Execute the LPE Tool Interface to generate a rule file with accurate parasitic capacitance coefficients in the syntax of the LPE tool.

• Input the rule deck to the LPE tool, along with information from the layout database, such as the GDS II stream.

• Complete the final SPICE models of the interconnects to be extracted.

See the Raphael Tutorial for a complete discussion of Raphael parasiticsextraction.

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Flow Diagram

Figure 1-2 illustrates the entire flow of parasitics extraction using Raphael.

Figure 1-2 Raphael parasitics extraction flow

Parasitics Database

Generic Regression Database

Field Solver

LPE Layer Information

LPE Rule Deck

LPE Tools

Technology Characteristics RegressionAnalysis

LPE Tool Interface

Layout Database GDS II

SPICE Models

RAPHAEL

Raphael 4.1or later only

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CHAPTER 2

R

RC2: 2D Resistance, Capacitance, and Inductance2

IntroductionRC2, a general-purpose, 2D program for solving Poisson’s equation, is based on the finite-difference method with an automatically adjustable rectangular mesh. The linear equations set up by the finite-difference method are solved by the Mod-ified Incomplete Cholesky Conjugate Gradient method (MICCG), which is a default method, or Incomplete Cholesky Conjugate Gradient method (ICCG). The combination of the automatic adjustment of mesh and the speed of the linear equa-tion solvers makes RC2 versatile and easy to use. RC2 can be used for any 2D Poisson problems, such as capacitance calculation and inductance calculation for transmission lines with the quasi-TEM approximation. This includes Laplace’s equation, which is a special case of Poisson’s equation.

RC2 uses dynamic memory allocation so the complexity of the structure to be analyzed is limited only by the available memory. In addition, RC2 allows a com-pletely arbitrary configuration of conductors and insulators. (Manhattan geome-tries are not required, and materials of different isotropic or anisotropic dielectric constant may be used.) RC2 uses two different models for discretization of initial Poisson’s equation. The first one is based on the nodal representation of the mate-rial properties (default model), as the second one is based on the elemental repre-sentation of the material properties. The last model is more accurate and useful for Manhattan geometry structures with strongly varying dielectric constants.

The syntax of the RC2 command line is:

The syntax of the input file is explained in the next section. If <file> is omitted, RC2 takes the input from the standard input. If no options are specified, RC2

raphael rc2 [OPTIONS] [FILE]

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Introduction Raphael Reference Manual

reads <file>, does calculations according to the input, and writes the results to the output file named <file>.out. One other file, <file>.pot is created that contains the input geometry information and the potential and electric field data after the potential calculation has been completed. The DPLOT program reads <file>.pot and plots the potential and the electric field. An additional graphics file, <file>.tdf may be created for visualization using the Taurus Visual program.

RC2 interprets the following options:

Option Definition

-b Invokes RC2-BEM. See Chapter 3.

-d Forces conductors to override the overlapping dielectrics.

-h Creates an additional graphics file in Technology Data Format (TDF) to be visualized by Taurus Visual.

-i Checks the input file for errors and creates a geometry graphics file; no calculation is performed.

-n No graphics file is created.

-o <file> Uses <file> as the output file name.

-p <file> Uses <file> as the potential file name.

-s Sends the output to the standard output.

-t Prints out a CPU time summary.

-u Appends an output summary to the output file. This summary is to be used by RIL for easy parsing of the RC2 output for post-processing.

-w <file> Specifies the name of a mesh file which contains the placement of grid lines in the x and y directions. This mesh file uses a sim-ple syntax:x <x1> <x2>... specifies the placement of grid lines perpendicular to the x-axis. y <y1> <y2>... specifies the placement of grid lines perpendicular to the y-axis. Positions of the grid lines are in the length units specified in the RC2 input file with the command OPTIONS.

-W Prints out the mesh distribution used in the final step of compu-tation to the output file.

-x Potential information is not written in <file>.pot or <file>.tdf.

-z The input file is not echoed in the output listing.

-j Skips the overlapping rule and considers all polygons in the input file during simulation. (Using this option will increase the simulation time)

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Raphael Reference Manual RC2 Input File

RC2 Input FileThe input file statements consist of comments and/or commands. The lines begin-ning with * or with $ are comment lines and are ignored. There are 1 available commands: PARAM, BOX, CIRC1, CIRC2, POLY, COPY, WINDOW, POTENTIAL, CAPACITANCE, INDUCTANCE, Z0, CURRENT, RESISTANCE, MERGE, SPICE, EXTRACT, and OPTIONS. A command may be followed by assignments and/or lists defined below and may occupy more than one line.

The input is in free format and uppercase and lowercase characters are interpreted differently, i.e., the input is case-sensitive. Any instruction is recognized when written with all lowercase or all uppercase letters. The syntax and usage of each command are described in this section.

The structures to be analyzed are first built up using the PARAM, BOX, CIRC1, CIRC2, POLY, COPY, and MERGE commands. The analysis is then performed by specifying POTENTIAL, CAPACITANCE, INDUCTANCE, CURRENT, RESIS-TANCE, Z0 (for impedance), or SPICE (for the generation of SPICE model).

Analyses performed by RC2 are divided into two categories:

1. Electrostatic analysis, indicated by either POTENTIAL, CAPACITANCE, INDUCTANCE, Z0, or SPICE commands,

2. Static resistance analysis (for planar resistive sheets only), indicated by either the CURRENT or RESISTANCE commands.

The EXTRACT command obtains the value of the current or the electric field and the potential at a given location. It can be used only with the POTENTIAL or the CURRENT commands.

CAUTIONElectrostatic analysis is intended for unit length parameters (capacitance, in-ductance, etc.) of transmission line structures (infinitely long), whereas static resistance analysis is intended to find a resistance of a thin planar conductor; hence, the commands specifying the electrostatic and resistance analyses are

-P “<param> = <value>;“

Replaces the value of the parameters defined in the input file with new values. For example, -P “w=0.5; h = 0.1” will override the values of parameters w and h defined in the input file with the new values, 0.5 and 0.1 respectively

-I Invokes ICCG method for solving linear equation system instead of the default MICCG method.

-E Invokes elemental model of the Poisson’s equation discretiza-tion instead of the default nodal model.

-N Invokes new linear solver (see Selection of Linear Solver, p. 2-21 for more details)

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RC2 Input File Raphael Reference Manual

incompatible and must not appear in the same input file. For instance, POTENTIAL and RESISTANCE commands can not be used together.

For electrostatic analysis, an object created by using geometry commands, such as BOX, CIRC1, CIRC2, and POLY, can be either a conductor or dielectric. This is designated by using either the VOLT (for a conductor) or DIEL (for a dielectric) keywords in the geometry command. Anisotropic dielectrics can also be specified, provided the dielectric constant is diagonal.

For static resistance analysis, an object can be either an electrical contact (elec-trode) or conductor. This is specified by the keywords VOLT (for an electrical con-tact) and RHO (for a conductor). An electric contact and conductor in static resistance analysis are analogous to a conductor and dielectric in electrostatic analysis.

Lines beginning with the special symbol (+) are considered a continuation line from the last entered command. Lines ending with the special symbol (+) indicate the following line is considered a continuation line. The symbol is not required and can be omitted.

Overlapping Rule: When geometries defined by BOX, CIRC1, CIRC2, POLY, and COPY commands overlap, the geometry defined later in the input overwrites the geometry previously defined in the input file. This overlapping rule makes it easy to input many structures such as hollow conductors and metal lines embed-ded in dielectric material. It is often convenient to make conductors override dielectrics by using the -d command-line option (see Introduction, p. 2-1 for command-line options).

Length Units: By default, all geometrical dimensions are in microns. The default can be changed with the OPTIONS command.

PARAM

This command defines variables and their values for later use in the input. The variable name must begin with an alphabetic character, but may include up to 300 alphanumeric characters. value can be any algebraic expression with numbers, predefined variables, and functions.

The supported functions are sin(), cos(), atan(), log(), log10(), exp(), sqrt(), int(), and abs(). The argument for any of the trigonomet-ric functions should be given in radians. The unit for the value used for the length dimension is specified in the OPTIONS command. This definition of value is valid on any value arguments in the other commands.

Example:

PARAM PARAMETER1=<value>; PARAMETER2=<value>;...;

PARAM A=2.0; B=4; Y=EXP(A+B);

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The command defines a value of 2.0 for parameter a, parameter b is assigned a value of 4, and y is defined as the exponential value of the addition of a and b.

BOX

This command defines a rectangular object.

BOX NAME=<NAME>; CX=<VALUE>; CY=<VALUE>; W=<VALUE>; H=<VALUE>;[ANG=<VALUE>;]{(VOLT=<VALUE>;[FLOAT=<VALUE>;]) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or(DIEL=<VALUE>,<VALUE>; [CHRG=<VALUE>;]) orRHO=<VALUE>;}[COLOR=<VALUE>;]

Parameter Data Type Definition

NAME character Name of the element. Names must begin with an alphabetic character, but may include up to 300 alphanumeric characters.

CX numeric x-coordinate of the center. default units: microns

CY numeric y-coordinate of the center. default units: microns

W numeric Width (dimension along local x' -axis). default units: microns

H numeric Height (dimension along local y'-axis). default units: microns

ANG numeric Optional rotation angle of the local coordinate system about the center.units: degreesdirection: counter clockwisereference: positive x-axisdefault value: 0

VOLT numeric Potential of electrode element. units: volt

FLOAT numeric Fixed charge or current of the floating electrode. units: coulomb/unit or ampere

DIEL numeric Relative dielectric constant of dielectric element. Use of two values denotes an anisotropic dielectric. The first value refers to the x component, the second, the y component.

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Example:

The command defines an electrode named square, with a width of 5 and a height defined as the addition of a and b, centered at coordinates (1,1) in a global coordi-nate system, rotated at an angle of 45 degrees with respect to the positive X axis, and biased at with 2.5 volts.

CIRC1

This command defines a full or partial circular element.

CHRG numeric Optional fixed charge density in dielectric element. units: coulomb/unit3 where unit is specified in the OPTIONS commanddefault value: 0

RHO numeric Sheet resistivity of material (used only for CURRENT or RESISTANCE calculations). units: ohm

COLOR numeric Optional color index to be used when plotting.

Figure 2-1 BOX type geometric element


hw

(cx,cy)

ang

x

x'y'y

BOX NAME=SQUARE; W=5; H=(A+B); CX=1; CY=1; ANG=45; VOLT=2.5;

CIRC1 NAME=<NAME>; CX=<VALUE>; CY=<VALUE>; R=<VALUE>; [ANG1=<VALUE>;] [ANG2=<VALUE>;] {(VOLT=<VALUE>;[FLOAT=<VALUE>;]) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>;} [COLOR=<VALUE>;]

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CX numeric x-coordinate of center. default units: microns

CY numeric y-coordinate of center. default units: microns

R numeric Radius. default units: microns

ANG1 numeric Optional starting angle of partial circle. units: degrees, default value: 0

ANG2 numeric Optional ending angle of partial circle. units: degrees, default value: 0



DIEL numeric Relative dielectric constant of dielectric element. Use of two values denotes an anisotropic dielectric. The first value refers to the x component, thesecond, the y component.

CHRG numeric Optional fixed charge density in dielectric element. units: coulomb/unit3 where unit is specified in OPTIONS command.



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Example:

The command defines an electrode named circle, centered at coordinates (1,2) in a global coordinate system, with a radius of 0.5, and biased at 1.0 volt. As neither ang1 nor ang2 has been defined, the electrode is a full circle.

CIRC2

This command defines a full or partial circular element.

Figure 2-2 CIRC1 type geometric element

(cx,cy)

ang1

ang2r

P

CIRC1 NAME=CIRCLE; CX=1.0; CY=2.0; R=0.5; VOLT=1.0;

CIRC2 NAME=<NAME>; CX=<VALUE>; CY=<VALUE>; PX=<VALUE>; PY=<VALUE>; [ANG=<VALUE>;]{(VOLT=<VALUE>; [FLOAT=<VALUE>;) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>;} [COLOR=<VALUE>;]



CX numeric x-coordinate of center. default units: microns

CY numeric y-coordinate of center. default units: microns

PX numeric x-coordinate of a point, P, on the periphery. default units: microns

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Example:

The command defines a sector of a circle as an electrode named c, centered at coordinates (1,1) in a global coordinate system, a point at coordinates (3,3) in the periphery. An angle of 270 degrees measured counter clockwise from the refer-

PY numeric y-coordinate of point P. default units: microns

ANG numeric Optional angle of partial circle starting at point P. units: degreesdirection: counter clockwisedefault value: 0



DIEL numeric Relative dielectric constant of dielectric element. Use of two values denotes an anisotropic dielectric. The first value refers to the x component, the sec-ond, the y component.

CHRG numeric Optional fixed charge density in dielectric element. units: coulomb/unit3 where unit is specified in OPTIONS commanddefault value: 0

RHO numeric Sheet resistivity of material (used only for CUR-RENT or RESISTANCE calculations). units: ohm


Figure 2-3 CIRC2 type geometric element


ang

(px,py)

(cx,cy)

CIRC2 NAME=C;CX=1;CY=1;PX=3;PY=3;ANG=270;VOLT=0.5;

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ence defined by the center and the node in the periphery marks the limits of the circular sector. The electrode is biased at 0.5 volts.

POLY

This command defines a polygon region with n vertices.

POLY NAME=<NAME>; COORD=<X1,Y1; X2,Y2; X3,Y3; [X4,Y4;] ... [XN,YN;]>{(VOLT=<VALUE>; [FLOAT=<VALUE>;]) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>;}[COLOR=<VALUE>;]



COORD numeric x1, y1;... x, y coordinates of vertices, a minimum of three vertices has to be specified. default units: microns



DIEL numeric Relative dielectric constant of dielectric element. Use of two values designates an anisotropic dielec-tric. The first value refers to the x component, the second, the y component.


RHO numeric Sheet resistivity of material (used only for CURRENT or RESISTANCE calculation). units: ohm


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Example:

The command defines a polygon as a conductor named vseven with seven vertices and a resistivity of .0001 ohm.

COPY

This command copies a predefined geometry onto a different place.

Figure 2-4 POLY type geometric element

(x1,y1)

(x2,y2)

(x3,y3)

(x4,y4)

(x5,y5)

(x7,y7)(x6, y6)

POLY NAME=VSEVEN; COORD=12.7,6; 12.8,7.9; 16.2,8.5; 16.7,8; 16.6,7.2; 16.7,6.7; 15.3,6.2; RHO=1E-4;

COPY FROM=<NAME>; TO=<NAME>; DX=<VALUE>; DY=<VALUE>; {(VOLT=<VALUE>; [FLOAT=<VALUE>;]) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>;} [COLOR=<VALUE>;]


FROM character Name of predefined geometry.

TO character Name of the new geometry.

DX numeric Translation in the x-direction. default units: microns

DY numeric Translation in the y-direction. default units: microns



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Example:

The command defines a new object named newpoly as a conductor copied from vseven and shifted by 50 in the X direction, with the same Y coordinate values, and a resistivity of .002 ohm.

WINDOW

This command sets the simulation window size and physical properties. A reflec-tive (Neumann) boundary condition is applied on the four sides of the simulation window. By using the reflective boundary condition, an input structure can be reduced to one half or one quarter of the whole structure. This symmetry can be exploited when both geometry and bias are symmetric around the same axis. This is not usually the case for capacitance simulations because when one electrode is biased all the others are grounded, thus imposing a nonsymmetric bias.

DIEL numeric Relative dielectric constant of dielectric element. Use of two values designates an anisotropic dielec-tric. The first value refers to the x component, the second, the y component.

CHRG numeric Optional fixed charge density in dielectric element.units: coulomb/unit3 where unit is specified in the OPTIONS command, default value: 0




COPY FROM=VSEVEN; TO=NEWPOLY; DX=50; DY=0; RHO=2E-3;

WINDOW X1=<VALUE>; Y1=<VALUE>; X2=<VALUE>; Y2=<VALUE>; {(DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>;}


X1 numeric x-coordinate of lower left corner of window default units: microns

Y1 numeric y-coordinate of lower left corner of window default units: microns

X2 numeric x-coordinate of upper right corner of windowdefault units: microns

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Example:

The command defines the simulation domain, any object outside the rectangle defined by the coordinate limits X1,Y1;X2,Y2 is ignored. All the spaces not filled with other objects is considered dielectric with a relative dielectric constant of 3.5.

MERGE

This command electrically connects the named electrodes to create a single one. The electrodes may be overlapping, touching, or separate. For capacitance calcu-lations, the new compound electrode may be referred to by any of its component electrode names. In case the values under VOLT=... or FLOAT=... were not speci-fied consistently among the merged electrodes, the first value overrides the others.

Note:Objects with the same name are automatically merged.

Example:

The command merges square and circle into a single object.

Y2 numeric y-coordinate of upper right corner of windowdefault units: microns

DIEL numeric Relative dielectric constant of dielectric element default value: 1. Use of two numbers specifies an anisotropic dielectric material. The first value refers to the x component, the second, the y component. Note that the default is an isotropic material with a dielectric coefficient of 1.


RHO numeric Sheet resistivity of window (used only for CURRENT or RESISTANCE calculations). units: ohmdefault value: 1e6 or 1e4*maximum rho of all structures, whichever is greater


WINDOW X1=0.0; Y1=0.0; X2=15.0; Y2=10.0; DIEL=3.5;

MERGE <NAME>; <NAME>; ... ;

MERGE SQUARE; CIRCLE;

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POTENTIAL

This command calculates the potential distribution and the amount of charge on each electrode for the given bias. It also creates the potential file <file>.pot and <file>.tdf, if specified. The file contains the potential and electric field data; unless the -n command flag is invoked.

CAPACITANCE

This command calculates the capacitance matrix for the listed electrodes. The short circuit capacitances are calculated. The diagonal term in the capacitance matrix always represent the total capacitance of the corresponding electrode with respect to all other electrodes (as if only that electrode is biased and all others are grounded). The off-diagonal terms represent the capacitances between both elec-trodes with the sign changed. Figure 2-5 represents a typical structure where capacitances may be required. The mathematical relationship between the capaci-tances expressed in the schematic and the corresponding short-circuit capaci-tances calculated by Raphael is expressed in Figure 2-6. The capacitance matrix should be symmetric. Due to regridding and/or the ICCG iteration tolerance, the final result may not be exactly symmetric.

When there is no name list following the CAPACITANCE command, the full capacitance matrix is calculated.

Example:

The command requests the simulation of the capacitance between electrodes elec1 and elec2 and all other electrodes.

POTENTIAL

CAPACITANCE <NAME>; <NAME>; ...;

CAPACITANCE ELEC1; ELEC2;

Figure 2-5 a) Structure formed by two electrodes parallel to a ground plane b) Corresponding schematic for the capacitance simulation

elec1 elec2

ground

elec1 elec2

ground ground

C10

C12

C20

a) b)

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CURRENT

This command calculates the current density distribution and total current flowing in each planar conductor. The current densities at every grid point are stored in the file <file>.pot.

RESISTANCE

This command calculates the equivalent resistive network among electrodes on a planar resistive sheet.

Example:

Note:CURRENT and RESISTANCE commands are incompatible with POTENTIAL and CAPACITANCE commands. They are intended to be used in different structures: CURRENT to calculate current distributions within a planar conductor and POTENTIAL to calculate potential distri-bution in transmission lines.

Figure 2-6 a) Short-circuit capacitance matrix calculated by Raphaelb) Relationship between these capacitances and those represented in Figure 2-5, where Σ represents the summation of all j elements with j ≠ i

Cs11 Cs12

Cs21 Cs22

elec1

elec2

elec1 elec2

C12 Cs12–=

Ci0 Csii Csij∑+=Cij Csij–=

C10 Cs11 Cs12+=

C20 Cs22 Cs21+=

C21 Cs21–=

a) b)

CURRENT

RESISTANCE <NAME>; <NAME>; ...;

RESISTANCE PROBE1; PROBE2;

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INDUCTANCE

This command calculates the inductance matrix for the listed elements. Since in 2D electrostatic analysis the inductance matrix is obtained by inverting the free-space capacitance matrix (calculated by removing all dielectric materials), at least one electrode should be grounded to provide a return path for the current not listed in the inductance calculation. Otherwise, the capacitance matrix is singular and cannot be inverted.

Example:

This command requests the self loop-inductance simulation for the electrode named circle1.

Z0

This command calculates the characteristic impedance assuming all but the named electrode are grounded. The characteristic impedance is calculated as follows:

Equation 2-1

Where L is the total inductance value for the named electrode and C is its total capacitance.

Example:

The command requests the impedance simulation for the electrode named elec1.

SPICE

This command causes a SPICE subcircuit model to be generated. Resistance, capacitance, and inductance are automatically calculated. Since the inductance calculation requires that a reference node be specified (in order that the capaci-tance matrix may be inverted), you must specify the GND node. The generated model is written to the file specified by FILE.

INDUCTANCE <NAME>; <NAME>; ...;

INDUCTANCE CIRCLE1;

Z0 <NAME>;

z0 L C⁄=

Z0 ELECT1;

SPICE GND=<NAME>; FILE=<NAME>; [RHO=<VALUE>;] [LENGTH=<VALUE>;]

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When creating a SPICE model, RC2 interprets each conductor as having uniform cross-section and being very long in the direction normal to the 2D plane where it was created. Then each conductor is assigned 3 nodes for the netlist construction. The electrode defined as ground is assigned only one node. The following exam-ple shows how to request a SPICE model for the structure presented in Figure 2-5; the corresponding schematic circuit is shown in Figure 2-7.

Example:

This command requests a SPICE subcircuit model to be generated; the model is written to the file spice1.out; the reference node is the electrode ground; all elec-


GND character The name of the reference node or electrode.

FILE character The name of the file to which the SPICE model is to be written.

LENGTH numeric The length in the Z dimension of the structure. The resistance, capacitance, and inductance are effec-tively multiplied by this dimension. default unit: micronsdefault value: 1.0

RHO numeric The resistivity of the conductors. This value is used to calculate the dc resistance of the conductors in a transmission-line problem, rather than in a planar resistive sheet. The unit is in ohm-meter, instead of ohm.units: ohm-meterdefault value: 3.*10-8, (aluminum)

SPICE GND=GROUND; FILE=SPICE1.OUT; RHO=3.0E-8; LENGTH=20.0;

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trodes are assigned a resistivity of 3.0*10-8 ohm-meter and a length normal to the plane of 20.0 mm.

For N transmission lines (N+1 conductors including a ground), the SPICE subcir-cuit model contains 2N+1 external nodes and N internal nodes. For each signal conductor, the external node names at the two terminals are constructed by attach-ing _x and _y to the conductor name. GROUND_RC2 is always used for the node name for a ground (reference).

EXTRACT

This command extracts the potential and the two Cartesian components of the electric field (or current) at the coordinate point specified by X1,Y1. The EXTRACT command can be used only with the POTENTIAL or CURRENT commands.

Example:

This command extracts the resulting potential and the electric field (or current) at the coordinate point (1.5,2.7).

Figure 2-7 Illustration of the automatic node assignment by RC2 when a SPICE subcircuit is created. Node GROUND_RC2 is assigned to the ground electrode

3

R1

R2

L1

L2

C_1_2

C_2_0

C_1_0K12

ground

elec1_y

elec2_x

elec1_x elec1

elec2 elec2_y

GROUND_RC2

EXTRACT X1=<VALUE>; Y1=<VALUE>;

EXTRACT X1=1.5; Y1=2.7;

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OPTIONS

This command sets the values of different options.

OPTIONS [SET_GRID=<VALUE>;][MAX_ITER=<VALUE>;][ITER_TOL=<VALUE>;][MAX_REGRID=<VALUE>;][REGRID_TOL=<VALUE>;][UNIT=<VALUE>;][GRID_SLIP=<VALUE>;][FAC_REGRID=<VALUE>;]


SET_GRID numeric Number of grid points automatically allocated for the first simulation. The default is dynamically set.

MAX_ITER numeric Maximum number of iterations. default value: 100 or 1% of the number of grid points, whichever is greater

ITER_TOL numeric Iteration tolerance with which the iteration stops.(The default is dynamically set between 10-4 and 10-12, depending on the values of VOLT, DIEL, and RHO.)

MAX_REGRID numeric Maximum number of regrid operations. (default value: 1) Setting MAX_REGRID=0 should be adequate for most applications.

REGRID_TOL numeric For each regrid, the calculated largest charge on a conductor is compared with the charge on the same conductor in the previous calculation. If the per-centage difference is less than this value, no more regridding is done. default value: 1.0

UNIT numeric Unit of geometrical dimensions with respect to meters. default value: 10-6 (in order to scale to microns)

GRID_SLIP numeric Maximum allowed distance between a polygon vertex and a mesh point divided by the correspond-ing window dimension. default value: 10-5

FAC_REGRID numeric Multiplication factor for increasing the number of grid points. For each regrid the number of grid points is increased FAC_REGRID times. default value: 1.0

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Theory of Floating Conductors Raphael Reference Manual

Example:

This command sets the dimensional unit system to be microns (10-6 with respect to meters) and requests the initial simulation to be done with 10,000 nodes.

Theory of Floating ConductorsA simple way to handle floating conductors (with nonzero fixed charges) is:

1. Consider floating conductors as regular signal conductors during the field-solver simulation, and generate the circuit model.

2. Float them during the circuit simulation.

Although this is a valid approach, it is numerically inefficient because it does not eliminate the electrical nodes associated with floating conductors, which may increase the CPU time and/or cause numerical instability during circuit simula-tion. A better approach is to remove the floating conductor nodes by preprocess-ing the circuit model before circuit simulation.

To demonstrate preprocessing of the circuit model, the short circuit capacitance matrix of a three-conductor system (two signal conductors with one ground con-ductor) is considered:

Equation 2-2

where V1 and V2 are the nodal voltages, and Q1 and Q2 are the free charges on the signal conductors. If you assume that conductor 2 has a fixed charge of Qo(=Q2), the above equation can be reduced to

Equation 2-3

and the voltage on the floating conductor, V2, is given by

Equation 2-4

Then, the capacitance Ceq of the reduced-circuit model, which relates Q1 and V1, can be expressed by

Equation 2-5

OPTIONS UNIT=1E-6; SET_GRID=10000;

C11 C12

C21 C22

V1

V2

Q1

Q2=

C11C12C21

C22-----------------–⎝ ⎠

⎛ ⎞ V1 Q1C12C22--------Qo–=

V21

C22-------- Qo C21V1–( )=

Ceq td

dV1 i1 td

dQ1=⎝ ⎠⎛ ⎞=

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Raphael Reference Manual Selection of Linear Solver

where

Equation 2-6

and the initial condition for the transient analysis is

Equation 2-7

If Qo is nonzero, a charge source need be included in a circuit simulator to account for the correct initial condition. When Qo is zero, as it often is, no extra term is needed.

Note:For the static sheet resistance analysis, floating electrodes correspond to those electrodes with fixed currents.

Although the above describes the relationship between the full and reduced circuit models, the actual implementation does not require the solution of an entire capacitance matrix. For better efficiency, the effect of fixed charge is accounted for directly during the field solver simulation.

Selection of Linear SolverRC2 filed solver includes a set of linear solvers for 2D elliptic problems. All these solvers are based on Preconditioned Conjugate Gradient (PCG) method with ILU(p) preconditioners (p denotes the level of fill-in for preconditioner). As the variant of the ILU(p) preconditioner, a set of linear solvers also includes Modified ILU preconditioner (MILU). The built-in system provides automatic selection of the proper linear solver. Manual selection of particular linear solver for RC2 can be done by using -I and/or -N options on the command line. Below is the descrip-tion of the effect of various command-line option on the selection of linear solver:

• Default behavior (neither -I nor -N options are specified on the command line) provides automatic selection between MILU(0) and ILU(0) preconditioners for PCG linear solver. This behavior is the default starting from Raphael ver-sion 2002.2. These solvers dramatically outperform old ICCG linear solver.

• -I option on the command line invokes old ICCG linear solver that was the only solver of choice before Raphael version 2002.2

• -N option on the command line invokes new PCG linear solvers with ILU(2) and MILU(2) preconditioners. These solvers outperform default linear solvers and use fewer iterations to converge. Choice of the particular solver will be done automatically in accordance with the type of the problem. This is the recommended choice.

• -N -I options on the command line force you to use ILU(2) preconditioner for PCG linear solver. This option provides manual control for linear solver

Ceq C11C12C21

C22-----------------–=

CeqV1 0-( ) Q1 0-( )C12C22--------Qo–=

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Examples Using RC2 Raphael Reference Manual

choice to obtain robust solution for linear problem with strongly varying dis-continuous coefficients (these linear problems may arise from DC resistance analysis and sometimes from electrostatic analysis).

Examples Using RC2The following examples illustrate the use of RC2:

1. The inductance simulation capability of RC2 is shown in the first example, and the results are compared with semi-empirical approximations.

2. The second example calculates the potential distribution, capacitance and inductance of a structure consisting of three metal lines and a ground plane.

3. The third example calculates the current density and resistance of a bent metal line.

4. The fourth example illustrates the automatic extraction of a SPICE model for a structure consisting of two metal lines above a ground plane.

5. The final example demonstrates the treatment of anisotropic dielectric materi-als using RC2. The program output is validated using analytic results.

All RC2 input files are echoed in the output listing for all examples.

Example 1: Inductance Simulation of a Line Above Ground Plane

A simple structure is selected to exemplify the inductance simulation and its com-parison with analytical results. Figure 2-8 shows a long, flat conductor parallel to a ground plane. Both conductors are very long in the direction normal to the simu-lation plane, and they form a loop such that the current flows in one direction through the conducting line and returns through the ground plane. An analytical solution for this case is presented by C. S. Walker in Capacitance, Inductance, and Cross-talk Analysis, 1990, pp. 97-99.

Figure 2-8 Structure used to demonstrate inductance simulations; current flows inward in the line and returns through the ground plane

w

h

ground plane

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Raphael Reference Manual Examples Using RC2

The input file, raexc21, used to calculate inductances using RC2 is presented in Figure 2-9. The PARAM command is used in line 6 to parametrically define the characteristic dimensions of the structure, and to ease the generation of several input files to compare the calculated inductance with Walker’s results. Lines 8 and 10 define the line and the ground plane. The simulation window is defined in line 12. In Line 15 the inductance simulation is requested.

A comparison of RC2 results with Walker’s results for this structure is presented in Figure 2-10 using the aspect ratio h/w as a parameter. A difference of less than 5% is observed in the whole range when the aspect ratio is changed from 0.5 to 5, corresponding to a variation in the wire-to-ground distance of 15 to 150 microns.

INPUT FILE : raexc21

1 * RC2 RUN OUTPUT=raexc21 2 $ Example RAEXC21 3 $ Inductance simulation for comparison with Walkers results 4 $ infinite line above a ground plane 5 $ parameter definition 6 PARAM mth=1.0; mwd=30.0; ratio=5.0; h=mwd*ratio; plth=2; plwd=6.0*h; 7 $ Metal line 1 8 BOX NAME=m1; CX=plwd/2.; CY=plth+h+mth/2; W=mwd; H=mth; VOLT=1; 9 $ Ground plane 10 BOX NAME=plane; CX=plwd/2.; CY=plth/2; W=plwd; H=plth; VOLT=0; 11 $ Define the simulation window 12 WINDOW X1=0; Y1=0.; X2=plwd; Y2=3.0*plwd; 13 OPTIONS SET_GRID=500; 14 $ Do calculations 15 INDUCTANCE m1;

*** INDUCTANCE CALCULATION [Henry / (1e-06*m)] m1 m1 7.155294e-13

==> SPICE Models for Inductance Matrix [Henry / (1e-06*m)] L_1 m1 m1_y 7.155294e-13

Figure 2-9 Partial listing of Example raexc21, that defines a line of rectangular cross-section running parallel to a ground plane

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Example 2: Three Lines Above a Plane

In the following example, a structure containing three metal lines and a ground plane are simulated. This example illustrates how structures can be created using the geometric elements that also take advantage of the Raphael overlapping rule. Once the structure has been generated, potential distribution, capacitance and inductance are calculated. The structure contains nonrectangular metal lines with round or beveled corners to illustrate how nonplanar geometries are created.

Refer to the output listing in Figure 2-11 for the statements. Refer to the graphical output in Figure 2-13 for visualization of the structure. The first step is to create the dielectric layer that contains the metal lines. This is done at Line 5 using a POLY command. To illustrate the ability of the program to handle nonplanar struc-tures, the oxide on the left half of the device is made thicker than the oxide on the right half. To create the polygon, the name dielect is assigned and then the vertices of the polygon are entered as x,y pairs. Finally, the relative dielectric constant of the layer is specified as 3, using the DIEL parameter.

The ground plane is created next using another POLY command. In this case, the polygon is designated as an electrode by specifying the voltage with the VOLT parameter. The voltage specified is used during the POTENTIAL calculation.

Refer to Figure 2-13; electrodes 1, 2, and 3 appear from left to right across the structure. Electrode number 1 is created using three geometric elements: a BOX and two CIRC1 at lines 9, 10, and 11. The BOX is used for the central square por-tion of the electrode and the CIRC1s are used to round off the ends. The BOX is defined by specifying its center at CX, CY and its length (in the x direction) and

Figure 2-10 Simulated inductance per unit length of the structure presented in Figure 2-8 compared with Walker’s results

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height (in the y direction) with l and h, respectively. The CIRC1s are defined by their centers (CX,CY), radius r, and starting and ending angles (ANG1 and ANG2, respectively). As all three geometric elements were given the same NAME, they are considered the same electrode.

Electrode number 2 is more complex. Another POLY is used to define its basic shape (Line 13), but a circular portion needs to be cut away from the left edge. To cut away the left edge, the left edge is overwritten with a dielectric using another CIRC1 (Line 14). Note that the left edge of the POLY is shifted to the right by 0.05 micron to make sure the CIRC1 fully covers the edge.

For the final electrode (number 3), a POLY is used to create a shape with one beveled edge.

Line 18 describes the simulation window, or region of the structure to be analyzed. The dielectric constant for all regions not filled by dielectric or electrode geome-tries is defined to be unity (or 1.0).

Line 19 defines options to be used during the simulation. In this case, the total number of grid points is set to 2000. The program chooses a mesh that fits the structure and contains as close to 2000 grid points as possible. Since no units are specified, the program uses 10-6 that corresponds to microns.

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1 * RC2 RUN OUTPUT=raexc22 2 $ Example RAEXC22 3 $ Three metal lines over a ground plane 4 $ Dielectric layer 5 POLY NAME=dielect; COORD=0,0;0,3.5;3,3.5;3.5,3;7.5,3;7.5,0; DIEL=3; COLOR=3; 6 $ Ground Plane 7 POLY NAME=grnd; COORD=0,0;0,1;7.5,1;7.5,0; VOLT=0; COLOR=2; 8 $ Metal line 1, with rounded corners 9 BOX NAME=m1; CX=2; CY=2; W=1; H=1; VOLT=1; COLOR=2; 10 CIRC1 NAME=m1; CX=1.5; CY=2; R=0.5; ANG1=90; ANG2=270; VOLT=1; COLOR=2; 11 CIRC1 NAME=m1; CX=2.5; CY=2; R=0.5; ANG1=270; ANG2=90; VOLT=1; COLOR=2; 12 $ Metal line 2, with 1 round and one beveled corner; 13 POLY NAME=m2; COORD=3.05,1.5;3.05,2.5;4.5,2.5;5,1.5; VOLT=2; COLOR=2; 14 CIRC1 NAME=m2a; CX=3; CY=2; R=0.5; ANG1=270; ANG2=90; DIEL=3; COLOR=3; 15 $ Metal line 3, with one beveled corner 16 POLY NAME=m3; COORD=5.5,1.5;5.0,2.5;6.5,2.5;6.5,1.5; VOLT=1; COLOR=2; 17 $ Define the simulation window 18 WINDOW X1=0; Y1=0; X2=7.5; Y2=5; DIEL=1; 19 OPTIONS SET_GRID=2000; MAX_REGRID=0; 20 $ Do calculations 21 POTENTIAL 22 CAPACITANCE m1; m2; m3; 23 INDUCTANCE m1; m2; m3;

*** POTENTIAL CALCULATION [Coulomb / (1e-06*m)] Charge on m1 = 4.205075e-17 Charge on m1__1 = 4.442372e-17 Charge on m1__2 = -8.224080e-17 Charge on grnd = -4.588865e-16 Charge on m2 = 4.397302e-16 Charge on m3 = 1.492262e-17 Maximum electric field: 7.331e+06 V/m at 3.017e-06, 1.500e-06 *** CAPACITANCE [Farad / (1e-06*m)] CALCULATION: (C)(V)=(Q) m1 m2 m3 m1 2.344432e-16 -1.141963e-16 -1.816808e-18 m2 -1.141963e-16 3.179355e-16 -8.194454e-17 m3 -1.816817e-18 -8.194457e-17 1.806286e-16 ==> SPICE Models for Entire Capacitance Matrix [Farad / (1e-06*m)] C_1_2 m1 m2 1.141963e-16 C_1_3 m1 m3 1.816812e-18 C_1_0 m1 GROUND_RC2 1.184301e-16 C_2_3 m2 m3 8.194455e-17 C_2_0 m2 GROUND_RC2 1.217946e-16 C_3_0 m3 GROUND_RC2 9.686724e-17 *** INDUCTANCE CALCULATION [Henry / (1e-06*m)] m1 m2 m3 m1 1.757890e-13 7.300506e-14 3.795363e-14 m2 7.300525e-14 1.474615e-13 6.959921e-14 m3 3.795371e-14 6.959916e-14 2.106375e-13 ==> SPICE Models for Inductance Matrix [Henry / (1e-06*m)] L_1 m1 m1_y 1.757890e-13 K_1_2 L_1 L_2 4.534385e-01 K_1_3 L_1 L_3 1.972378e-01 L_2 m2 m2_y 1.474615e-13 K_2_3 L_2 L_3 3.949089e-01 L_3 m3 m3_y 2.106375e-13

Figure 2-11 Partial output of Example raexc22. Three nonrectangular metal lines run parallel to a ground plane with a nonplanar dielectric

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Lines 21, 22, and 23 specify that Raphael perform POTENTIAL, CAPACITANCE and INDUCTANCE calculations. For the POTENTIAL calcula-tion, the voltages specified on the electrodes is used and the results are written to the potential file (which has a .pot extension). For the CAPACITANCE and INDUCTANCE calculations, the electrodes to be used for the capacitance and inductance matrices are specified. For this part of the calculation, the voltage val-ues specified on the electrodes and charges assigned to the insulator regions are ignored. The net result of the CAPACITANCE calculation is the capacitance matrix, which in this case has the unit of Farads/micron. (The micron comes from the unmodeled z dimension). The INDUCTANCE calculation results in the induc-tance matrix with the unit of Henry/micron.

Note:Synopsys will continue to include DPLOT as part of the Raphael release; however, DPLOT is no longer being developed. Please use Taurus Visual to visualize output from Raphael. Taurus Visual is easier to use, has higher capacity, and better quality graphics

The DPLOT input file that generated the plots of Figure 2-13 and Figure 2-14 is shown in Figure 2-12. The file contains only five command lines. The first line reads in the data created by RC2 from the file raexc22.pot. The data file contains both the potential distribution and a description of the device structure. The key-word RAPHAEL informs DPLOT that one of the Raphael programs created the file. The second line creates a window for the plot 8 cm high, with a title above the plot. The third line plots the device boundaries (electrodes and dielectrics) as well as the grid. The fourth line creates a new plot frame (again 8 cm high), and the last line generates the color-filled contours of potential.

$ Read in the geometry and the potential dataDATA RAPHAEL FILE=raexc22.pot

$ Plot the framPLOT.2D TITLE="Example RAEXC22: Structure & Grid" Y.LENGTH=8

$ Plot the structure and gridSTRUCTURE BOUND GRID

$ Plot a new framePLOT.2D TITLE="RAEXC22: Potential Plot" Y.LENGTH=8

$ Plot the potential contoursCONTOUR POTENTIAL FILL

$ add the structure and gridSTRUCTURE GRID

Figure 2-12 DPLOT input file dpraexc22 for Example raexc22

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To generate graphics for Taurus Visual, when invoking Raphael Rc2 use the -h option:

raphael rc2 - h raexc22

Figure 2-13 DPLOT output showing structure and grid for Example raexc22

Figure 2-14 DPLOT output showing potential contours for Example raexc22

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Raphael then generates a file raexc22.tdf which can be read using Taurus Visual. Simply start Taurus Visual and open the TDF file. By clicking the field and grid selection buttons, you can generate a figure similar to Figure 2-15.

Example 3: Current Density and Resistance Analysis

In this example RC2 calculates the current density in a bent metal wire. Such an analysis might be useful for calculating the probability of electromigration. The output file for this simulation is shown in Figure 2-16. Also refer to Figure 2-18 and Figure 2-19 for the graphical output generated.

1. Create the wire itself using a single polygon (see line 5). The resistivity of the material for the wire is set using the RHO parameter.

2. Create contacts to the wire using two BOXes (lines 7 and 8) at each end and set the bias (voltage) for these two points.

3. Perform the analysis using the CURRENT and RESISTANCE statements.

From the current density plot of Figure 2-19, it is apparent that the highest current density is at the concave corners where the current must bend sharply. You can compare the resistance with the value obtained by counting the squares: R=rho * L /W where L is the length of the path, and W is the width. Given that the path is 1 micron wide, and the length L is approximately 4 microns, it gives a resistance of

Figure 2-15 Taurus Visual output showing potential contours and simulation mesh

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4.0*10-8ohms. The difference between this value and the simulated value is due to the inaccuracy of the square-counting methods for a bent line.

1 * RC2 RUN OUTPUT=raexc23 2 $ Example RAEXC23 3 $ Calculates the current distribution and the resistance of a metal line 4 $ Define the wire (use low resistance) 5 POLY NAME=wire; COORD=0,2;2,2;2,1;5,1;5,2;3,2;3,3;0,3; RHO=1e-8; COLOR=2; 6 $ Define two contacts to the wire 7 BOX NAME=c1; CX=0.5; CY=2.5; W=1; H=1; VOLT=.001; COLOR=3; 8 BOX NAME=c2; CX=4.5; CY=1.5; W=1; H=1; VOLT=0; COLOR=3; 9 $ Define the simulation window (as an insulator) 10 WINDOW X1=0; Y1=0; X2=5; Y2=4; RHO=1e5; 11 OPTIONS SET_GRID=2000; 12 $ Calculate the currents 13 CURRENT 14 RESISTANCE

*** CURRENT CALCULATION [Amps]

Current at c1 = 3.134035e+04 Current at c2 = -3.134035e+04 Maximum current density: 5.925e+10 A/m at 2.000e-06, 2.000e-06

*** CONDUCTANCE [Mho] CALCULATION: (G)(V)=(I)

c1 c2 c1 3.134032e+07 -3.134032e+07 c2 -3.134033e+07 3.134033e+07

==> SPICE Models for Total Resistance [Ohm]

R_1_1 c1 OTHERS 3.190777e-08 R_2_2 c2 OTHERS 3.190776e-08

==> SPICE Models for Entire Resistance Matrix [Ohm]

R_1_2 c1 c2 3.190777e-08

Figure 2-16 Partial output file of raexc23 that computes current density in a bent line

$ Read in the geometry and the potential dataDATA RAPHAEL FILE=raexc23.pot

$ Plot the framPLOT.2D TITLE="Example RAEXC23: Structure & Grid" Y.LENGTH=8

$ Plot the device structure with electrodes and gridsSTRUCTURE BOUND GRID

$ Plot a new framePLOT.2D TITLE="RAEXC23: Current Density" Y.LENGTH=8

$ Plot the current densityCONTOUR CURRENT FILL

$ Add gridsSTRUCTURE GRID


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Figure 2-18 Device structure for Example raexc23, with left and right dark gray areas representing electrical contacts

Figure 2-19 Graphical output showing current density distribution for Example raexc23

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To use Taurus Visual to display results, use the -h option when running Raphael RC2:

Raphael rc2 -h raexc23

A TDF file raexc23.tdf will be produced. Simply start Taurus Visual and load this file and select the mesh and current density field.

Example 4: SPICE Model Extraction

In this example, the SPICE command automatically extracts a SPICE model for an interconnect structure that consists of two metal lines above a ground plane. The output file from RC2 is shown in Figure 2-21. The two metal lines are defined by the BOX commands. The ground plane is defined using a POLY element. The dielectric constant for the lower portion of the structure (containing the conduc-tors) is set to 3.0, which represents silicon dioxide. The upper portion, above 4 microns, is left as air (with dielectric constant equal to 1).

To calculate the inductance of the structure, you need to invert the capacitance matrix, which requires that the capacitance matrix be nonsingular. To ensure a nonsingular capacitance matrix, you must define one of the electrodes as ground by using the GND parameter in the SPICE statement. The SPICE statement also defines the file where the SPICE model is stored, using the FILE parameter. Finally, the resistivity of the conductors is defined using the RHO parameter in the SPICE statement. The resulting SPICE model is presented in Figure 2-22, and a graphical representation of this net list is shown in Figure 2-23. RC2 also prints the calculated capacitance and inductance matrices (per unit length) to the output

Figure 2-20 Mesh and current density for Example RAEXC23 visualized using Taurus Visual

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file. Therefore, the SPICE statement is equivalent to performing CAPACITANCE and INDUCTANCE analyses.

1 * RC2 RUN OUTPUT=raexc24 2 $ Example RAEXC24 3 $ Two metal lines over a ground plane 4 $ Generation of a spice model 5 $ Dielectric layer 6 BOX NAME=dielect; CX=3; CY=2; W=6; H=4; DIEL=3; 7 $ Ground Plane 8 POLY NAME=grnd; COORD=0,0;0,.5;6,.5;6,0; VOLT=0; 9 $ Metal line 1 10 BOX NAME=m1; CX=2; CY=2; W=2; H=1; VOLT=1; 11 $ Metal line 2 12 BOX NAME=m2; CX=4.5; CY=2; W=1; H=1; VOLT=1; 13 $ Define the simulation window 14 WINDOW X1=0; Y1=0; X2=6; Y2=6; DIEL=1; 15 OPTIONS SET_GRID=2000; 16 $ Assume aluminum lines, 20 microns long 17 SPICE GND=grnd; FILE=spice.out; RHO=3e-8; LENGTH=20;

*** CAPACITANCE [Farad / (1e-06*m)] CALCULATION: (C)(V)=(Q) m1 m2 m1 1.331765e-16 -4.913080e-17 m2 -4.912432e-17 1.061383e-16

==> SPICE Models for Total Capacitance [Farad / (1e-06*m)] C_1_1 m1 OTHERS 1.331765e-16 C_2_2 m2 OTHERS 1.061383e-16

==> SPICE Models for Entire Capacitance Matrix [Farad / (1e-06*m)] C_1_2 m1 m2 4.912756e-17 C_1_0 m1 GROUND_RC2 8.404877e-17 C_2_0 m2 GROUND_RC2 5.701073e-17

*** INDUCTANCE CALCULATION [Henry / (1e-06*m)] m1 m2 m1 3.005874e-13 1.425585e-13 m2 1.424953e-13 3.756231e-13

==> SPICE Models for Inductance Matrix [Henry / (1e-06*m)] L_1 m1 m1_y 3.005874e-13 K_1_2 L_1 L_2 4.241655e-01 L_2 m2 m2_y 3.756231e-13

Figure 2-21 Output file generated by RC2 for Example raexc24

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Example 5: Floating Conductors

In this example, the simulation of a floating conductor is considered. Where the capacitances of the three-conductor system in Figure 2-22 are computed with the center conductor, m2, possessing zero fixed charges. The partial listing of the out-put file is shown in Figure 2-23.

The computed voltage on the floated conductor, m2 was 0.498 V, which is very close to the exact value of 0.5 V. The equivalent capacitance between m1 and m3 is 11.33pF/m. To verify this capacitance value, the same system is simulated with-out floating conductors, and the resulting SPICE model is shown in Figure 2-26.

.SUBCKT intercon m1_x m1_y m2_x m2_y GROUND_RC2 C_1_2 m1 m2 9.825512e-16 C_1_0 m1 GROUND_RC2 1.680979e-15 C_2_0 m2 GROUND_RC2 1.140215e-15 L_1 m1 m1_y 6.011747e-12 K_1_2 L_1 L_2 4.241655e-01 L_2 m2 m2_y 7.512462e-12 R_1 m1_x m1 5.000000e-01 R_2 m2_x m2 8.000000e-01.ENDS intercon

Figure 2-22 SPICE subcircuit created by Example raexc24

Figure 2-23 Graphical representation of the SPICE subcircuit created by Example raexc24

m1_yR1

R2

L1

L2

C_1_2

C_2_0

C_1_0 K12

m2_x

m1_x m1

m2 m2_y

GROUND_RC2

Figure 2-24 Three equal-sized rectangular conductors running parallel to each other with equal spacing

10 μm

2 μm

10 μm

m1 m2 m3(floating Conductor)

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The equivalent capacitance between the nodes m1 and m3 after floating the node m2 from Figure 2-24 is 11.352 pF/m, which is very close to the previous capaci-tance value.

Note:The command option -n was used to obtain Figure 2-23. If a potential graphics file is desired or the EXTRACT command is present, the charge on m2 may not read as equal exactly to 0. The reason is: an extra poten-tial run would be executed using the derived floating conductor voltage, and the iterative matrix solver would lead to a small tolerance. That little deviation can be tightened by setting a smaller ITER_TOL.

1 * RC2 RUN OUTPUT=raexc25 2 $ Example RAEXC25 3 $ Floating conductor example 4 $ Three metal lines in free space 5 $ Define microstrip spacing, width and other parameters 6 PARAM s=10 ws=10 w2=ws/2 windx=6*s+3*ws ybox=6*windx t=2 7 $ Metal line 1 8 BOX NAME=m1; CX=2*s+w2; CY=ybox/2; W=ws; H=t; VOLT=1; 9 $ Metal line 2 10 BOX NAME=m2; CX=3*s+ws+w2; CY=ybox/2; W=ws; H=t; VOLT=1; FLOAT=0.0; 11 $ Metal line 3 12 BOX NAME=m3; CX=4*s+2*ws+w2; CY=ybox/2; W=ws; H=t; VOLT=0; 13 $ Define the simulation window 14 WINDOW X1=0; Y1=0; X2=windx; Y2=ybox; 15 OPTIONS SET_GRID=3000; 16 $ Do calculations 17 POTENTIAL 18 CAPACITANCE m1; m2; m3;

*** Voltages of Floating Conductors: Voltage on m2 = 4.965557e-01

*** POTENTIAL CALCULATION [Coulomb / (1e-06*m)]

Charge on m1 = 1.129814e-17 Charge on m2 = 00000000e+00 Charge on m3 = -1.130452e-17 Maximum electric field: 8.565e+04 V/m at 3.065e-05, 2.690e-04

*** CAPACITANCE [Farad / (1e-06*m)] CALCULATION: (C)(V)=(Q)

m1 m3 m1 1.129814e-17 -1.130452e-17 m3 -1.130355e-17 1.129802e-17

==> SPICE Models for Total Capacitance [Farad / (1e-06*m)]

C_1_1 m1 OTHERS 1.129814e-17 C_2_2 m3 OTHERS 1.129802e-17

==> SPICE Models for Entire Capacitance Matrix [Farad / (1e-06*m)]

C_1_2 m1 m3 1.130403e-17

Figure 2-25 Output file generated by RC2 for Example raexc25

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Example 6: Anisotropic Dielectric Materials

As a final example, RC2 simulates a system with anisotropic dielectrics. While anisotropy is not supported by either RC2-BEM or RC3-BEM, it can be used to model the behavior of other devices.

• On-chip interconnects. The epitaxy of the dielectric deposition process causes the process to be nonisotropic.

• FeRAM devices. Ferroelectric materials used in FeRAM devices possess very large dielectric constants. Great differences exist between the dielectric con-stant in the c-axis direction versus the a- or b-axis directions.

To test the accuracy of RC2 for a system with an anisotropic dielectric, the geom-etry shown in Figure 2-28 is used. This geometry corresponds to a conventional stripline structure with a sapphire filler. The principal axes-relative dielectric con-stants for sapphire are 11.6 for ε || and 9.4 for . For comparison, see analytical results from conformal mapping theory in Reference [1]. The resulting program output listing is shown in Figure 2-27.

Figure 2-26 The SPICE model of the three-conductor system shown in Figure 2-22 without floating conductors

m1 m2

m3

12.875 pF/m

12.880 pF/m4.922 pF/m

ε ⊥

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In lines 6 and 7, variables are defined to clarify the structure definition in the sub-sequent BOX commands. Line 10 defines the properties of the sapphire dielectric. Two values for the x and y components of the dielectric constant are used: 11.6 for the x-component and 9.4 for the y-component. The coordinate system of the prob-lem must be aligned with the sapphire axes to render the dielectric coefficient ten-sor diagonal.

As designated by the variable gw, the dielectric is as wide as the groundplanes are, The window width, however, as used in line 15, is set to be 1 micron smaller than the groundplanes. At the right and left window edge, this setting enforces a Neu-mann boundary condition and sets the x component of the E field at zero.

The resulting total capacitance from the strip to its groundplanes is 503.96 af/micron. Conformal mapping theory gives a value of 496 af/micron, for a relative error of 1.6%. Discrepancies may be attributed to the limitations of the theory, which assumes an infinitesimally thin center strip.

1 $ RC2 example 2 $ for comparision with conformal mapping results in 3 $ IEEE MTT-30, No. 8 Aug 1982, pp 1264-1267 4 $ dieletric used is sapphire. 5 $ 6 param st=0.2;sw=20.0;wx=sw+10.0*20+10.0*st;sx=wx/2.0;gw=wx+1.0; 7 param tgy=20+0.3+st;wy=20+0.4+st;sy=0.2+st/2.0+20/2.0;dh=st+20; 8 $ 9 $ Next line defines the sapphire dielectric 10 box name=diel_0;2 cx=sx; cy=sy; w=wx; h=dh; diel=11.6,9.4; 11 $ 12 box name=top_gnd; cx=sx; cy=tgy; w=gw; h=0.2; volt=0; 13 box name=bot_gnd; cx=sx; cy=0.1; w=gw; h=0.2; volt=0; 14 box name=strip; cx=sx; cy=sy; w=sw; h=st; volt=1; 15 window x1=0.0; x2=wx; y1=0; y2=wy; diel=1.0; 16 capacitance 17 OPTIONS MAX_REGRID=0; 18 OPTIONS SET_GRID=20000;


C_1_1 top_gnd OTHERS 1.041508e-15 C_2_2 bot_gnd OTHERS 1.041507e-15 C_3_3 strip OTHERS 5.039636e-16


C_1_2 top_gnd bot_gnd 7.895237e-16 C_1_3 top_gnd strip 2.519830e-16 C_2_3 bot_gnd strip 2.519828e-16

Figure 2-27 Output file generated by RC2 for Example aniso.rc2

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References Raphael Reference Manual

References[1] H. Shibata, et al, IEEE MTT-30, vol. MTT-42, No. 8, pp. 1264-1267, August

1992.

Figure 2-28 Stripline structure with sapphire dielectric for simulation in Example aniso.rc2

20 um

20 um 0.2 um

εx = 11.6εy = 9.4

y

x

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CHAPTER 3

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RC2-BEM: 2D Field Solver by Boundary Element Method3 3

IntroductionThe three most widely used numerical techniques for 2D static field solution are the:

1. Boundary-Element method (BEM)

2. Finite-Difference method (FD)

3. Finite-Element method (FEM)

All three of these methods have been successfully applied to resistance, capaci-tance, and inductance analysis. However, each method has its advantages and dis-advantages depending on particular applications. For instance, even though the FD and FEM methods are, in general, more versatile than BEM and can be applied to a wide range of applications, they tend to be slower than BEM. To over-come some limitations of the Finite-Difference method in RC2, RC2-BEM pro-vides a viable alternative based on the Boundary-Element method. In this chapter, the terms RC2 and RC2-BEM are used to denote the Finite-Difference and Boundary-Element solvers, respectively.

The Boundary-Element method used in RC2-BEM is optimized in terms of the computational efficiency by employing Galerkin’s method in conjunction with the recently-developed closed-form Green’s function for stratified dielectric media. In the following section, a theoretical background of BEM implemented in RC2-BEM is briefly introduced, along with a short comparison of the FD and FEM methods.

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Theoretical Background Raphael Reference Manual

Theoretical BackgroundIn the Finite-Difference and Finite-Element methods, the potential distribution must be solved before the charge (or current) distribution can be computed. Because the unknown potential distribution needs to be solved over the entire space, a large (but sparse) number of linear system equations is needed. These methods generally have a poor computational efficiency for structures in open space or small conducting objects with large spacing.

The Boundary-Element method is based on an integral equation instead of a dif-ferential equation, and is commonly known as the Method of Moments (MoM). Unlike the FD or FEM method, BEM directly solves the charge (or current) distri-bution on the surface of conductors. Unknowns in the BEM method lie only on the surface of conductors, and the resulting system of linear equations is small (but dense) compared to the FD or FEM case.

Green’s Function

In a conventional BEM, the Green’s function for stratified dielectric media is used as a kernel for an integral equation, and the open boundaries of geometry are han-dled in an exact manner. The expression of this function is obtained using the image theory, which consists of slowly-converging nested infinite series. Because the function’s evaluation is computationally burdensome, it becomes a major bot-tleneck of the conventional BEM approach. Alternatively, the free-space Green’s function can be used, but additional unknown charges on dielectric interfaces must be added, resulting in a larger system of linear equations.

In RC2-BEM, a closed-form Green’s function (Reference [2]) is used for stratified dielectric media. This closed-form Green’s function does not introduce additional unknowns, and it avoids the evaluation of the nested infinite series by using a finite number of weighted images. To further improve the performance, Garler-kin’s method (Reference [3]) is used in RC2-BEM.

FD, FEM, BEM Differences

Perhaps the most important difference among FD, FEM, and BEM is that both FD and FEM result in a large and sparse matrix, whereas BEM results in a small and dense matrix. FD and FEM are suitable for complex 3D problems, and BEM is more suitable for 2D or simple 3D problems.

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Raphael Reference Manual RC2-BEM Command Line Options

R

A detailed comparison of FD and BEM based on the nature of geometry is pre-sented in the following table:

Note:The comparison is based on the general FD and BEM methods. Limita-tions of the current version of RC2-BEM are shown in the later section.

RC2-BEM Command Line OptionsThe syntax of the RC2-BEM command line is as follows:

raphael rc2 -b “[BEM-OPTIONS]” [RC2-OPTIONS] [FILE]

where -b flag in RC2 is used to invoke the BEM solver.

CAUTIONThe double quotation marks for the BEM-options are always needed, even if no additional BEM options are desired. Example:

raphael rc2 -b “” [RC2-OPTIONS] [FILE]

The same input file syntax is used for RC2 and RC2-BEM except for a few minor differences. These differences are listed in the following section. Please refer to Chapter 2 for the general input syntax and options.

Options related to RC2-BEM are:

BEM FD (FEM)

Modeling of Open Region Excellent Poor

Planar Dielectrics Average Good

General Inhomogeneous Dielectrics Average Good

Unknowns Charge Potential

Modeling of Non-Manhattan Geometry Excellent Poor (Excellent)

Matrix Small & Dense Large & Sparse

Visualization of Fields Poor Excellent

Numerical Efficiency (2D) Excellent Average

Numerical Efficiency (3D) Poor Good

Parallelism Low High

Option Definition

-s m Places a perfect magnetic plane at the maximum x-coordinate of the window boundaries. This magnetic plane can be used to model the symmetric geometry with even-mode (symmetric) excitations.

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Notes on the Current Version of RC2-BEM Raphael Reference Manual

A proper choice of these boundary conditions, which are discussed in more detail in Appendix B, can improve the accuracy and/or performance of simulation.

Notes on the Current Version of RC2-BEM

Note:Read this section before using RC2-BEM.

This section lists the limitations of the current version of RC2-BEM and any dis-crepancies between the RC2 and RC2-BEM solvers. The potential and capaci-tance computations are analogous to the current and resistance computations. Limitations mentioned in this section related to the potential and capacitance anal-yses apply equally to the current and resistance analyses.

• Nonplanar dielectrics are ignored. The planar dielectrics are defined by box or rectangular-shaped poly objects whose boundaries are parallel to the x- and y-axis and whose widths (in x-direction) are at least equal to the width of the window.

• The graphics files (i.e., <file>.pot and <file>.tdf) for potential or field plots are not supported.

• The EXTRACT keyword is ignored.

• The keyword CHRG (i.e., fixed charge density in a dielectric element) is ignored.

• Regridding is not needed. Thus, the keywords MAX_REGRID, REGRID_TOL, and FAC_REGRID are ignored.

• As in RC2, non-Manhattan boundaries are staircased.

• Because of fast convergence, RC2-BEM uses a smaller default grid (which can be overridden by the keyword SET_GRID) than RC2.

• When conducting traces are located either above or below a conducting plane (layer), the conducting plane is modeled as an infinitely wide ground plane (no unknowns are placed at this plane).

-s M Places a perfect magnetic plane at the maximum x-coordinate of the window boundaries.

-s e Places a perfect electric plane at the maximum x-coordinate of the window boundaries. The electric plane is simply a ground plane and can be used to model the symmetric geometry with odd-mode (asymmetric) excitations.

-s E Places a perfect electric plane at the maximum x-coordinate of the window boundaries.

-s B Places perfect magnetic planes at both the minimum and the maximum x-coordinates of the window boundaries. To better correlate the results between RC2 and RC2-BEM, this flag can be turned on.

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Raphael Reference Manual Comparison of RC2-BEM and RC2

• When conducting traces are sandwiched by two conducting planes (layers), the bottom conducting plane is modeled by an infinitely wide ground plane (no unknowns are placed at this plane), and the top conducting plane is mod-eled by a regular conducting trace with its width equal to the window width. Because of this truncation of the top ground plane to a finite width, the cou-pling capacitance between the top and bottom ground planes may not be accu-rate.

• RC2-BEM cannot directly handle geometries where a conducting plane is located between traces. Since the conducting plane electrically isolates the traces from one side of the plane to the other side of the plane, this type of problem can be indirectly solved by formulating two separate problems for each side of the plane.

Most of these limitations are identical to those of RC3-BEM.

Comparison of RC2-BEM and RC2Three models are constructed in this section to compare the accuracy and simula-tion time of RC2-BEM with RC2.

• In the first example, three microstrip lines above a ground plane are consid-ered. Various circuit parameters are computed using RC2-BEM and RC2, and the results are compared with each other. Then, the short-circuit capacitance is computed and compared with the measurement data.

• In the second example, three microstrip lines embedded in stratified dielectric media are simulated, and the results are compared with published results.

• In the last example, the equivalent resistive network among vias is calculated to verify the current computation of RC2-BEM.

For all examples considered in this chapter, RC2-BEM resulted in more stable solutions (in terms of accuracy and symmetry), and it ran 3-7 times faster than RC2. The CPU time given in this section is measured on a SPARC-10 worksta-tion.

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Comparison of RC2-BEM and RC2 Raphael Reference Manual

Example 1: Microstrip Lines Above a Ground Plane

The three-microstrip line structure shown in Figure 3-1 compares the results from RC2 and RC2-BEM. The structure represents three equidistant rectangular wires running parallel to a ground plane. The homogeneous dielectric layer (defined by the window statement in the input file) is considered for this example. In the sim-ulations, h1=h2=10 microns, t=1.7 microns, w=10.8 microns, and s varies from 5 to 60 microns.

The capacitance, inductance and characteristic impedance are computed at 5-micron spacing. The partial output file listing is shown in Figure 3-2 for the RC2-BEM case, and the simulation results are compared in Figure 3-3.

During the simulation, SET_GRID is set to 3000, and regridding is performed only once in RC2. The solution times were 1.2 sec for RC2-BEM and 8.5 sec for RC2. The short-circuit characteristic impedance is a vector instead of a matrix, calculated by grounding all but one conductors at a time.

The maximum difference between the two results, excluding the very small cou-pling terms, is about 10%. The major source of this difference is due to the trunca-tion of the open geometry in the Finite-Difference method. This difference can be reduced by increasing the window size. The window truncation error can be observed in Figure 2-4. The m2 conductor is farther away from the window boundaries than the other two conductors, so the matrix elements corresponding to the m2 conductor match very closely.

The short-circuit capacitance of m2 (grounding all conductors except m2) is cal-culated with the spacing varied from 5 to 60 microns. Then, the simulation results from RC2-BEM and RC2 are compared with the published measurement data Reference [4] in Figure 3-4. As shown in the figure, the simulation results of both RC2-BEM and RC2 match well with the measurement. Note that the fringing capacitance plays an important role in this example, which is illustrated by the dramatic dependency of the capacitance on the interline distance.

Figure 3-1 Three-microstrip line structure

h2 s w

t

diel =3.5

m1 m2 m3

Ground

h1

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1 * RC2 RUN OUTPUT=raexc26 2 $ Example RAEXC26 3 $ Simulation to compare with Lin’s measured results 4 $ Three metal lines over a ground plane 5 $ Define microstrip spacing, width and other parameters 6 PARAM s=5 ws=10.8 w2=ws/2 windx=6*s+3*ws ybox=3.0*windx 7 $ Ground Plane 8 POLY NAME=grnd; COORD=0,0;0,1;windx,1;windx,0; VOLT=0; 9 $ Metal line 1 10 BOX NAME=m1; CX=2*s+w2; CY=11.85; W=ws; H=1.7; VOLT=1; 11 $ Metal line 2 12 BOX NAME=m2; CX=3*s+ws+w2; CY=11.85; W=ws; H=1.7; VOLT=1; 13 $ Metal line 3 14 BOX NAME=m3; CX=4*s+2*ws+w2; CY=11.85; W=ws; H=1.7; VOLT=1; 15 $ Define the simulation window 16 WINDOW X1=0; Y1=0; X2=windx; Y2=ybox; DIEL=3.5; 17 OPTIONS SET_GRID=3000; 18 $ Do calculations 19 CAPACITANCE m1; m2; m3; 20 INDUCTANCE m1; m2; m3; 21 z0 m1; m2; m3;

*** CAPACITANCE CALCULATION [Farad / (1e-06*m)] m1 m2 m3 m1 1.223114e-16 -4.020077e-17 -3.955884e-18 m2 -4.020077e-17 1.388647e-16 -4.020077e-17 m3 -3.955884e-18 -4.020077e-17 1.223114e-16

==> SPICE Models for Total Capacitance [Farad / (1e-06*m)] C_2_2 m1 OTHERS 1.223114e-16 C_3_3 m2 OTHERS 1.388647e-16 C_4_4 m3 OTHERS 1.223114e-16

==> SPICE Models for Entire Capacitance Matrix [Farad / (1e-06*m)] C_2_3 m1 m2 4.020077e-17 C_2_4 m1 m3 3.955884e-18 C_2_0 m1 GROUND_RC2 7.815473e-17 C_3_4 m2 m3 4.020077e-17 C_3_0 m2 GROUND_RC2 5.846316e-17 C_4_0 m3 GROUND_RC2 7.815473e-17

*** INDUCTANCE CALCULATION [Henry / (1e-06*m)] m1 m2 m3 m1 3.589990e-13 1.185723e-13 5.058283e-14 m2 1.185723e-13 3.490899e-13 1.185723e-13 m3 5.058283e-14 1.185723e-13 3.589990e-13

==> SPICE Models for Inductance Matrix [Henry / (1e-06*m)] L_2 m1 m1_y 3.589990e-13 K_2_3 L_2 L_3 3.349407e-01 K_2_4 L_2 L_4 1.408996e-01 L_3 m2 m2_y 3.490899e-13 K_3_4 L_3 L_4 3.349407e-01 L_4 m3 m3_y 3.589990e-13

==> Z0 Calculation [Ohm] Z0 of m1 = 5.102078e+01 Z0 of m2 = 4.493887e+01 Z0 of m3 = 5.102078e+01

Figure 3-2 RC2-BEM partial output listing of raexc26

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Figure 3-3 Comparison of RC2 and RC2-BEM for the equidistant three-microstrip line structure

RC2-BEM RC2 (FD)

C (pF/m)

L (nH/m)

Zo (Ω/m)

122.31 40.201– 3.956–

40.201– 138.86 40.201–

3.956– 40.201– 122.31

112.85 44.154– 7.6164–

44.383– 138.89 44.701–

7.6184– 44.158– 112.85

359.00 118.57 50.583118.57 349.09 118.5750.583 118.57 359.00

416.10 162.07 92.606162.53 384.26 163.4192.166 161.99 416.36

51.02144.93951.021

55.36744.93355.361

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Example 2: Inhomogeneous Dielectric Layers

In this example, inhomogeneous dielectric media are considered. The geometry of the simulated structure is shown in Figure 3-5, and the partial listing of the output file is shown in Figure 3-6. The computed capacitance values are compared with the published results in Figure 3-7.

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The solution time is 1.9 sec for RC2-BEM and 5.3 sec for RC2 on a SPARC-10 workstation. Overall, the simulation data agree well with the published results.

Figure 3-4 Short circuit capacitance of line m2 plotted against interwire distance

Figure 3-5 Three rectangular wires immersed in stratified dielectric media

Capacitance (nF/m) vs. Spacing

0.09

0.10

0.11

0.12

0.13

0.14

5 15 25 35 45 55

Spacing (microns)

Cap

acita

nce

(nF/

m)

RC2-BEM

RC2 (FD)

Measurement

diel = 4.3

Ground

200 μm

diel = 1.0

diel = 3.2

m2 m3

m1100 μm

70 μm

350 μm

150 μm150 μm

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Figure 3-6 RC2-BEM partial output listing of raexc27, which simulates three-microstrip lines embedded in stratified dielectric media


1 * RC2 RUN OUTPUT=raexc27 2 $ Example RAEXC27 3 $ Simulation to compare with Oh’s and Delbare’s simulated data 4 $ Three metal lines in a multilayered dielectric medium. 5 $ Define microstrip spacing, width and other parameters 6 PARAM s=150 ws=350 w2=ws/2 h=70 h2=h/2 7 PARAM windx=6*s+3*ws ybox=3.0*windx 8 $ Dielectric layers 9 POLY NAME=diel1 COORD=0,10;0,210;windx,210;windx,10; DIEL=4.3; 10 POLY NAME=diel2 COORD=0,210;0,310;windx,310;windx,210; DIEL=3.2; 11 $ Ground Plane 12 POLY NAME=grnd; COORD=0,0;0,10;windx,10;windx,0; VOLT=0; 13 $ Metal line 1 14 BOX NAME=m1; CX=2*s+w2; CY=210+h2; W=ws; H=h; VOLT=1; 15 $ Metal line 2 16 BOX NAME=m2; CX=3*s+ws+w2; CY=310+h2; W=ws; H=h; VOLT=1; 17 $ Metal line 3 18 BOX NAME=m3; CX=4*s+2*ws+w2; CY=310+h2; W=ws; H=h; VOLT=1; 19 $ Define the simulation window 20 WINDOW X1=0; Y1=0; X2=windx; Y2=ybox; DIEL=1.0; 21 OPTIONS SET_GRID=3000; 22 $ Do calculations 23 CAPACITANCE m1; m2; m3;

*** CAPACITANCE CALCULATION [Farad / (1e-06*m)]

m1 m2 m3 m1 1.418150e-16 -2.171567e-17 -9.024094e-19 m2 -2.171567e-17 9.349853e-17 -1.806908e-17 m3 -9.024094e-19 -1.806908e-17 8.793788e-17


C_2_2 m1 OTHERS 1.418150e-16 C_3_3 m2 OTHERS 9.349853e-17 C_4_4 m3 OTHERS 8.793788e-17


C_2_3 m1 m2 2.171567e-17 C_2_4 m1 m3 9.024094e-19 C_2_0 m1 GROUND_RC2 1.191969e-16 C_3_4 m2 m3 1.806908e-17 C_3_0 m2 GROUND_RC2 5.371378e-17 C_4_0 m3 GROUND_RC2 6.896639e-17

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Figure 3-7 Comparison of RC2 and RC2-BEM for Figure 3-5

Capacitance (pF/m)

RC2 (FD)

RC2-BEM

Reference [2]

Reference [5]

139.48 23.461– 1.8939–

23.685– 94.598 19.890–

1.8188– 19.516– 85.477

141.82 21.716– 0.9024–

21.716– 93.499 18.069–

0.9024– 18.069– 87.938

141.41 21.492– 0.8952–

21.491– 92.951 17.859–

0.8952– 17.859– 87.494

142.09 21.765– 0.8920–

21.733– 93.529 18.098–

0.8900– 18.087– 87.962

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Example 3: Modeling of the Power-Plane Resistance

In this example, the resistances among three vias on a power-plane are calculated. To simplify the problem, the vias are modeled by rectangular bars. As shown in Figure 3-8, each via’s cross section is an equipotential area. All three vias are equally spaced with center-to-center spacing of 100 mm. The conductivity of the resistive sheet is 57.6 MS/m (copper). The 2D representation of this problem is shown in Figure 3-9. (Note that the drawing is not properly scaled) A partial list-ing of the output file is given in Figure 3-10. The simulation results from RC2-BEM and RC2 are compared in Figure 3-11.

Again, the results match well with each other. The simulation time is 0.33 sec for RC2-BEM and 17.5 sec for RC2.

Figure 3-8 Three rectangular vias on a planar resistive sheet

Figure 3-9 2D modeling of the three-via geometry shown in Figure 3-8

Thickness = 1 μm

Width, Length = 2 μm

100 μm

RHO = 0.0173611

m2 m3

m1

4 μm

4 μm

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Figure 3-10 Partial output listing of raexc28 that simulates the resistances among three vias

Figure 3-11 Comparison of RC2 and RC2-BEM for the resistance computation

1 * RC2 RUN OUTPUT=raexc28 2 $ Example RAEXC28 3 $ Simulation to compare the resistance values from the finite-difference 4 $ method and the boundary element method. 5 $ The ground resistances between three retangular vias. 6 PARAM width=2 s=100 s2=50 offset=2*s windx=2*offset+s windy=windx 7 $ rectangular vias 8 BOX NAME=m1; cx=offset+s2; cy=offset+sqrt(s*s-s2*s2); w=width; h=width; volt=1.0; 9 BOX NAME=m2; cx=offset; cy=offset; w=width; h=width; volt=0.0; 10 BOX NAME=m3; cx=offset+s; cy=offset; w=width; h=width; volt=1.0; 11 $ Define the simulation window 12 WINDOW X1=0; Y1=0; X2=windx; Y2=windy; RHO=0.01736111; 13 OPTIONS SET_GRID=3000; ITER_TOL=1e-8; 14 $ Do calculations 15 POTENTIAL 16 RESISTANCE

*** CURRENT CALCULATION [Amps] Current on m1 = 2.717057e+01 Current on m2 = -5.434114e+01 Current on m3 = 2.717057e+01 *** CONDUCTANCE CALCULATION [Mho] m1 m2 m3 m1 5.434114e+01 -2.717057e+01 -2.717057e+01 m2 -2.717057e+01 5.434114e+01 -2.717057e+01 m3 -2.717057e+01 -2.717057e+01 5.434114e+01 ==> SPICE Models for Total Resistance [Ohm] R_1_1 m1 OTHERS 1.840226e-02 R_2_2 m2 OTHERS 1.840226e-02 R_3_3 m3 OTHERS 1.840226e-02 ==> SPICE Models for Entire Resistance Matrix [Ohm] R_1_2 m1 m2 3.680453e-02 R_1_3 m1 m3 3.680453e-02 R_2_3 m2 m3 3.680453e-02

RC2-BEM RC2 (FD)

ConductanceMatrix (Mho)

54.341 27.171– 27.171–

27.171– 54.341 27.171–

27.171– 27.171– 54.341

55.431 27.715– 27.716–

27.964– 55.287 27.323–

27.970– 27.324– 55.294

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Raphael Reference Manual References

R

References[2] K. S. Oh, D. B. Kuznetsov, and J. E. Schutt-Aine, “Capacitance Computa-

tions in a Multi-layered Dielectric Medium Using the Closed-form Spatial Green’s Functions,” IEEE Trans. Microwave Theory Tech., vol. MTT-42, pp. 1443-1453, August 1994.

[3] C. C. Huang, “Two-dimensional Capacitance Calculation in Stratified and/or Arbitrary Dielectric Media,” IEEE Trans. Microwave Theory Tech., vol. MTT-42, pp. 501-504, March 1994.

[4] M. S. Lin, IEEE Trans. Comp., Hybrids, Manufact. Technol., vol. 13, no. 4, pp. 1050-1054, Dec. 1990.

[5] W. Delbare and D. D. Zutter, “Space-domain Green’s Function Approach to the Capacitance Calculation of Multi-conductor Lines in Multi-layered Dielectrics With Improved Surface Charge Modeling,” IEEE Trans. Micro-wave Theory and Tech., vol. MTT-37, pp. 1562-1568, October 1989.

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CHAPTER 4

R

RC3: 3D Resistance, Capacitance, and Thermal Resistance4

IntroductionRC3, a general-purpose, 3D program for solving Poisson’s equation, is based on the finite-difference method with an automatically adjustable rectangular mesh. The linear equations set up by the finite-difference method are solved by the Mod-ified Incomplete Cholesky Conjugate Gradient method (MICCG), which is a default method, or by the Incomplete Cholesky Conjugate Gradient method (ICCG). The combination of the automatic adjustment of mesh and the speed of linear equation solvers makes RC3 versatile and user friendly. You can use RC3 for any 3D Poisson problem, such as the capacitance and resistance calculation and steady-state thermal analysis. This includes Laplace’s equation, which is a special case of Poisson’s equation. RC3 uses two different models for discretiza-tion of initial Poisson’s equation:

• The first one is based on the nodal representation of the material properties (default model).

• The second one is based on the elemental representation of the material prop-erties. The last model is more accurate and useful for Manhattan geometry structures with strongly varying dielectric constants.

Both 32-bit and 64-bit versions of this solver are available.

The syntax of the RC3 command line is as follows:

raphael rc3 [OPTIONS] [FILE]

The syntax of the input file is explained in the next section, RC3 Input File on page 4-3. If <file> is omitted, RC3 takes the input from the standard input. If no options are specified, RC3 reads <file>, makes calculations according to the input, and writes the results to the output file named <file>.out. One other file,

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Introduction Raphael Reference Manual

<file>.pot, is also created and contains the calculated potential or current distribu-tion as well as a description of the original structure. An additional graphics file, <file>.tdf, may be created for visualization using the Taurus Visual program.

The following options are interpreted by RC3:

-b Invokes RC3-BEM. See Chapter 5.

-d Forces conductors to always override the overlapping dielectrics.


-i Checks the input file for errors and generates the geometry file if no errors are found. No calculation is performed.



-p <file> Uses <file> as the potential file name.

-s Sends output to the standard output.

-t Prints out CPU time summary.

-u Appends an output summary to the output file. This summary is used by RIL for easy parsing of the RC3 output.

-v Invokes Taurus Topography interface. See Appendix E for Taurus Topography interface.

-w <file> Specify the name of a mesh file which contains the placement of grid lines in the x, y, and z directions. This mesh file uses a rather simple syntax:x <x1> <x2>... specifies the placement of grid lines perpendicular to the x-axis. y <y1> <y2>... specifies the placement of grid lines perpendicular to the y-axis. z <z1> <z2>... specifies the placement of grid lines perpendicular to the z-axis.The position of the grid lines are in the length units specified in the RC3 input file with the command OPTIONS UNIT.

-W Prints the mesh distribution used in the final step of computation to the output file.

-x The potential information is not written in <file>.pot.


-j Skip the overlapping rule and considers all polygons in the input file during the simulation. (Using this option increases simulation time.)

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RC3 Input FileThe input file statements consist of comments and/or commands. Lines beginning with * or with $ are comment lines and can be ignored. Sixteen commands are available in RC3: PARAM, BLOCK, CYLINDER, SPHERE, POLY3D, COPY3D, MERGE, WINDOW3D, POTENTIAL, CAPACITANCE, CURRENT, RESISTANCE, THERMORES, TEMPERATURE, EXTRACT, and OPTIONS. A command may be followed by assignments and/or lists defined in this section and may occupy more than one line. The input is in free format. Uppercase and lowercase characters are interpreted differently; i.e., the input to RC3 is case sensitive, meaning any

-P “<param > = <value>;”

Replaces the value of the parameters defined in the input file with new values. For example, -P “w=0.5; h = 0.1” overrides the values of parameters w and h defined in the input file with the new val-ues, 0.5 and 0.1, respectively

-I Invokes ICCG method for solving linear equation system instead of the default MICCG method.

-E Invokes elemental model of the Poisson’s equation discretization instead of the default nodal model.

-N Invokes new RC3 solver with enhanced speed. It does not work with -w and/or -v options.

-L Prints the content of nets (complex electrodes) in the following format:ELECTRODE <Electrode Name> <Electrode Type> <Value><Object Name><Object Name>....ELECTRODE <Electrode Name> <Electrode Type> <Value><Object Name><Object Name>....Where <Electrode Name> is the name assigned to particular com-plex electrode (net), <Electrode Type> is the type of particular electrode (COND or FLOAT), <Value> is the value (voltage or charge) assigned to particular electrode, and <Object Name> is the name of objects from input file that form the electrode with name <Electrode Name>.

-H Creates graphic file in tdf format to be visualized by Taurus Visual (see Creating Graphics Files on page 4-22 for more details).This option must be used with -N option.

-T Converts input file into Taurus-Device format for transient ther-mal analysis (see Interface to Taurus Device on page 4-23 for more details).

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instruction is recognized when written with all lowercase or all uppercase letters. The syntax and usage of each command are described in this section.

The simulation commands in RC3 can be divided into three categories depending on the types of analyses:

• Electrostatic analysis—POTENTIAL and CAPACITANCE commands.

• DC resistance analysis—CURRENT and RESISTANCE commands.

• Static thermal analysis—TEMPERATURE and THERMORES commands.

The EXTRACT command allows the value of the potential, current, temperature, or electric field at a given location. It can be used only with the POTENTIAL, TEMPERATURE, or CURRENT commands.

CAUTIONEach input file can specify only one type of analysis. Raphael allows only one simulation WINDOW3D and one OPTIONS command. Commands associat-ed with two different analyses should not be used in the same input file. For instance, the POTENTIAL and RESISTANCE commands cannot appear at the same time.

Lines beginning with the special symbol (+) are considered a continuation line from the last entered command. When the special symbol (+) is at the end of a line, the following line is considered a continuation line. The symbol is not required and can be omitted.

Overlapping Rule: When the geometries defined by BLOCK, CYLINDER, SPHERE, POLY3D, and COPY3D commands overlap, the geometry defined later in the input overwrites the geometry previously defined in the input file. This overlapping rule makes it easy to input many structures, such as hollow conduc-tors and metal lines embedded in dielectric material. In some cases, it is desirable to make conductors always override dielectrics by using the -d command line option (see Introduction on page 4-1 for command-line options).

Length Units: By default, all geometrical dimensions are in microns. The default can be changed with the OPTIONS command.

PARAM

The PARAM command defines variables and their values for later use in the input file. Variable names must begin with an alphabetic character, but may include any number of alphanumeric characters. value can be any algebraic expression with numbers, predefined variables, and functions. The supported functions are: sin(), cos(), atan(), log(), log10(), exp(), sqrt(), int(), and abs(). The unit for the values used for the length dimension is specified in the OPTIONS command. The argument for all trigonometric functions is given in

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radians. This definition of value is valid when used in value arguments in other commands.

Example:

The command defines a value of 2.0 for parameter a, parameter b is assigned a value of 4, and y is defined as the exponential of the sum of a and b.

BLOCK

The BLOCK command defines a rectangular box element (right prism with rectan-gular base). Here vector is the collection of three numbers X, Y, Z, separated by commas. These numbers are coordinates of a vector on a global coordinate system (See Figure 4-1).

PARAM PARAMETER1=<VALUE>; PARAMETER2=<VALUE>


BLOCK NAME=<NAME>;{(V1=<VECTOR>; [DIRECTION=<VECTOR>;] HEIGHT=<VALUE>;) or (V1=<VECTOR>; V2=<VECTOR>;)}[PERP=<VECTOR>;] WIDTH=<VALUE>; LENGTH=<VALUE>;{(VOLT=<VALUE>; [FLOAT=<VALUE>;]) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>; or(TEMP=<VALUE>; [FLOAT=<VALUE>;]) or(CTC=<VALUE>; [HEAT=<VALUE>;])}[COLOR=<VALUE>;]


NAME character Name of the element. NAME must begin with an alphabetic character, but may include up to 300 alphanumeric characters.

V1 vector Coordinates of the center of the bottom of the block. Default units: microns

DIRECTION vector Direction vector of the height axis. This vector is always perpendicular to the bottom and the top of the block. Default value: 0,0,1; or parallel to the Z axis

HEIGHT numeric Height of the block. Default units: microns

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V2 vector Coordinates of the center of the block top. The fol-lowing vector equation connects the geometrical data.v2=HEIGHT * (normalized DIRECTION) + v1Default units: microns

PERP vector Defines a vector along which the length of the block is calculated. This vector is always perpendic-ular to DIRECTION. (If not, RC3 automatically projects this vector to the bottom plane, causing PERP to be perpendicular to DIRECTION.) Default value: 0,1,0

WIDTH numeric Width of the block. default units: microns

LENGTH numeric Length of the block along the PERP vector. Default units: microns

VOLT numeric Potential of an electrode element. units: volt

FLOAT numeric Fixed charge, current, or heat of floating electrode.Units: coulomb, ampere, or watt

DIEL numeric Relative dielectric constant of a dielectric element.Use of three values denotes an anisotropic dielec-tric, with the numbers referring to the X, Y, and Z components, respectively.

CHRG numeric Optional fixed charge density in a dielectric ele-ment. Units: coulomb/unit3 where unit is specified in the OPTIONS commandDefault value: 0

RHO numeric Resistivity of the material comprising the block (used only for CURRENT or RESISTANCE calcula-tions).Units: ohm-meter

TEMP numeric Temperature of thermal electrode (used only for thermal analysis)Units: degrees K

CTC numeric Thermal conductivity of thermal conductor (used only for thermal analysis).Units: Watt/degrees K-meter


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The recommended method to define a BLOCK geometry is:

BLOCK NAME=A;V1=VECTOR; [DIRECTION=VECTOR;] HEIGHT=VALUE;

[PERP =VECTOR;]WIDTH =VALUE; LENGTH=VALUE;

Figure 4-1 clarifies the meanings of these parameters.

Note:DIRECTION and PERP have default values (DIRECTION= 0,0,1; PERP= 0,1,0), so by default WIDTH, LENGTH, and HEIGHT are aligned with X-, Y-, and Z-axis, respectively.

HEAT numeric Optional heat source density in thermal conductor (used only for thermal analysis).Units: Watt/unit3 where unit is specified in the OPTIONS commandDefault value: 0

COLOR numeric Optional color index to be used when plotting this block.

Figure 4-1 BLOCK type geometric element (DIRECTION and PERP are arbitrarily oriented)


width

height

X

Y

Z

direction

length

O

perp

v1

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A second method to define a BLOCK geometry uses V2 instead of DIRECTION and HEIGHT: is:

BLOCK NAME=A;V1=VECTOR; V2=VECTOR;[PERP =VECTOR;]WIDTH =VALUE; LENGTH=VALUE;

In this way, the direction of the block results as the direction of the vector differ-ence V2 - V1, and its height as the magnitude of this difference.

The first method is recommended because it is more intuitive, especially when the BLOCK has an arbitrary direction with respect to the global coordinate system.

Examples:

Each one of the following four commands generates a block 5 units long in the X direction, 3 units long in the Y direction, and 4 units long in the Z direction with the center of the lower face at (1,2,3).

Note:The first two, the default directions of (0,0,1) and (0,1,0) are used for DIRECTION and PERP, respectively.

1. An electrode, biased at 1 volt:

BLOCK NAME=a;V1=1,2,3; HEIGHT=4;WIDTH=5; LENGTH=3;VOLT=1;

2. A conductor:

BLOCK NAME=a;V1=1,2,3; V2=1,2,7;WIDTH=5; LENGTH=3;RHO=2.5*10-4;

3. A thermal conductor:

BLOCK NAME=a;V1=1,2,3; HEIGHT=4;PERP=1,0,0;WIDTH=3; LENGTH=5;CTC=1; HEAT=0.3;

4. A thermal electrode:

BLOCK NAME=a;V1=-0.5,2,5; DIRECTION=1,0,0; HEIGHT=5;PERP=0,0,1;WIDTH=4; LENGTH=3;TEMP=310;

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CYLINDER

The CYLINDER command defines a cylinder element.

CYLINDER NAME=<NAME>;{(V1=<VECTOR>; [DIRECTION=<VECTOR>;] HEIGHT=<VALUE>;) or (V1=<VECTOR>; V2=<VECTOR>;)}RADIUS=<VALUE>;{(VOLT=<VALUE>; [FLOAT=<VALUE>;]) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>; or (TEMP=<VALUE>;[FLOAT=<VALUE>;]) or (CTC=<VALUE>; [HEAT=<VALUE>;])}[COLOR=<VALUE>;]



V1 vector Coordinates of the center of the cylinder bottom. Default units: microns

DIRECTION vector Direction vector of the height axis. It is perpendicu-lar to the bottom of the cylinder. Default value: 0,0,1; or parallel to the Z axis

HEIGHT numeric Height. Default units: microns

RADIUS numeric Radius of the cylinder. Default units: microns

V2 vector Coordinates of the center of the top of the cylinder. If V2 is not specified, it is calculated as: V2 = HEIGHT * (normalized DIRECTION) + V1 Default units: microns

VOLT numeric Potential of the electrode element.Units: volt


DIEL numeric Relative dielectric constant of dielectric element. Use of three values designates an anisotropic dielectric, with the numbers referring to the X, Y, and Z components of the dielectric constant, respec-tively.

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The recommended method to define a CYLINDER geometry is:

CYLINDER NAME=NAME;V1=VECTOR; [DIRECTION=VECTOR;] HEIGHT=VALUE;RADIUS =VALUE;

Figure 4-2 shows these parameters.

CHRG numeric Optional fixed charge density in dielectric element.Units: coulomb/unit3 where unit is specified in OPTIONS command

RHO numeric Resistivity of material (used only for CURRENT or RESISTANCE calculations). Units: ohm-meter

TEMP numeric Temperature of thermal electrode (used only for thermal analysis). Units: degrees K

CTC numeric Constant of thermal conductivity of thermal con-ductor (used only for thermal analysis).Units: Watt/degrees K-meter




Figure 4-2 CYLINDER geometric element

direction

v1radius

height

z

y

xO

v2

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Example:

CYLINDER NAME=BBB; V1=1,2,3; HEIGHT=4; RADIUS=1; VOLT=1;

The command defines a cylinder named bbb as an electrode; the base is centered at (1,2,3); it has a height of 4, a radius of 1, and an applied bias of 1 volt.

Note:The default direction (0,0,1) is used, so the cylinder is aligned with the Z-axis.

SPHERE

This command defines a sphere element.

SPHERE NAME=<NAME>; CENTER=<VECTOR>; RADIUS=<VALUE>;{(VOLT=<VALUE>; [FLOAT=<VALUE>;]) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>; or (TEMP=<VALUE>; [FLOAT=<VALUE>;]) or (CTC=<VALUE>; [HEAT=<VALUE>;])}[COLOR=<VALUE>;]



CENTER vector Coordinates of center of the sphere. Default units: microns

RADIUS numeric Radius of the sphere. Default units: microns

VOLT numeric Potential of electrode element. Units: volt


DIEL numeric Relative dielectric constant of dielectric element. Use of three values denotes an anisotropic dielec-tric, with the values referring to the X, Y, and Z components of the dielectric constant, respectively.

CHRG numeric Optional fixed charge density in dielectric element. Units: coulomb/unit3 where unit is specified in the OPTIONS command

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Example:

SPHERE NAME=a431; CENTER=4,5,6; RADIUS=4; DIEL=3.9;

The command defines a dielectric sphere named a431, centered at (4,5,6) with a radius of 4, and a relative dielectric constant of 3.9.

POLY3D

The POLY3D command, which can also be abbreviated as POLY, defines a right prism with n vertices in the polygon base (bottom of the prism) and by extrusion of this base.

RHO numeric Resistivity of material (used only for CURRENT or RESISTANCE calculations). Units: ohm-meter

TEMP numeric Temperature of thermal electrode (used only for thermal analysis).Units: degrees K

CTC numeric Thermal conductivity of thermal conductor (used only for thermal analysis). Units: Watt/K-meter

HEAT numeric Optional heat source density in thermal conductor (used only for thermal analysis). Units: Watt/unit3 where unit is specified in the OPTIONS commandDefault value: 0



POLY3D NAME=<NAME>; COORD =< X1LOC, Y1LOC; X2LOC, Y2LOC; X3LOC, Y3LOC;

[X4LOC, Y4LOC;] .... [XNLOC, YNLOC;]>{(V1=<VECTOR>; [DIRECTION=<VECTOR>;] HEIGHT=<VALUE>;)or (V1=<VECTOR>; V2=<VECTOR>;)}[PERP=<VECTOR>;] {(VOLT=<VALUE>; [FLOAT=<VALUE>;]) or (DIEL=<VALUE>; [CHRG=<VALUE>;]) or (DIEL=<VALUE>,<VALUE>,<VALUE>; [CHRG=<VALUE>;]) or RHO=<VALUE>; or (TEMP=<VALUE>; [FLOAT=<VALUE>;]) or (CTC=<VALUE>; [HEAT=<VALUE>;])}[COLOR=<VALUE>;]

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COORD vector x1loc, y1loc;x2loc, y2loc;x3loc, y3loc;...:

X,Y coordinates of the vertices of the polygon base expressed in the local coordinate system with origin v1 and axis (iloc, jloc, kloc), where: iloc = jloc x klocjloc = normalized PERPkloc = normalized DIRECTION (see Figure 4-3)

The polygon base lies in the XY plane of this local coordinate system. A minimum of three coordinate points should be given. Default units: microns

V1 vector Coordinates of the origin of the local coordinate system used to define the vertices of the polygon base. Default units: microns

DIRECTION vector Direction vector of the height axis. This vector defines the perpendicular of the polygon base. Default value: 0,0,1, (parallel to the Z-axis)

HEIGHT numeric Height of prism. Default units: microns

V2 vector V2 = HEIGHT * (normalized DIRECTION) + V1.Represents a shift in the local coordinate system of a magnitude HEIGHT in the direction specified by DIRECTION. Default units: microns

PERP vector Used to determine the local coordinate system for the polygon base. This vector must always be per-pendicular to DIRECTION. (If not, RC3 automati-cally projects this vector to the polygon base, which causes PERP to be perpendicular to DIRECTION) Default value: 0,1,0 (parallel to the Y-axis)

VOLT numeric Potential of electrode element. Units: volts


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The recommended method to define a POLY3D geometry is:

POLY3D NAME=NAME;COORD=X1LOC, Y1LOC; ... ;XNLOC, YNLOC;V1=VECTOR; [DIRECTION=VECTOR;] HEIGHT=VALUE;[PERP =VECTOR;]

Figure 4-3 explains these parameters.

• The coordinates of vectors V1, PERP, and DIRECTION are expressed on the global coordinate system XYZ.

• The defaults are DIRECTION = 0,0,1; and PERP = 0,1,0; so that the X-, Y-, and Z-axes of the local coordinate system are parallel to the X-, Y-, and Z-axes of the global coordinate system, respectively.

• When the structure is not aligned with the axis of the global coordinate sys-tem, the definition of the geometry using the vector V2 (instead of using

DIEL numeric Relative dielectric constant of dielectric element. Use of three values denotes an anisotropic dielec-tric, with the numbers pertaining to the X, Y, and Z components of the dielectric constant, respectively.

CHRG numeric Optional fixed charge density in dielectric element Units: coulomb/unit3 where unit is specified in the OPTIONS command

RHO numeric Resistivity of material (used only for CURRENT or RESISTANCE calculations). Units: ohm-meters


CTC numeric Constant of thermal conductivity of thermal con-ductor (used only for thermal analysis) Units: Watt/degrees K-meter




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DIRECTION and HEIGHT) should be avoided, because it is difficult to have clear geometrical feeling.

Example:

POLY3D NAME=POLY11; COORD=1,0;10,0;10,2;1,2; HEIGHT=5.; V1=0,0,0; DIRECTION=0,0,1; PERP=0,1,0; DIEL=3;

The command defines an extruded polygon named poly11 as a dielectric with a relative dielectric constant of 3. The polygon is defined by four vertices, with a height of 5. The origin of the local coordinate system is the same as the origin of the global coordinate system (V1=0,0,0); the direction perpendicular to the plane of the polygon is along the global Z axis (DIRECTION=0,0,1); and the PERP direction is coincident with the global Y direction.

COPY3D

The COPY3D command, which can also be abbreviated as COPY, copies a pre-defined element to a different location

Figure 4-3 POLY3D type element with 4 vertices in polygon base

LOCAL COORDINATE SYSTEM

perp

v1

x

yloc

loc

1

2

3

4

XZ

1

2

3

4

direction

yloc

x loc

height

Y

GLOBAL COORDINATE SYSTEM


FROM character Name of a predefined element.

TO character Name of a new element.

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Example:

COPY3D FROM=POLY11; TO=POLY123; DIRECTION=1,2,3; VOLT=5.;

The command defines an electrode named poly123, with the same dimensions as the object poly11. It is translated from the position of poly11 with the translation vector (1,2,3). The electrode is biased at 5.0 volts.

MERGE

The MERGE command electrically connects the named electrodes (or thermal electrodes) to create a single electrode. The electrodes may be overlapping, touch-

DIRECTION vector Translation vector. Default units: microns

VOLT numeric Potential of electrode element. Units: volts


DIEL numeric Relative dielectric constant of dielectric element. Use of three values specifies that the new element be made of an anisotropic dielectric material. The three values refer to the X, Y, and Z components of the dielectric constant, respectively.

CHRG numeric Optional fixed charge density in dielectric element.Units: coulomb/unit3 where unit is specified in the OPTIONS command

RHO numeric Resistivity of material (used only for CURRENT or RESISTANCE calculations).Units: ohm-meters


CTC numeric Constant of thermal conductivity of thermal con-ductor (used only for thermal analysis).Units: Watt/degrees K-meter

HEAT numeric Optional heat source density in thermal conductor (used only for thermal analysis).Units: Watt/unit3 where unit is specified in the OPTIONS command Default value: 0


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ing, or separate. For capacitance calculations, the new compound electrode may be referred to by any of its component names.

Example:

The following command merges the two named electrodes:

MERGE POLY11; POLY123;

Note:Objects with the same name are automatically merged.

WINDOW3D

The WINDOW3D command, which can also be abbreviated as WINDOW, sets the simulation window size and physical properties. A reflective (or, Neumann) boundary condition is applied on the four sides of the simulation window. If the structure and excitations are symmetric, an input structure can be reduced to one half or one quarter of the whole structure by using the reflective boundary condi-tion. This is not usually the case for capacitance simulations because when one electrode is biased, all the others are grounded causing nonsymmetric bias condi-tions.

MERGE NAME1; NAME2;...;

WINDOW3D V1=<VECTOR>; V2=<VECTOR>;{([DIEL=<VALUE>;] [CHRG=<VALUE>;]) or ([DIEL=<VALUE>,<VALUE>,<VALUE>,;] [CHRG=<VALUE>;]) or [RHO=<VALUE>;] or ([CTC=<VALUE>;] [HEAT=<VALUE>;])}


V1 vector X, Y, Z coordinates of one corner of the simulation window.Default units: microns

V2 vector X, Y, Z coordinates of the opposite corner of the simulation window.Default units: microns

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Example:

WINDOW3D V1=0,0,0; V2=10,5,5; DIEL=3.5;

This command defines the simulation domain to be between coordinates (0,0,0) and (10,5,5). All space that is not occupied by defined objects is filled with a dielectric with a relative dielectric constant equal to 3.5.

DIEL numeric Optional relative dielectric constant in the dielectric element filling the simulation window. Use of three values causes the window to be filled with an anisotropic dielectric. The numbers pertain to the X, Y, and Z components of the dielectric con-stant, respectively.

Note: The window dielectric defaults to being isotropic.

default value: 1.0

CHRG numeric Optional fixed charge density in dielectric element.units: coulomb/unit3 where unit is specified in the OPTIONS commandDefault value: 0.0

RHO numeric Resistivity of the electrical conductor filling the simulation window (used only for CURRENT or RESISTANCE calculations).Units: ohm-meterDefault value: 1.0e6 or 1.0e4*maximum rho of all structures, whichever is greater

CTC numeric Optional constant of thermal conductivity of ther-mal conductor filling the simulation window (used only for thermal analysis).Units: Watt/K-meterDefault value: 0.026 for air

HEAT numeric Optional heat source density in thermal conductor filling the simulation window (used only for ther-mal analysis).Units: Watt/unit3 where unit is specified in the OPTIONS commandDefault value:0.0


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POTENTIAL

This command calculates the potential distribution and the amount of charge on each electrode for the given bias. It also creates the potential file <file>.pot, which contains the potential and electric field data.

CAPACITANCE

This command calculates the capacitance matrix for the listed electrodes. If no names are listed, the full capacitance matrix is calculated.

CURRENT

This command calculates the current density distribution and total current flowing at each conductor. The current densities are stored in the file <file>.pot.

RESISTANCE

This command calculates the equivalent resistive network among electrodes.

Example:

RESISTANCE PROBE1; PROBE2;

Note:CURRENT and RESISTANCE commands are incompatible with POTENTIAL and CAPACITANCE commands. They are intended to be used in different structures:

• CURRENT calculates current distributions within conductors.

• POTENTIAL calculates potential distribution outside the conductors.

TEMPERATURE

This command calculates the temperature distribution and the amount of heat on each thermal electrode. It also creates the potential file <file>.pot, which contains the temperature data.

POTENTIAL

CAPACITANCE <NAME1>; <NAME2>;...;

CURRENT

RESISTANCE <NAME>; <NAME>;...;

TEMPERATURE

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THERMORES

This command calculates the equivalent thermal resistive network among the listed electrodes. (Previously, this keyword was called THERMOCAP.)

EXTRACT

This command extracts the potential (or temperature) and the three Cartesian components of the electric field (or current or temperature gradient) at the coordi-nate point specified by (X1,Y1,Z1). The EXTRACT command can be used only with the POTENTIAL, TEMPERATURE, or CURRENT commands.

Example:

EXTRACT X1=1.0; Y1=2.5; Z1=1.3;

This command extracts the resulting potential (or temperature) and the electric field (or current or temperature gradient) at the coordinate point (1.0,2.5,1.3).

OPTIONS

This command sets the values of the options that drive the calculation process.

THERMORES <NAME1>; <NAME2>;...;

EXTRACT X1=<VALUE>; Y1=<VALUE>; Z1=<VALUE>;

OPTIONS [SET_GRID=<VALUE>; ][GRID_SLIP=<VALUE>; ][MAX_ITER=<VALUE>; ][ITER_TOL=<VALUE>;] [MAX_REGRID=<VALUE>;] [REGRID_TOL=<VALUE>;] [UNIT=<VALUE>;][FAC_REGRID=<VALUE>;]


SET_GRID numeric Total number of grid points to be used in the simu-lation. Default: Dynamically set.

GRID_SLIP numeric Maximum allowed distance between a POLY vertex and a grid point divided by the maximum WINDOW dimension.Default value: 10-5

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Raphael Reference Manual Theory of Floating Conductors

The most important parameter is SET_GRID. This value directly affects the time and accuracy of calculations. To confirm that a result is accurate, vary this param-eter and compare the difference in results. If necessary, increase the number of grid points until the results become independent of this number.

In general, the default values should be acceptable for accuracy.

Theory of Floating ConductorsRefer to Theory of Floating Conductors on page 2-20 in Chapter 2.

Selecting Linear SolverRC3 filed solver includes set of linear solvers for 3D elliptic problems. All these solvers are based on Preconditioned Conjugate Gradient (PCG) method with ILU(p) preconditioners (p denotes the level of fill-in for preconditioner). As the variant of the ILU(p) preconditioner, a set of linear solvers also includes Modified

MAX_ITER numeric Maximum number of iterations for the ICCG method.Default value: 100 or 1% of the number of grid points, whichever is greater

ITER_TOL numeric Iteration tolerance with which the iteration stops.The default is dynamically set between 10-4 and 10-12, depending on the values of VOLT, DIEL, RHO, and CTC.

MAX_REGRID numeric Maximum number of regrid operations. default value: 1 In many cases, setting MAX_REGRID=0 should be sufficiently accurate.

REGRID_TOL numeric Regrid tolerance. For each regrid step, the amount of the charge on the conductor that has the largest charge is compared with the amount of the charge on the same conductor obtained with the previous grid. If the difference is less than this percentage value, no more regridding is done. Default value: 1.0

UNIT numeric Unit of geometrical dimensions. Units: metersDefault value: 10-6 in order to scale to microns

FAC_REGRID numeric Multiplication factor for increasing the number of grid points. For each regrid the number of grid points is multiplied by FAC_REGRID.Default value: 1.0


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Creating Graphics Files Raphael Reference Manual

ILU preconditioner (MILU). The built-in system provides automatic selection of the proper linear solver. Manual selection of particular linear solver for RC3 can be done by using -I and/or -N options on the command line. Below is the description of the effect of various command-line option on the selection of linear solver:

• Default behavior (neither -I nor -N options are specified on the command line) provides automatic selection between MILU(0) and ILU(0) precondi-tioners for PCG linear solver. This behavior is the default starting from Raphael version 2002.2. These solvers dramatically outperform old ICCG lin-ear solver.

• -I option on the command line invokes old ICCG linear solver that was the only solver of choice before Raphael version 2002.2

• -N option on the command line invokes new PCG linear solvers with ILU(1) and MILU(0) preconditioners. These solvers outperform default linear solvers and use fewer iterations to converge. Choice of the particular solver will be done automatically in accordance with the type of the problem. This option also invokes new meshing algorithm that dramatically outperforms the old one (before release 2003.03). This is recommended choice.

• -N -I options on the command line force to use ILU(0) preconditioner for PCG linear solver as well as new meshing algorithm. This option provides backward compatibility (with releases 2002.2-2003.09) and adds manual con-trol for linear solver choice to obtain robust solution for linear problem with strongly varying discontinuous coefficients (these linear problems may arise from DC resistance analysis, from steady-state thermal analysis and some-times from electrostatic analysis).

Creating Graphics FilesRC3 filed solver saves structures and distributions of the physical fields in tdr for-mat used by TecPlot™ and both “simplified” and “extended” Technology Data Format (tdf) formats. To save graphic file in “simplified” tdf format, specify -h option on the command line. This format provides compact graphic files that con-tain geometry objects, mesh, and distributions of the physical fields (electrostatic potential, electric field, temperature, current, etc.).

Note:The main drawback of this format is that when visualizing these files in Taurus Visual, you cannot hide field distributions in specific regions to see the detail picture of the field distribution in the region of interest.

To overcome this drawback, the RC3 field solver can create output graphic files in “extended” tdf format. This format is completely compatible with tdf files gener-ated by other tools of the Taurus family. Size of “extended” tdf file is an about 4X bigger than size of “simplified” tdf file.

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Raphael Reference Manual Interface to Taurus Device

Creation of the “extended” tdf file and tdr file is available only with -N option on the command line and works only for the input files that contain POTENTIAL, CURRENT or TEMPERATURE commands.

The default extension of the graphic files in tdr format is .tdr, while the “simpli-fied” tdf format and the “extended” tdf file is saved with extension .tdf and .TDF, respectively.

Interface to Taurus DeviceThe interface between Raphael RC3 field solver and Taurus Device allows you to perform transient thermal analysis. RC3 field solver is used to convert RC3 input file for steady-state thermal analysis into Taurus Device input files, including translation of the material thermal properties to format recognized by Taurus Device program. Taurus Device is another Synopsys product and it requires a spe-cial license.

Converter is available only in the batch mode and can be invoked by the -T option on the command line:

raphael rc3 -T input_file.rc3

where input_file.rc3 is the name of the input file in RC3 format. All other command line options are available with -T option except -v command line option. Some of the command line option will not affect the program. RC3 auto-matically detects type of the problem and will not convert input file if electrostatic or DC resistance analyses are requested in the input file.

RC3 creates two output files with the predefined names RC3_geom_file and RA2TD. The first file contains description of the same geometry as was specified in input_file.rc3 and is written in Taurus Device input format. In addition to geometry specification this file contains specification for physical properties of the different objects, which are described in Taurus Device format and correspond to original RC3 input file.To user’s convenience all geometry objects have the ref-erences to objects from original RC3 input file.

Note:As a rule, you should not change anything in this file.

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Examples Raphael Reference Manual

RA2TD file is a short file that contains mainly specifications for thermal contacts. You should modify this file (you have to “uncomment” some lines and reset val-ues for thermal contacts). Below is a simple example of the RA2TD file:

Taurus {device}Include (RC3_geom_file)# Put regrid criterion into regrid command below:Regrid## Below is specification for contact regions.#DefineContact (Region=R_0, isContactRegion=True, Type=ThermalCon-tact,) # RC3_NAME: contact_BDefineContact (Region=R_1, isContactRegion=True, Type=ThermalCon-tact,) # RC3_NAME: contact_A## Below is specification for contact values.## SetValue (value=400) { Contact(Name=R_0, Type=ContactTemperature) } # RC3_Name: contact_B# SetValue (value=300) { Contact(Name=R_1, Type=ContactTemperature) } # RC3_Name: contact_A## Below is the list of objects, which have to be# modified for transient temperature simulation.### Below is a place for proper SOLVE/SAVE commands#Save

RC3_geom_file is already included into RA2TD file via INCLUDE command. Regrid command in RA2TD file requires additional adjustments that cannot to be done automatically by RC3 field solver. For more details about Taurus Device input file format, see the Taurus Device User Guide.

To run Taurus Device, you must use RA2TD file as an input file.

ExamplesThe following examples illustrate the use of RC3 and the DPLOT plotting pack-age:

• Example 1 calculates the current density and total resistance of a cylindrical via.

• Example 2 calculates the capacitance and potential distribution in a simple crossover structure.

• Example 3 demonstrates the floating conductor modeling feature of RC3 by computing the capacitance of a floating-gate transistor.

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Raphael Reference Manual Examples

Example 1: Current Density and Resistance of a Cylindrical Via

This example calculates the current density and resistance of a cylindrical via con-necting two metal lines. Analysis of this type is useful in determining if vias add substantial series resistance. The possibility of electromigration also can be exam-ined, because the electromigration worsens with higher current density.

.

The output file generated by RC3 is shown in Figure 4-4. To understand the struc-ture, refer to the graphical output of Figure 4-6- and 4-7. The two metal lines are


1 * RC3 RUN OUTPUT=raexc31 2 $ Example RC3EX1 Current density in a via 3 $ Lower metal line 4 POLY3D NAME=conductor_A; COORD=1,0;10,0;10,2;1,2; V1=0,0,0; V2=0,0,5; + 5 PERP=0.0,1.0,0.0; COLOR=0; RHO=1e-6; 6 $ Upper metal line 7 POLY3D NAME=conductor_B; COORD=1,0;10,0;10,2;1,2; V1=0,7,0; V2=0,7,5; + 8 PERP=0.0,1.0,0.0; COLOR=0; RHO=1e-6; 9 $ Cylindrical via 10 CYLINDER NAME=conductor_C; COLOR=0; V1=7,2,2.5; V2=7,7,2.5; RADIUS=1.5; + 11 RHO=1e-6; 12 $ Contact to lower line 13 POLY3D NAME=contact_A; COORD=0,0;1,0;1,2;0,2; V1=0,0,0; V2=0,0,5; + 14 PERP=0.0,1.0,0.0; COLOR=7; VOLT=1; 15 $ Contact to upper line 16 POLY3D NAME=contact_B; COORD=0,0;1,0;1,2;0,2; V1=0,7,0; V2=0,7,5; + 17 PERP=0.0,1.0,0.0; COLOR=7; VOLT=0; 18 $ Simulation window 19 WINDOW3D V1=0.0,0.0,0.0; V2=10.0,9.0,5.0; 20 OPTIONS UNIT=1e-6; SET_GRID=10000; 21 $ Calculate the current density 22 CURRENT 23 RESISTANCE

*** CURRENT CALCULATION [Amps]

Current at contact_A = 5.084032e-01 Current at contact_B = -5.084032e-01 Maximum current density: 1.153e+11 A/m^2 at 5.667e-06, 2.000e-06, 2.125e-06


contact_A contact_B contact_A 5.084032e-01 -5.084032e-01 contact_B -5.084032e-01 5.084032e-01

==> SPICE Models for Total Resistance (in Ohm)

R_0_0 contact_A OTHERS 1.966943e+00 R_1_1 contact_B OTHERS 1.966943e+00

==> SPICE Models for Entire Resistance Matrix (in Ohm)

R_0_1 contact_A contact_B 1.966943e+00

Figure 4-4 Output generated by Example raexc31

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defined first. Lines 4 and 5 of the output file define the lower metal line. A POLY3D type element is used to define a simple rectangle with corners at the points (1,0), (10,0), (10,2), (1,2). Since the PERP vector is (0,-1,0), the rectangle is in the X-Y plane. Using V1 and V2, the rectangle is extruded 5 units into the Z direction. The first element consists of a simple box 9 units long in the X direc-tion, 2 units long in the Y direction, and 5 units long in the Z direction. The origin of the box is located at (1,0,0). Note that the resistivity of the metal is specified as 10-6 ohm-meter.

The upper metal line is created in a similar way. Only the Y coordinates of the V1 and V2 points have changed. The second metal line is also a box of 9 x 2 x 5 units, with the origin at the point (1,7,0).

The cylindrical via that connects the metal lines is created at Lines 10 and 11. The radius of the cylinder is 1.5 units (microns), and the axis of the cylinder is defined by the line connecting the points V1 (7,2,2.5) and V2 (7,7,2.5). The cylinder, like the metal lines, has a resistivity of 10-6 ohm-meter.

Lines 13 through 14 define the contact to the lower metal line. This contact has zero resistance and is necessary to apply the bias voltage to the metal line. Like the metal lines, a POLY3D type element is used. You see that V1, V2, and PERP are the same as for the lower metal line. However, the points of the polygon have been changed to (0,0), (0,1), (1,2), (0,2). The contact is in effect a box of 1x2x5 microns in size with the origin at the point (0,0,0). The contact abuts the first metal line on the left side. The COLOR of the contact is set to 7 to distinguish itself from the metal components defined earlier. The potential is set to 1.0 volts

Lines 16 and 17 define the contact to the upper line. This contact abuts the upper metal line at the left side and is biased at zero volts. The actual values of the bias voltage used are of no consequence as long as there is some difference in voltage between the two contacts. This is the case since RC3 conducts a linear analysis with respect to voltage.

Line 19 defines the simulation window large enough to enclose all the elements and all significant electric fields. Finally, at Lines 22 and 23, the analysis is per-formed. The total current is 0.5084 amps and the resistance is 1.967 ohms. The current densities are written to the file raexc31.pot and are plotted using DPLOT.


After the current density has been calculated, plots using DPLOT are produced. The DPLOT input file dpraexc31 is shown in Figure 4-5. First, Line 2 reads in the data and device structure using the DATA statement. The keyword RAPHAEL tells DPLOT that one of the Raphael programs (RC2, RC3, or RI3) generated the file. Next, to generate a plot of the input geometries, at Line 7 all geometric objects (i.e., the metal lines, via, and contacts) must be selected. The LIT.FLAT lighting

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mode must also be specified. Finally, at Line 9, the objects using the PLOT.3D statement are plotted. The final plot is shown in Figure 4-6.

In the second plot, you plot a plane through the device with current density con-tours. For this plot, first (at Line 13) all the geometric objects are selected. It is specified that they be plotted as WIRE frames. (The device structure must not obscure the contour plane.) Next, at Line 15, the contour plane using an EXTRACT statement is generated. The plane is located at z=2.5 microns, and CURRENT density is specified as the plot quantity.

In Line 17, the contour slice using another SELECT statement is selected. Then, at Line 19, the two contacts are selected and the LIT.FLAT lighting model is speci-fied. (The contacts are not large and do not significantly obscure the contour slice.) Finally, at Line 21, all the objects using a PLOT.3D statement are plotted. The final plot can be seen in Figure 4-7.

1... $ Read in all data 2... DATA RAPHAEL FILE=raexc31.pot

3... $ plot the input geometry 4... $ Select all objects and use ”LIT.FLAT” surface model 5... $ This is necessary since conductors with non-zero resistance 6... $ are treated as dielectrics 7... SELECT ALL LIT.FLAT

8... $ Plot the objects 9... PLOT.3D TITLE=”Example RAEXC31: Via Structure”

10... $ Plot a current density slice through the object 11... $ Use wire frame model for structure so the slice won’t be 12... $ obscured 13... SELECT ALL WIRE

14... $ Generate the contour slice at Z=2.5 microns 15... EXTRACT CONTOUR CURRENT Z.SLICE=2.5

16... $ Select the slice you just created 17... SELECT TYPE=CONTOUR

18... $ Use the ”LIT.FLAT” mode for the two contacts 19... SELECT TYPE=CONDUCTOR LIT.FLAT

20... $ Plot 21... PLOT.3D TITLE=”RAEXC31: Current Density in Via”

22... QUIT


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Figure 4-6 Structure for Example raexc31

Figure 4-7 Current density in 3D via structure for Example raexc31

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Taurus Visual can also be used to visualize results from Raphael RC3. When run-ning RC3, specify the -h option to generate a TDF file which can be read in Tau-rus Visual. For example:

raphael rc3 -h raex3c1

The file raex3c1.tdf is generated. Next start Taurus Visual and open the file and use the mouse to rotate the structure to the desired viewing angle. Figure 4-8 shows example graphics from Taurus Visual for this example.

Example 2: Capacitance and Potential Analysis of a Cross-Over Structure

In this Example, the potential and capacitance for a cross-over structure composed of three metal lines are calculated. The output file generated by the simulation is found in Figure 4-10. Graphical output showing the device structure and two dif-ferent representations of the potential distribution is found in Figure 4-9-, 4-12, and 4-13. Referring to Figure 4-10, the structure is created using POLY3D type elements as in the preceding example.

Figure 4-8 Current density in via shown using Taurus Visual

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In this example a dielectric block is created at Line 15 with a relative dielectric constant of 3.0. The rest of the structure has a relative dielectric constant of 1.0 (the default for the background material).

Figure 4-9 Structure for Example raexc32 generated by Line 6 of dpraexc32

The second and third conductors are examples of more complicated polygons. In this case each polygon has 12 vertices. In Raphael, the memory is configured dynamically, so there is no limit on the maximum number of vertices a polygon may contain. Raphael, however, tries to assign a mesh point to each vertex, so a large number of polygon vertices may result in long run times. At Line 21, the potential distribution is calculated. The results are stored in the file raexc32.pot and are plotted using DPLOT. At Line 22, the capacitance with respect to conduc-tors CONDUCTOR_A and CONDUCTOR_C is calculated.

Note:It is possible to calculate portions of the capacitance matrix. (This saves time for structures with many conductors.) The diagonal terms of the matrix represent the total capacitance from the named conductor to all other conductors in the system.

Figure 4-11 shows the DPLOT input file dpraexc32 for Example raexc32. The graphical output generated by Figure 4-11 is presented in Figure 4-9- to 4-13.

As in the previous examples, Raphael begins by reading in the potential file cre-ated by RC3. Next, all the objects (at Line 4) are selected and plotted at Line 6.

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Note that the viewing angle has been altered at Line 6 using the angles THETA and PHI. The finished plot appears in Figure 4-9.

For the next plot, a potential contour slice is created through the middle of the device. By extracting the contour at Line 11, contours in the plane Z=1.5 are extracted. The extracted slice becomes a new graphics object named object00. At Line 13, the newly created contour slice is selected. Finally, at Line 15, the plot, which appears in Figure 4-12, is generated.

For the final plot, a contour surface equal to 0.5 volts is generated and shown along with the device structure. The surface (at Line 19) must first be extracted. As before, the parameters CONTOUR and POTENTIAL are specified to indicate the type of object and the plot quantity. The contour value (0.5 volt) using the MINIMUM parameter is specified, and only one surface is extracted by specifying N.VALUE=1. The 3D parameter indicates a 3D surface to be extracted as opposed to the contour slices used earlier.

At Line 21, all objects (including the two extracted contours generated by Lines 11 and 19) are selected. The PRINT parameter allows a listing on screen of all the objects that have been extracted. In this plot, the potential slice extracted by Line 11 must not be deplotted. The first contour object should be deselected in Line 23 by selecting object OBJECT00 and specifying REMOVE.

At Line 26 the representation of the conductors is changed by specifying WIRE EDGE.VIS and EDGE.COLOR. This causes only the outline of the conductors to be plotted in color number 2. The final plot is generated by Line 28; it appears in Figure 4-13.

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1 * RC3 RUN OUTPUT=raexc32 2 $ Example RAEXC32 two conductors crossing over two others 3 $ First conductor, a simple block 4 POLY3D NAME=conductor_a; COORD=4,0;9,0;9,1;4,1; V1=0,0,0; V2=0,0,5; + 5 PERP=0.0,1.0,0.0; VOLT=0; 6 $ Second conductor crosses over the first 7 POLY3D NAME=conductor_b; + 8 COORD=0,0;2,0;4,2;9,2;11,0;12,0;12,1;11.5,1;9,3;4,3;2,1;0,1; V1=0,0,0; + 9 V2=0,0,2; PERP=0.0,1.0,0.0; VOLT=1; 10 $ Third conductor crosses over the first, next to the second 11 POLY3D NAME=conductor_c; + 12 COORD=0,0;2,0;4,2;9,2;11,0;12,0;12,1;11.5,1;9,3;4,3;2,1;0,1; V1=0,0,3; + 13 V2=0,0,5; PERP=0.0,1.0,0.0; VOLT=0; 14 $ Nitride block (shows how multiple dielectrics are used) 15 POLY3D NAME=nitride2; COORD=4,1;9,1;9,2;4,2; V1=0,0,0; V2=0,0,5; + 16 PERP=0.0,1.0,0.0; DIEL=3; 17 $ Simulation window 18 WINDOW3D V1=0.0,0.0,0.0; V2=12.0,5.0,5.0; 19 OPTIONS UNIT=1e-6; SET_GRID=8000; 20 $ Do the analysis 21 POTENTIAL 22 CAPACITANCE conductor_a; conductor_c;

*** POTENTIAL CALCULATION [Coulombs]

Charge on conductor_a = -3.948775e-16 Charge on conductor_b = 6.316414e-16 Charge on conductor_c = -2.367658e-16 Maximum electric field: 1.356e+06 V/m at 9.500e-06, 1.500e-06, 2.000e-06

*** CAPACITANCE [Farad] CALCULATION: (C)(V)=(Q)

conductor_a conductor_c conductor_a 7.916002e-16 -3.978118e-16 conductor_c -3.948772e-16 6.316384e-16

==> SPICE Models for Total Capacitance (in Farad)

C_0_0 conductor_a OTHERS 7.916002e-16 C_1_1 conductor_c OTHERS 6.316384e-16

==> SPICE Models for Entire Capacitance Matrix (in Farad)

C_0_1 conductor_a conductor_c 3.963445e-16 C_0_0 conductor_a GROUND_RC3 3.952557e-16 C_1_0 conductor_c GROUND_RC3 2.352939e-16

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1... $ Read in the data 2... DATA RAPH FILE=raexc32.pot

3... $ Plot the device structure, first select all the objects 4... SELECT ALL

5... $ Plot the objects 6... PLOT.3D THET=70 PHI=30 TITLE=”Example RAEXC32: Device Structure”

7... $ Plot the structure with a color-filled contour slice 8... $ showing potential within the device 9... $ First extract the contour plane at Z=1.5 10... $ The extracted contour will be named ”object00” 11... EXTRACT CONTOUR POT Z.SLICE=1.5

12... $ Now select the contour (previously selected objects remain) 13... SELECT TYPE=CONTOUR 14... $ Plot all objects 15... PLOT.3D THET=80 PHI=30 TITLE=”RAEXC32: Structure & Contour Slice”

16... $ Plot the structure with a potential surface 17... $ First extract the surface with potential=0.5 Volts 18... $ This surface will be named ”object01” 19... EXTRACT CONTOUR POT MIN=.5 N.VAL=1 3D

20... $ Select all objects and get a listing of them 21... SELECT ALL PRINT

22... $ De-select the contour slice 23... SELECT REM NAME=OBJECT00

24... $ Change the plotting mode for the conductors so that only 25... $ their edges are visible 26... SELECT TYPE=CONDUCTOR WIRE EDGE.VIS EDGE.COL=2

27... $ Plot all objects 28... PLOT.3D PHI=110 TITLE=”RAEXC32: Structure & 0.5V Surface”

Figure 4-11 DPLOT file dpraexc32 for Example raexc32

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Figure 4-13 Structure and 0.5 volt surface for Example raexc32

Figure 4-12 DPLOT graphical output generated by Line 15 of file dpraexc32

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Example 3: Floating-Gate Transistor

Figure 4-14, shows a floating-gate transistor example, which is often used in a flash memory design. Although the simulation accounts for the existence of the floating poly, the goal is to find the equivalent capacitance between the metal and ground.

A partial listing of the output file is shown in Figure 4-15. The computed capaci-tance between the metal and ground is 0.3620 fF. The simulation has been per-formed using the -n option. Without using this option, the output value of the charge on the floating poly differs slightly from the specified value due to resimu-lation and the tolerance in the ICCG routine

Figure 4-14 Floating-gate transistor

metal

poly (floating)

ground (silicon)

0.5

0.5

0.4

0.8

2Medium

(Oxide:DIEL=3.9)

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.

To verify this result, the same simulation is performed without floating the poly. Figure 4-16 shows the resulting circuit model with three capacitors. After floating the poly node, these three capacitors can be reduced to a single capacitor from the metal node to the ground node, and the value of this equivalent capacitor is 0.3620 fF.

1 * RC3 RUN OUTPUT=raexc33 2 $ Example RAEXC33 a floating-gate transistor 3 PARAM w=0.8; t=0.4; l=2; gndt=0.1; h1=0.5+gndt; h2=h1+t+0.5; 4 PARAM offx=3*w; windx=2*offx+w; offy=3*l; windy=2*offy+l; windz=5*h2; 5 $ poly 6 BLOCK NAME=poly1; VOLT=1; FLOAT=0.0; + 7 v1=offx+w/2,offy+l/2,h1; + 8 HEIGHT=0.4; WIDTH=w; LENGTH=l; 9 $ metal 10 BLOCK NAME=metal; VOLT=1; + 11 V1=offx+w/2,offy+l/2,h2; + 12 HEIGHT=0.4; WIDTH=w; LENGTH=l; 13 $ ground 14 BLOCK NAME=gnd; VOLT=0; + 15 V1=windx/2,windy/2,0; + 16 HEIGHT=gndt; WIDTH=windx; LENGTH=windy; 17 $ Simulation window 18 WINDOW3D V1=0,0,0; V2=windx,windy,windz; DIEL=3.9; 19 $ Do the analysis 20 POTENTIAL 21 CAPACITANCE

*** Ref: Default set_grid is 10440.

*** Voltages of Floating Conductors:

Voltage on poly1 = 3.373413e-01

*** POTENTIAL CALCULATION [Coulombs]

Charge on poly1 = 0.000000e+00 Charge on metal = 3.622643e-16 Charge on gnd = -3.622511e-16


metal gnd metal 3.622643e-16 -3.622511e-16 gnd -3.617351e-16 3.617258e-16


C_0_0 metal OTHERS 3.622643e-16 C_1_1 gnd OTHERS 3.617258e-16


C_0_1 metal gnd 3.619931e-16


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Example 4: Anisotropic Dielectric Materials

As a final example, RC3 simulates a system with anisotropic dielectrics. While anisotropy is not supported by either RC2-BEM or RC3-BEM, it can be used to model the behavior of other devices.

• On-chip interconnects: The epitaxy of the dielectric deposition process causes the process to be nonisotropic.

• FeRAM devices: Ferroelectric materials used in FeRAM devices possess very large dielectric constants. Great differences exist between the dielectric con-stant in the c-axis direction versus the a- or b-axis directions.

To test the accuracy of RC3, the geometry shown in Figure 4-18 is used. This geometry corresponds to a conventional stripline structure with sapphire as a filler. Typical principal axes-relative dielectric constants for this material are 11.6 for ε || and 9.4 for . (For comparison, see analytical results from conformal mapping theory in Reference [1]. A partial program output list is shown in Figure 4-17.

Figure 4-16 The circuit model of Figure 4-14:a) before floating the polyb) after floating the poly

metal

poly

a) b)ground

0.2343 fF

0.3811 fF

0.192 fF

metal

ground

0.3620 fF

ε ⊥

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.

In Lines 6 and 7, variables are defined so as to clarify the structure definition in the subsequent BLOCK commands. Line 10 defines the properties of the sapphire dielectric block. Three values for the X, Y, and Z components of the dielectric constant are required. A randomly chosen value of 20 was used for the dielectric constant in the y direction which is along the axis of the stripline. By symmetry, εy must always be zero and hence the resulting capacitance must be independent of the y component of the dielectric constant. The use of a large value for εyy thus serves as an additional test of correct program operation.

Note:The coordinate system of the problem must be aligned with the axes of the sapphire so as to render the dielectric coefficient tensor diagonal.

Also note that although the dielectric block has identical size as the simula-tion window, the strip and groundplanes are defined to be 1 micron longer in the y direction than the simulation window. Similarly, the window width in the x direction as used in Line 16 is set to be 1 micron narrower than the groundplanes. These overhanging conductors have the effect of enforcing a

1 $ RC3 Anisotropic stripline 2 $ for comparision with conformal mapping results in IEEE MTT 3 $ MTT-30, No. 8 Aug 1982, pp 1264-1267 4 $ dieletric used is sapphire. 5 $ 6 param st=0.2;sw=20.0;wx=sw+10.0*20+10.0*st;sx=wx/2.0;gw=wx+1.0; 7 param tgz=20+0.2+st;wz=20+0.4+st;sz=0.2+20/2.0;dh=st+20; 8 $ 9 $ define the dielectric filler to be sapphire, which is anisotropic 10 block name=diel_0; v1=sx,0.5,0.2; width=wx;length=1;height=dh; diel=11.6,20,9.4; 11 $ 12 $ Now define groundplanes and the strip 13 block name=top_gnd; v1=sx,0.5,tgz; width=gw;length=2;height=0.2;volt=0; 14 block name=bot_gnd; v1=sx,0.5,0.0; width=gw;length=2;height=0.2;volt=0; 15 block name=strip; v1=sx,0.5,sz; width=sw;length=2;height=st; volt=1; 16 window3d v1=0,0,0; v2=wx,1,wz; diel=1.0; 17 capacitance 18 OPTIONS SET_GRID=40000; 19 OPTIONS MAX_REGRID=0;


C_0_0 top_gnd OTHERS 1.041500e-15 C_1_1 bot_gnd OTHERS 1.041498e-15 C_2_2 strip OTHERS 5.039391e-16


C_0_1 top_gnd bot_gnd 7.895221e-16 C_0_2 top_gnd strip 2.519737e-16 C_1_2 bot_gnd strip 2.519730e-16

Figure 4-17 Output generated by Example aniso.rc3

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Neumann boundary condition at the front, back, right and left window edges and forcing the normal components of the E field to be zero there.

The resulting total capacitance from the strip to its groundplanes is 503.94 af/micron. Conformal mapping theory gives a value of 496 af/micron, for a relative error of 1.6%. Discrepancies may be attributed to the limitations of the theory which assumes as infinitesimally thin center strip.

Figure 4-18 3D anisotropic stripline geometry for Example aniso.rc3

strip20

0.5

0.2

20

1

groundplane

zy

x

εx = 11.6εy = 20εz = 9.4

groundplane

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CHAPTER 5

R

RC3-BEM: 3D Field Solver by Boundary Element Method5

IntroductionRC3-BEM provides a viable alternative field solver to RC3 for 3D static field analysis. The physics of a 3D static field are governed by Possion’s equation. RC3 finds the solution to the Possion’s equation using the finite-difference (FD) method in which the continuous differential operators are approximated by the discrete finite-difference operators. RC3-BEM solves Possion’s equation by the boundary element method (BEM), where a differential equation is first converted into an equivalent integral equation, and then the integral equation is solved by the weighted residual method.

The theoretical background of BEM and the comparison of FD and BEM are pre-sented in the first two sections of Chapter 3, and you are encouraged to review these sections. In this chapter, the terms RC3 and RC3-BEM are used to denote the finite-difference and boundary element 3D solvers, respectively.

3D field analysis often involves a large number of unknowns compared to the 2D case. Because BEM uses a dense matrix requiring much memory and CPU time in a complicated structure in the 3D case, BEM does not hold a computational advantage over the FD method. However, as demonstrated in this chapter, RC3-BEM does perform significantly better than RC3 in simple 3D structures.

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RC3-BEM Command-Line Options Raphael Reference Manual

RC3-BEM Command-Line OptionsThe syntax of the RC3-BEM command line is as follows:

raphael rc3 -b “[BEM-OPTIONS]” [RC3-OPTIONS] [FILE]

where -b flag in RC3 is used to invoke the BEM solver.

CAUTIONThe double quotation marks for the BEM-options are always needed, even if no additional BEM options are desired. Example:

raphael rc3 -b “” [RC3-OPTIONS] [FILE]

The same input file syntax is used for RC3 and RC3-BEM except for a few minor differences, and these differences are listed in the following section. For the gen-eral input syntax and options, refer to Chapter 4.

Currently, only one option is related to RC3-BEM:

Option Definition

-s flag The symmetric plane option. The flag consists of two charac-ters: the first character defines the boundary condition at the window’s minimum and maximum x-coordinates, and the sec-ond character defines the boundary condition at the window’s minimum and maximum y-coordinates. The character can be either e, m, M, B, or x:• e places a perfect electric plane at the minimum x- or y-

coordinate.

• E places a perfect electric plane at the maximum x- or y-coordinate.

• m places a perfect magnetic plane at the minimum x- or y-coordinate.

• M places a perfect magnetic plane at the maximum x- or y-coordinate.

• B places two perfect magnetic planes at both the maximum and the minimum of x- or y-coordinate.

• x indicates that no additional boundary condition is imposed.

The perfect magnetic and electric planes can be used to model the symmetric geometry with even-mode (symmetric) and odd-mode (asymmetric) excitations, respectively. The magnetic plane is the default condition at all window boundaries for the finite-difference solver RC3.

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Raphael Reference Manual Notes on the Current Version of RC3-BEM

A proper choice of these boundary conditions, which are discussed in more detail in Appendix B, can improve the accuracy and/or performance of simulation.

Example:

raphael rc3 -b “-s BB” [RC3-OPTIONS] [FILE]

means that magnetic planes are placed at all minimum and maximum x- and y-coordinates of the window boundaries.

Notes on the Current Version of RC3-BEM

Note:Read this section before using RC3-BEM.

This section lists the limitations of the current version of RC3-BEM and discrep-ancies between the RC3 and RC3-BEM solvers. The electrostatic analysis (POTENTIAL and CAPACITANCE keywords) is analogous to the static resistance analysis (CURRENT and RESISTANCE keywords) and the static thermal analysis (TEMPERATURE and THEMORES keywords). Thus, limitations mentioned in this section that are related to the electrostatic (capacitance) analysis apply equally to the resistance and thermal analyses.

• Nonplanar dielectrics are ignored. The planar dielectrics are defined by box-shaped or rectangular-shaped poly objects whose boundaries are parallel to the x-, y- and z-axes, with widths (in the xy plane) at least equal to the size of the window in the xy plane.

• The graphics files (i.e., <file>.pot and <file>.tdf) for potential or field plots are not supported.

• The keyword EXTRACT is ignored.

• The keyword CHRG (i.e., fixed charge density in a dielectric element) is ignored.

• Regridding is not needed. Keywords MAX_REGRID, REGRID_TOL, and FAC_REGRID are ignored.

• As in RC3, non-Manhattan boundaries are staircased.

• Because of fast convergence, RC3-BEM uses a smaller default grid than RC3. (The default grid can be overridden by the keyword SET_GRID.)

• When conducting traces are located either above or below a conducting plane, the conducting plane is modeled as an infinitely wide ground plane, and VOLT=0 is assumed on that ground plane.

• When conducting traces are sandwiched in between two conducting planes (layers), the bottom conducting plane is modeled as an infinitely wide ground plane, and the top conducting plane is modeled as a normal conducting trace, with its area equal to the xy plane of the window. Because of this truncation of the top ground plane to a finite width, RC3-BEM may not give accurate results for the coupling term between the top and bottom ground planes.

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• RC3-BEM cannot directly handle geometries where a conducting plane is located between traces. Since the conducting plane electrically isolates the traces from one side of the plane to the traces on the other side of the plane, geometries can be indirectly solved by formulating a separate problem for each side of the plane.

Note that most of the limitations shown above are identical to the 2D case (RC2-BEM).

Comparison of RC3-BEM and RC3In this section, four models are constructed to compare the accuracy and simula-tion time of RC3-BEM to RC3:

• In the first example, a single plate over a ground plane is considered. The con-vergence of the capacitance value and CPU time versus the grid size is stud-ied, and the results from both RC3-BEM and RC3 simulations are compared with published results.

• In the second example, the crossover capacitances of two rectangular conduc-tors are simulated and compared.

• In the third example, a trapezoidal conductor between two ground planes is considered to verify the RC3-BEM result for non-Manhattan geometry.

• In the last example, the substrate resistance is simulated to verify the resis-tance computation of RC3-BEM.

It should be noted that RC3 uses the magnetic plane as a default condition at all window boundaries. In some cases, applying the magnetic plane to exploit the symmetry of geometry can both reduce the problem size and remove the effects of an artificial magnetic plane at the open boundary. For better comparison between RC3 and RC3-BEM, geometrical symmetry is modeled by magnetic planes. The CPU time given in this section is measured on a SPARC-20 workstation.

Example 1: Single Plate Above a Plane

A single plate above a ground plane shown in Figure 5-1 is simulated for this example. This model considers the homogenous dielectric layer through the win-dow statement in the input file. The RC3-BEM partial output listing is shown in Figure 5-2. In the RC3 simulation, regridding is performed once without increas-ing the grid points (i.e., FAC_REGRID=1.0).

For both RC3-BEM and RC3, the convergence of the capacitance value versus the grid size is analyzed, and shown in Figure 5-3. In this simulation, the Galerkin’s method allows RC3-BEM to converge much faster than RC3. The fast conver-gence speed of RC3-BEM validates the use of the small default SET_GRID value for the RC3-BEM solver.

The CPU time versus the grid size is shown in Figure 5-4. Although for a small number of grid points, the RC3-BEM performed much better than RC3, as the number of grid points increased, the simulation time for RC3-BEM increased

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more rapidly than that for RC3. For this reason, simulation time must be consid-ered carefully when applying RC3-BEM.

For this structure, RC3-BEM and RC3 use a default SET_GRID of 441 and 21,112, respectively. The capacitance value converged to 2.554 fF. FASTCAP gave 2.550 fF (K. Nabors and J. White, IEEE Trans. Circuits Syst.-I, November 1992), and Sun et. al showed 2.408 fF (33rd DAC, 1996).

Figure 5-1 Single plate above a ground plane (all units are in microns)

l = 10

w = 5

thickness = 1

diel =3.9

Ground Planeh = 2

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Figure 5-2 Partial listing of output file raexc34 for RC3-BEM


1 * RC3 RUN OUTPUT=raexc34 2 $ Example RAEXC34 3 $ Single plate over a ground plane 4 5 PARAM w=5.0 l=10 t=1.0 h=2.0 offx=2*w offy=2*l 6 PARAM windx=2*offx+w windy=2*offy+l windz=5*windy gndthick=0.1 7 $ Ground Plane 8 block name=grnd; volt=0.0; 9 width=windx; length=windy; 10 v1 = 0.5*windx, 0.5*windy, 0.0; 11 v2 = 0.5*windx, 0.5*windy, gndthick; 12 perp = 0.0, 1.0, 0.0; 13 $ Metal line 1 14 block name=box; volt=1.0; 15 width=w; length=l; 16 v1 = 0.5*windx, 0.5*windy, h+gndthick; 17 v2 = 0.5*windx, 0.5*windy, h+gndthick+t; 18 perp = 0.0, 1.0, 0.0; 19 $ Define the simulation window 20 window diel=3.9; 21 v1 = 0.0, 0.0, 0.0; 22 v2 = windx, windy, windz; 23 $ Do calculations 24 options set_grid=5000; 25 capacitance

*** CAPACITANCE CALCULATION [Farad]

box box 2.548814e-15


C_2_2 box OTHERS 2.548814e-15


C_2_0 box GROUND_RC3 2.548814e-15

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Figure 5-3 Convergence of capacitance value versus grid size for single plate structure in Figure 5-1

Figure 5-4 CPU time versus grid size for single plate structure in Figure 5-1

Capacitance vs. Grid Size

2.5

2.55

2.6

2.65

2.7

2.75

2.8

4.4E

+2

1.0E

+3

3.0E

+3

7.0E

+3

1.0E

+4

1.5E

+4

2.1E

+4

2.5E

+4

3.5E

+4

6.0E

+4

set_grid

C (f

F)

RC3-BEM

RC3

Default set_gridfor RC3

Default set_gridfor RC3-BEM

CPU Time vs. Grid Size

0.1

1

10

100

1000

4.4E

+02

1.0E

+03

3.0E

+03

7.0E

+03

1.0E

+04

1.5E

+04

2.1E

+04

2.5E

+04

3.5E

+04

6.0E

+04

set_grid

Seco

nds

RC3--BEM

RC3

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Example 2: Crossover Capacitance

In Figure 5-5, the cross-over capacitance of two equal-sized rectangular conduc-tors above a ground plane is computed. To take advantage of the symmetric struc-ture, two magnetic planes are placed at the maximum x and y window boundaries (i.e., the command option -s MM is used). Thus, the calculated capacitance values must be quadrupled to obtain the true capacitance. The width, thickness, and length of the conductor are 2, 0.5, and 30 microns, respectively. The relative dielectric constants of the superstrate and substrate are 4.3 and 3.9. The RC3-BEM partial output listing is shown in Figure 5-6.

The default grid size is used in the simulation, and the default SET_GRID value is 637 for RC3-BEM and 13200 for RC3. Again, during the RC3 simulation, regrid-ding is used once. The difference between the results with and without regridding is about 6%. The simulation time is 2.9 sec for RC3-BEM and 80.3 sec for RC3. The calculated crossover capacitance matrix is:

Figure 5-5 Two crossover rectangular conductors above a ground plane

4 μm

m2

m1

Substrate

Superstrate

Ground Plane

Free Space

8 μm

m1m2

0.77450 0.08527–0.08527– 0.38695

(fF) for RC3-BEM

m1m2

0.80492 0.10458–0.09679– 0.412848

(fF) for RC3

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1 * RC3 RUN OUTPUT=raexc35 2 $ Example RAEXC35 3 $ A rectangular conductor crossover above a ground plane 4 $ NOTE: Symmetric planes are used at the xmax and ymax boundaries 5 $ of window 6 PARAM w=2.0 w2=w/2 l=30.0 l2=l/2 t=0.5 h1=4.0 h2=8.0 gndthick=1 7 PARAM off=3.0*l2 windx=off+l2 windy=windx windz=4*windy 8 9 $ Ground Plane 10 block name=grnd; volt=0.0; 11 width=windx; length=windy; 12 v1 = 0.5*windx, 0.5*windy, 0.0; 13 v2 = 0.5*windx, 0.5*windy, gndthick; 14 perp = 0.0, 1.0, 0.0; 15 16 $ Substrate 17 block name=subtrate; diel=3.9; 18 width=windx; length=windy; 19 v1 = 0.5*windx, 0.5*windy, gndthick; 20 v2 = 0.5*windx, 0.5*windy, h1+gndthick; 21 perp = 0.0, 1.0, 0.0; 22 23 $ Superstrate 24 block name=superstrate; diel=4.3; 25 width=windx; length=windy; 26 v1 = 0.5*windx, 0.5*windy, h1+gndthick; 27 v2 = 0.5*windx, 0.5*windy, h1+h2+gndthick; 28 perp = 0.0, 1.0, 0.0; 29 30 $ Metal line 1 31 block name=m1; volt=1.0; 32 width=w2; length=l2; 33 v1 = windx-w2/2.0, windy-l2/2.0, h1+gndthick; 34 v2 = windx-w2/2.0, windy-l2/2.0, h1+gndthick+t; 35 perp = 0.0, 1.0, 0.0; 36 37 $ Metal line 2 38 block name=m2; volt=1.0; 39 width=l2; length=w2; 40 v1 = windx-l2/2.0, windy-w2/2.0, h1+h2+gndthick; 41 v2 = windx-l2/2.0, windy-w2/2.0, h1+h2+gndthick+t; 42 perp = 0.0, 1.0, 0.0; 43 44 $ Define the simulation window 45 window diel=1.0; 46 v1 = 0.0, 0.0, 0.0; 47 v2 = windx, windy, windz; 48 $ Do calculations 49 capacitance

*** CAPACITANCE CALCULATION [Farad] m1 m2 m1 7.745911e-16 -8.514228e-17 m2 -8.514228e-17 3.871475e-16 ==> SPICE Models for Entire Capacitance Matrix (in Farad) C_2_3 m1 m2 8.514228e-17 C_2_0 m1 GROUND_RC3 6.894488e-16 C_3_0 m2 GROUND_RC3 3.020052e-16

Figure 5-6 RC3-BEM partial listing output file raexc35

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Example 3: A Trapezoidal Conductor Between Two Ground Planes

This example considers a trapezoidal conductor between two ground planes, as shown in Figure 5-7. To simplify the problem, the symmetry in the longitudinal direction is exploited by using a perfect magnetic plane (i.e., the command option -s Mx is used). Because of this change, the calculated capacitance values should be doubled to find true answers. The RC3-BEM partial output listing is shown in Figure 5-8.

In the simulation, to accurately model the trapezoidal shape, SET_GRID is set to 15000 for both RC3-BEM and RC3. The simulation time is 95.4 sec for RC3-BEM and 52.3 sec for RC3. The simulation time for RC3-BEM significantly increases because of the finer grid and modeling of the top ground plane. The computed capacitance matrix is:

It should be noted that the top plate capacitance term, C22, for RC3-BEM is larger than for RC3 due to the fringing effect. The fringing effect is caused by modeling the infinite-long top ground plane by a finite, window-sized plate.

Figure 5-7 Trapezoidal conductor between two ground planes

4 μm

m1

diel = 3.9

Bottom Ground Plane

8 μm

Top Ground Plane

Free Space

m1TopGnd

1.5427 0.1642–

0.1642– 4.6774(fF) for RC3-BEM

m1TopGnd

1.6147 0.1820–

0.1829– 2.1580(fF) for RC3

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1 * RC3 RUN OUTPUT=raexc36 2 $ Example RAEXC36 3 $ A trapezoidal conductor between two ground planes 4 $ NOTE: Symmetric planes are used at the xmax boundary of window 5 6 PARAM w1=1.0 w2=2.0 l=20.0 l2=l/2 t=1.0 h1=4.0 h2=8.0 gndthick=1 7 PARAM offx=5.0*l2 windx=offx+l2 offy=8.0*w2 windy=offy+w2 8 9 $ Bottom Ground Plane 10 block name=BottomGnd; volt=0.0; 11 width=windx; length=windy; 12 v1 = 0.5*windx, 0.5*windy, 0.0; 13 v2 = 0.5*windx, 0.5*windy, gndthick; 14 perp = 0.0, 1.0, 0.0; 15 16 $ Top Ground Plane 17 block name=TopGnd; volt=0.0; 18 width=windx; length=windy; 19 v1 = 0.5*windx, 0.5*windy, gndthick+h1+h2; 20 v2 = 0.5*windx, 0.5*windy, 2.0*gndthick+h1+h2; 21 perp = 0.0, 1.0, 0.0; 22 23 $ Substrate 24 block name=subtrate; diel=3.9; 25 width=windx; length=windy; 26 v1 = 0.5*windx, 0.5*windy, gndthick; 27 v2 = 0.5*windx, 0.5*windy, gndthick+h1; 28 perp = 0.0, 1.0, 0.0; 29 30 $ Conductor 31 poly name=m1; volt=1.0; 32 coord=-0.5*w2,0;0.5*w2,0;0.5*w1,t;-0.5*w1,t; 33 v1 = windx-l2, 0.5*windy, h1+gndthick; 34 height=l2; 35 direction = 1.0, 0.0, 0.0; 36 37 $ Define the simulation window 38 window diel=1.0; 39 v1 = 0.0, 0.0, 0.0; 40 v2 = windx, windy, 2.0*gndthick+h1+h2; 41 42 $ Do calculations 43 options set_grid=15000; 44 capacitance

*** CAPACITANCE CALCULATION [Farad] TopGnd m1 TopGnd 2.338676e-15 -8.211564e-17 m1 -8.211564e-17 7.713394e-16 ==> SPICE Models for Entire Capacitance Matrix (in Farad) C_2_3 TopGnd m1 8.211564e-17 C_2_0 TopGnd GROUND_RC3 2.256560e-15 C_3_0 m1 GROUND_RC3 6.892238e-16

Figure 5-8 RC3-BEM partial listing of raexc36

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Example 4: Substrate Resistance

In this example, the resistances among three contacts are simulated. Figure 5-9 shows the simulated structure after applying the symmetric plane at the center of contacts (i.e., the command option -s xM is used). All contacts have the same size: width =5, length = 5, thickness =1.4, and spacing = 6 microns. The RC3-BEM partial output listing is shown in Figure 5-10.

In this simulation, SET_GRID is set to 20,000 for RC3-BEM and 60,000 for RC3. The simulation time is 11.8 sec for RC3-BEM and 17.7 min for RC3. The conduc-tance matrix is:

Figure 5-9 Three contacts over two substrates

7 μm

c2c1

Ground Plane

297 μmRHO = 5

RHO = 1.0e6

Bulk

Contacts

Epi-layerRHO = 1500

c3

c1c2c3

9272.8 436.26– 12.127–

436.26– 9299.8 436.26–

12.127– 436.26– 9272.8(pMho) for RC3-BEM

c1c2c3

9930.8 482.18– 15.493–

483.76– 9945.4 493.36–

16.040– 482.02– 9929.8(pMho) for RC3

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1 * RC3 RUN OUTPUT=raexc37 2 $ Example RAEXC37 3 $ Three contacts above two substrate and a ground plane 4 $ Doping resistance computation 5 $ NOTE: A symmetric plane is used at the ymax window boundary. 6 7 PARAM w=5.0 w2=w/2 l=5.0 l2=l/2 t=1.4 s=6.0 h1=297.0 h2=7.0 gndthick=2 8 PARAM off=1.5*(h1+h2) windx=off+3.0*w*2.0*s windy=off windz=4*windy 9 $ Ground Plane 10 block name=grnd; volt=0.0; 11 width=windx; length=windy; 12 v1 = 0.5*windx, 0.5*windy, 0.0; 13 v2 = 0.5*windx, 0.5*windy, gndthick; 14 perp = 0.0, 1.0, 0.0; 15 $ Substrate 16 block name=Bulk; rho=5.0; 17 width=windx; length=windy; 18 v1 = 0.5*windx, 0.5*windy, gndthick; 19 v2 = 0.5*windx, 0.5*windy, h1+gndthick; 20 perp = 0.0, 1.0, 0.0; 21 $ Superstrate 22 block name=EpiLayer; rho=1500.0; 23 width=windx; length=windy; 24 v1 = 0.5*windx, 0.5*windy, h1+gndthick; 25 v2 = 0.5*windx, 0.5*windy, h1+h2+gndthick; 26 perp = 0.0, 1.0, 0.0; 27 $ Contacts 28 block name=c1; volt=1.0; 29 width=w; length=l2; 30 v1 = off+w2, windy-l2/2.0, h1+h2+gndthick; 31 v2 = off+w2, windy-l2/2.0, h1+h2+gndthick+t; 32 perp = 0.0, 1.0, 0.0; 33 block name=c2; volt=1.0; 34 width=w; length=l2; 35 v1 = off+w+s+w2, windy-l2/2.0, h1+h2+gndthick; 36 v2 = off+w+s+w2, windy-l2/2.0, h1+h2+gndthick+t; 37 perp = 0.0, 1.0, 0.0; 38 block name=c3; volt=1.0; 39 width=w; length=l2; 40 v1 = off+2.0*w+2.0*s+w2, windy-l2/2.0, h1+h2+gndthick; 41 v2 = off+2.0*w+2.0*s+w2, windy-l2/2.0, h1+h2+gndthick+t; 42 perp = 0.0, 1.0, 0.0; 43 $ Define the simulation window 44 window rho=1.0e6; 45 v1 = 0.0, 0.0, 0.0; 46 v2 = windx, windy, windz; 47 $ Do calculations 48 options set_grid=20000 fac_regrid=1.0; max_regrid=1; 49 resistance

*** CONDUCTANCE CALCULATION [Mho]

c1 c2 c3 c1 4.636389e-09 -2.181296e-10 -6.063534e-12 c2 -2.181296e-10 4.649880e-09 -2.181296e-10 c3 -6.063534e-12 -2.181296e-10 4.636389e-09

Figure 5-10 RC3-BEM partial listing of output file raexc37

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In general, the conductance values are very low because of the lightly doped epi-layer. The conductance between C1 and C3 exhibits the largest difference between RC3 and RC3-BEM. This difference is mainly due to the window bound-aries. Fortunately, because this value is so small, the difference can be ignored for most applications.

The SPICE model representation of the conductance matrix obtained from RC3-BEM is shown in Figure 5-11. As the figure illustrates, the resistance between C1 and C3 is so large that it is effectively an open circuit.

Figure 5-11 SPICE model of the substrate resistance using RC3-BEM

C1 C2

C3

0.1133 GΩ

164.92 GΩ

2.292 GΩ

2.292 GΩ

0.1187 GΩ

0.1133 GΩ

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CHAPTER 6

R

RI3: 3D Resistance and Inductance with Skin Effect6

IntroductionRI3 is a computer program for calculating 3D inductances. For a given set of con-ductors, RI3 calculates the impedance matrix that contains an inductance matrix and resistance vector. Since the impedance matrix is usually too large for practical applications, the matrix can be:

• Saved for RI3 post-processing.

• Immediately reduced for specified external nodes at given frequencies. The advantage of this setup is that the full inductance and resistance matrix is cal-culated only once, allowing different grounding and power strategies to be studied in a post-processing step.

This chapter describes the theoretical aspects of RI3, program usage, and simula-tion examples.

TheoryFor inductance problems, a clear understanding of self- and mutual-loop induc-tance, as well as self- and mutual-partial inductance is required.

Figure 6-1 shows an example of loop inductances. In this example, L11 and L22 represent self inductances, and L12 represents the mutual inductance between the two loops.

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Theory Raphael Reference Manual

Figure 6-2 shows partial inductances. In this example, L11 and L22 are the partial self inductances of both wires, and L12 is their mutual partial inductance. Once the partial inductances are defined, they can be treated as loop inductances because the voltage and current relationships are the same for both cases. For this reason, Raphael can use the same circuit model for loop inductance and partial inductance.

If the current density is uniform in conductors k and m, the partial mutual induc-tance, is Reference [8]:

Equation 6-1

where and are the cross-sectional areas of each conductor, and are the start and end points of each conductor, and is the distance between and

, which represent differential elements of length of conductors k and m.

Thus, the partial inductance can be approximated by inductances of thin conduc-tors (filaments), , with P filaments in conductor and Q filaments in conduc-tor

Equation 6-2

Figure 6-1 Loop inductances

Figure 6-2 Partial inductances

L11 L22L12

L12

L11

L22

Loop = L11 + L22 -2 * L12

Lpkm

Lpkmμ

4π------ 1

akam------------

ld k ld mrkm

--------------- ad k ad m

bm

cm

∫bk

ck

∫am

∫ak

∫=

ak am bi cirkm dlk

dlm

Lpfij km

Lpkm1

PQ--------

P ∞→ Q ∞→,lim Lpfij

j 1=

Q

∑i 1=

P

∑=

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Raphael Reference Manual Theory

where:

. Equation 6-3

All the preceding calculations assume that the current density in the conductor is uniform. If a conductor has a nonuniform current distribution, with skin effect for example, the conductor must be divided into small pieces in which the assumption of uniform current density can be satisfied in each segment Reference [9].

RI3 has a basic current cell called SINGLE_BAR in which the current distribution is assumed to be uniform. When two current cells are neither parallel nor orthogo-nal to each other, the current-filament formulation of Equations 6-2 and 6-3 is used to compute the mutual inductances. For greater accuracy and efficiency, a current-sheet formulation is used to compute the mutual inductances among paral-lel conductors where many current filaments can be grouped together into a few current sheets. An analytic formula of mutual inductances among current sheets is evaluated directly.

Complex problems with many current cells yield very large matrices. However, usually only the impedance matrix seen from the external nodes is needed. With the following procedure, a full impedance matrix with an order the same as the number of SINGLE_BARs can be reduced to an external impedance matrix with a smaller order the same as the number of external nodes.

The voltage (Velem) and current (Ielem) of elements are dependent on the frequency ( ), and they have the relationship:

Equation 6-4

where the partial impedance matrix (Zp) is a function of the resistance (R) and the partial inductance (Lp) by the equation that follows:

. Equation 6-5

The above equations relate potentials, currents, and impedances for conductive elements. In order to express these values on the nodes that define the elements, the equations must be converted into nodal equations as follows:

Equation 6-6

where, if A is the incidence matrix:

. Equation 6-7

The incidence matrix stores geometrical information that relates conductive ele-ments with the nodes used to define them.

Lpfijμ

4π------=

lid ljd⋅rij

-------------------

bj

cj

∫bi

ci

∫

ω

Velem ω( )[ ] Zp ω( )[ ] Ielem ω( )[ ]=

Zp ω( )[ ] R[ ] jω Lp[ ]+=

Vnode ω( ) Znode ω( )[ ] Inode ω( )[ ]=

Znode ω( )[ ] AZp ω( ) 1– AT[ ]1–

=

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Program RI3 Raphael Reference Manual

Program RI3To help understand the operation of the program, the data flow for RI3 is shown in Figure 6-3.

The syntax of the RI3 command line is as follows:

raphael ri3 [OPTIONS] [FILE]

The syntax of the input file is explained in the next section, “RI3 Input File.” If <file> is omitted, RI3 takes the input from the standard input.

When no options are specified, RI3 reads <file> which contains geometry and connectivity information, and creates three output files:

• <file>.out contains a copy of the input file and the error messages, if any.

• <file>.mat contains the names of the current elements and nodes, the imped-ance matrix, and the incidence matrix in binary form. RI3 takes file.mat as an input for post-processing analysis.

• <file>.geo contains geometry information for plotting.

The following options are interpreted by RI3:

Figure 6-3 Data flow for RI3

GEOMETRY

RI3

.mat

RI3

.out

.out

.geo DPLOT

EXT. NODESREF NODESFREQUENCY

Option Definition

-c <file> Generates an RC3 input <file> to be used in 3D capacitance simulations.

-g <file> Uses <file> as the geometry file name.


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Raphael Reference Manual RI3 Input File

f

Example:

raphael ri3 -c file1.rc3 -i file1

This command checks the input file file1. If there are no errors, it generates the file1.geo geometry file and the file1.rc3 RC3 input file.

RI3 Input FileThe input file statements consist of comments and/or commands. Lines beginning with * or with $ are comment lines and can be ignored. Twelve commands are available: PARAM, NODE, PLANE_NODE, SINGLE_BAR, MULTI_BAR, PLANE, OPTIONS3I, MATRIX, MERGE3I, EXT, REF, and FREQUENCY. An input file consists of four sections:

• Node definitions with NODE and PLANE_NODE.

• Current element definitions with SINGLE_BAR, MULTI_BAR and PLANE.

• Definition of options for simulation control with the command OPTIONS3I.

• Post-processing commands MATRIX, MERGE3I, EXT, REF, and FREQUENCY.

A command is followed by assignments defined in this section and may occupy more than one line. The input is in free format. Uppercase and lowercase charac-ters are interpreted differently, meaning any instruction is recognized when writ-

-i Checks the input file for errors and generates the geometry file ino errors are found. No calculation is performed.

-m <file> Uses <file> as the matrix file name.



-p Prints “original” resistance and partial inductance values.

-s Sends output to the standard output.

-t Prints CPU time summary.

-w <number> Override the adjacent cell ration in MULTI_BAR to <number>.


-P “<param > = <value>;”

Replaces the value of the parameters defined in the input file with new values. For example, -P “w=0.5; h = 0.1” will over-ride the values of parameters w and h defined in the input file with the new values, 0.5 and 0.1 respectively

-N Invokes new solver with increased efficiency (see Selection of Extraction Algorithm, p. 6-17 for more details).

Option Definition

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RI3 Input File Raphael Reference Manual

ten with all lowercase or all uppercase letters. The syntax and usage of each command are described in this section.

Lines beginning with the special symbol (+) are considered a continuation line from the last entered command. When the (+) special symbol is at the end of a line, the following line is considered a continuation line. The symbol is not required and it can be omitted.

PARAM

This command defines variables and their values for later use in the input file. Variable names must begin with an alphabetic character, but may include up to 300 alphanumeric characters.

value can be any algebraic expression with numbers, predefined variables, and functions. The supported functions are sin(), cos(), atan(), log(), log10(), exp(), sqrt(), int(), and abs(). The argument for all trigono-metric functions should be given in radians. The unit for the values used for length dimensions is specified in the OPTIONS command. This definition of value is also valid when used in value arguments in the following commands.

Example:


The command defines a value of 2.0 for parameter a, parameter b is assigned a value of 4, and y is defined as the exponential of the sum of a and b.

NODE

This command defines a node. A vector is the collection of three values separated by commas.

PARAM PARAMETER1=<VALUE>; PARAMETER2=<VALUE>; …;

NODE NAME=<NAME>; POSITION=<VECTOR>; [PLANE_NAME=<PLANE_NAME>;]

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Example:

NODE NAME=N1; POSITION=0,1,1;

The command defines a node named n1 located at (0,1,1) in the global coordinate system.

PLANE_NODE

This command defines a set of nodes forming a plane.


NAME character Name of the node. name must begin with an alphabetic character, but may include up to 10 alphanumeric charac-ters. This rule applies to all name arguments in other commands.

POSITION0 vector Position of the node. Nodes are used to define current elements. Coordinates of the node should be in the center of the cross-section of the current element defined by the node.

PLANE_NAME

character Name of the plane to be connected by a SINGLE_BAR with this NODE. This optional parameter is used for easy connection of PLANEs with BARs. When PLANE_NAME is defined in this instruction, a SINGLE_BAR is created between the coordinates of this NODE and the nearest node of the PLANE plane_name.

PLANE_NODE NAME=<NAME>; NORMAL=<VECTOR>; CENTER=<VECTOR>; L1=<VALUE>; L2=<VALUE>; N1=<VALUE>; N2=<VALUE>;


NAME character Base name of the nodes that form a plane. The plane is discretized and each node in the plane is named automatically. The name of each node is defined by this base name with an additional suffix, as shown in Figure 6-4.

NORMAL vector Normal vector of the plane. The normal vector must be parallel to the X-,Y-, or Z-axis. When the normal vector is parallel to the X-axis (Y-axis, Z-axis), the horizontal axis in Figure 6-4 is parallel to Y-axis (Z-axis, X-axis) and the vertical axis is parallel to Z-axis (X-axis, Y-axis).

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Example:

PLANE_NODE NAME=PN1; NORMAL=0,0,1; CENTER=0,0,0; L1=3.0; L2=5.0; N1=3; N2=4;

The command defines a set of plane nodes named pn1 with its normal oriented in the global Z-axis, at a width of 3.0 units along the X-axis and 3 subdivisions in that direction, and a length of 5.0 units along the Y-axis and 4 subdivisions in this direction.

CENTER vector Coordinates of the center of the plane. This refers to the true center of the plane, even in the middle of its thickness.

L1 numeric Horizontal length in Figure 6-4 (from the first node to the last node). See Table 6-1.

L2 numeric Vertical length in Figure 6-4 (from the first node to the last node). See Table 6-1.

N1 numeric Number of sections in the horizontal direction in Figure 6-4. There are (n1+1) nodes in this direc-tion.

N2 numeric Number of sections in the vertical direction in Fig-ure 6-4. There are (n2+1) nodes in this direction.


Figure 6-4 Discretization of the plane and the suffixes defined by RI3 (n1=3 and n2=4)

000_000

000_002

000_003

000_004

001_000

001_002

001_003

001_004

002_000

002_002

002_003

002_004

003_000

003_002

003_004

003_001002_001001_001000_001

003_003

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SINGLE_BAR

This command defines a single bar, the basic current cell unit in RI3. The current in the cell is assumed to be uniform. A single bar is defined by its width and height, and by the two nodes that can be arbitrarily oriented. The width direction should be always parallel to the XY-plane. As an example, if the axis defined by the two nodes is parallel to the Z-axis, the width direction is parallel to the X-axis. See Table 6-2 for examples of orthogonal directions. This rule is the same for MULTI_BAR elements.

Table 6-1 Orientation of L1 and L2

normal L1 L2

X Y Z

Y Z X

Z X Y

SINGLE_BAR NAME=<NAME>; NODE1=<NAME>; NODE2=<NAME>; W=<VALUE>; H=<VALUE>; [RHO=<VALUE>;]


NAME character Name of the single bar.

NODE1 character Name of the start node.

NODE2 character Name of the end node.

W numeric Width. If not defined, the width of the last defined SINGLE_BAR is used.

H numeric Height. If not defined, the height of the last defined SINGLE_BAR is used.

RHO numeric Resistivity of the single bar.Units: Ohm-metersDefault: If not defined, it is assumed to be 0. How-ever, if RHO is defined for another element ahead of this element, the last RHO is used.

Table 6-2 Orientation of w and h

bar direction w h

X Y Z

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Example:

SINGLE_BAR NAME=SB1; NODE1=1; NODE2=10; w=.8; h=1.0;

The example presents a current element of a rectangular cross-section named sb1 that extends between nodes 1 and 10, with a width of 0.8 and a height of 1.0 units. As the resistivity was not defined, the previously defined value is used. (If it is not defined previously, it is assumed to be zero.)

MULTI_BAR

This command creates a set of SINGLE_BARs for skin-effect calculation. The width and height of the outer current cells are smaller than those of the inner cells. The width and height ratio between adjacent cells is , as shown in Figure 6-5, to allow for finer discretization in areas of higher current density. The multi_bar can be arbitrarily oriented, but the width should be con-tained in the XY-plane. (See SINGLE_BAR, p. 6-9.)

Y X Z

Z X Y

Table 6-2 Orientation of w and h

bar direction w h

1 6 n⁄ n NWorNH=( )+

MULTI_BAR NAME=<NAME>; NODE1=<NAME>; NODE2=<NAME>; W=<VALUE>; H=<VALUE>; NW=<VALUE>; NH=<VALUE>; or(FRCTN=<VALUE>)[RHO=<VALUE>;]


NAME character Base name of the MULTI_BAR. Separate names are assigned internally to each current cell using suf-fixes.

NODE1 character Name of the start node.

NODE2 character Name of the end node.

W numeric Width. Default: If not defined, the width of the last defined MULTI_BAR is used.

H numeric Height. Default: If not defined, the height of the last defined MULTI_BAR is used.

NW numeric Number of divisions in width. Default: If it is an even integer, it increments by 1.

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Example:

MULTI_BAR NAME=MB1; NODE1=11; NODE2=21;

W=1.8; H=1.0; FRCTN=0.05; RHO=1.0E-8;

In this example, the granularity of the current elements will be determined by FRCTN such that the percentage estimated current change ratio between any two neighbored current elements to the total current in the bar is less than 5 percent.

Example:

MULTI_BAR NAME=MB1; NODE1=11; NODE2=21;W=1.8; H=1.0; NW=5; NH=3; RHO=1.0E-8;

The example presents a current element of a rectangular cross- section used for skin effect simulations named mb1 that extends between nodes 11 and 21, with a width of 1.8 and a height of 1.0 units. In the width direction, 5 cell elements were

NH numeric Number of divisions in height. Default: If it is an even integer, it increments by 1.

FRCTN numeric Estimated current change ratio between two neigh-bored current elements. If FRCTN is specified, the current elements are generated such that the esti-mated current change between two neighbored cur-rent elements is equal to the total current in the bar times FRCTN. The value of END_FREQ is used in calculating the estimated currents. If parameter NW or NH is specified, NW or NH takes the precedence in the corresponding direction, respectively.Default: FRCTN has a lower precedence than NH and NW.

RHO numeric Resistivity of each single bar comprising the multi-bar. Units: Ohm-meters.Default: If not defined, it is assumed to be 0. How-ever, if RHO is defined for another element ahead of this element, the last RHO is used.


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requested, and 3 cell elements were defined for the height direction. The resistiv-ity of the material is 1.0*10-8 Ohm-meter.

PLANE

This command creates a set of current cells linking the set of nodes created by a PLANE_NODE command.

Example:

PLANE NAME=PL1; BASE_NODE=BN10; H=.4;

The command defines a plane named pl1, with a height of 0.4 units and with its position, orientation, and dimensions defined by the PLANE_NODE bn10.

Figure 6-5 Discretization of the cross-section of MULTI_BAR

h (nh = 3)

w (nw = 5)

PLANE NAME=<NAME>; BASE_NODE=<NAME>; H=<VALUE>; [RHO=<VALUE>;]


NAME character Base name of the plane. Separate names are assigned internally to each current cell using suf-fixes.

BASE_NODE character Base name of the PLANE_NODE that defines the plane.

H numeric Height (thickness) of the plane.

RHO numeric Resistivity of the single bar. Units: Ohm-metersDefault: If not defined, it is assumed to be 0. How-ever, if RHO is defined for another element ahead of this element, the last RHO is used.

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OPTIONS3I

This command sets optional values for calculations.

Example:

OPTIONS3I UNIT=1.0E-6; FILAMENT=2; SEGMENT=6;

In the example, the OPTIONS3I command defines the length unit as microns. Each SINGLE_BAR is discretized with 2 filaments in the cross section and 6 seg-ments along the direction defined by the nodes.

MATRIX

This command opens the matrix file and reads its contents into memory. The matrix file is created by RI3.

OPTIONS3I UNIT=VALUE; FILAMENT=VALUE; SEGMENT=VALUE; CRIT_ANGLE=VALUE;


UNIT numeric Conversion factor for geometrical dimensions with respect to meters. Default value: 1.0

FILAMENT numeric For nonparallel current cells, the specified number of filaments is applied to the larger side of each cell’s cross-section, and the number of filaments in the smaller side is determined proportionally. For example, if the width-to-height ratio is 2, and 4 fila-ments are defined, then 4 filaments are assigned to the width direction and 2 to the height direction. For parallel current cells, the aggregate of current fila-ments on the larger side of the cross-section is replaced by a current sheet.Default value: 2

SEGMENT numeric Number of segments in the line integral used in the inductance calculation for the current cells that are neither parallel nor orthogonal. Default value: 6

CRIT_ANGLE numeric When two current cells are almost orthogonal (90 o +/- crit_angle o), the mutual inductance is taken to be 0.0 without calculation. Units: degreesDefault value: 0.0

MATRIX MATRIX_FILE=<FILENAME;>

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Example:

MATRIX MATRIX_FILE=LEAD_FRAME1.MAT;

The example indicates that the file lead_frame.mat was obtained with a previous run of RI3 and the information contained in it is used for post-processing analysis.

MERGE3I

This command causes two nodes to be electrically connected or merged. Once merged, from_node cannot be used in the input file. This command is used to cre-ate a single ground or reference node.

Example:

MERGE N33 N55

The command merges node n33 into node n55. Both nodes are now electrically connected and node n33 cannot be referenced again in the input file.

EXT

This command defines external nodes at which the inductance is calculated (i.e., defines the nodes that are the leads of the inductor). node is included in the exter-nal node list for external impedance matrix calculation.

To reference a particular node that belongs to a PLANE element, the numbering notation defined in PLANE_NODE, p. 6-7 can be used. A less cumbersome approach uses the optional plane field in the NODE command defined in NODE, p. 6-6. The advantage of this method is that it is not necessary to make reference to a particular node in the plane, because it may be difficult to determine the correct suffix. Reference should be made to the name of the corresponding node. RI3 establishes the proper connection between NODE and the nearest node in the PLANE.

Example:

EXT POWER_NODE

The command defines the node power_node as one of the nodes were the induc-tance and resistance is calculated.

MERGE3I <FROM_NODE> <TO_NODE;>

EXT <NODE; >

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REF

This command defines at least one ground node.

There should be at least one reference node (ground node) for each set of contigu-ous conductors. Since the input allows only one reference node, all the reference nodes in different conductors should be merged into one node before the reference node command is executed.

Example:

REF GROUND_NODE

In the example, the node ground_node is grounded to define a return path for cur-rent.

FREQUENCY

This command defines the start and end frequencies.

The frequency assignment defines the frequencies for which the impedance matri-ces are calculated. When LINEAR is specified, the frequency range between START and END is divided into n linear steps, and the impedance matrix is calcu-lated at each frequency. When DECADE is specified, the impedance matrix is cal-culated for m frequency points in each decade using a log scale.

Example:

FREQUENCY START_FREQ=1.E9; END_FREQUENCY=100.E9; DECADE=3;

The command defines the start frequency as 1 GHz, and the end frequency as 100 GHz. Since three decades were defined, the inductance matrix and resistance vector is calculated at 1, 10, and 100 GHz.

ADMITTANCE

This command calculates the admittance matrix:

ADMITTANCE

If this command is specified, the admittance matrix will be calculated and output.

REF NODE;

FREQUENCY START_FREQ=<START>; END_FREQ=<END>; {LINEAR=<N>; OR DECADE=<M>;}

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Equivalent Circuit Raphael Reference Manual

SPARAMETER

This command calculates the s-parameter matrix:

SPARAMETER <EXT_NODE1>=<value> <EXT_NODE2>=<value>;<EXT_NODE1> and <EXT_NODE2> are the name of the

External nodes are defined by command EXT. <value> is the characteristic resis-tance of the corresponding external node in Ohm. The default value is 50 Ohm. SPARAMETER calculates and outputs the s-parameter matrix.

Example:

EXT n1

EXT n2

EXT n3

SPARAMETER n1=55 n2=45

This command defines the characteristic resistance for node n1 and node n2 to be 55 Ohm and 45 Ohm, respectively. The characteristic resistance for node n3 takes the default value, 50 Ohm.

OUTPUT

This command outputs the resistance and inductance data in the ASIC TecPlot™ format to a .dat file. If the admittance or s-parameter matrix is calculated by ADMITTANCE or SPARAMETER, respectively, the corresponding data will be output along with the resistance and inductance data.

Equivalent CircuitTo use an external impedance matrix in circuit simulation, a proper circuit model must be built. The imaginary parts of the impedance matrix repre-sent self- and mutual- inductances when they are divided by . The real parts rep-resent resistances. However, there can be nonzero real parts in the off-diagonal entries. Therefore, current-dependent voltage sources should be used rather than resistors. (In most cases, however, the real off-diagonal components are small, and a circuit composed of inductors, resistors and mutual inductors can be used). If the mutual inductance between two lines is , then the coupling constant K required by SPICE is , where and are the self inductances of the two lines.

When the inductance and the resistance do not change with respect to frequency, the equivalent circuit can be used for other frequencies than the frequency at which the model is built. In other words, the circuit can be used for transient sim-ulation where the signal has a wide spectrum of frequencies. However, if the inductance and the resistance are frequency dependent, the equivalent circuit must be used only over a limited frequency range.

Zext ω( )( )Zext ω( )

ω

L12K L12 L1L2⁄= L1 L2

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Two variations of SPICE models are provided in RI3. Please refer to Appendix D for theoretical treatments.

Selection of Extraction AlgorithmRI3 field solver contains two different algorithms for inductance and resistance extractions. Command-line option -N is used to select particular algorithm. Default behavior (-N option is not specified on the command line) uses old algo-rithm (before release 2004.06) and provides backward compatibility of the pro-gram with previous releases. Activation of -N option on the command line invokes new extraction algorithm that demonstrates up to 4x speed improvement with respect to release 2003.09. This is the recommended choice.

ExamplesThis section presents four examples to illustrate the use of RI3. Since the output files of RI3 contain complete copies of the corresponding input files, only the out-put files are shown:

• Example 1 introduces the use of these programs by showing a comparison of the simulated results with semi-empirical results.

• Example 2 introduces the use of these programs by showing a comparison of the simulated results with a skin effect simulation.

• Example3 calculates the inductance of a circular ring above a ground plane.

6 The last example calculates the self and mutual inductance of four wire bond leads, which might be used in an IC package.

Example 1(raexi31): Inductance of Two Parallel Microstrips

The first example presented in this section corresponds to the inductance calcula-tion of two parallel microstrips carrying current in opposite directions. The struc-ture is presented in Figure 6-6. The conductors and the plane that separates them are very long in the direction normal to the paper, and they transport current in opposite directions. This structure is of particular interest because it can be used to compare the numerical value of the inductance obtained by RI3 to Reference [10].

Figure 6-6 Inductance calculation of two parallel microstrips

d

w

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For a structure similar to the structure simulated by Raphael, Reference [10] pre-sents the following approximate formula to calculate the inductance per unit length ( ) in [μH/m]:

. Equation 6-8

In Equation 6-8, is a coefficient that depends on the aspect ratio . Refer-ence [10] also presents the coefficient in graphical form.

Figure 6-7 is the RI3 output file raexi31.out showing an example of inductance calculation of a two parallel-line structure in comparison with Walker’s results. To facilitate the generation of several similar input files, on Line 6 dith is defined parametrically to represent the distance between traces. To generate an input file with a different aspect ratio, only this parameter has to be modified. Lines 8-11 define the nodes, andLines 13-15 define 3 conductors to represent the structure. In Line 16, two filaments per bar are requested and the units are defined to be microns.

To obtain the inductance for this case, post-processing is performed with the post processing instructions in the same input file. The inductance for the whole struc-ture is 0.72 μH/m (after the proper unit conversions). A comparison for different aspect ratios is presented in Figure 6-8, where a difference of less than 10 percent is observed.

L l⁄

Ll---

1.26KL1----------- d

W-----⎝ ⎠

⎛ ⎞×=

KL1 L l⁄

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1 * RI3 RUN 2 $ Example RAEXI31: 3 $ Inductance of parallel lines with current flowing in opposite 4 $ directions 5 $ Geometry definition 6 PARAM dith=2 7 $ Define the nodes 8 NODE NAME=n0 POSITION=0,0,0; 9 NODE NAME=n1 POSITION=100,0,0; 10 NODE NAME=n2 POSITION=100,0,dith; 11 NODE NAME=n3 POSITION=0,0,dith; 12 $ Define the current elements which connect the nodes 13 SINGLE_BAR NAME=b0 NODE1=n0 NODE2=n1 W=1 H=0.5 14 SINGLE_BAR NAME=b1 NODE1=n1 NODE2=n2 W=1 H=0.5 15 SINGLE_BAR NAME=b2 NODE1=n2 NODE2=n3 W=1 H=0.5 16 OPTIONS3I UNIT=1e-6 FILAMENT=2 17 $ Post-processing instructions 18 $ Define signal node 19 EXT n0 20 $ Define reference (ground) node 21 REF n3 22 $ inductance 23 FREQUENCY START_FREQ=1e4 END_FREQ=1e4 DECADE=1

IMPEDANCE MATRIX

External Node 1 : n0

freq = 1.000e+04 [Hz]

Resistance Vector [Ohms]n0 : -0.000e+00

Inductance Matrix [Henry] ext. node n0 n0 7.222e-11

Figure 6-7 Partial listing of RI3 output file raexi31.out showing an example of inductance calculation

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Example 2 (raexi32): Skin Effect Simulation

A simple structure was selected to illustrate the use of MULTI_BARs to study the variation of resistance and inductance due to skin effects. This structure consists of a single MULTI_BAR. The output file raexi32.out showing a partial listing for RI3 is presented in Figure 6-9 performing skin effect simulations using MULTI_BARs.

InLines 5 and 6 two nodes are defined, and in Lines 9 and 10 a single multi-bar is defined. 9 subdivisions were selected in the width direction and 7 subdivisions for the height. The material for the multi-bar was assumed to be copper, and the cross-sectional dimensions are 20.0 and 7.0 microns.

Figure 6-8 Results obtained with RI3 compared with Walker’s results for the inductance per unit length

Inductance per Unit Length [microH/m] Walker [Ref. 4]

Aspect Ratio (d/W) 0.00 5.00 10.00 15.00 20.00

1.80

1.40

1.00

0.60

0.20

2.00

1.60

1.20

0.80

0.40

RI3 simulated results

0. 00

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To have an appreciable skin effect, the skin depth must be of the order of the cross-sectional dimensions. For copper, Equation 6-9 shows the dependence of the skin depth (d) expressed in meters as a function of the frequency (f) expressed in Hz:

. Equation 6-9

As reference values for the skin depth of copper, for a frequency of 0.1 GHz (108 Hz) this depth is 6 microns, and at 10 GHz the corresponding value is 0.6

1 * RI3 RUN 2 $ Example RAEXI32: 3 $ Simulation for Skin effect analysis 4 $ conductor 5 NODE NAME=n1 POSITION=0,50,23.5; 6 NODE NAME=n2 POSITION=100,50,23.5; 7 $ Using multibars to study skin effects 8 MULTI_BAR NAME=n1_2 NODE1=n1 NODE2=n2 W=20. H=7. NW=9 NH=7 RHO=1.72e-8 9 $ As the dimensions are in microns define the unit options 10 OPTIONS3I UNIT=1e-6 FILAMENT=3 11 $ Post-processing commands 12 $ of ground and signal 13 $ Signal input 14 EXT n1 15 $ ground 16 REF n2 17 $ Selecting the frequencies for output 18 FREQUENCY START_FREQ=1e8 END_FREQ=1e11 DECADE=3

IMPEDANCE MATRIX


freq = 1.000e+08 [Hz]

Resistance Vector [Ohms]n1 : 1.256e-02


freq = 2.154e+08 [Hz]



Figure 6-9 Partial listing of RI3 output file raexi32.out

d 0.0661f

----------------=

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microns. Based on these frequencies and dimensions, both the discretization of the multi-bar and the frequency range for the post-processing analysis were decided.

The variation of the resistance with frequency is presented in Figure 6-10. At fre-quencies below 0.1 GHz, the resistance is approximately equal to the DC value for the microstrip, but above this frequency, its value increases dramatically according to the reduction in the conduction area. The data for this plot was calculated using the following parameters of MULTI_BAR statement: NW=18 and NH=9.

Note:You may notice that the resistance values differ slightly between Raphael v98.4 and v4.2. The reason is due to the change of default adjacent cell ratio in MULTI_BAR. In general, the new default (of ) gives a more desirable result. To revert back to the old default, you can invoke RI3 with -w2 command option.

Figure 6-10 Resistance dependency with frequency for a microstrip

Resistance per Unit Length [Ohm/m] x 103

1e-01 3 1e+00 3 1e+01 3 1e02

1.80

1.40

1.00

0.60

0.20

2.00

1.60

1.20

0.80

0.40

RI3 results

0.00Frequency [GHz]

1 6 N⁄+

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Example 3 (raexi33): Circular Ring Above Ground Plane

This example illustrates the basic use of RI3 to calculate the inductance of a circu-lar conductor above a ground plane. The output file for the simulation can be found in Figure 6-11, and a rendering of the structure generated by DPLOT can be found in Figure 6-13.

Starting at line 5, the NODEs are defined. As with most numerical simulation tools, the more nodes that are used the greater the accuracy and CPU require-ments. In this example the circle is constructed from eight straight segments. Nine nodes, each spaced 2p/8 radians apart, are used. The circle is 2 units or 2*10-4 meters (200 microns) in diameter. Each node receives its own unique name. The starting node (n0) and the ending node (n8) are located at the same spatial posi-tion. This makes it possible to completely close the circle, and while the two nodes share the same spatial position, they remain electrically isolated.

At Line 14 a PLANE_NODE, that is really a set of nodes, is defined. The plane is centered at (0,-1,0) or one unit below the center for the circle. Since the normal to the plane is in the (0,1,0) direction, the PLANE_NODE lies in the X-Z plane. The plane has a length and width of 2 units (as specified by l1 and l2), and there are four rows and four columns of nodes (as specified by n1 and n2). The PLANE_NODE therefore has a total of 16 nodes.

The eight sides of the circle are defined in Lines 16-23 using SINGLE_BARs. These are the current pipes that connect the nodes. The width W and height H per-pendicular to the direction of current flow are also defined here as 0.1 unit. Each SINGLE_BAR receives its own name and that its two terminating nodes are also specified at NODE1 and NODE2.

Line 24 uses a PLANE to define the ground plane. The PLANE command gener-ates the current elements that connect the 16 nodes that were generated by the PLANE_NODE command. Each current element has a height (in the direction of the normal) and resistivity specified by H and RHO, respectively. The BASE_NODE command is used to refer back to the PLANE_NODE.

Line 25 specifies options that are used to control the simulation. The first is the number of filaments used within each conductor. The second is the set of UNITS to be used. RI3 reads the input deck and generates the output file (as shown in Fig-ure 6-11) and a matrix file named raexi33.mat. The matrix file is a binary file that is used later for post-processing with RI3 to calculate the inductance.

Figure 6-12 shows the DPLOT output file that generated the graphics of Figure 6-13. The DPLOT input file is very simple and only contains six lines (only three of which are required). The file simply requests that DPLOT read the geometry file created by RI3, select all the objects and plot them.

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Note:The geometry file has the extension .geo. RI3 does not calculate plotta-ble quantities; therefore, the geometry file contains only a description of the original input structure.


Figure 6-14 shows the output file generated after post-processing the file raexi33.mat with RI3. The actual user input is very short (just 17 lines, most of which are comments). Line 5 reads in the matrix file that was generated previ-ously. Line 8 merges the ending node (n8) of the circle with the base node of the ground plane. This step is necessary, since there can be only one reference node in

1 * RI3 RUN 2 $ Example RAEXI33 circle above a ground plane 3 $ Geometry definition 4 $ Define the nodes 5 NODE NAME=n0; POSITION=cos(2*PI*0/8),0,sin(2*PI*0/8); 6 NODE NAME=n1; POSITION=cos(2*PI*1/8),0,sin(2*PI*1/8); 7 NODE NAME=n2; POSITION=cos(2*PI*2/8),0,sin(2*PI*2/8); 8 NODE NAME=n3; POSITION=cos(2*PI*3/8),0,sin(2*PI*3/8); 9 NODE NAME=n4; POSITION=cos(2*PI*4/8),0,sin(2*PI*4/8); 10 NODE NAME=n5; POSITION=cos(2*PI*5/8),0,sin(2*PI*5/8); 11 NODE NAME=n6; POSITION=cos(2*PI*6/8),0,sin(2*PI*6/8); 12 NODE NAME=n7; POSITION=cos(2*PI*7/8),0,sin(2*PI*7/8); 13 NODE NAME=n8; POSITION=cos(2*PI*8/8),0,sin(2*PI*8/8); 14 PLANE_NODE NAME=gnd; NORMAL=0,1,0; CENTER=0,-1,0; L1=2; L2=2; N1=4; N2=4; 15 $ Define the current elements which connect the nodes 16 SINGLE_BAR NAME=b0; NODE1=n0; NODE2=n1; W=0.1; H=0.1; 17 SINGLE_BAR NAME=b1; NODE1=n1; NODE2=n2; W=0.1; H=0.1; 18 SINGLE_BAR NAME=b2; NODE1=n2; NODE2=n3; W=0.1; H=0.1; 19 SINGLE_BAR NAME=b3; NODE1=n3; NODE2=n4; W=0.1; H=0.1; 20 SINGLE_BAR NAME=b4; NODE1=n4; NODE2=n5; W=0.1; H=0.1; 21 SINGLE_BAR NAME=b5; NODE1=n5; NODE2=n6; W=0.1; H=0.1; 22 SINGLE_BAR NAME=b6; NODE1=n6; NODE2=n7; W=0.1; H=0.1; 23 SINGLE_BAR NAME=b7; NODE1=n7; NODE2=n8; W=0.1; H=0.1; 24 PLANE NAME=gndplane; BASE_NODE=gnd; H=.05; RHO=1e-8; 25 OPTIONS3I UNIT=1e-4; FILAMENT=2;

Figure 6-11 Output file for Example raexi33

1... $ Read in all the data2... DATA RAPHAEL FILE=raexi33.geo

3... $ Select all the objects4... SELECT ALL

5... $ Plot them6... PLOT.3D TITLE=”Example RAEXI33: Ring Above GND Plane”

Figure 6-12 DPLOT input file dpraexi33 for Example raexi33

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a structure. Line 11 defines the external node(s) at which the impedance (induc-tance and resistance) is calculated. In this case, the external node is the starting node (n0) of the circular ring. It is possible to select as many nodes as desired for external nodes. Each external node yields one row and one column of the impedance matrix.

Figure 6-13 Device structure for Example raexi33

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Line 14 defines the reference node gnd000_000 (which is actually both gnd000_000 and n8). Note that the entire node name (including the suffix) must be used. Every structure must have one, and only one, reference node.

Line 17 defines, respectively, the start frequency, end frequency and the number of frequency points per decade. The output indicates that the inductance of the ring is 0.3728 nanoHenries. The analytical formula for the inductance is:

Equation 6-10

Equation 6-11

In this equation, L is the inductance in microHenries, a is the radius of the ring in centimeters, and c is the cross-sectional thickness of the ring in centimeters. In the example, a = 10-2 and c = 10-3 giving the inductance as 0.4 nanoHenries, which is

1 * Example RAEXI33P post-processing por RAEXI33 2 * circle above a ground plane 3 4 * Read in the matrix file 5 matrix matrix_file=raexi33.mat 6 7 * Merge ground nodes 8 MERGE3I n8 gnd000_000 9 10 * Define signal node 11 EXT n0 12 13 * Define reference (ground) node 14 REF gnd000_000 15 16 * inductance 17 FREQUENCY START_FREQ=1e4; END_FREQ=1e4; DECADE=1;

IMPEDANCE MATRIX


freq = 1.000e+04 [Hz]



Figure 6-14 Output from RI3 after post-processing the file raexi33.mat with the file raexi33p

z c2a------⎝ ⎠

⎛ ⎞ 2=

L 0.004aπ 12---

112------z+

⎩ ⎭⎨ ⎬⎧ ⎫ 8

z---⎝ ⎠

⎛ ⎞ 0.84834– 0.2041z+( )ln=

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close to the value found by RI3. If more segments were used, the result would be closer. In this example the ground plane is sufficiently far away so as not to greatly impact the inductance.

Raphael RI3 structures also can be visualized using Taurus Visual. To save output for Taurus Visual, use the -h option when running Raphael.

raphael ri3 -h raexi33

The TDF file raexi33.tdf will be generated. Next start Taurus Viaual and load in the TDF file. You can then rotate the structure using the mouse to get the desired viewing angle. No fields are saved as part of the analysis.The example output is shown in Figure 6-15 below.

Example 4 (raexi34): Inductance of Four Bond Wires

This example calculates the self- and mutual- inductance of a set of bond wires that might be used to tie the bond pads of a chip to the package. The output file from RI3 can be found in Figure 6-16, and a rendering of the structure from DPLOT can be found in Figure 6-18.

Line 5 of Figure 6-16 begins by defining some parameters for future use. The use of parameters makes it easy to change the dimensions of certain features of the structure. The templates used by the Interconnect Library program (RIL) make use of parameters in this way. Note, in particular, how the parameter PITCH is defined and used to control the spacing between the individual bond wires.

At Line 13, the node definition begins. Each bond wire has three straight segments and therefore, four nodes. In addition, the X and Y positions of the nodes do not

Figure 6-15 Inductance ring structure shown using Taurus Visual

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change from wire to wire. Hence, the predefined parameters X1, X2, X3, X4, Y1 and Y2 may be used to define the X and Y locations of the nodes.

The definition of the current elements that connect the nodes begins at Line 30. Each segment has width and height (perpendicular to the direction of current flow) of 1.4 mils and 1.1 mils. The resistivity of the current elements is 2.5x10-6 ohm-meter. The OPTIONS statement at line 42 defines the unit used to 25.4x10-6 meters, which corresponds to mils and sets the number of filaments used within each element to 5. Executing RI3 generates the binary file raexi34.mat, which can be used later if a power and grounding strategy different from the one specified in Lines 49 to 54 is desired.

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Figure 6-17 shows the DPLOT output file that generated the graphics of Figure 6-18. The DPLOT input file is very simple, containing just 6 lines (only 3

2 $ Example RAEXI34 Four Bond Wires3 $ Geometery definition file4 $ Define some parameters5 PARAM x1 = 5;6 PARAM x2 = 10.5;7 PARAM x3 = 19.5;8 PARAM x4 = 24.1;9 PARAM y1 = 30.7;10 PARAM y2 = 5.7;11 PARAM pitch = 8;12 $ Define Nodes13 NODE NAME=n1a; POSITION=x1,y1,0*pitch; 14 NODE NAME=n1b; POSITION=x2,y1,0*pitch; 15 NODE NAME=n1c; POSITION=x3,y2,0*pitch; 16 NODE NAME=n1d; POSITION=x4,y2,0*pitch; 17 NODE NAME=n2a; POSITION=x1,y1,1*pitch; 18 NODE NAME=n2b; POSITION=x2,y1,1*pitch; 19 NODE NAME=n2c; POSITION=x3,y2,1*pitch; 20 NODE NAME=n2d; POSITION=x4,y2,1*pitch; 21 NODE NAME=n3a; POSITION=x1,y1,2*pitch; 22 NODE NAME=n3b; POSITION=x2,y1,2*pitch; 23 NODE NAME=n3c; POSITION=x3,y2,2*pitch; 24 NODE NAME=n3d; POSITION=x4,y2,2*pitch; 25 NODE NAME=n4a; POSITION=x1,y1,3*pitch; 26 NODE NAME=n4b; POSITION=x2,y1,3*pitch; 27 NODE NAME=n4c; POSITION=x3,y2,3*pitch; 28 NODE NAME=n4d; POSITION=x4,y2,3*pitch; 29 $ Connect nodes with current elements30 SINGLE_BAR NAME=bar1a; NODE1=n1a; NODE2=n1b; W=1.4; H=1.1; RHO=2.5e-6; 31 SINGLE_BAR NAME=bar1b; NODE1=n1b; NODE2=n1c; W=1.4; H=1.1; 32 SINGLE_BAR NAME=bar1c; NODE1=n1c; NODE2=n1d; W=1.4; H=1.1; 33 SINGLE_BAR NAME=bar2a; NODE1=n2a; NODE2=n2b; W=1.4; H=1.1; 34 SINGLE_BAR NAME=bar2b; NODE1=n2b; NODE2=n2c; W=1.4; H=1.1; 35 SINGLE_BAR NAME=bar2c; NODE1=n2c; NODE2=n2d; W=1.4; H=1.1; 36 SINGLE_BAR NAME=bar3a; NODE1=n3a; NODE2=n3b; W=1.4; H=1.1; 37 SINGLE_BAR NAME=bar3b; NODE1=n3b; NODE2=n3c; W=1.4; H=1.1; 38 SINGLE_BAR NAME=bar3c; NODE1=n3c; NODE2=n3d; W=1.4; H=1.1; 39 SINGLE_BAR NAME=bar4a; NODE1=n4a; NODE2=n4b; W=1.4; H=1.1; 40 SINGLE_BAR NAME=bar4b; NODE1=n4b; NODE2=n4c; W=1.4; H=1.1; 41 SINGLE_BAR NAME=bar4c; NODE1=n4c; NODE2=n4d; W=1.4; H=1.1; 42 OPTIONS3I UNIT=25.4e-6; FILAMENT=5; 43 $ Post-processing commands44 $ Merge ground nodes45 MERGE3I n2d n1d46 MERGE3I n3d n1d47 MERGE3I n4d n1d48 $ Define external (signal) nodes49 EXT n1a50 EXT n2a51 EXT n3a52 EXT n4a53 $ Define reference (ground) node54 REF n1d55 $ Calculate inductance56 FREQUENCY START_FREQ=1e4; END_FREQ=1e5; DECADE=2;

Figure 6-16 RI3 input file for Example raexi34

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of which are required). The file simply requests that DPLOT read the geometry file created by RI3, select all the objects and plot them.

Note:The geometry file has the extension .geo. RI3 does not calculate any field or potential distribution. Therefore, the geometry file contains only a description of the original input structure.


Lines 45-47 merge the four ground nodes (at the lower end of each wire) into a single node named n1d. Lines 49-52 define the set of external nodes where the

1... $ Read the input file2... DATA RAPHAEL FILE=raexi34.geo3... $ select all the objects4... SELECT ALL5... $ plot them all6... PLOT.3D TITLE=”Example RAEXI34: Four Bond Leads”

Figure 6-17 DPLOT input file dpraexi34 for Example raexi34

Figure 6-18 Device structure for Example raexi34, which simulates four bond leads

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Raphael Reference Manual References

impedance matrix is calculated. Line 54 defines the reference node as n1d, (the lower ends of all bond wires). Finally, Line 56 sets the range of frequencies for the analysis. In this case, the range starts at 10kHz and ends at 100kHz. It performs two frequency points per decade (for a total of three frequency points).

Figure 6-19 shows the impedance matrix.

The resistance of the bond wires has a numerical value of 2.344 ohms. This value is frequency invariant. The self-inductance of each wire is given in the diagonal terms of the matrix. The mutual inductance between the wires is found on the off-diagonal terms.

Note:The matrix is symmetric and that there is virtually no frequency depen-dence over the specified range. This frequency independence is due to low frequency. In addition, only multi-bars can reproduce skin effects. Because only single-bars were used in this example, there were no skin effects.

References[7] E. Weber, Electromagnetic Theory, Dover, New York, 1962

[8] A.E. Ruehli, “Inductance Calculations in a Complex Integrated Circuit Envi-ronment,” IBM J. Res. Develop., September 1972, pp. 470-481.

[9] C.C. Huang, “Computation of Resistance and Inductance Matrices in a Sym-metric Structure,” IEEE Trans. Components, Packaging, and Manufacturing Technology-Part A, vol. 18, no. 3, September 1995, pp. 674-676.

[10] C. S. Walker, Capacitance, Inductance, and Cross-talk Analysis, 1990, pp. 92-94.

freq = 1.000e+05 [Hz]

Resistance Vector [Ohms]n1a : 2.344e+00n2a : 2.344e+00n3a : 2.344e+00n4a : 2.344e+00

Inductance Matrix [Henry] ext. node n1a n2a n3a n4a n1a 6.429e-10 2.095e-10 1.302e-10 9.412e-11n2a 2.095e-10 6.429e-10 2.095e-10 1.302e-10n3a 1.302e-10 2.095e-10 6.429e-10 2.095e-10n4a 9.412e-11 1.302e-10 2.095e-10 6.429e-10

Figure 6-19 Portion of the output generated by RI3 for Example raexi34

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CHAPTER 7

R

Raphael Interconnect Library 7

IntroductionThe Raphael Interconnect Library (RIL) utility, is a database program that gener-ates and stores electrical model parameters for interconnect elements. The param-eters are based on dimensions, geometries, and material properties. The elements are simple geometric structures that can be joined together to form interconnects for various electrical designs. The electrical parameters for each element are cal-culated using either one or two of the numerical simulation programs, RC2, RC3, RI3. RIL then generates the SPICE subcircuit definitions, which are visually inspected with STUDIO Visualize, Synopsys TCAD’s visualization tool.

Raphael Interconnect LibraryThe Raphael Interconnect Library (RIL) is an excellent tool for creating variations of the parameters that characterize the library elements. RIL also analyzes the effect of these variations on the parasitics associated with the interconnect struc-tures.

Figure 7-1 presents the information flow for RIL and its main components. You interact with RIL by first selecting an element for the library and then specifying a set of values that define a set of structures for the element. After this step is com-pleted, RIL generates the input files, runs the corresponding solvers, and stores the parasitic values in the database. When desired, the Results Inspectors can inspect any of the Parasitics Databases for each of the structures in the Parametric Library.

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Raphael Interconnect Library Raphael Reference Manual

Under this option, you can generate listings, printouts, or you can interactively plot the simulated parasitics versus the parameters that describe the structure.

The Parametric Library of RIL consists of a set of 30 structures that represent most of the common interconnect structures found in printed circuit boards, multi-chip modules, packaging and on-chip interconnects. All of these template library elements are listed in Appendix A of this manual. For each library element a default Parasitics Database exists under the default name data. Other databases to store specific sets of parameters can be created by user-defined database names. Each one of the parametric structures is defined by a set of support files that are used by RIL for the generation of input files and parsing the outputs generated by the solvers. These files are also used in the generation of the Parasitics Databases. Figure 7-2 shows the structure of the Parametric Library.

Figure 7-1 Main components and information flow of the Raphael Interconnect Library (RIL)

USER

RIL

RC2 RC3 RI3

ParametricLibrary

InterconnectDatabases

ResultsInspectors(Listing,STUDIOVisualize)

Figure 7-2 The Parametric Library

Interconnect Database (data user

defined)

Support Files

Element 12 Pins/Vias ...

InterconnectDatabase(data user

defined)

SupportFiles

Element 30Level 2 ...

ParametricLibrary

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Raphael Reference Manual Running RIL

Running RILRIL can be invoked by issuing the command:

raphael ril

The program displays a short message and then prompts you to choose a unit for the input geometries of the input parameters. There are three possible choices:

• Millimeters (mm)

• Microns

• Mils

Once a choice has been made, it remains the default unit for the remainder of the session.

RIL operates only on the structure types defined in its Parametric Library. To begin the program, you must first choose a structure from a defined list. A dia-gram of the structure you have chosen and its equivalent electrical circuit is then displayed in two separate windows. Figure 7-3 shows the structure for element 20 of the Parametric Library (crossover of two conductive layers above a ground plane). In Figure 7-4, the associated SPICE subcircuit is presented.

Each time you select a new structure, you are prompted to select an output file name for the Parasitics Database to be used for storing and inspecting the results of your simulations. This file is created or opened in the subdirectory associated with the new structure. By default, RIL stores all new data in a file called data in the subdirectory associated with the newly selected structure.

Figure 7-3 Structure corresponding to element 20 of the Parametric Library

s1 w1

Cross Over of Structures in 2 LevelsAbove a Ground PlaneZ

h3

Z

X Y

h2

h1

E3

E2

E1

s2 w2

t2

t1

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Figure 7-4 SPICE subcircuit associated with element 20 of the Parametric Library

Trace 6

Trace 5

Trace 4

Trace3

Trace 2

Trace 1

11

9

7

R6

R5

R4

12

10

8

C66

C55

C44

995

3

1

6

4

2

R3

R2

R1

C11C22

C33

Note: Cij in the output represents all the cross capacitances between different traces.

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Raphael Reference Manual RIL Interactive Commands

RIL Interactive CommandsRIL uses the following commands. The commands can be abbreviated by their initial letters.

ADD

The add command adds input parameters to a file in order to generate data. An editor defined by you is employed to modify the file (see Customization, p. 7-8). The file contains two types of lines: comments (which begin with an asterisk *) and data lines. Comment lines may be used to comment on the data items or to list their sequence in the file. Comments are ignored by the program.

When you select the add command, you are prompted to define the input file name for the new structures to be simulated. By default, RIL opens a file called “sets” that contains the current library element in the RIL subdirectory (see Direc-tory Structure and Files, p. 7-8).

Data lines contain geometry and material parameters. Each line represents a spe-cific input parameter set associated with the structure type. The input geometry parameters use the unit selected earlier (mils, microns, or millimeters). The rela-tive dielectric constants (E1 and E2) have no units, and the resistivity of the con-ductor (Z1) is always specified in units of Ohm-meter. RIL automatically checks to see if the data set has been previously simulated. In such cases, RIL ignores the input set and informs you that the set has been simulated.

Note:There is no limit on the number of data sets that can be added to a file, however, EACH DATA SET MUST OCCUPY A SINGLE LINE.

CHECK

This command checks to determine if a specific data set exists in the database. The program prompts for the input parameters.

When this command is selected, you are prompted to define the input file name for the new structures to be checked. By default, RIL opens a file named sets in the RIL subdirectory that contains the current library element (see the Directory Structure and Files, p. 7-8).

GENERATE

This command starts the numerical simulation to extract the electrical parameters. This process may take from a few seconds to several minutes depending on the structure type that has been chosen and the number of data sets. Completion of the process is marked by the appearance of the command prompt =>. Interruption of

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the generation process by pressing CNTL-C, or CNTL-BREAK may result in the loss of newly generated results.

When generating results for a new data set, RIL reads the parameters from the file that was defined by the execution of the command add. RIL stores the results in the file defined when the last element was selected.

INPUT CHECK / PLOTS

This command runs RC2 or RC3 in input check mode to check the validity of the input data. DPLOT is also executed to produce plots of the actual input data.

When this command is selected, you are prompted to define the input file name for the new structures to be plotted. By default, RIL opens a file named sets in the RIL subdirectory that contains the current library element (see Directory Structure and Files, p. 7-8).

LIST

This command lists all of the electrical parameter sets stored in the database for each structure type. The parameter sets are listed in ascending order based on the value of their geometric dimensions. If the structure type is 2D, RIL prompts for the length in the Z dimension before displaying the electrical data (which is scaled appropriately). RIL also generates SPICE subcircuit models at this point. For 2D structures, RIL calculates the number of RLC sections based on the length in the Z dimension and the estimated rise time of the electrical signal as supplied by you.

When listing the results for all the calculated data sets, RIL lists the results that are stored in the file defined when the last element was selected.

The parasitic values and the SPICE netlist generated by RIL are stored in your current directory. RIL prompts you for the file names. The default file names are Your_data for the parasitic values and Your_subckt for the SPICE subcircuit.

PRINT

This command produces copies of the electrical data and input parameters on a specified printer. The printer device name is configured in the start_up file (see Customization, p. 7-8).

NEW

This command selects a new structure type. After the new type is chosen, its phys-ical structure and equivalent circuit model replace the ones in the display win-dows.

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When this command is selected, you are prompted to define the input file name for the new structures to be simulated. By default, RIL opens a file named sets in the RIL subdirectory that contains the current library element. See the Directory Structure and Files, p. 7-8.

SAVE

This command allows you to save the recently generated data into the database. The updated data sets are also saved automatically when a new element is selected or when exiting RIL.

VISUALIZE

The VISUALIZE command generates a STUDIO Visualize input file (.dvm) that can later be inspected with STUDIO Visualize, Synopsys TCAD’s visualization tool. This tool allows you to interactively select a specific dependent variable (e.g., capacitance to ground) and plot it against an independent variable (e.g., spacing of metal 2 lines.) The default file name for this option is Your_SV.dvm, and you are prompted to define your own file name. The .dvm suffix is automati-cally added by RIL.

TABLE

With this option the complete set of simulated results and parameters that describe a given RIL element is saved in a standard ASCII file in tabular format. The first row of the table lists the independent parameter names first (geometrical and elec-trical) and the names of the dependent parameters at the end. The first row also lists the column names. The following rows list the values corresponding to each entry in the database for that library element. The default file name for this com-mand is Your_ascii.asc. You can select your own file name (the .asc suffix is auto-matically added).

QUIT

This command terminates the execution of RIL and closes the display windows. All the values generated when working with library elements are automatically saved.

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Directory Structure and Files Raphael Reference Manual

Directory Structure and FilesRIL uses a separate subdirectory for each structure type. Each subdirectory con-tains the templates for RC2, RC3, RI3, and DPLOT as well as a data file contain-ing the electrical parameters for that structure only. The 2D structures, such as trace and wire, use a second level directory to store the templates and data (sub-subdirectory). The base directory is pointed to by the RILDIR environment vari-able.

Each directory contains some or all of the following files:

• pic.xbm - Bitmap of the structure.

• ckt.xbm - Bitmap of the circuit diagram.

• plot3.in - DPLOT input file for 3D device.

• plot2.in - DPLOT input file for 2D device.

• rc2.in - RC2 template file.

• rc3.in - RC3 template file.

• ri3.in - RI3 template file

• rc2.fmt - Awk control file used to extract data from RC2 output.

• rc3.fmt - Awk control file used to extract data from RC3 output.

• ri3.fmt - Awk control file used to extract data from RI3 output.

• sets - Data sets file created by the ADD command.

• data - Stored input parameters and extracted electrical data.

• title - Structure type name and parameter list.

• tmplt - Template used to create SPICE models.

In addition, many temporary files are generated by RIL and the solvers during a run. Since RIL and the solvers must be able to write to the data base (and create temporary files), it is important that ALL USERS WHO WISH TO RUN RIL MUST HAVE WRITE PRIVILEGE TO THE RIL SUBDIRECTORIES AND THEIR CONTENTS.

CustomizationThe following methods help you to customize RIL.

start_up File

The file start_up in the RIL base directory (as indicated by the RILDIR environ-ment variable) contains a number of parameters that control the operation of RIL.

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Raphael Reference Manual Customization

The start_up file supplied with RIL is given in Figure 7-5. The parameters are as follows:

To define the environment variable RILDIR, type the following:

setenv RILDIR <path>

where <path> represents the complete path of the RIL library.

RIL Environment Variables

RIL uses two environment variables to find the starting point for its file system and the solvers. RILDIR points to the directory containing the start_up file and the RIL structure subdirectories. RILTOOLS points to the directory containing the executables for RC2, RC3, RI3 and DPLOT.


UNIT number Sets default units (1 for meters, 2 for microns, 3 for mils).

EDITOR character Defines the editor for adding parameters.

PRINTER character Defines the printer command for printing data.

DIR character Defines the structure type directories (should not be modified).

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SPICE Netlist Generation and Naming Convention Raphael Reference Manual

SPICE Netlist Generation and Naming ConventionThe rules used to automatically number the nodes of the SPICE netlist in the 2D RIL models usually follow the rules used by the RC2 solver, but in this case the generation of several sections for each electrode adds another level of complexity.

* start_up file for RIL* default units 1=millimeters 2=microns 3=millsunit 2* editor to use when generating data setseditor vi* printer to print dataprinter lpr* template directory namesdir pin_via1dir pin_via2dir pin_via3dir bend.g1dir bend.g2dir widen.g1dir widen.g2dir narrow.g1dir narrow.g2dir pad.g1dir wiredir trace.eq.g0dir trace.eq.g1dir trace.eq.g2dir trace.ue.g0dir trace.ue.g1dir trace.ue.g2dir lev1.g2dir cross.1tracedir cross.lev1.lev2tdir cross.lev1.lev2t.lev3dir conf.3diel.eq.g1dir conf.4diel.eq.g1dir conf.2diel.2met.eq.g1dir conf.3diel.2met.eq.g1dir spacer.eq.g1dir spacer.2met.eq.g1dir overlapdir par.lev1.lev2dir par.lev2.lev1

Figure 7-5 RIL start_up file

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Raphael Reference Manual SPICE Netlist Generation and Naming Convention

R

Node numbers are assigned using the following rules:

• Each electrode is assigned an ordinal number that corresponds to its position in the RIL structure (i.e., the first electrode to the left is assigned n=1, the sec-ond electrode to the right n=2, and so on).

• The terminal node numbers for electrode n, when there are N electrodes, are assigned using the following rules:

Equation 7-1

. Equation 7-2

• The ground (reference) node (when present in the model) is always assigned the number 99.

• Assuming that there are M (RCL lumped sections), the intermediate nodes for electrode n are assigned sequentially between the first_intermediate_node and the last_intermediate_node that are calculated according to the following rules:

Equation 7-3

. Equation 7-4

In Figure 7-6, this numbering system is applied to an RIL model with two elec-trodes and a ground plane. The model also has two sections per electrode. In this example, when a SPICE subcircuit is requested for a 2D model, two RCL sections per electrode are generated by RIL.

The naming convention for the lumped element follows the rules below:

• For resistance of section s, electrode n: Rsn

Figure 7-6 Automatic node assignment by RIL

elec2

elec11

3

R1L1

ground

2

4

R1L1

1 1

C111

C122

C211

C2222 2

L21 R21

L22 R22

C112 C212K112 K212

106 107

101 102 103

105

99

99

99

99

first_node 2 n 1–( )=

last_node 2n=

first_intermediate_node 101 2 n 1–( )M[ ]+=

last_intermediate_node 99 2nM+=

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RIL Example Sessions Raphael Reference Manual

• For self -inductance of section s, electrode n: Lsn

• For coupling term of inductances of section s, electrodes n and m: Ksnm

• For capacitance to ground of section s, electrode n: Csnn

• For mutual capacitance of section s, electrodes n and m: Csnm

RIL Example SessionsSince RIL is an interactive program, a simple example file cannot be created. To illustrate the use of RIL, however, the following interactive sessions were created. In these sessions, you generate data and SPICE models for a “bend over a ground plane” and “two lines over a ground plane.” In the last session, you explore the use of RIL with STUDIO Visualize.

In the following examples:

• User input is indicated by bold face type.

• Comments that explain what is happening are indicated by italics.

• All other text indicates responses from RIL.

Session 1: 3D Example

This example illustrates the use of RIL in the generation of parasitic values for a 3D structure. In this case, the structure corresponds to a bend running on top of a ground plane.

herman [3]> raphael ril

****************************************************************************** ** RAPHAEL INTERCONNECT MODEL LIBRARY ** Synopsys Corp. ** All Rights Reserved ** ** The InterConnect Model (RIL) Library generates and stores electrical model ** parameters (R, C and L’s) for packaging interconnect elements based on ** their dimensions, geometries and material properties. ******************************************************************************

Choose one from the following units for your geometries by its index number (1 - 3): 1) mm 2) micron 3) mil Default unit is micron. --> 2 You have chosen micron for the geometries.

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Here you should press RETURN. The simulated results are stored in the file called data in the subdirectory RILDIR/bend.g1.

At this point, two windows appear containing a diagram of the structure and a schematic of the SPICE equivalent circuit.

Following are the elements available in the library.You can choose one by entering its index number:

9) 2 Pins/Vias with Pads in Antipad Holes of a Plane10) 2 Pins/Vias with Pads above a Solid Plane11) 2 Pins/Vias with Pads above a Plane with Antipads12) Trace Bend above a Plane13) Trace Bend between 2 Planes14) Trace Widening above a Plane15) Trace Widening between 2 Planes16) Trace Narrowing above a Plane17) Trace Narrowing between 2 Planes18) A Pad above a Plane19) Parallel, Equally Spaced Bonding Wires20) Parallel, Equal Width/Equal Spacing Traces without Plane21) Parallel, Equal Width/Spacing Traces Above a Plane22) Parallel, Equal Width/Equal Spacing Traces between 2 Planes23) Parallel, Unequal Width/Spacing Traces without Plane24) Parallel, Unequal Width/Spacing Traces above a Plane25) Parallel, Unequal Width/Spacing Traces between 2 Planes26) Array above substrate below dense array27) Cross Over Above a Single Trace Above a Plane28) Cross Over of Two Conductive Levels Above a Plane29) Cross Over of Three Conductive Levels Above a Plane30) One Conformal Dielectric Layer on Top of Parallel Lines31) Two Conformal Dielectric Layers on Top of Parallel Lines32) One Conformal Dielectric Layer, Conformal Metal Layer on Top of Parallel Lines33) Two Conformal Dielectric Layers, Conformal Metal Layer on Top of Parallel Lines34) One Conformal Dielectric Layer on Top of Parallel Lines with Sidewall Spacers35) One Conformal Dielectric Layer, Conformal Metal Layer on Top of Parallel Lines

with Sidewall Spacers36) Overlap Conductor Above Metal Trace and Ground PLane37) Level 1 Array Under Parallel Level 2 Array above Substrate38) Level 2 Array Above Parallel Level 1 Array Above Substrate

Index [1 - 30] --> 4Current element is: Trace Bend above a Plane

Input the filename to save your data set (default: data):

CHOOSE A COMMAND FROM THE FOLLOWING MENU

Add more input for data generation [a]Check whether a data set is available [c]Generate data (using RC2, RC3 or RI3) [g]List existing data [l]Get a new element [n]Print out data [p]Save data [s]

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Generate plots to check input [i]Generate file for STUDIO Visualize [v]Generate standard ascii table [t]Quit [q]

=> a

This command at present invokes the editor vi. The file you open may have some input left by previous data generation. You can delete them all and put in new ones.

Press <Return> key to start with the default data set or type your “ready to use” file name.

Press RETURN to use the mask file provided with RIL.

Press <Return> key to start

The following appears in the editor window

Trace Bend above a Plane** h1 h2 t1 w1 th E1 E2 Z1 2.0 3.0 1.0 1.0 135 8.4 9.4 15.2e-8~~~~~~”/usr1/raphael/ril/bend.g1/sets” 4 lines, 128 characters

You can change the parameters as desired. Now generate plots to view your structure.

=> i

Press <Return> key to start with the default data setor type your “ready to use” file name:

Press RETURN to use the file stored in RILDIR/bend.g1.

Generating plot for the set:h1 = 2 h2 = 3 t1 = 1 w1 = 1 th = 135 E1 = 8.4 E2 = 9.4 Z1 = 1.52e-07

3D graphics scene group readRead data from file: rc3.pot

At this point, a plot prepared by DPLOT appears in a new window showing the structure to scale.

=> g

Now calculate the parameters. RC3, and RI3 are run automatically.

Generating data for the set:h1 = 2 h2 = 3 t1 = 1 w1 = 1 th = 135 E1 = 8.4 E2 = 9.4 Z1 = 1.52e-07

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Begin RC3

POTENTIAL CALCULATION [Coulomb]

Charge on trc.bnd = 6.743239e-16Charge on trival = 9.587848e-18Charge on plane = -6.839153e-16

End of RC3

The file /usr1/raphael/ril/bend.g1/ri3.mat contains the inductance matrix. Process this file with RI3 to obtain resistance and inductance. RI3 completed.

Size of the complex matrix is: 4 X 4. Calculating inductance at 1.000000e+03, Hz

Resistance Vector [Ohms]n1 : 1.349e+00


Inductance calculations are complete. Refer to the output file /usr1/raphael/ril/bend.g1/ri3.out for results.

Data generated are stored in the library.

Now examine the data generated and create a SPICE model using the list command.

=> l

Trace Bend above a Plane

Input Parameters for the Structure: h1 h2 t1 w1 th E1 E2 Z1

Electrical Parameters for the Model: C L R

Data Available:

index Input Parameters ----- ----- ---------- 1 h1=2.00e+00 h2=3.00e+00 t1=1.00e+00 w1=1.00e+00 th=1.35e+02 E1=8.40e+00 E2=9.40e+00 Z1=1.52e-07

Do you want to look at a specific data set [y/n] ? y Input the filename to save your data (default: Your_data):

Press RETURN to use the default name to store the results.

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Session 2: 2D Example

In this session, you select a 2D model and generate the parasitics for two different structures.

Status 1: If you are continuing from where you left off in the previous example, request a new element from the library as follows:

Status 2: If you are running RIL from the beginning, follow the example of Ses-sion 1 until you are presented with the listing of all the structures in the Parametric Library as shown below:

Type in the index number [1 - 1]: 1

Input Parameters: h1=2.00e+00 h2=3.00e+00 t1=1.00e+00 w1=1.00e+00 th=1.35e+02 E1=8.40e+00 E2=9.40e+00 Z1=1.52e-07

Electrical Model Parameters: C = 6.84e-16 Farads L = 1.18e-12 Henrys R = 1.35e+00 Ohms

Do you want to generate a SPICE circuit model with these values [y/n] ? y Input the filename to save your circuit (default: Your_subckt):

Press RETURN to use the default name to store the SPICE subcircuit.

.subckt icmsb 1 2 99 c1 101 99 6.83915e-16l1 1 101 1.177e-12r1 101 2 1.349.ends icmsb

Another set [y/n] ? n

At this point you have finished the analysis of the “trace bend over a ground plane.” Now select the sec-ond structure “two traces over a ground plane.”

=> n

The data for the element on which you have been working are automatically saved.

After this point, the complete listing of RIL library elements are displayed.

Following are the elements available in the library. You can choose one by typ-ing in its index number:

1) 2 Pins/Vias with Pads in Antipad Holes of a Plane2) 2 Pins/Vias with Pads above a Solid Plane3) 2 Pins/Vias with Pads above a Plane with Antipads

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4) Trace Bend above a Plane5) Trace Bend between 2 Planes6) Trace Widening above a Plane7) Trace Widening between 2 Planes8) Trace Narrowing above a Plane9) Trace Narrowing between 2 Planes10) A Pad above a Plane11) Parallel, Equally Spaced Bonding Wires12) Parallel, Equal Width/Equal Spacing Traces without Plane13) Parallel, Equal Width/Spacing Traces Above a Plane14) Parallel, Equal Width/Equal Spacing Traces between 2 Planes15) Parallel, Unequal Width/Spacing Traces without Plane16) Parallel, Unequal Width/Spacing Traces above a Plane17) Parallel, Unequal Width/Spacing Traces between 2 Planes18) Array above substrate below dense array19) Cross Over Above a Single Trace Above a Plane20) Cross Over of Two Conductive Levels Above a Plane21) Cross Over of Three Conductive Levels Above a Plane22) One Conformal Dielectric Layer on Top of Parallel Lines23) Two Conformal Dielectric Layers on Top of Parallel Lines24) One Conformal Dielectric Layer, Conformal Metal Layer on Top of Parallel Lines25) Two Conformal Dielectric Layers, Conformal Metal Layer on Top of Parallel Lines26) One Conformal Dielectric Layer on Top of Parallel Lines with Sidewall Spacers27) One Conformal Dielectric Layer, Conformal Metal Layer on Top of Parallel Lines

with Sidewall Spacers28) Overlap Conductor Above Metal Trace and Ground PLane29) Level 1 Array Under Parallel Level 2 Array above substrate30) Level 2 Array Above Parallel Level 1 Array Above SubstrateIndex [ 1 - 30 ] --> 13

In this category there are more elements listed as following.You can choose one by typing in its index number:

1) One trace2) 2 traces3) 3 traces4) 4 traces5) 5 traces6) 6 traces7) 7 traces8) 8 traces9) 9 traces

Index [1 - 9] --> 2

Current element is: Parallel, Equal Width/Spacing Traces Above a Plane2 Traces


The simulated results are stored in the database called data, in the subdirectory RILDIR/trace.eq.g1/trace2.

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At this point the two windows showing the structure diagram and the schematic is updated to show the new structure.

=> a

This command at present invokes the editor vi. The file you’ll open may have some input left by previous data generation. You can delete them all and put in new ones.

Press the <Return> key to start

The editor window comes up at this time. You must modify the first line to read like the first data line that follows and add the second line to the file.

* 2 Traces above a Plane** h1 h2 t1 w1 s0 s1 E1 E2 Z1 1.1 10.0 1.0 2.0 10. 2.0 1.0 1.0 2.24e-8 1.2 10.0 1.0 2.0 10. 2.0 1.0 1.0 2.24e-8

”/usr1/raphael/ril/trace.eq.g1/trace2/sets” 4 lines, 126 characters

Once again generate the data. Only RC2 is run this time since the structure is 2D. RC2 is run twice, once for each parameter set.

=> g

Press <Return> key to start with the default data set or type your “ready to use” file name:

Press RETURN to use the file stored in RILDIR/trace.eq.g1/trace2.

Generating data for the set:h1 = 1.1 h2 = 10 t1 = 1.0 w1 = 2.0 s0 = 10 s1 = 2.0 E1 = 1.0 E2 = 1Z1 = 2.24e-08

Begin RC2

CAPACITANCE CALCULATION [Farad / unit length] (matrix [C] in [Q] = [C][V])

trace1 trace2 trace1 4.617088e-11 -9.396888e-12 trace2 -9.396888e-12 4.617088e-11

INDUCTANCE CALCULATION [Henry / unit length]

trace1 trace2 trace1 2.518085e-07 5.116122e-08 trace2 5.116122e-08 2.518085e-07

End of RC2

Generating data for the set: h1 = 1.2 h2 = 10 t1 = 1.0 w1 = 2.0 s0 = 10 s1 = 2.0 E1 = 1.0 E2 = 1 Z1 = 2.24e-08

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Begin RC2

CAPACITANCE CALCULATION [Farad / unit length] (matrix [C] in [Q] = [C][V])

trace1 trace2 trace1 4.432387e-11 -9.648827e-12 trace2 -9.648827e-12 4.432387e-11

INDUCTANCE CALCULATION [Henry / unit length]

trace1 trace2 trace1 2.636717e-07 5.716674e-08 trace2 5.716674e-08 2.636717e-07

End of RC2

Data generated are stored in the library.

Now examine the data using the list command.

=> l

Parallel, Equal Width/Spacing Traces Above a Plane 2 Traces

Input Parameters for the Structure: h1 h2 t1 w1 s0 s1 E1 E2 Z1

Electrical Parameters for the Model: C11 C12 L11 L12 R1

Data Available:

index Input Parameters ----- ----- ---------- 1 h1=1.10e+00 h2=1.00e+01 t1=1.00e+00 w1=2.00e+00 s0=1.00e+01 s1=2.00e+00 E1=1.00e+00 E2=1.00e+00 Z1=2.24e-08 2 h1=1.20e+00 h2=1.00e+01 t1=1.00e+00 w1=2.00e+00 s0=1.00e+01 s1=2.00e+00 E1=1.00e+00 E2=1.00e+00 Z1=2.24e-08

There are two sets of data from the run just completed.

Do you want to look at a specific data set [y/n]? y Input the filename to save your data (default: Your_data):

Press RETURN to use the default name to store the results.

Type in the index number [1 - 2]: 2

How long is this structure (in microns)? 250

Input Parameters: h1=1.20e+00 h2=1.00e+01 t1=1.00e+00 w1=2.00e+00 s0=1.00e+01 s1=2.00e+00 E1=1.00e+00 E2=1.00e+00 Z1=2.24e-08 Electrical Model Parameters: (length = 250 microns) C11 = 8.669e-15 Farads C12 = 2.412e-15 Farads L11 = 6.592e-11 Henrys L12 = 1.429e-14 Henrys R1 = 2.800e+00 Ohms

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Do you want to generate a SPICE circuit model with these values [y/n]? y Input the filename to save your circuit (default: Your_subckt):

Press RETURN to use the default name to store the SPICE sub-circuit.

Input the rise-time of the circuit waveform in seconds Or enter zero to use a single section: 1e-10

1 sections will be used.

.subckt icmsb 1 2 3 4 99 ** Line 1, Zo=8.720146e+01, Td=7.559277e-13 **** Line 2, Zo=8.720146e+01, Td=7.559277e-13 **** Section 1 **L11 1 101 6.591800e-11 C111 101 99 8.668750e-15 R11 101 2 2.80000e+00 C112 101 103 2.412208e-15 K112 L11 L12 2.168099e-01 L12 3 103 6.591800e-11 C122 103 99 8.668750e-15 R12 103 4 2.80000e+00 .ends icmsb

Another set [y/n] ? n => q


herman [4]>

At this point the analysis of the “two traces over a ground plane” is finished. Now continue with the example of generating a data file for STUDIO Visualize.

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Session 3: RIL and STUDIO Visualize

Status 1: If you are continuing from where you left off in the previous example, request a new element from the library as follows:

Status 2: If you are running RIL from the beginning, follow the example of Session 1 until you are presented with the listing of all the structures in the Parametrics Library.

=> n The data for the element on which you have been working are automatically saved.

After this point, the complete listing of RIL library elements are displayed.

Index [1 - 30] --> 18

Current element is: Array above substrate below dense array


The simulated results are stored in the database called m18s1, in the subdirectory RILDIR/lev1.g2. This name was selected to reflect model 18 (m18) and you must change the parameter s1 (s1).

At this point, the two windows showing the structure diagram and the schematic is updated to show the new structure. Figure 7-7 shows the structure for the simulation, the corresponding SPICE subcircuit is shown in Figure 7-8. Note that for this element an RC circuit is generated in which the capacitance corre-sponds to the total capacitance for the central trace, but that the output lists the coupling capacitances to the nearest line in the same level and to the substrate and upper level.

Figure 7-7 Parametric structure used to calculate capacitances of a central trace between two ground planes

9 TracesEqual Width / Equal SpacingBetween 2 Planes

t1

t1

h2

h1

w1

E1

t1

s1

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=> a

This command at present invokes the editor vi. The file you’ll open may have some input left by previous data generation. You can delete them all and put in new ones.

Press the <Return> key to start

Figure 7-8 SPICE subcircuit corresponding to structure 18 of the parametric library

1

R1

C1

99

All capacitances in this modelare for the central trace

C11: Total capacitanceC1N: Capacitance to left traceC1B: Capacitance to substrateC1T: Capacitance to level 2

2

Here a set of simulations is generated by changing only one parameter, i.e., the spacing between traces (s1), so the parameter table looks like:

* 9 Traces between 2 Planes** h1 h2 t1 w1 s1 E1 E2 Z1 0.80 2.0 0.8 0.9 0.2 3.9 3.9 3.34e-8 0.80 2.0 0.8 0.9 0.4 3.9 3.9 3.34e-8 0.80 2.0 0.8 0.9 0.8 3.9 3.9 3.34e-8 0.80 2.0 0.8 0.9 1.2 3.9 3.9 3.34e-8 0.80 2.0 0.8 0.9 1.6 3.9 3.9 3.34e-8 0.80 2.0 0.8 0.9 2.0 3.9 3.9 3.34e-8

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As in the previous sessions, the runs are activated by selecting the option generated from the main menu:

=> g Press <Return> key to start with the default data set or type your “ready to use” file name:

Press RETURN to use the file stored in RILDIR/lev1.g2.

After running the complete set of simulations, select the visualize option to generate a STUDIO Visualize input file:

=> v

Input the file name to save your data in Visualize formatthe”.dvm” suffix will be added (default: Your_SV.dvm): m18s11

All the following data sets will be listed in the file m18s11.dvm Array above substrate below dense array

Input Parameters for the Structure: h1 h2 t1 w1 s1 E1 E2 Z1

Electrical Parameters for the Model: C11 C1N C1T C1B R1

Data Available:

index Input Parameters ----- ----- ---------- 1 h1=8.00e-01 h2=2.00e+00 t1=8.00e-01 w1=9.00e-01 s1=2.00e-01 E1=3.90e+00 E2=3.90e+00 Z1=3.34e-08 2 h1=8.00e-01 h2=2.00e+00 t1=8.00e-01 w1=9.00e-01 s1=4.00e-01 E1=3.90e+00 E2=3.90e+00 Z1=3.34e-08 3 h1=8.00e-01 h2=2.00e+00 t1=8.00e-01 w1=9.00e-01 s1=8.00e-01 E1=3.90e+00 E2=3.90e+00 Z1=3.34e-08 4 h1=8.00e-01 h2=2.00e+00 t1=8.00e-01 w1=9.00e-01 s1=1.20e+00 E1=3.90e+00 E2=3.90e+00 Z1=3.34e-08 5 h1=8.00e-01 h2=2.00e+00 t1=8.00e-01 w1=9.00e-01 s1=1.60e+00 E1=3.90e+00 E2=3.90e+00 Z1=3.34e-08 6 h1=8.00e-01 h2=2.00e+00 t1=8.00e-01 w1=9.00e-01 s1=2.00e+00 E1=3.90e+00 E2=3.90e+00 Z1=3.34e-08 => q


At this point, you have generated a STUDIO Visualize input file. Refer to the STUDIO Visualize Manual to learn how to interactively inspect data with STUDIO Visualize. Figure 7-9 has been generated with STUDIO Visualize loading the just created “m18s1.dvm” file. Note in this figure the dependency of the line to ground capacitance and the coupling capacitance with the spacing of the lines.

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Figure 7-9 Capacitances for level array between 2 ground planes

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CHAPTER 8

R

Advanced Parser of Interconnect Structures8

The Advanced Parser of Interconnect Structures (APIS) utility is a program for building 3D interconnect structures using mask layout files, written in the indus-try-standard GDS II format. This program also allows you to specify 3D confor-mal dielectric layers and dielectric stacks. The APIS program provides 3D interconnect structure in the format, available for the RC3 and RI3 field solver program.

OverviewThe APIS tool allows you to create complicated 3D interconnect structures from the mask layout files of the GDS II format. The structure is composed of the right prisms. Each prism has a polygon from the mask layer as its base. All prisms, which are connected in space, compose a net.

The program has the following advanced features:

• Boolean operations on mask layers

• Oversize/undersize operations on mask layers

• Attachment of labels from the internal text layers of the GDS II file for RC3

• Attachment of labels from the external labeling file

• Ability to specify dielectric stack

• Automatic building of conformal dielectric layers

• Ability to merge selected nets, which are separated in space

• Ability to create RI3 bars along the current direction and build RI3 nets

• Ability to create RI3 planes

• Ability to process geometries with any angle in RI3 bar creation,

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Overview Raphael Reference Manual

Figure 8-1 shows the information flow for the APIS program. You provide the mask file in GDS II stream format, the APIS command file, and labeling file (optional). The APIS command file is an ASCII file that describes 3D geometry of the interconnect structure, input and output files, labeling method, and so on in terms of the APIS commands. The APIS program generates a file with 3D inter-connect structure. The structure is written in the format used by the RC3/RI3 field solver.

The interconnect structure is constructed as many-storied building (see Figure 8-2). Each floor is a layer from the GDS II layout file. Metal polygons of the layer are spanned in Z direction as right prisms. All prisms that touch each other are combined into the net.

Figure 8-1 Main components and information flow of the APIS utility program

APIS

APIS

RC3/RI3 input file

APIS labeling

User

RC3 field solver

command file fileGDS II

stream file

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Raphael Reference Manual Running APIS

Running APISAPIS can be invoked by issuing the command:

raphael apis <command file name>

The argument of the command is a name of the APIS command file.

The following example illustrates the usage of the APIS.

1. Copy a file examples/topography/sample.tl2 into your working directory.

2. Run Taurus Layout and load a file sample.tl2

3. Save this file layout in GDS II format as myfile.gds

4. Prepare the APIS command file myfile.inp as in Figure 8-3.

Figure 8-2 Constructing the nets from the polygons, located at 2D layers. Polygons at the layers are shown by bold lines, and prisms, grown from these polygons are shown by thin lines. Grey rectangles include separated nets.

Layer Metal1

Layer Via

Layer Meal2

Net metal_1Net metal_2

Z

X

1 μ

m0.6

μm

0.5

μm

GroundPlane

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Running APIS Raphael Reference Manual

5. Run raphael apis myfile.inp. A file myfile.rc3 should appear when program is finished.

6. Run raphael rc3 -i -z -N -h myfile.rc3. A file myfile.rc3.tdf appears.

7. Run the Taurus Visual program; load a file myfile.rc3.tdf. You should see 3D interconnect structure as in Figure 8-4.

HEADER { INLIB = myfile.gds BLOCK = myfile RC3_DESIGN_FILE=myfile.rc3 }

ASSIGN { Met1(0) Via (2) Met2 (4) }

GEOMETRY { Met1 { LOCATION=0 THICKNESS=0.5 } Via { LOCATION=0.5 THICKNESS=0.5 } Met2 { LOCATION=1.0 THICKNESS=0.5 } }

Figure 8-3 Example of the APIS input file

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Raphael Reference Manual Using the APIS Program

Using the APIS ProgramThis section describes methods of building 3D interconnect structure with the APIS program.

APIS Input and Output Files

To build interconnect structure, you need the following input files:

• The APIS command file that includes references to other input files, the description of the 3D geometry of the interconnect structure, and some other commands needed for building a structure.

• Mask layout file in GDS II format that contains 2D layers description.

• Labeling file with labels which defines pins and attach to the selected nets for RI3 and RC3, respectively. This file is optional for RC3.

The APIS program writes interconnect structure into the output file in RC3/RI3 field solver format as a set of metal and dielectric right prisms.

Names of GDS II file, labeling, and output files are specified in the HEADER state-ment in the command file (see example in Figure 8-3). In addition, the BLOCK parameter specifies a cell which should be extracted from the GDS II file. If the layout is not hierarchical (plain), it has only single cell that can be extracted, and its name should be assigned to the BLOCK parameter. In the case of hierarchical layout, you can choose any cell from any hierarchical level.

Figure 8-4 Example of the interconnect structure, defined in Figure 8-3

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Using the APIS Program Raphael Reference Manual

Assigning Metal and Text Layers

In the GDS II stream file, layers are recognized by their numbers. The ASSIGN command specifies which layers will be extracted from the GDS II file and assigns names to them. Two different names can be assigned to the same mask file layer. In this case the APIS program considers it as two different layers. A text layer can be attached to the metal layer to be used for net labeling. If a net con-tains a prism, grown from the current metal layer, its label is seeking among the text strings, located at the text layer attached. For RI3, text layers will be ignored for pin name definition. pin name is defined in the label file.

The example of the ASSIGN command is presented in Figure 8-5. Here GDS II layers 0, 2 and 4 are assigned by the names Met1, Via, Met2. Labels of poly-gons at Met1 layer are searched in the text layer 100, and labels for Met2 layer are searched at the same layer which is supposed to contain polygons and text.

Boolean and Size Operations at Layout Layers

The APIS program can produce new layers from those defined in the GDS II file using Boolean and size operations.

BOOLEAN command is used for merging, intersecting, subtracting, and finding unique data from the specified layers of data to create new layers of data. The APIS boolean operators are AND, NOT, OR, and XOR. Results of the action of each Boolean command in Figure 8-6 and Figure 8-7.

ASSIGN { Met1(0) TEXT (100) Via (2) Met2 (4) TEXT (4) }

Figure 8-5 An example of the ASSIGN command

BOOLEAN Layer1 AND Layer2 TEMP=Layer3 BOOLEAN Layer2 NOT Layer1 TEMP=Layer4 BOOLEAN Layer1 OR Layer2 TEMP=Layer5 BOOLEAN Layer1 XOR Layer2 TEMP=Layer6

Figure 8-6 The BOOLEAN commands with main Boolean operations

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The SIZE command, which is used for changing a size of all polygons of the layer, has two options:

• The OVERSIZE option expands all polygons, adding the specified amount to each edge of each polygon of the selected layer.

• The UNDERSIZE option shrinks polygons, subtracting the specified amount from each edge of each polygon.

Figure 8-7 Results of the action of main boolean operations from Figure 8-6: a) two initial layers; b) Layer1 AND Layer2; c) Layer2 NOT Layer1; d) Layer1 OR Layer2; e) Layer1 XOR Layer2.

Layer1 Layer2

a)

Layer3

b)

Layer4

c)

Layer5

d)

Layer6

e)

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For example, the action of the SIZE commands, presented in Figure 8-8, is shown in Figure 8-9

Both BOOLEAN and SIZE commands produce resulting layers which names are specified by the TEMP parameter. These layers can be used below in the APIS command file in the same way, as layers, specified by the ASSSIGN command.

Specifying 3D Arrangement of Metal Layers

To specify the arrangement of metal layers, use the GEOMETRY command. For each layer the location and thickness along Z axis are specified by the LOCATION and THICKNESS parameters. It is assumed that metal prisms are grown from the polygons of the layer in positive Z direction. Their height are equal to value of the THICKNESS parameter. Resistivity of metal layers is defined by parameter RHO.

Special meaning is given to the GROUNDPLANE layer. This layer contains only one polygon of the size of simulation window in XY plane. Everything below this layer is ignored by the RC3 field solver. Prism from the GROUNDPLANE is spanned in negative Z direction by the value of the THICKNESS parameter.

An example of the ASSIGN command is presented in Figure 8-10. This command corresponds to the structure from Figure 8-2.

SIZE met1 {OVERSIZE=1. } TEMP=met2 SIZE met1 {UNDERSIZE=1.} TEMP=met3

Figure 8-8 An example of the SIZE command

Figure 8-9 The result of the SIZE command action: met1 is original layer, met2 is expanded layer, met3 is shrunk layer

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Net Constructing

All prisms that touch each other at least in one point are included into the same net. All nets that touch ground plane are merged into the special ground net. So after performing the GEOMETRY command, the number of nets is equal to the number of unconnected sets of metal prisms.

The CONNECTING command allows you to make farther connections, assuming that nets are connected outside the simulating structure. An example of the CON-NECTING command is presented in Figure 8-11.

In this example net Met1 will be merged with net VDD, and Met2 will be merged with VSS and the ground net.

To be connected by the CONNECTING command, nets should have names. Regarding unnamed nets, there is a special macro name UNNAMED, which allows you to connect all unnamed nets with particular selected net. In the example above, all unnamed nets are connected to the ground net.

GEOMETRY { Metal1 { LOCATION=0 THICKNESS=1.0 } Via { LOCATION=1.0 THICKNESS=0.6 } Metal2 { LOCATION=1.6 THICKNESS=0.5 } GROUNDPLANE { LOCATION=-0.6 THICKNESS=0.5 } }

Figure 8-10 An example of the GEOMETRY command

CONNECTING { {VDD Met1} {VSS Met2 GROUND} {UNNAMED GROUND} }

Figure 8-11 An example of the CONNECTING command

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Being connected, nets are merged into a single net. The name of this net is chosen among the names of connected nets, using rules of precedence (see Specifying Net Names, p. 8-12).

The WINDOW command is used to select a part of interconnect structure for simu-lation. It defines a simulation region as 3D box. All objects out of this box are removed from the interconnect structure. If some prism sets are not connected inside the box, but are connected outside the box, they are considered as belong-ing to the single net. The default window includes all prisms of the structure. The example of full 3D interconnect structure and a part of it, selected by the WINDOW command is shown in Figure 8-12.

Specifying Dielectric Stack

To specify the dielectric stack, use the DIELECTRIC command. This command defines location, thickness, and permittivity of dielectric layers. The DEFAULT keyword specifies the permittivity of unfilled space. If dielectric layers overlap each other, next layer hides previous. The example of the DIELECTRIC com-mand and corresponding structure are presented in Figures 8-13 and 8-14.

Figure 8-12 An example of the full 3D interconnect structure and part of it, selected by the WINDOW command.

DIELECTRIC { { EPS=3.9 DEFAULT } { EPS=12 LOCATION=0.0 THICKNESS=1.0 } { EPS=4.4 LOCATION=0.5 THICKNESS=1.0 } { EPS=2.4 LOCATION=2.0 THICKNESS=1.0 } }

Figure 8-13 An example of the DIELECTRIC command

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Specifying Conformal Dielectric

The APIS program can automatically build conformal dielectric structures. Speci-fying conformal dielectric for a given layer means that this dielectric covers all sides of prisms, grown at this layer. Thickness of the covering is defined by the inflation parameters. Multi-layer covering is available also. In this case, conformal dielectric covers other conformal dielectric. To specify conformal dielectric lay-ers, use the CONFORMAL command. An example of the CONFORMAL command and results of its action are presented in Figures 8-15 and 8-16.

Figure 8-14 Dielectric layers, defined by the DIELECTRIC command in Figure 8-13. All all unfilled space has ε=3.9.

ε=3.9

ε=2.4

ε=3.9

ε=4.4

ε=12

ε=3.9

0.5μ

m1.

0 μm

0.5μ

m1.

0 μm

CONFORMAL { NAME=conformal1 PARENT_NAME=poly EPS=2.8 XY_INFLATION=0.5 ZN_INFLATION=0.6 ZP_INFLATION=0.3 } { NAME=conformal2 PARENT_NAME=conformal1 EPS=3.4 XY_INFLATION=0.4 ZN_INFLATION=0.8 ZP_INFLATION=0.2 }

Figure 8-15 An example of the CONFORMAL command

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Specifying Net Names

You can specify net name by two methods:

• Extracting text string from the GDS II text layer:

To extract net names from the text layer, attach it to the interconnect layer by the ASSIGN command. Processing the ASSIGN command, the APIS pro-gram finds all pairs of polygon in the interconnect layer and text rectangle in the text layer, such that a central point of the text rectangle belongs to the polygon. A name corresponding to a given polygon, is attached to the net which includes prism based on this polygon. If different names are attached to the single net, the net name is chosen from them using precedence rules.

In hierarchical GDS II files labels in text layers can belong to different levels of hierarchy. The USE_EXPLODED_TEXT parameter of the TEXT_OPTIONS statement specifies the behavior of the APIS program in this case. If it is set to FALSE, only a top hierarchy level is used, and if TRUE - text is sought at all hierarchy levels.

Figure 8-16 Conformal dielectric layers, built by the CONFORMAL command (see Figure 8-15). Dashed line specify location and thickness of the poly layer.

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• Specifying a name in the label file:

A labeling file can be used for net name specifying. The example of this file is presented in Figure 8-17.

The only command LABEL is used in this file. Its parameters define name, layer and 2D point. The name is attached to the polygon at the layer specified which contains 2D point specified. Layer can be defined by number in the GDS II file or by Z coordinate value. The value of Z should be not less than LOCATION, and not greater then LOCATION+WIDTH, where LOCATION and WIDTH are parameters of given layer from the GEOMETRY command.

If few different text labels can be attached to the single net, the following rules of precedence are used to choose net name:

• Labels from the labeling file have higher precedence than labels from the text layers.

• Labels from the same interconnect layer have higher precedence than la-bels from other text layers.

• The first alphabetically label has higher precedence.

• Longer label has higher precedence.

If equal labels are attached to different nets, the behavior of the program is defined by the MERGE_EQUAL_NETS parameter of the TEXT_OPTIONS command. If this parameter is set to FALSE, net names are appended by the unique postfix “_<n>”, where n is integer number, for example A_1, A_2,.... If it is set to TRUE, all nets with equal names are merged into the single net.

All unnamed nets obtain default names metal_1, metal_2,....

Defining Pins

For RI3 and RC3 current calculation, pins must be defined in the labeling file. A pin definition is composed from pin name, pin location, pin layer number, and pin type. While a pin name is defined by parameter NAME, pin type is defined by parameter VOLTAGE. There are four types of pins:

• External Pin: Specifies the external node for RI3 and the voltage for RC3 cur-rent calculation. The external pin type is defined by assigning parameter VOLTAGE to a non-zero value.

LABEL NAME=A X=4.2 Y=3.4 Z=3.3 LABEL NAME=B X=-2.4 Y=-1.4 Z=3.3 LABEL NAME=SUM X=-2.4 Y=-1.4 Z=3.3 LABEL NAME=VDD X=3.8 Y=3.2 LAYER=4 LABEL NAME=VSS X=-2.4 Y=-1. LAYER=4 LABEL NAME=pin1 X=-3.4 Y=-2 LAYER=4 VOLTAGE=1.0 LABEL NAME=pin2 X=-1.4 Y=-3. LAYER=4

Figure 8-17 An example of the labeling file

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• Reference Pin: Defines the reference node in RI3. For RC3 current calcula-tion, it merely defines voltage 0. The reference pin type is defined by assign-ing parameter VOLTAGE to zero.

• Float Pin: Specifies the connection point in a net for connecting a net to a plane in RI3. The float pin type is defined by not specifying parameter VOLT-AGE in the pin definition. This type of pins will be treated as reference pins in RC3 current calculation.

• Connection Pin: Defines a connection point on a plane for connecting a plane to a net. The connection pin type is defined by assigning parameter VOLT-AGE to a float pin name. This type of pin is treated as reference pins in RC3 current calculation.

In RC3 current calculation, if the layer is not a pin layer, the voltage value will be ignored. If the voltage value is not specified in the label file for a pin, zero volts will be assigned to the pin. In capacitance calculation, both pins and their voltage will be ignored.

Generating RI3 Bars and Planes

APIS generates two types of RI3 nets: bar nets and plane nets. APIS treats physi-cal connected metal shapes from GDSII as a net and translates them into either a collection of connected bars to form a bar net or a collection of connected planes to form a plane net. The generation of a bar net or a plane net is determined by the pin types and the number of pins attached to the net.

• A net can have either pins with float pin type or pins with connection pin type, but not both. Bar nets and plane nets are generated for the nets with float pins and connection pins, respectively. If a net has neither float pins nor connection pins attached to it, the types of net generated are determined by the number of external pins and reference pins in the net as described below.

• Bar net generation: Float pins define the generation of bar nets. For the nets that have neither a float pin nor a connection pin, bar nets are generated for those nets that have more than one external/reference pin.

• Plane net generation: Connection pins define the generation of plane nets. For the nets that have neither a float pin nor a connection pin, plane nets are gen-erated for those nets that have zero or one external/reference pin.

• A connection between a bar net and a plane net can be established by refer-encing a float pin through the VOLTAGE parameter in the pin definition of the connection pin.

• A pin geometry shape cannot overlap two disconnected metal shapes.

• The original representation of polygon geometries in GDSII data is not pre-served, but is fractured into rectangles and trapeziods along the x-direction. Bars are generated based on the fractured geometries.

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The following is an example of generating two connected nets, one is a bar net and one is a plane net. In this example, an inductor is connected to a ground plane as shown in Figure 8-18.

The “bridge layer” is used to wire the current from the inside of the inductor to the outside. The “via layer” is used to make the physical connection between “induc-tor layer” and “bridge layer”. APIS recognizes the geometries of “via layer” in Figure 8-18 as a vertical connection and automatically generates vertical bars for the connection. An external pin is defined in the labeling file and the layout on the inductor. This pin is generated as an external node in RI3. A reference pin is defined for the ground plane and is generated as a reference node in RI3.

Having a connection pin defined on the ground plane shape causes APIS to gener-ate a plane net for this shape. A float pin attached to the inductor makes APIS to generate a bar net for the inductor. The connection between the ground plane and the inductor is made by having the connection pin reference the float pin on the

Figure 8-18 GDSII Layout. An inductor is built from rectangles marked by the red area. The green rectangle is on "bridge layer" and used for wring currect across the inductor. The bule rectangles are on the "via layer" for the connection between the inductor and the green rectangle.

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inductor. Figure 8-19 shows the generated structure by APIS from Figure 8-18. The RI3 data in 3D is generated from the GDSII layout data in Figure 8-18.

The details of APIS commands and pin definition in the labeling file for this example follow.

In the ASSIGN block:

ASSIGN { inductor (38) /* inductor */ bridge (33) /* a high layer metal for connection*/ via (39) /* connection via between inductor and bridge */ pin (10) /*pin layer for inductor layer*/ gplane (160) /*ground plane layer*/ gplane_pin (22) /*pin layer for ground plane*/ }

Figure 8-19 Translated RI3 inductor

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In the GEOMETRY command:

GEOMETRY { gplane { LOCATION=0.0 THICKNESS=1.0 RHO=2.0e-8 } gplane_pin { PIN=160 } /*define the pin layer for gplane layer*/ inductor { LOCATION=10.0 THICKNESS=1.0 RHO=2.0e-08 } via { LOCATION=10.85 THICKNESS=6.0 RHO=2.0e-08 } /*for the physical connection between inductor and bridge*/ bridge { LOCATION=16.85 THICKNESS=4.0 RHO=2.0e-08 } pin { PIN=38 } /*define the pin layer for inductor layer*/ }

In the labeling file:

LABEL NAME=drv X=-182.2 Y=52.15 LAYER=10 VOLTAGE=1.0 /*an external pin on inductor*/LABEL NAME=net2plane X=195.0 Y=52.3 LAYER=10/*a float pin for the connection between inductor and the ground plane*/LABEL NAME=plan2net X=195.0 Y=52.3 LAYER=22 VOLT-AGE=net2plan/*make the connection by creating a connection pin to reference a float pin*/LABEL NAME=grnd X=-175.0 Y=190.0 LAYER=22 VOLTAGE=0.0/*a reference pin on the ground plane*/

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APIS Program Commands Raphael Reference Manual

APIS Program CommandsThis section contains a guide to the commands and statements used in the APIS command file.

HEADER

The HEADER statement specifies input GDSII file, input block name, output files for RANXT and RC3 for capacitance and resistance calculation, RI3 inductance calculation, and labeling file.

HEADER HEADER{ NLIB=<NAME> BLOCK=<NAME> [RANXT_TECH_FILE=<NAME> ] [RC3_DESIGN_FILE=<NAME>] [RC3_R_DESIGN_FILE=<NAME>] [RI3_DESIGN_FILE=<NAME>] [RANXT_DESIGN_FILE=<NAME>] [RANXT_RUN_FILE=<NAME>] [RANXT_GRIDS_PER_METER] = <NUMBER>] [LABELING_FILE=<NAME>] [ITF_FILE=<NAME>] [EPS_DEFAULT=<NUMBER>] [GROUND_LOC=<NUMBER>]}


INLIB character Name of the input GDSII stream file

BLOCK character Name of the block in the GDSII file, which should be processed

RC3_DESIGN_FILE character Name of the output file in RC3 format for capacitance calcula-tion

RC3_R_DESIGN_FILE character Name of the output file in RC3 format for inductance calcula-tion

RI3_DESIGN_FILE character Name of the output file in RI3 format for resistance calcula-tion

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Raphael Reference Manual APIS Program Commands

The HEADER statement is mandatory. INLIB and BLOCK are mandatory param-eters. The RANXT_DESIGN_FILE, RC3_DESIGN_FILE, RC3_R_DESIGN_FILE, and RI3_DESIGN_FILE parameters determine the type of output, NXT, RC3 for capacitance calculation, RC3 for resistance calcula-tion, and RI3 for inductance calculation, respectively. If none of those parameters are specified by default, only NXT data will be generated. LABELING_FILE is an optional parameter for RC3 calculation. But it is mandatory for RI3 calcula-tion. The value of the BLOCK parameter should be a name of a cell which has to be extracted from the GDSII file. Plain GDS II file contains the only cell, while hierarchical one can contain a particular number of cells which can be extracted

The RANXT_GRIDS_PER_METER parameter is used to specify the grid density that Raphael NXT uses to represent the structure. Typically this parameter should be chosen so that there are approximately 100 grid points for the smallest feature size. Thus, if you have a 90-nm proxies, RANXT_GRIDS_PER_METER=1e9 would be a good choice. RANXT_GRIDS_PER_METER is used by APIS in con-verting both the GDS2 data and the layer thickness in the geometry section into appropriate integer values for Raphael NXT. Using too small a value for RANXT_GRIDS_PER_METER results in inaccuracy because geometries may not be represented accurately. Using too large a value for RANXT_GRIDS_PER_METER may cause longer runtimes, although usually this is a weak effect. For example, using RANXT_GRIDS_PER_METER=1e10 when RANXT_GRIDS_PER_METER=1e9 is sufficient, results in about a 2X increase in runtime

APIS can obtain information about the dielectric stack and metal layers from an interconnect technology format (ITF) file. The name of the file is specified with ITF_FILE. In addition, the GROUND_LOC parameter is used to set the vertical position of the ground plan and dielectric stack. The EPS_DEFAULT file can be

RANXT_TECH_FILE character Name of output file for Raphael NXT technology file

RANXT_DESIGN_FILE character Name of output file for Raphael NXT design file

RANXT_RUN_FILE character Name of output file for Raphael NXT control

RANXT_GRIDS_PER_METER number Grid specification for Raphael NXT

LABELING_FILEdefault value: none

character Name of labeling file

ITF_FILE character ITF file to be processed

GROUND_LOCdefault value: 0.0

number Location of ground plane and bottom of the dielectric stack.

EPS_DEFAULTdefault value: 1.0

number Default permittivity outside of the dielectric stack


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used to set the default relative permittivity for the simulation window. Since the ITF file does not contain information about how layers assigned, it is still neces-sary to use the ASSIGN command to specify the required layer information. More information about the ITF file format is provided in the Star-RCXT Refer-ence Manual.

CAUTIONThe parser in APIS only understands a subset of the ITF commands. Process variation, air gaps, and trapezoidal dielectrics are not supported. In addi-tion, conformal dielectrics specified using the MEASURED_FROM = TOP_OF_CHIP option are handled correctly. When multiple conformal di-electric layers are stacked, the lower dielectrics must have THICKNESS = 0.0 to be processed correctly. Check the generated RC3 and Raphael NXT files as well as the screen output.

Example:

HEADER {

INLIB = ad4ful1.gdsBLOCK = ADFULAHRC3_DESIGN_FILE=adfulah.rc3RC3_R_DESIGN_FILE=adfulah_r.rc3}

HEADER {

INLIB = ad4ful1.gdsBLOCK = ADFULABITF_FILE = adfulab.itfRANXT_TECH_FILE=adfulah.tecRANXT_DESIGN_FILE=adfulah.des}

ASSIGN

The ASSIGN command names the layers which are loaded from the GDS II file.

ASSIGN { <LAYER NAME> (<NUMBER>)[TEXT (<NUMBER>)] [<LAYER NAME> (<NUMBER>)[TEXT (<NUMBER>)]] ... }

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Layer names, defined in the ASSIGN command, identify layers in BOOLEAN, SIZE, GEOMETRY and CONFORMAL commands. At least one layer should be assigned in the input file. TEXT is optional parameter. If text layer is attached to the interconnect layer, it will be sought for net labels.

Example:

ASSIGN {

poly (5)

met1(8) text (100)

met2(10) text (120)

pin_met1 (300)

}

BOOLEAN

The BOOLEAN command performs Boolean operations with layers

Boolean expression is composed of the elementary expressions of the form:

<layer name 1> <boolean operator> <layer name 2>

where <layer name 1> and <layer name 2> are any valid layer names, defined in the ASSIGN BOOLEAN or SIZE commands above the current command;

<boolean operator> is one of the following Boolean operators:

AND - the intersection of data from layer 1 and layer 2

NOT - the subtraction of data from layer 1 and layer 2


<LAYER NAME> character Name of the layer

<NUMBER> number Number of the layer in the GDSII file

TEXT character Attaches text layer to interconnect layer

BOOLEAN <BOOLEAN EXPRESSION> TEMP=<NAME>


<BOOLEAN EXPRESSION> Boolean expression (see below)

TEMP character A name of the resulting layer

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OR - the merge of data from layer 1 and layer 2

XOR - the unique data of layer 1 and layer 2

Boolean expressions are calculated using the following operator precedence (from highest to lowest): OR, XOR, NOT, AND. To override these rules, parentheses can be used.

Value of the TEMP parameter is a name of the output layer. This name can be used in BOOLEAN, SIZE, CONFORMAL and GEOMETRY commands.

Example:

BOOLEAN ( layer1 NOT layer2 ) AND layer3 TEMP=layer4

SIZE

The SIZE command changes size of all polygons of the layer

<LAYER NAME> should be defined above in the current command file in the ASSIGN BOOLEAN or SIZE commands.

The UNDERSIZE statement subtracts the specified amount (in μm) from each edge of each polygon on the selected layer. Thin areas of polygons may be removed depending on the polygon width and undersize value.

The OVERSIZE operator adds the specified amount (in μm) to each edge of each polygon on the selected layer. Notches in polygons may be filled in depending on the notch width and oversize value.

Value of the TEMP parameter is a name of the output layer. This name can be used in BOOLEAN, SIZE CONFORMAL and GEOMETRY commands.

Example:

SIZE met1 { OVERSIZE=1.0 } TEMP=met2

SIZE <LAYER NAME> {UNDERSIZE=<VALUE> |OVERSIZE=<VALUE>} TEMP=<NAME>


<LAYER NAME> character Any layer name, defined above in this command file

UNDERSIZE <VALUE> Performs a shrinking of each polygon in the selected layer

OVERSIZE <VALUE> Performs a expanding of each polygon in the selected layer

TEMP character A name of the resulting layer

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SIZE met1 { UNDERSIZE=1.0 } TEMP=met3

Results of operations are shown in Figure 8-9.

GEOMETRY

The GEOMETRY command specifies a 3D geometry of interconnect structure.

GEOMETRY { <LAYER NAME> { LOCATION=<NUMBER> THICKNESS=V [RHO=<VALUE>] }<LAYER NAME>{ PIN=<LAYER NUMBER> }][GROUNDPLANE { LOCATION=<NUMBER> THICKNESS=<NUMBER> }][ <LAYER NAME> { LOCATION=<NUMBER> THICKNESS=<NUMBER> [RHO=<VALUE>][ <LAYER NAME> { PIN=<LAYER NUMBER> }]} ...]


<LAYER NAME> character Any layer name, defined above in this command file

LOCATION <VALUE> Specifies Z coordinate of the bottom of the layer (in μm)

PIN <NUMBER> Specifies the layer number to which the pin layer is attached. This parameter defines a layer as a pin layer.

RHO <VALUE> Specifies the resistivity of the layer. This value is used to generate the RC3 input file for current and resistance calcula-tions.

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At least one layer should be defined in the GEOMETRY command.

It is assumed that each polygon on the layer given is a base of right prism. Height of the prism is equal to the parameter THICKNESS, and the prism spans in the positive Z direction.

<LAYER NAME> should be defined above in the current command file in the ASSIGN BOOLEAN or SIZE commands, and the GEOMETRY command should follow after all above.

The ground plane is a special layer in the structure and is not contained in the GDSII file. The LOCATION parameter specifies top coordinate of the ground plane. It is supposed that the ground plane parallelepiped fills the entire simulation window in the XY plane, and spans in the negative Z direction by the THICK-NESS value.

Example:

GEOMETRY {

GROUNDPLANE {LOCATION=0. THICKNESS=1.}

met1 {LOCATION=3.2 THICKNESS=0.8} RHO=1.0e-06}

via {LOCATION=4. THICKNESS=0.8}

met2 {LOCATION=4.8 THICKNESS=0.8}

pin_met1 {PIN=8}

}

DIELECTRIC

The DIELECTRIC command specifies a dielectric stack

THICKNESS <VALUE> Specifies thickness of the layer in Z direction (in μm)

GROUNDPLANE Specifies location and thickness of the ground plane


DIELECTRIC { EPS=<VALUE> DEFAULT EPS=<VALUE> LOCATION=<VALUE> THICKNESS=<VALUE> [EPS=<VALUE> LOCATION=<VALUE> THICKNESS=<VALUE> ...]}

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The DIELECTRIC is an optional command. By default it is supposed that metal nets are placed into the uniform dielectric with ε=1.

Each group of parameters in the DIELECTRIC command defines plane dielectric layer that spans along Z coordinate from the LOCATION to the LOCA-TION+THICKNESS. Permittivity of its material is equal to EPS.

If layers with different permittivity overlap, next layer hides previous.

The DEFAULT parameter defines permittivity of dielectric material that fills the whole unfilled space.

Example:

DIELECTRIC {

{EPS=3.2 DEFAULT}

{ EPS=2.6 LOCATION=1. THICKNESS=0.8}

{ EPS=1.5 LOCATION=2.5 THICKNESS=2.}

}


EPS <VALUE> Permittivity of dielectric material

LOCATION <VALUE> Specifies location of the layer in Z direction (in μm)

THICKNESS <VALUE> Specifies thickness of the layer in Z direc-tion (in μm)

DEFAULT Specifies the default material permittivity

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CONFORMAL

The CONFORMAL command specifies conformal dielectric layers.

Name of the conformal dielectric can be used below in the same CONFORMAL statement, if it is covered by next conformal dielectric. PARENT_NAME can be a layer name, defined in the ASSIGN, BOOLEAN and SIZE commands as well as a conformal dielectric name defined in the CONFORMAL command above.

CONFORMAL { { NAME=<CHARACTER> PARENT_NAME=<CHARACTER> EPS=<VALUE> XY_INFLATION=<VALUE> ZP_INFLATION=<VALUE> ZN_INFLATION=<VALUE> } [{ NAME=<CHARACTER> PARENT_NAME=<CHARACTER> EPS=<VALUE> XY_INFLATION=<VALUE> ZP_INFLATION=<VALUE> ZN_INFLATION=<VALUE> }...] }


NAME character Name of conformal dielectric

PARENT_NAME character Name of parent interconnect layer or conformal dielectric.

EPS <VALUE> Permittivity of the material of conformal dielectric

XY_INFLATION <VALUE> Specifies the extension of the dielectric out of the parent box in the XY plane in μm (see Figure 8-20)

ZN_INFLATION <VALUE> Specifies the extension of the dielectric below the parent box in μm (see Figure 8-20)

ZP_INFLATION <VALUE> Specifies the extension of the dielectric above the parent box in μm (see Figure 8-20)

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Positive ZN_INFLATION means spanning of dielectric to negative Z direction.

Example:

CONFORMAL {

{

NAME=conformal1

PARENT_NAME=mat1

EPS=2.8

XY_INFLATION=0.5

ZN_INFLATION=0.6

ZP_INFLATION=0.3

}

{

NAME=conformal2

PARENT_NAME=conformal1

EPS=3.4

XY_INFLATION=0.4

ZN_INFLATION=0.8

ZP_INFLATION=0.2

}

}

Figure 8-20 Parameters of the CONFORMAL command

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RI3_OPTIONS

The RI3_OPTIONS command specifies the ri3 parameters for RI3 calculation. For details, please refer to “RI3: 3D Resistance and Inductance with Skin Effect.

RI3_OPTIONS { FRCTN=<VALUE> START_FREQ=<VALUE> END_FREQ=<VALUE> LINEAR=<NUMBER> DECADE=<NUMBER> PLANE_NX=<NUMBER> PLANE_NY=<NUMBER> PLANE_DX=<VALUE> PLANE_DY=<VALUE> SPARAMETER=<BOOLEAN> IMPEDANCE=<BOOLEAN> ADMITTANCE=<BOOLEAN> OUTPUT=<BOOLEAN> }


FRCTN <VALUE> The value of FRCTN in RI3 for multi-bar calculation. If this parameter is specified, multi-bars are generated. Otherwise, sin-gle bars are generated.Default value:

START_FREQ <VALUE> Specifies the value of START_FREQ parameter in FREQUENCY command in RI3.Default value:

END_FREQ <VALUE> Specifies the value of END_FREQ parameter in FREQUENCY command in RI3.Default value:

LINEAR <NUMBER> Specifies the value of LINEAR parame-ter in FREQUENCY command in RI3.Default value:

DECADE <NUMBER> Specifies the value of DECADE parame-ter in FREQUENCY command in RI3.Default value:

PLANE_NX <NUMBER> Specifies the value of N1 parameter in PLANE_NODE command in RI3.Default value:

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WINDOW

The WINDOW command specifies simulation box:

PLANE_NY <NUMBER> Specifies the value of N2 parameter in PLANE_NODE command in RI3.Default value:

PLANE_DX <VALUE> Specifies the interval in μm for calculat-ing N1 parameter in PLANE_NODE command in RI3.Default value:

PLANE_DY <VALUE> Specifies the interval in μm for calculat-ing N2 parameter in PLANE_NODE command in RI3.Default value:

SPARAMETER <BOOLEAN> Generates PARAMETER commands for RI3.Default value:

IMPEDANCE <BOOLEAN> Generates IMPEDANCE commands for RI3.Default value:

ADMITTANCE <BOOLEAN> Generates ADMITTANCE commands for RI3Default value:

OUTPUT <BOOLEAN> Generates OUTPUT commands for RI3Default value:


WINDOW { LEFT=<VALUE> RIGHT=<VALUE> TOP=<VALUE> BOTTOM=<VALUE> ZMIN=<VALUE> ZMAX=<VALUE> }

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All metal or dielectric boxes out of the window is ignored by the RC3 field solver.

By default, the window box is chosen as a minimal box which includes all metal and dielectric boxes and the upper half of the ground box.

Example:

WINDOW {

LEFT=0.

RIGHT=10.

BOTTOM=0.

TOP=5.

ZMIN=-10.

ZMAX=5.

}


LEFT <VALUE> Minimal X coordinate of the simulating box (in μm)Default value: minimal value of X coor-dinate of whole structure

RIGHT <VALUE> Maximal X coordinate of the simulating box (in μm)Default value: maximal value of X coor-dinate of whole structure

BOTTOM <VALUE> Minimal Y coordinate of the simulating box (in μm)Default value: minimal value of Y coor-dinate of whole structure

TOP <VALUE> Maximal Y coordinate of the simulating box (in μm)Default value: maximal value of Y coor-dinate of whole structure

ZMIN <VALUE> Minimal Z coordinate of the simulating box (in μm)Default value: minimal value of Z coor-dinate of whole structure or Z coordinate of the middle of the ground plane

ZMAX <VALUE> Maximal Z coordinate of the simulating box (in μm)Default value: maximal value of Z coor-dinate of whole structure

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TEXT_OPTIONS

The TEXT_OPTIONS statement specifies the behavior of the APIS program in the case of hierarchical label structure of the GDS II file and in the case when dif-ferent nets obtain equal names.

If the USE_EXPLODED_TEXT parameter is set to TRUE, a text from all levels of hierarchy is used for labelling. FALSE means that only the highest hierarchy level text is used. This option works only in the case of hierarchic GDS II file. By default, labels can be chosen from only the top hierarchy level of the extracted cell. If design consists of few identical cells, nets can have different labels depend-ing while they considered as a part of the whole design or they are extracted from the whole design. For example, a cell being extracted from the design can have a net with a label SUM, but in the whole structure this net can have a label SUM2. In the case of plain GDS II files, this option has no action.

If the MERGE_EQUAL_NETS parameter is set to TRUE, nets with equal names are merged into the single net. If it is set to FALSE, equal net names are modified to make them different. For example, if nets have the same name Met1, their names will be Met1, Met1_1, Met1_2. By default, nets with equal names are merged into the single net.

Example:

TEXT_OPTIONS {

USE_EXPLODED_TEXT=TRUE

MERGE_EQUAL_NETS=FALSE

}

TEXT_OPTIONS { USE_EXPLODED_TEXT=TRUE|FALSE MERGE_EQUAL_NETS=TRUE|FALSE }


USE_EXPLODED_TEXT <LOGICAL> Specifies behavior of the APIS program in the case of hierarchi-cal label structure of the GDS II fileDefault value: FALSE

MERGE_EQUAL_NETS <LOGICAL> Specifies behavior of the APIS program in the case when differ-ent nets obtain equal namesDefault value: TRUE

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VOLTAGE

The VOLTAGE command specifies the electric potential at the net

This command defines the VOLT parameter in RC3 file.

Example:

VOLTAGE {

{

NET_NAME=SUM

VALUE=1.

}

}

VOLTAGE { { NET_NAME=<STRING> VALUE=<VALUE> } [ { NET_NAME=<STRING> VALUE=<VALUE> } ... ]}


NET_NAME <STRING> Name of the net

VALUE <VALUE> Value of the voltage (in V)default value: 0.

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CONNECTING

The CONNECTING command merges unconnected nets

This command merges different nets into a single net. This command is used when it is assumed that some nets are connected out of the simulating structure. All nets of the each group are merged into the single net. If the GROUND keyword is specified in the group, all nets are connected to ground. The UNNAMED keyword means that all unnamed nets are merged with other nets in the group.

Example:

CONNECTING {

{ VSS VDD GROUND }

{ UNNAMED GROUND }

}

CONNECTING { { <NET NAME> [<NET NAME> ] [GROUND] [UNNAMED] } [ { <NET NAME> [<NET NAME> ] } ... ]}


<NET NAME> <STRING> Name of the net

UNNAMED Specifies all unnamed nets to be merged with other nets of the group

GROUND Specifies all nets of the group to be grounded

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Labeling File Format Raphael Reference Manual

Labeling File FormatThe labeling file is used for defining names of nets, defining pins, and assigning a voltage value to a pin. Voltage value defines pin types in RI3 and is used to gener-ate a RC3 input file for current and resistance calculations. Mechanism of labeling and voltage assignment follow. For each label, a text string and its location are defined. The location specification consists of layer identifier and 2D point. The APIS program finds polygon at the specified layer which contains the point and then attaches label and voltage values to this polygon. During nets constructing, labels from the polygons are attached to the corresponding nets. If net obtains more than one label, the name of the net is chosen from the labels using prece-dence rules. See Specifying Net Names, p. 8-12.

Labeling file consists of strings with the LABEL command.

LABEL NAME=<NAME> X=<VALUE> Y=<VALUE> LAYER=<NUM-BER>|Z=<VALUE>[<VOLTAGE>=<VALUE/STRING>]........LABEL NAME=<NAME> X=<VALUE> Y=<VALUE> LAYER=<NUM-BER>|Z=<VALUE>[<VOLTAGE>=<VALUE>]

Default: If the layer is a pin layer, the default value for pin is zero.

The LABEL command has following parameters:

Default: If the layer is a pin layer, the default value for pin is zero.

Layer can be specified by its number in the GDS II file or by its Z location. The value of Z should fall between LOCATION and LOCATION+WIDTH values as they are defined in the GEOMETRY command (see GEOMETRY, p. 8-23).If the layer is a pin layer, as an option, VOLTAGE specifies the voltage value of the pin.

Example:

LABEL NAME=net1 X=-1.6 Y=3.4 LAYER=0

LABEL NAME=net2 X=3.0 Y=-2.0 LAYER=2

LABEL NAME=net3 X=3.4 Y=2.8 Z=4.6


<NAME> <STRING> Label text

X <VALUE> X coordinate of the label (in μm)

Y <VALUE> Y coordinate of the label (in μm)

LAYER <NUMBER> Layer number in the GDS II file

Z <VALUE> Z location of the layer (in μm)

<VOLTAGE> <VALUE> or <STRING>

The value of voltage on a pin or a float pin name

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Raphael Reference Manual Labeling File Format

LABEL NAME=pin1 X=1.4 Y=3.8 LAYER=300 VOLTAGE=1.0

LABEL NAME=pin2 X=3.9 Y=8.6 LAYER=200

LABEL NAME=pin3 X=3.9 Y=8.6 LAYER=100 VOLTAGE=pin2

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Labeling File Format Raphael Reference Manual

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APPENDIX A

RIL TemplatesA

This appendix contains drawings of the templates currently available under RIL. There are 30 templates allowing up to 9 variations of a given template. Only one representation of each template is presented in this appendix.

3D StructuresThe 3D structures are the following:

Figure A-1 2 Pins/Vias with pads in antipad holes of a plane....................... A-4Figure A-2 Equivalent circuit for 2 Pins/Vias with pads in antipad

holes of a plane ........................................................................... A-4Figure A-3 2 Pins/Vias with pads above a solid plane .................................. A-5Figure A-4 Equivalent circuit for 2 Pins/Vias with pads above a

solid plane................................................................................... A-5Figure A-5 2 Pins/Vias with pads above a plane with antipads .................... A-6Figure A-6 Equivalent circuit for 2 Pins/Vias with pads above a

plane with antipads ..................................................................... A-6Figure A-7 Trace bend above a plane............................................................ A-7Figure A-8 Equivalent circuit for trace bend above a plane.......................... A-7Figure A-9 Trace bend above a plane............................................................ A-8Figure A-10 Equivalent circuit for trace bend above a plane.......................... A-8Figure A-11 Trace widening between 2 planes ............................................... A-9Figure A-12 Equivalent circuit for trace widening between 2 planes ............. A-9Figure A-13 Trace widening between 2 planes ............................................. A-10Figure A-14 Equivalent circuit for trace widening between 2 planes ........... A-10Figure A-15 Trace narrowing above a plane ................................................. A-11Figure A-16 Equivalent circuit for trace narrowing above a plane ............... A-11Figure A-17 Trace narrowing between 2 planes ........................................... A-12Figure A-18 Equivalent circuit for trace narrowing between 2 planes.......... A-12Figure A-19 A pad above a plane.................................................................. A-13Figure A-20 Equivalent circuit for a pad above a plane................................ A-13Figure A-21 A crossover of two traces above a ground plane ...................... A-14Figure A-22 Equivalent circuit for a crossover of two traces above

a ground plane........................................................................... A-14Figure A-23 A two-level crossover above a ground plane ............................ A-15

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2D Structures Raphael Reference Manual

Figure A-24 Equivalent circuit for a two level cross-over above a5ground plane..............................................................................A-16

Figure A-25 A three-level crossover above a ground plane...........................A-16Figure A-26 Equivalent circuit for a three level cross-over above a6

ground plane..............................................................................A-17

2D StructuresThe 2D structures are the following:

Figure A-27 3 parallel bonding wires ............................................................A-17Figure A-28 Equivalent circuit for 3 parallel bonding wires .........................A-17Figure A-29 Equal width/spacing traces without ground plane.....................A-18Figure A-30 Equivalent circuit for equal width/spacing traces

without ground plane.................................................................A-18Figure A-31 Equal width/spacing traces with one ground plane...................A-19Figure A-32 Equivalent circuit for equal width/spacing traces with

one ground plane .......................................................................A-19Figure A-33 Equal width/spacing traces between two ground planes ...........A-20Figure A-34 Equivalent circuit for equal width/spacing traces

between two ground planes .......................................................A-20Figure A-35 Array above substrate below dense array ..................................A-21Figure A-36 Equivalent circuit for array above substrate below

dense array.................................................................................A-21Figure A-37 Unequal width/spacing traces without ground plane ................A-22Figure A-38 Equivalent circuit for unequal width/spacing traces

without ground plane.................................................................A-22Figure A-39 Unequal width/spacing traces with one ground plane...............A-23Figure A-40 Equivalent circuit for unequal width/spacing traces

with one ground plane ...............................................................A-23Figure A-41 Unequal width/spacing traces between two ground planes.......A-24Figure A-42 Equivalent circuit for unequal width/spacing traces

between two ground planes .......................................................A-24Figure A-43 Conformal dielectric layer on top of parallel lines....................A-25Figure A-44 Equivalent circuit for conformal dielectric layer on

top of parallel lines....................................................................A-25Figure A-45 Two conformal dielectric layers on top of parallel lines ...........A-26Figure A-46 Equivalent circuit for two conformal dielectric layers

on top of parallel lines...............................................................A-26Figure A-47 Conformal dielectric layer and metal layer on top of

parallel lines ..............................................................................A-27Figure A-48 Equivalent circuit for conformal dielectric layer and

metal layer on top of parallel lines ............................................A-27Figure A-49 Two conformal dielectric layers and metal layer on

top of parallel lines....................................................................A-28Figure A-50 Equivalent circuit for two conformal dielectric layers

and metal layer on top of parallel lines .....................................A-28Figure A-51 Conformal dielectric layer on top of parallel lines

with sidewall spacers.................................................................A-29

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Raphael Reference Manual 2D Structures

Figure A-52 Equivalent circuit for conformal dielectric layer on top of parallel lines with sidewall spacers ................................ A-29

Figure A-53 Conformal dielectric layer and metal layer on top of parallel lines with sidewall spacers........................................... A-30

Figure A-54 Equivalent circuit for conformal dielectric layer and metal layer on top of parallel lines with sidewall spacers ........ A-30

Figure A-55 Overlap conductor above metal trace and ground plane........... A-31Figure A-56 Equivalent circuit for overlap conductor above metal

trace and ground plane.............................................................. A-31Figure A-57 Level 1 array under parallel level 2 array above substrate........ A-32Figure A-58 Equivalent circuit for level 1 array under parallel level 2

array above substrate ................................................................ A-32Figure A-59 Level 2 array under parallel level 1 array above substrate........ A-33Figure A-60 Equivalent circuit for level 2 array under parallel level 1

array above substrate ................................................................ A-33

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Figure A-1 2 Pins/Vias with pads in antipad holes of a plane

Figure A-2 Equivalent circuit for 2 Pins/Vias with pads in antipad holes of a plane

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Figure A-3 2 Pins/Vias with pads above a solid plane

Figure A-4 Equivalent circuit for 2 Pins/Vias with pads above a solid plane

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Figure A-5 2 Pins/Vias with pads above a plane with antipads

Figure A-6 Equivalent circuit for 2 Pins/Vias with pads above a plane with antipads

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Figure A-7 Trace bend above a plane

Figure A-8 Equivalent circuit for trace bend above a plane

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Figure A-9 Trace bend above a plane

Figure A-10 Equivalent circuit for trace bend above a plane

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Figure A-11 Trace widening between 2 planes

Figure A-12 Equivalent circuit for trace widening between 2 planes

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Figure A-13 Trace widening between 2 planes

Figure A-14 Equivalent circuit for trace widening between 2 planes

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Figure A-15 Trace narrowing above a plane

Figure A-16 Equivalent circuit for trace narrowing above a plane

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Figure A-17 Trace narrowing between 2 planes

Figure A-18 Equivalent circuit for trace narrowing between 2 planes

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Figure A-19 A pad above a plane

Figure A-20 Equivalent circuit for a pad above a plane

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Figure A-21 A crossover of two traces above a ground plane

Figure A-22 Equivalent circuit for a crossover of two traces above a ground plane

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Figure A-23 A two-level crossover above a ground plane

Figure A-24 Equivalent circuit for a two-level crossover above a ground plane

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Figure A-25 A three-level crossover above a ground plane

Figure A-26 Equivalent circuit for a three-level crossover above a ground plane

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Figure A-27 Three parallel bonding wires

Figure A-28 Equivalent circuit for three parallel bonding wires

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Figure A-29 Equal width/spacing traces without ground plane

Figure A-30 Equivalent circuit for equal width/spacing traces without ground plane

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Figure A-31 Equal width/spacing traces with one ground plane

Figure A-32 Equivalent circuit for equal width/spacing traces with one ground plane

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Figure A-33 Equal width/spacing traces between two ground planes

Figure A-34 Equivalent circuit for equal width/spacing traces between two ground planes

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Figure A-35 Array above substrate below dense array

Figure A-36 Equivalent circuit for array above substrate below dense array

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Figure A-37 Unequal width/spacing traces without ground plane

Figure A-38 Equivalent circuit for unequal width/spacing traces without ground plane

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Figure A-39 Unequal width/spacing traces with one ground plane

Figure A-40 Equivalent circuit for unequal width/spacing traces with one ground plane

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Figure A-41 Unequal width/spacing traces between two ground planes

Figure A-42 Equivalent circuit for unequal width/spacing traces between two ground planes

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Figure A-43 Conformal dielectric layer on top of parallel lines

Figure A-44 Equivalent circuit for conformal dielectric layer on top of parallel lines

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Figure A-45 Two conformal dielectric layers on top of parallel lines

Figure A-46 Equivalent circuit for two conformal dielectric layers on top of parallel lines

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Figure A-47 Conformal dielectric layer and metal layer on top of parallel lines

Figure A-48 Equivalent circuit for conformal dielectric layer and metal layer on top of parallel lines

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Figure A-49 Two conformal dielectric layers and metal layer on top of parallel lines

Figure A-50 Equivalent circuit for two conformal dielectric layers and metal layer on top of parallel lines

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Figure A-51 Conformal dielectric layer on top of parallel lines with sidewall spacers

Figure A-52 Equivalent circuit for conformal dielectric layer on top of parallel lines with sidewall spacers

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Figure A-53 Conformal dielectric layer and metal layer on top of parallel lines with sidewall spacers

Figure A-54 Equivalent circuit for conformal dielectric layer and metal layer on top of parallel lines with sidewall spacers

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Figure A-55 Overlap conductor above metal trace and ground plane

Figure A-56 Equivalent circuit for overlap conductor above metal trace and ground plane

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Figure A-57 Level 1 array under parallel level 2 array above substrate

Figure A-58 Equivalent circuit for level 1 array under parallel level 2 array above substrate

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Figure A-59 Level 2 array under parallel level 1 array above substrate

Figure A-60 Equivalent circuit for level 2 array under parallel level 1 array above substrate

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APPENDIX B

Boundary Conditions for the BEM SolversB

IntroductionWhile the finite-difference solvers, RC2 and RC3, always assume a Neumann condition on the window boundaries, the boundary-element solvers, RC2-BEM and RC3-BEM, allow the user to choose among three different boundary condi-tions: open-space, Neumann, or Dirichlet. A proper choice of these boundary con-ditions can improve the accuracy and/or performance of simulation. These sections discuss the possible choices.

Open-Space Boundary ConditionBy default, RC2-BEM and RC3-BEM assume the open-space condition on the window boundaries. Effectively, the window boundaries are non-existent and the “radiation” condition is implied that the electric field at infinity is zero. The bot-tom ground plane and planar dielectric layers that cover the entire window are extended to infinity. Otherwise, the free space is assumed beyond the boundaries. The open-space boundary condition is ideal for simulating isolated structures. For the on-chip application, however, caution is to be exercised because an isolated structure, without the influence of neighboring conductors, can hardly be guaran-teed.

Note:To correlate the results between RC2-BEM and RC2 (or, RC3-BEM and RC3), both using the default boundary conditions (i.e., the former with open-space and the latter with Neumann), the window size needs to be

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Neumann Boundary Condition (or, Magnetic Ground Plane) Raphael Reference Manual

made large enough so that the electric field on the window boundaries is nearly negligible.

Neumann Boundary Condition (or, Magnetic Ground Plane)The Neumann boundary condition is the default of RC2 and RC3, and it forces the normal component of the electric field at the window boundaries to be zero (i.e., ∂V/∂n=0). Effectively, both the geometry and potential distribution are reflected with respect to the boundary (see Figure B-1). Thus, the following terms are often used synonymously: Neumann boundary condition, reflective boundary condition, magnetic ground plane, even mode, and “hard” boundary condition.

When the structures are symmetric, and the biasing condition is such that the potential distribution is also symmetric, the property of magnetic ground planes can be exploited to reduce the storage and CPU time of simulation. The setup files of the Raphael Parasitic Database (RPD) fully utilized this symmetric property, so the command option -s M (or -s BB) need be specified, if RC2-BEM (or RC3-BEM) is to be used.

In general, the Neumann boundary condition results in a better approximation of an open region than the Dirichlet boundary condition.

Figure B-1 A conductor next to a magnetic plane and the corresponding equivalent structure

Magnetic Ground Plane

1V 1V 1V

imageconductor

∂V/∂n=0

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Raphael Reference Manual Dirichlet Boundary Condition (or Electric Ground Plane)

Dirichlet Boundary Condition (or Electric Ground Plane)The Dirichlet boundary condition forces the tangential component of the electric field at the window boundaries to be zero. Geometrically, it is equivalent to having an electric ground plane at the boundary. This electric ground plane is conve-niently chosen to be the reference conductor at 0 volt (i.e., V=0). From the image theory, an image conductor of the same voltage but different polarity can be used in place of this electric ground plane (see Figure B-2). Thus, the following terms are often used synonymously: Dirichlet boundary condition, electric ground plane, odd mode, and “soft” boundary condition.

Currently, RC2-BEM and RC3-BEM allow electric ground planes to be specified at either the minimum or the maximum window boundary, but not both. The verti-cal electric ground planes are not often used for on-chip application.

Two Magnetic Planes Facing Each OtherAll of the previously mentioned boundary conditions are implemented in the BEM solver in an exact manner, except for the case where two magnetic planes are facing each other (note that the BEM solvers do not allow two electric planes to face each other). A conductor between these two magnetic planes introduces an infinite number of images (Figure B-3). Currently, the BEM solvers uses only four images (two images on each side), and these images are enclosed in the rectangu-lar box in Figure B-3.

These four images represents a good approximation of the original problem with infinite many images since all images have the same voltage value. However, if your simulation window is relatively tall, four images may not be sufficient enough to model infinite many images, and the FD solvers are recommended for such problems.

Figure B-2 A conductor next to an electric plane and the corresponding equivalent structure

Electric Ground Plane

1V -1V 1V

imageconductor

V = 0

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Two Magnetic Planes Facing Each Other Raphael Reference Manual

Figure B-3 Equivalent representation of a conductor between two magnetic planes

Magnetic Ground Planes

1V 1V 1V1V1V

actual conductor image conductor

1V 1V

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APPENDIX C

R

Capacitance TheoryC

Maxwell’s EquationsFaraday’s law:

Equation C-1

Ampere’s law:

Equation C-2

Gauss’ law:

Equation C-3

Equation C-4

In electrostatics, use

Equation C-5

Equation C-6

Because

Equation C-7

let

Equation C-8

From and Equations C-6 and C-8, you have Poisson’s Equation.

∇ E× ∂B∂t------–=

∇ H× ∂D∂t------- J+=

∇ B• 0=

∇ D• ρ=

∇ E× 0=

∇ D• ρ=

∇ ∇ φ( )× 0≡

E ∇ φ–=

D ε E=

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Poisson’s Equation Raphael Reference Manual

Poisson’s Equation

Equation C-9

In a homogeneous medium, you must solve Equation C-9 with the boundary con-dition that vanishes at infinity.

Green’s FunctionSolve the following instead:

Equation C-10

with the boundary condition:

Equation C-11

Here, , and G(r) is called Green’s function.

The reason for using Green’s function is that if you know G(r), , then

Equation C-12

The primed coordinates are used for the source points, and the unprimed coordi-nates are used for the field (or observation, or testing) points.

For , Equation C-10 is reduced to

Equation C-13

The solutions for Equations C-13 and C-11 are

Equation C-14

and

Equation C-15

The constant c2 can be found by integrating Equation C-10 over a small spherical (in 3D) or cylindrical (in 2D) volume around . For a 3D case, from the LHS and divergence theorem,

∇ 2φ ρε---–=

E

∇ 2G r( ) δ r( )ε

----------–=

∇ G r( ) 0= as r ∞→

r R R'–=

G R R',( )=( )

R( ) G

V'

∫ R R',( ) ρ R'( )dV=

r 0>

∇ 2G r( ) 0=

G r( ) c1 c21r---+= for 3D,

G r( ) c1 c2 ln r+= for 2D.

R'

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Raphael Reference Manual Green’s Function

R

Equation C-16

and, from the RHS,

Equation C-17

Thus,

Equation C-18

Merging c1 into a constant potential term, you can now write Equation C-12 as

Equation C-19

where, for a 3D homogeneous case,

Equation C-20

Similarly, for a 2D homogeneous case,

Equation C-21

Equations C-20 and C-21 are also known as the free-space Green’s functions.

In the 3D case, since G(r) = 0 as , you can choose the reference at infinity and let . In the 2D case, however, a local reference is needed and is to be determined.

Ground Planes

When there is a ground plane, , and the Green’s function is obtained from the image theory. For example, Equation C-21 now becomes

Equation C-22

where is the image of .

∇ ∇ G( )• V'd

ΔV'

∫ΔV' 0→lim ∂G

∂r-------

r a=a 0→lim 4πa2

4πc2–=

=

δ r( )ε

----------– V'd

ΔV'

∫ΔV' 0→lim 1

ε---–=

c21

4πε---------=

R( ) φ0 G R R',( )ρ R'( ) Vd

V'

∫+=

G R R',( ) 14πε R R'–---------------------------=

G R R',( ) 12πε--------- ln R R'––=

r ∞→φ0 0= φ0

φ0 0=

G R R',( ) 12πε--------- ln R R'–

R R'η–------------------–=

+

R'+ R'

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Inhomogeneous Cases

If dielectric boundaries exist, you must also enforce the following conditions on these boundaries:

1. The tangential is continuous, a result of Equation C-5:

Equation C-23

2. The normal is continuous, a result of Equation C-6 with :

Equation C-24

2D vs. 3D

Consider a unit line source that extends from (0,0, − l/2) to (0,0,l/2). For a field point at , you have

Equation C-25

Thus, to apply a 2D solution to a system of parallel conductors, it is required that

< < 1 Equation C-26

Here, l is the length, and d is the maximum transverse distance between any two points on the conductor surfaces.

Capacitance Matrix

If there are M + 1 conductors, in the 3D case,

Equation C-27

with Vj being the potential difference between the jth conductor and the reference at infinity; and, in the 2D case,

E

n̂ E + E––( )× 0=

D ρ 0=

n̂ D + D––( )• 0=

R d 0 0, ,( )=

φ R( ) 14πε--------- 1

d2 z'2

+--------------------- z'd

l 2⁄–

l 2⁄

∫=

12π ε--------- ln d ln l

2---– ln 1 2d

l------⎝ ⎠

⎛ ⎞ 2+ 1+⎝ ⎠

⎛ ⎞–⎩ ⎭⎨ ⎬⎧ ⎫

–=

φ01

2πε---------ln d–≅

2dl

------⎝ ⎠⎛ ⎞ 2

Qi C ijVj

j 1=

M 1+

∑= ; i 1 … M 1+, ,=

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R

Equation C-28

with Vj being the potential difference between the jth conductor and the reference conductor (M + 1th). The following relationships exist:

Equation C-29

and

Equation C-30

The capacitance matrix [Cij] in Equation C-28 is called the short-circuit capaci-tance matrix.

Let

Equation C-31

then

Equation C-32

Because

Equation C-33

where li is the surface area of the ith conductor, you must solve for ρ numerically from Equation C-19 with the condition that φ is a constant on each conductor sur-face.

Two-Terminal Capacitances

Equation C-28 can also be written as

Equation C-34

Qi CijVj

j 1=

M

∑= ; i 1 … M, ,=

Cij 0< ; if i j≠

Cij Cji=

Vj1 if j k=

0 if j k≠⎩⎨⎧

=

Qi Cik= ; i 1 … M, ,=

Qi ρ R'( ) ld

li

∫=

Qi C ′i0 Vi C ′ij Vi V– j( )

j 1=j i≠

M

∑+= ; i 1 … M, ,=

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where

Equation C-35

and

Equation C-36

The capacitance matrix defined in Equation C-34 is referred to as the two-terminal capacitance matrix.

Method of Moments

Write ρ as

Equation C-37

where is called the expansion function. The most often used expansion functions are pulse and triangular (or rooftop) functions.

The coefficients are solved by integrating both sides of Equation C-19 with a testing (or, weighting) function . That is,

Equation C-38

N + 1 linear equations must be solved.

When the testing function is an impulse function, Equation C-38 is called the col-location (or point-matching) method. When the testing is the same as the expan-sion function, it is called the Galerkin’s method.

Variational Method

A functional maps a function to a number. If there exists a function y = y(x) that minimizes the functional Ψ, where

Equation C-39

C 'i0 Cij

j 1=

M

∑=

C 'ij Cij–= ; i j≠

C'ij[ ]

ρ R'( ) ρnPn R'( )n 1=

N

∑=

Pn R'( )

ρnWm R'( ) m 1 to N=,

V R( )Wm R'( ) ld

l

φ0Wm R'( ) ld

l

∫ +=

ρn Wm R( ) G R R',( )

l'

∫l

∫ Pn R'( )dl1=

N

∑

Ψ y( ) f x y y', ,( )d∫=

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R

then y(x) satisfies the associated Euler-Lagrange equation:

Equation C-40

Thomson’s Theorem

Charges placed on a system of fixed conductors embedded in a dielectric distrib-ute themselves on the surface of these conductors such that the energy of the resultant electrostatic field is a minimum.

Boundary-Element Method

Consider the functional Ψ1, where

Equation C-41

You can verify that Ψ1 is stationary, and the Euler-Lagrange equation is reduced to the integral equation

Equation C-42

Applying the Rayleigh-Ritz technique by letting

Equation C-43

and

Equation C-44

you obtain the same set of linear equations as the Galerkin’s method.

∂f∂y-----

ddx------ ∂f

dy'-------⎝ ⎠

⎛ ⎞– 0=

ρ( ) V R( )ρ R( ) ld

l

∫ 12--- φ R( )ρ R( )

l

∫–=

V R( ) φ0 G R R',( )ρ R'( )d

l'

∫+=

∂Ψ1∂ρn--------- 0= , n 1 to N=

∂Ψ1∂φ0--------- 0=

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Finite-Element Method

Let

Equation C-45

then the Euler-Lagrange equation is reduced to the Laplace’s equation:

Equation C-46

Note that

Equation C-47

and

Equation C-48

Minimizing Equations C-41 and C-45 therefore results in the lower and upper bounds of estimated capacitances, respectively.

Finite-Difference Method

The Method of Moments (MOM), Boundary Element Method (BEM), and Finite Element Method (FEM) are all derived from an integral expression, and they are frequently referred to as the Integral Equation (IE) method.

The Finite Difference (FD) Method, on the other hand, deals with the Laplace’s equation (i.e., the differential form) directly. The derivatives are evaluated numer-ically by finite-difference equations.

Raphael’s RC2 and RC3 use the Finite Difference Method, and RC2-BEM and RC3-BEM use the Boundary Element Method.

Relationship Between [L] and [C]

For 2D perfect conductors in a homogenous medium of ( ), you have

Equation C-49

where [I] is an identity matrix.

Ψ2 φ( ) ∇ φ Sd2

S

∫=

∇ 2φ 0=

1Ctrue------------ k1

ψ1

Q2------≤

Ctrue k2ψ2

V2------≤

μ ε,

L[ ] C[ ] με I[ ]=

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R

For 2D perfect conductors in an inhomogeneous medium of (μn,εn), n = 1 to N, you can obtain [L] through the following:

1. let ˜ εn = 1/μn, n = 1 to N

2. compute [C]

3. the desired [L] is then given by [L] = [C].

The above relationships can be easily seen by examining the linear equations in solving for charge and current densities on the conductor surfaces. By perfect (or, lossless) conductors, it has been implied that either charges or currents can exist on the conductor surfaces only.

Resistance Calculation

To compute capacitances, start with

Equation C-50

Equation C-51

and = ε .

To compute resistances, use

Equation C-52

Equation C-53

and = σ .

Note that

Equation C-54

Thus, if you run RC2 with ε replaced by σ, you get the conductance matrix [G] between nodes on a planar resistive sheet.

∇ E× 0=

∇ D• ρ=

D E

∇ E× 0=

∇ J• ∂ρ∂t------–=

J E

G[ ] V[ ] I[ ]=

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Analytic Formulae Raphael Reference Manual

Analytic FormulaeExact solutions are available for some geometries:

1. Sphere of radius a:

Equation C-55

2. Concentric sphere of inner radius a and outer radius b:

Equation C-56

3. Concentric sphere of inner radius a, outer radius b and ε(r) = ε1 for a≤ r ≤ t and ε(r) = ε2 for t ≤ r ≤ b.

Equation C-57

4. Disk of radius a:

Equation C-58

5. Coaxial cable of inner radius a and outer radius b:

Equation C-59

6. Coaxial cable of inner radius a, outer radius b and ε(r) = ε1 for a ≤ r ≤ t and ε(r) = ε2 for t ≤ r ≤ b:

Equation C-60

7. Coaxial cable of inner radius a, outer radius b and ε(r) for a ≤ r ≤ b:

Equation C-61

8. Coaxial elliptic cylinder of inner semi-axes a1, b1 and outer semi-axes a2, b2:

C 4πε a=

C ---4πε1a--- 1

b---–

---------------=

C 4π1ε 1----- 1

a--- 1

t---–⎝ ⎠

⎛ ⎞ 1ε 2----- 1

t--- 1

b---–⎝ ⎠

⎛ ⎞+

------------------------------------------------------=

C 8ε a=

C 2πε

ln ba---⎝ ⎠

⎛ ⎞--------------=

C 2π1ε 1-----ln t

a---⎝ ⎠

⎛ ⎞ 1ε 2----- ln b

t---⎝ ⎠

⎛ ⎞+

----------------------------------------------=

C 2πb

∫ drε r( )r-------------

----------------------------=

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Equation C-62

9. Circular cylinder of radius a above a ground plane with h = center of cylinder to ground plane:

Equation C-63

10. Two circular cylinders, each of radius a, with 2h = center-to-center spacing:

Equation C-64

C 2πε

lna2 b2+

a1 b1+-----------------⎝ ⎠

⎛ ⎞----------------------------=

C 2πε

ln h h2 a2–+

a------------------------------

⎝ ⎠⎜ ⎟⎛ ⎞

-----------------------------------------=

C πε

ln h h2 a2–+

a------------------------------

⎝ ⎠⎜ ⎟⎛ ⎞

-----------------------------------------=

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APPENDIX D

R

Inductance TheoryD

Maxwell’s EquationsFaraday’s law:

Equation D-1

Ampere’s law:

Equation D-2

Gauss’ law:

Equation D-3

Equation D-4

In magnetostatics, use

Equation D-5

Equation D-6

Because

Equation D-7

let

Equation D-8

where is called the magnetic vector potential.

From , Equations D-5 and D-8, and the following vector identity

∇ E× ∂B∂t------–=

∇ H× ∂D∂t------ J+=

∇ B• 0=

∇ D• ρ=

∇ H× J=

∇ B• 0=

∇ ∇ A×( )• 0≡

B ∇ A×=

A

B μH=

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Equation D-9

you have Poisson’s equation.

Poisson’s Equation

Equation D-10

provided that

Equation D-11

From Equation D-8 and

Equation D-12

you can properly choose ψ and add ∇ψ to so that Equation D-11 is satisfied and is uniquely defined.

Again, can be expressed as

Equation D-13

where, for a 3D homogeneous case,

Equation D-14

and, for a 2D homogeneous case,

Equation D-15

Magnetic EnergyThe magnetic energy T can be expressed as

Equation D-16

where the integral extends over all space.

∇ ∇ A×× ∇ ∇ A•( ) ∇ 2A–=

∇ 2A μ– J=

∇ A• 0=

∇ ∇ ψ× 0≡

AA

A

R) G

V′∫ R R′,( )J R′( )dV=

G R R′,( ) μ4π R R′–--------------------------=

G R R′,( ) μ2π-----– ln R R– ′=

T 12--- H B•

V∞∫ dV=

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Raphael Reference Manual Effective Inductance

Using

Equation D-17

you have

Equation D-18

where the surface integral is evaluated at infinity. In a 3D case, the second integral always vanishes. In a 2D case, the second integral vanishes if the constraint that the sum of total currents is zero is imposed. Then, Equation D-18 is reduced to

Equation D-19

Effective InductanceComparing Equation D-19 with

Equation D-20

you have

Equation D-21

where Lij is called the partial inductance.

If you now choose 0 as the reference conductor, and let

Equation D-22

H ∇ A×• A ∇ H ∇ A H×(•+×•=

12--- A ∇•

V∞∫ H× dV 1

2--- A H×

S∞∫ d+=

T 12--- A J•

V∞∫ dV=

12---= Aj Ji•

Vi

∫j 0=

N

∑i 0=

N

∑ dV

T 12--- Lij

j 0=

N

∑i 0=

N

∑ IiIj=

Lij

Aj Ji•Vi

∫ dV

IiIj--------------------------------=

I0 Ij

j 1=

N

∑–=

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Equation D-20 can then be written as

Equation D-23

with

Equation D-24

Here, is known as the effective (or, loop) inductance.

Internal and External InductancesThe volume integral in Equation D-16 extends over all space. The self inductance associated with the field energy inside/outside a conductor is called the internal/external self inductance. Apparently, the external self inductance of a single 2D conductor is infinity.

Current FilamentsFollowing Equation D-21, the mutual inductance L12 between two conductors (1 and 2) is given by

Equation D-25

where the integral is carried out over V1, the volume of conductor 1. I1 and I2 are the currents in conductors 1 and 2, respectively; is the current density in con-ductor 1; is the magnetic vector potential at conductor 1 due to the current at conductor 2.

If the two conductors are straight, and the current is flowing uniformly along the conductor length, Equation D-25 can then be written as

Equation D-26

where is the angle between two conductors; S1 and S2 are the cross-sectional areas of conductors 1 and 2, respectively; and at DC,

T 12--- L

j 1=

N

∑i 1=

N

∑ eij

IiIj=

L eij

Lij Li0– Lj0– L00+=

L eij

L12

A2 J1•∫ dV

I1I2------------------------------------=

J1A2

2

Lf∫∫ J1J2 θcos dS1dS

J1∫ dS1 J2∫ dS2

----------------------------------------------------------------=

θ

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Raphael Reference Manual Skin Effect

Equation D-27

Here, l1 and l2 are the length of filament l and 2, respectively; and r is the distance between two minute segments of the current filaments. Lf is also known as the mutual inductance between two current filaments in the conductors.

At DC, if uniform current distribution in the conductor’s cross section is also assumed, Equation D-26 is then reduced to

Equation D-28

In effect, L12 can be interpreted as Lf averaged over two cross-sectional areas, S1 and S2.

Skin EffectRI3 uses a volume-integral approach to model the skin effect where the conduc-tor’s cross-section is subdivided into many segments and each segment is assumed to have uniform current. After the resistance and partial inductance is computed for each segment, an equivalent circuit of frequency-dependent resistance and inductance is derived from the circuit theory.

Equivalent CircuitsIf [IB] and [VB] are the branch current and voltage of each conductor segment, then

Equation D-29

where

Equation D-30

and [Rp] is a diagonal resistance matrix and [Lp] is the partial inductance matrix. Let [A] be the incidence matrix and [IN] be the nodal current vector, then

Equation D-31

fμ

4π----- 1

r---∫∫ dl1d=

2

Lf∫∫ dS1dS2

S1S2--------------------------------------------- cos=

Zp IB⋅ VB=

Zp Rp jω Lp⋅+=

A IB⋅ IN=

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Equivalent Circuits Raphael Reference Manual

and

Equation D-32

Here, T denotes the transpose of matrix. Thus,

Equation D-33

Let

Equation D-34

where x denotes the external nodes and i denotes the internal nodes, then

Equation D-35

and

Equation D-36

Thus,

Equation D-37

where [Zxx] corresponds to the first type of SPICE model in RI3. In this case, the reference node is connected to all other nodes in a radial shape.

If you define a new incidence matrix [B] where

Equation D-38

and

Equation D-39

then

AT

VN⋅ VB=

A Zp1–

AT

⋅ ⋅⎭⎬⎫

1–

IN⋅ VN=

Zxx Zxi

Zxi ZiiA Zp

1–A

T⋅ ⋅

⎩ ⎭⎨ ⎬⎧ ⎫

–

=

ININx

INi

INx

0= =

VNVNx

VNi

=

Zxx INx⋅ VNx=

B IBx⋅ INx=

BT

VNx⋅ VBx=

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Equation D-40

Here, [B]T[Zxx][B] corresponds to the second type of SPICE model, which is more desirable in general. In RI3, [B] is automatically derived to preserve the topology.

BT

Zxx B⋅ ⋅⎭⎬⎫

IBx⋅ VBx=

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APPENDIX E

R

Taurus Topography InterfaceE

Note:Taurus Topography is no longer sold by Synopsys.

Note:The Raphael Taurus Topography interface is fully operational only with Taurus Topography/Terrain v1.4 or later. In the 98.4 release, Terrain has been renamed to Taurus Topography.

The interface between Raphael RC3 and Taurus Topography allows you to evalu-ate how changes of fabrication processes or design rules affect the parasitic capac-itance and resistance of 3D interconnects. This appendix explains the flow of the Raphael RC3 to the Taurus Topography interface and presents an example that demonstrates its use.

Taurus Topography, another Synopsys TCAD product, simulates topography modification processes such as deposition and etching. One important feature of Taurus Topography is its accurate characterization of the deposition of dielectrics and conductors and the etching of different materials. With Taurus Topography you can address complex 2D- and 3D local interconnect problems.

Refer to the Taurus Topography Reference Manual for more details and applica-tions.

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Methodology Raphael Reference Manual

MethodologyThe complete simulation flow from Taurus Topography to Raphael RC3 is shown in Figure E-1 and explained in the following sections.

Taurus Layout

The starting point is a cell layout saved in the .tl2 file format. In principle, you may use a layout with an arbitrary number of transistors. However, due to the amount of detailed physical analysis, you may want to limit the layouts to circuits of about 20 transistors. You can create layouts in the .tl2 file format by using the IWB module in Taurus Layout. In the IWB module, each polygon defined in the layout is given an unique name, which you may overwrite. The default name, if none is specified, is derived from the polygon’s number. In the same module, you can specify two additional layer properties for each material layer: Eps for defin-ing relative dielectric constant, and Rho for defining resistivity in the units of ohm-meter. The values of Eps and Rho can be negative. The following conven-tions apply for the definitions of Eps and Rho.

• For capacitance computation, a layer is considered a conducting layer if its Eps value is negative or zero, and as a dielectric layer if its Eps value is pos-itive. The dielectric constant of a dielectric layer is assigned to it by its Eps value.

• For resistance computation, a layer is considered an electrode layer if its Rho value is negative or zero, and is considered a dielectric or conductor layer if its Rho value is positive. A positive Rho value represents the resistivity of the layer.

Note:The IWB module is available only in Taurus Layout v1. 5 and higher.

For example, assume there are three different material layers: frog, bird, and fish. Assume the values of Eps and Rho are 3.9 and 1e-6, respectively, for the frog layer. Any polygon belonging to that layer is dielectric with a dielectric constant of 3.9 and resistivity of 1e6. Similarly, if the values of Eps and Rho are -1 and 1e-6, respectively, for the bird layer, any polygon belonging to that layer is a conduc-tor with a resistivity value of 1e-6. Finally, if the values of Eps and Rho are -1

Figure E-1 Flow of Taurus Topography to Raphael interface

Taurus LayoutTL2 File

Taurus Topography 3D

TDF File

Raphael RC3

Capacitance Command

Resistance Command

OR

Input File

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and -1, respectively, for the fish layer, any polygon belonging to the layer is an electrode. Refer to the Taurus Layout Tutorial for more details.

Note:The polygon names are preserved during all further simulation used in Taurus Topography and Raphael RC3. In contrast, the material proper-ties may be overwritten in the input file for Raphael RC3. For more details, refer to REGION on page E-5 and Step 3: Computing Capaci-tance, Resistance with Raphael RC3 on page E-11.

Taurus Topography 3DTaurus Topography is used to generate a 3D structure of the interconnects for the selected layout. You may use the Taurus Topography GUI to specify the deposi-tion and etch process flow. The simulated results of the structure are stored in a Technology Data Format (TDF) file, which can be visualized with Taurus Visual. The generated TDF file is used as input for Raphael.

Note:You must use Taurus Topography v1.4 and higher versions to produce the TDF file. TDF files produced by Taurus Topography (Terrain) v1.3 cannot be used as input for Raphael.

Raphael RC3

Raphael RC3 is used to perform capacitance and resistance analysis for 3D inter-connects. The input for Raphael RC3 is a TDF file which stores the simulated structure generated by Taurus Topography.

Command-Line SyntaxThe command line syntax to invoke the Taurus Topography interface is:

where [Tdf_file] is the TDF file name generated by Taurus Topography. The file <input file> may contain commands to overwrite material properties, to establish electrical connectivity between different polygons in the layout, or to specify the type of computation (resistance or capacitance). The results are written to the output file named Tdf_file.out.

raphael rc3 -v “-t [Tdf_file] <OPTIONS>” <input file>

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Note:Since a finite difference solver is the only available solver in the Taurus Topography interface, the -b option is ignored if used for invoking the BEM solver.

The following options are interpreted by the Raphael Taurus Topography Inter-face:

You may use the -i option to obtain a list of the polygon names defined in the .tl2 file used by Taurus Topography. This is useful if you need to overwrite the mate-rial properties of some of the polygons.

Taurus Topography uses a uniform rectangular mesh throughout the entire domain of interest, which often leads to a large number of grid points. If the -l option is specified, Raphael uses the same mesh distribution as used in the Taurus Topography simulation. Consequently, you may experience an unrealistic requirement for a large amount of memory and long CPU time to complete com-putation. Without -l, Raphael RC3 adopts an adaptive rectangular mesh based on the complexity of the topography described in the TDF file. Use the adaptive mesh since it is more computationally efficient.

Input File Format The input file format for the Raphael RC3 to Taurus Topography interface is simi-lar to the format for RC3. The <input file> consists of commands and com-ments. Lines beginning with * or $ are comment lines and can be ignored. Seven commands are available for use in the input file: REGION, MERGE, POTENTIAL, CAPACITANCE, CURRENT, RESISTANCE, and OPTION. The REGION com-mand lets you refer to a unique polygon defined in the .tl2 file and overwrite its material properties. The MERGE command allows you to establish connectivity between various polygons (i.e., regions). The commands POTENTIAL, CAPAC-ITANCE, CURRENT, and RESISTANACE indicate the types of computation you may specify. The OPTION command allows you to overwrite the default values of the numerical parameters that drive the calculation process. Except for the REGION and OPTION commands, the remaining commands are explained in Chapter 4 of this reference manual.

Option Definition

-i Print the polygon names defined in the .tl2 file; no computation is performed.

-l Raphael adopts the same uniform mesh as used in Taurus Topography.

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Note:The CURRENT and RESISTANCE commands are incompatible with the POTENTIAL and CAPACITANCE commands. They are intended to be used in different structures: CURRENT is used to compute current distri-butions within conductors, whereas POTENTIAL is used to compute potential distribution outside the conductors.

Note:The other commands available in RC3, BLOCK, POLY3D, CYLINDER, SPHERE, COPY3D, and WINDOW3D, are illegal input file formats for the RC3 to Taurus Topography interface.

REGION

The REGION statement is a new command introduced only in the Taurus Topog-raphy interface. Each region corresponds to a unique polygon defined in the .tl2 file for Taurus Topography and inherits the name and material properties of the polygon. This command allows you to overwrite the material properties defined in the .tl2 file without needing to repeat the Taurus Topography computation.

Note:The <NAME> must match one of the polygon names defined in the .tl2 file used by Taurus Topography; otherwise, the command is ignored. For

REGION NAME=<NAME>;{(VOLT=<VALUE>; [FLOAT=<VALUE>;]) or (DIEL=<VALUE>; RHO=<VALUE>;


NAME character Name of the region.

VOLT numeric Potential of an electrode region. units: volt

FLOAT numeric Fixed charge, current, or heat of floating electrode.units: coulomb, ampere, or watt

DIEL numeric Relative dielectric constant of a dielectric region. (Used only for POTEN-TIAL or CAPACITANCE calculations).

CHRG numeric Optional fixed charge density in a dielectric region. units: coulomb/unit3 where unit is specified in the OPTIONS commanddefault value: 0

RHO numeric Resistivity of the material comprising the region. (used only for CURRENT or RESISTANCE calculations).units: ohm-meter

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your convenience, you may use the -i option to print a list of the avail-able polygon names.

For example, you may have a .tl2 file that contains two polygons, spring and sum-mer, defined on the same layer. If the layer is defined to have a resistivity of 1e-6 ohm-meter according to the .tl2 file, the default material properties of the two polygons have the same resistivity value. However, if the polygon spring is an electrode, you can use the following command line

REGION NAME=spring; VOLT=0;

to assign the region spring as an electrode.

On the other hand, if the resistivity value of the polygon summer is 1e-5 instead of 1e-6, then you can use the following command:

REGION NAME=summer; RHO=1e-5;

to assign a resistivity value of 1e-5 ohm-meter to the region summer.

OPTIONS

This command sets the values of the options that drive the calculation process.

In general, the default values should be acceptable for accuracy. However, you can reset the values by defining the options. For instance,

OPTIONS max_iter = 100; iter_tol=1e-5;

sets the max number of iteration steps to 100 and the iteration tolerance to 1e-6.

OPTIONS [MAX_ITER=<VALUE>;][ITER_TOL=<VALUE>;]


MAX_ITER numeric Maximum number of iterations for the ICCG method.default value: 100 or 1% of the number of grid points, whichever is greater

ITER_TOL numeric Iteration tolerance with which the iteration stops.The default is dynamically set between 10-4 and 10-12, depending on the values of VOLT, DIEL, and RHO.

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Raphael Reference Manual Example Using the Taurus Topography Interface

Example Using the Taurus Topography InterfaceThe following three-step example illustrates use of the Taurus Topography inter-face and the usage of input files to overwrite material properties for some of the polygons.

Step 1: Creating a Mask Layout with Taurus Layout

A sample layout is shown in Figure E-3- and E-4.

1. Specify a unique name for each polygon and specify the values of Eps and Rho for each layer.

For this example, two polygons, m4 and p4, are defined on the layer Met1; one polygon, via2, is defined on the layer Via2;two polygons, p1 and m1, are defined on the Met2 layer. The values of Eps and Rho for each layer are specified as follows:

The polygons defined on Met1, Via2, and Met2 are defaulted conductors during capacitance computation and dielectrics during resistance computation.

2. Save the layout using the TL2 file format.

Layer Name Eps Rho

Diel3 3.9 1e6

Met2 -1 1e-6

Via2 -1 1e-6

Diel2 3.9 1e6

Met1 -1 1e-6

Diel1 3.9 1e6

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The file, sample.tl2, Figure E-2, is the tl2 file created for this example.

TL2 0004/ These will be user-generated comments./%% ---------------- Taurus Layout ------------%% Mask layout file for 2-dimensional data %% ----------------------------------------7Diel3 1.0 1.0 3.9 1e+06 6 1 0 1 0 8 0 0Met2 1.0 1.0 -1 1e-06 4 1 0 1 0 24 21 1Via2 1.0 1.0 -1 1e-06 2 1 0 1 0 31 28 2Diel2 1.0 1.0 3.9 1e+06 8 1 0 1 0 8 0 3Met1 1.0 1.0 -1 1e-06 0 1 0 1 0 16 9 4Diel1 1.0 1.0 3.9 1e+06 7 1 0 1 0 8 0 5groundp 1.0 1.0 -1 1000 5 1 0 1 0 5 0 61.000000E+03-1250 2350 -2500 70005 0 4 0 m4 -250 -1450 -250 -1200 1300 -1200 1300 -1450 2 4 1 via2 -200 -1400 -200 -1250 50 -1250 50 -1400 4 4 2 m1 -266 -1448 -266 -500 97 -500 97 -1448 4 4 3 p1 -273 -500 -273 -240 97 -240 97 -500 0 4 4 p4 1300 -1450 1300 -1200 1550 -1200 1550 -1450151 0 2 4 3

Figure E-2 The mask file in TL2 format, sample.tl2

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Step 2: Creating an Input File for Taurus Topography

A Taurus Topography input file specifies all the details of a process flow, includ-ing machine types and process parameters. Refer to the Taurus Topography Refer-ence Manual for a detailed description of the Taurus Topography input file.

Figure E-3 Top views of the mask saved in sample.tl2 (visualized by Taurus Layout)

Figure E-4 Side view of the mask saved in sample.tl2 (visualized by Taurus Layout)

p1m1

via2

m4 p4

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An example of a Taurus Topography input file, tma.inp, is shown in Figure E-5. The .tl2 file, sample.tl2, created in Step 1 is used as the mask file in the Taurus Topography simulation.

You must pay particular attention to the layername parameter when using the Taurus Topography/Raphael interface. Because it does not have a detailed descrip-tion in the Taurus Topography Reference Manual, it is fully described here.

The parameter layername helps to define the related material properties for every newly deposited layer. The most important material properties for parasitics extraction are dielectric constants (of dielectrics) and resistivity (of conductors). All of these properties are defined in a .tl2 file, where each different layer of dielectrics or conductors has its unique layer name and its own material proper-ties. In a Taurus Topography simulation, different layers of the same material that are deposited at different times have the same material name. Specifying the layername parameter assigns the corresponding material properties defined in the .tl2 file to every newly deposited layer, and distinguishes the layers of the same material type from each other.

If no layername is specified for a deposition process, default values of material properties are used. As in specifying the names of mask layers, the layer name of each deposition command should be the same as defined in the .tl2 file. The out-put tdf file is sample.tdf as specified by the SAVE command in the input file. The command to run Taurus Topography is:

topography 3d tma.inp.

Initialize mask.file=sample.tl2 material=Elec thick=0.2+ delta.x=0.025 delta.y=0.025 delta.z=0.025deposit material=silicon thickness=0.2 layername=groundp

deposit material=oxide thickness=0.2 layername=Diel1

deposit material=Ti thickness=0.3 layername=Met1etch mask=Met1 material=Ti positive

mach.depo name=isodepo isotropi material=oxide rate=1.0depo machine=isodepo material=oxide time=0.3 dt.max=0.1 layername=Diel2etch mask=Via2 material=Oxide negative

depo material=Polysilicon thickness=0.1 layername=Via2etch mask=Via2 material=Polysilicon positive

mach.depo name=isodepo isotropi material=Aluminum rate=0.3depo machine=isodepo material=Aluminum time=1.0 dt.max=0.1 layername=Met2

etch mask=Met2 material=Aluminum positive

depo machine=isodepo material=oxide time=0.7 dt.max=0.1 layername=Diel3

save filename=sample.tdfStop

Figure E-5 Input file for Taurus Topography

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The simulated structure, sample.tdf is shown in Figure E-6

Step 3: Computing Capacitance, Resistance with Raphael RC3

Since resistance computation is incompatible with capacitance computation, two separate input files have been created for this example, one for capacitance com-putation and the other for resistance computation.

For capacitance computation, this example assumes all of the polygons are elec-tronically connected as a single net. The MERGE statement is specified to merge those polygons together. The input file, cap.in, contains the following statement:

merge p4; p1; m1; m4; v2;capacitance

Figure E-6 Final structure after Taurus Topography simulation (visualized by Taurus Visual)

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Figure E-7 shows part of the output using the following commands to execute RC3:

raphael rc3 -v “-t sample.tdf” cap.in

For resistance computation, this example assumes the polygons p1and p4 are elec-trodes, while the others remain as resistors. Therefore, the REGION command is specified to overwrite the material properties of the two polygons. The input file, resist.in, contains the following statement:

region name=p1; volt=0;region name=p4; volt=0;resistance

1 merge p4; p1; m1; m4; v2;2 capacitance3 options iter_tol=1e-6;

$ Polygon names defined in the .tl2 file.$ m4 $ v2 $ m1 $ p1 $ p4


p4 groundp p4 3.921056e-16 -3.921056e-16 groundp -3.921056e-16 3.921056e-16

regrid nx ny nz total iter lin_tol. rgd_tol 0 65 61 41 162565 102 8.168e-07 N.A.



C_0_0 p4 OTHERS 3.921056e-16 C_1_1 groundp OTHERS 3.921056e-16


C_0_1 p4 groundp 3.921056e-16

Figure E-7 Capacitance values calculated by the RC3-Taurus Topography interface

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Figure E-8 shows part of the output using the following commands to execute RC3:

raphael rc3 -v “-t sample.tdf” resist.in

To demonstrate how topographic effects impact the capacitance and resistance values, Table E-1 compares the results with those results obtained with respect to the original mask structure. Part of the RC3 output files are shown in Figure E-9- and E-10, respectively.

1 region name=p1; volt=0;2 region name=p4; volt=0;3 resistance

$ Polygon names defined in the .tl2 file.$ m4 $ via2 $ m1 $ p1 $ p4


p1 p4 p1 3.313506e-02 -3.313506e-02 p4 -3.313506e-02 3.313506e-02




R_0_0 p1 OTHERS 3.017951e+01 R_1_1 p4 OTHERS 3.017951e+01


R_0_1 p1 p4 3.017951e+01

Figure E-8 Resistance values calculated by the RC3-Taurus Topography interface

Table E-1 Comparative capacitance and resistance results

NodeIdealized structure

(default grid)After topographic effect

(without -l option)

Capacitance (fF) P4 0.425 0.392

Resistance (ohm) p4 37.9 30.1

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1 $ RC3 RUN OUTPUT=/tmp/sample.cap.rc3 2 $ 3 poly3d name=m4; + 4 coord=”1,1.05; 1,1.3; 2.55,1.3; 2.55,1.05;” + 5 v1=0,0,0.2; height=0.3; volt=0; 6 poly3d name=via2; + 7 coord=”1.05,1.1; 1.05,1.25; 1.3,1.25; 1.3,1.1;” + 8 v1=0,0,0.5; height=0.4; volt=0; 9 poly3d name=m1; + 10 coord=”0.984,1.052; 0.984,2; 1.347,2; 1.347,1.052;” + 11 v1=0,0,0.9; height=0.3; volt=0; 12 poly3d name=p1; + 13 coord=”0.977,2; 0.977,2.26; 1.347,2.26; 1.347,2;” + 14 v1=0,0,0.9; height=0.3; volt=0; 15 poly3d name=p4; + 16 coord=”2.55,1.05; 2.55,1.3; 2.8,1.3; 2.8,1.05;” + 17 v1=0,0,0.2; height=0.3; volt=0; 18 poly3d name=groundp; + 19 coord=”0,0; 3.6,0; 3.6,3.2; 0,3.2;” + 20 v1=0,0,-1; height=1; volt=0; 21 window3d v1=0,0,-1; v2=3.6,3.1,1.9; diel=3.9; 22 merge p4; via2; m4; m1; p1; 23 capacitance 24



p4 groundp p4 4.253618e-16 -4.253625e-16 groundp -4.253618e-16 4.253625e-16


C_0_0 p4 OTHERS 4.253618e-16 C_1_1 groundp OTHERS 4.253625e-16


C_0_1 p4 groundp 4.253621e-16

Figure E-9 Part of the RC3 output file and capacitance values with respect to the original mask structure

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1 $ RC3 RUN OUTPUT=/tmp/sample.resist.rc3 2 $ 3 poly3d name=m4; + 4 coord=”1,1.05; 1,1.3; 2.55,1.3; 2.55,1.05;” + 5 v1=0,0,0.2; height=0.3; rho=1e-6; 6 poly3d name=via2; + 7 coord=”1.05,1.1; 1.05,1.25; 1.3,1.25; 1.3,1.1;” + 8 v1=0,0,0.5; height=0.4; rho=1e-6; 9 poly3d name=m1; + 10 coord=”0.984,1.052; 0.984,2; 1.347,2; 1.347,1.052;” + 11 v1=0,0,0.9; height=0.3; rho=1e-6; 12 poly3d name=p1; + 13 coord=”0.977,2; 0.977,2.26; 1.347,2.26; 1.347,2;” + 14 v1=0,0,0.9; height=0.3; volt=0; 15 poly3d name=p4; + 16 coord=”2.55,1.05; 2.55,1.3; 2.8,1.3; 2.8,1.05;” + 17 v1=0,0,0.2; height=0.3; volt=0; 18 poly3d name=groundp; + 19 coord=”0,0; 3.6,0; 3.6,3.2; 0,3.2;” + 20 v1=0,0,-1; height=1; rho=1e3; 21 window3d v1=0,0,-1; v2=3.6,3.1,1.9; rho=1e6; 22 resistance 23



p1 p4 p1 2.638028e-02 -2.638028e-02 p4 -2.638028e-02 2.638028e-02


R_0_0 p1 OTHERS 3.790710e+01 R_1_1 p4 OTHERS 3.790710e+01


R_0_1 p1 p4 3.790710e+01

Figure E-10 Part of the RC3 output file and resistance values with respect to the original mask structure

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APPENDIX F

R

Capacitance Post-ProcessingF

IntroductionCpost is a utility program for the manipulation of Raphael and Raphael-NES out-put data. Cpost accepts SPICE capacitance matrix data and calculates the simpli-fied equivalent capacitance matrix with selected nets either electrically grounded or floating. In addition, per-unit-length capacitance matrices may be inverted to yield the inductance matrices. Calculation results are saved in standard SPICE for-mat.

Cpost supports batch mode processing as well as interactive. Full logging of all commands and data is provided.

For easy manipulation of systems with large numbers of nets, a default capability is included wherein all nets may be set to float or ground and only those of interest retained in the final output.

Cpost is invoked as follows:

cpost [options] [spicefilename]

or

cpost [options] [commandfilename]

or

cpost [options] [commandfilename] [spicefilename]

Cpost accepts two types of input files: SPICE data and cpost commands that direct the reduction of that data. An output file is opened with the name <spice-filename>.spice or <commandfilename>.spice, depending on which type of file is specified on invocation. In the event both are specified, the output file name derives from the command file name, i.e. <commandfilename>.spice. The default output file may be overridden if desired.

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If no file names are specified, the program starts in interactive mode. You may manually enter the name of the file to be processed.

The following command options are interpreted by Cpost:

Cpost SPICE Input FilesThe SPICE data file format accepted by Cpost conforms to conventional SPICE specifications with the limitation that only capacitance components are read in. Cpost supports the usual physical unit specifiers for capacitive components.

Additional enhancements have been included to facilitate easy usage with Raphael. The netnames GROUND_RC2 and GROUND_RC3 are reserved words and are equivalent to the use of zero as a designation for ground. Finally, Cpost treats both ‘;’ and ‘*’ as comment characters. Thus, anything on a given line after a ‘;’ or ‘*’ character is treated as a comment and ignored.

Cpost Command Input FilesCpost command files consist of comments and/or commands. Again, anything on a given line after a ‘;’ or ‘*’ character is treated as a comment and ignored. Com-mand input is in free format, and all command keywords are case insensitive. Component and net names are case sensitive, however, unless this option is over-ridden with the -case switch. If this option is used, all such names are mapped into their uppercase equivalents in the output file. Statements may be given in any order, with the exception of END or QUIT, which serve as terminators for batches of statements to treat as a single job. A command file may contain multiple jobs, each delimited by END statements. A job is then examined by the program and all reduction commands processed in the following order: INPUT (if any), OUTPUT,

-noquery Suppresses user queries and use defaults when necessary. Used only in batch mode.

-case Turns off case sensitivity. All netnames converted to uppercase. Note: Commands are always case-insensitive.

-O #.# Lowers cutoff for output of SPICE components. Default value is 1.0e-18. This is relative to the chosen engineering scale in which the rest of the components are printed. If capacitance values are output in μf, then capacitances smaller than 1.0e-18 μf are not printed.

-p* Changes interactive mode prompt character to *.

-spiceoff Turns *off* SPICE file outputting of all results

-toobig # Suppresses screen output of matrix if more than # nets in use.

-verbose Uses long messages.

-z Suppresses listing of input file in SPICE output

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DEFAULT TO GROUND, DEFAULT TO FLOAT, GROUND, FLOAT, and SIGNAL. These commands are capable of directing capacitance matrix reduction. All reduc-tions are completed and the results output before matrix inversions are done. Inversion is performed executing inductance-specific commands in the following order: UNIT, INDUCTANCE. Note that the UNIT command has no effect on any capacitance matrix reduction command.

The specifics of syntax and usage for each command are described in the next sec-tions.

INPUT

Specifies the SPICE data file to open. All subsequent reduction commands apply to the contents of this file.

Note:This command is optional in the sense that Cpost may be invoked with the SPICE file name given on the invocation line.

A SPICE output file with the name: <filename>.spice opens to receive all results unless the -spiceoff option is given.

OUTPUT

Specifies the file name to be opened to receive output. This overrides any defaults.

This command operates for Batch Mode only.

Note:The OUTPUT command is optional. If not specified, the default output file name is the input file name with the suffix “.spice”.

FLOAT

Specifies which nets to consider as electrically floating which is equivalent to requiring that the total charge on each conductor be zero.

Net names are by default case-sensitive unless overridden with the -case option. If overridden, all names are translated to all uppercase.

INPUT <filename>

OUTPUT <filename>

FLOAT <net1> <net2>...

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GROUND

Specifies which nets to fix at ground potential.

Net names are by default case-sensitive unless overridden with the -case option. If overridden, all names are translated to all uppercase.

DEFAULT GROUND

Specifies all nets not explicitly overridden by the SIGNAL command to be grounded.

Only those nets specified in SIGNAL command(s) are retained in the reduced out-put file.

DEFAULT FLOAT

Specifies all nets not explicitly overridden by the SIGNAL command to be floated.

Only those nets specified in SIGNAL command(s) are retained in reduced output file.

SIGNAL

Permits the overriding of a previous status choice for a given net or nets.

Specifies which nets to be retained in the output file. Nets retained correspond to electrodes available from outside of the circuit network. For a given net, this over-rides the effect of any previous GROUND, FLOAT, or DEFAULT statement.

INDUCTANCE

Requests that the reduced capacitance matrix be inverted to obtain the inductance matrix.

GROUND <net1> <net2>...

DEFAULT [TO] GROUND

DEFAULT [TO] FLOAT

SIGNAL <net1> <net2>...

INDUCTANCE

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The inductance matrix is computed by the following equation:

where μ0 and ε0 are respectively, the permeability and permittivity of free space and and are the inductance and capacitance matrices. Thus, for this calcula-tion to be valid, the capacitance data must be in per-unit-length values, derived from parallel conductors in free space.

Note:If the INDUCTANCE command is applied to data with no units or units of farads, the program pauses and gives you the choice of aborting the calculation or setting the units to farads per micron. This pause may be overridden by the -noquery option, as described above. If the -noquery option is given, the default is to set units to farads per micron. This is useful for batch mode operation on groups of files of varying units. You may also explicitly set the units of input data using the UNIT command.

UNIT

Specifies the units of the input capacitance matrix and/or output inductance matrix. This command only affects the results of INDUCTANCE calculations. Capacitance calculations are unchanged by the UNIT command.

The followings can be used for capacitanceScale, inductanceScale, and length-Scale, respectively: f, mf, uf, nf, pf, ff, and af; h, mh, uh, nh, ph, fh, and ah; m, mm, um, nm, pm, fm, and am.

Example:

UNIT f/um

or

UNIT f/1.0e-6 m

indicates that the input capacitance matrix is in farads per micron.

Example:

UNIT f/um, ph/um

indicates that the input capacitance matrix is in farads per micron and the output inductance matrix is in picohenries per micron.

If the capacitance data were obtained using RC3 or Raphael-NES then the length of the structure must be specified so that the data can be converted to per-unit-

L C• μ0ε 0=

L C

UNIT [capacitanceScale/[#] lengthScale] [,] [inductanceScale/[#] lengthScale]

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length values. For instance, if a 3D simulation is performed on a structure of uni-form cross-section one inch long, the corresponding UNIT command is:

UNIT f/2.54e-2 m, h/um

This form provides Cpost with correct dimensions for the input capacitance data and additionally requests the inductance data be printed in units of henries per micron.

Note:For this calculation to be valid, the 3D structure simulated must have uniform cross-section along at least one dimension.

END

Denotes the end of a group of commands to be considered as one job.

In batch mode, END allows a single command file to process multiple input files or to process a single input file multiple ways. In interactive mode, END requests that processing be done and the results output.

QUIT

Requests the program to terminate. Used in interactive mode only.

Any outstanding processing is done and the matrix output before the program exits.

END

QUIT

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ExamplesFor a series of examples of program operation, consider the simple capacitor net-work shown in Figure F-1.

Example 1: Batch Mode

The first example demonstrates batch mode where one command file directs the reduction of a SPICE file in several ways.

Figure F-2 lists the command file and Figure F-3 lists the resulting output.

The first command specifies the SPICE file to open and subsequent commands direct reductions. Use of the END statements to group the reduction commands into separate jobs.

The resulting simplified capacitance matrices, with net1 grounded as in the first case and net2 floated, result in capacitances of 6 and 2.333 femtofarads. These are in accordance with expectations from basic circuit theory.

Figure F-1 Circuit model for example SPICE file cpost1.spice

input cpost1.spiceground net1endfloat net2end

Figure F-2 Cpost command file mcsi.cmd

net1 net24 fF

2 fF1 fF

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Example 2: Multiple Input Files

In the second example, a single command file directs reductions for separate input files cpost1.spice and cpost2.spice. The second circuit model for cpost2.spice is shown in Figure F-4.

The command file mcmi.cmd as listed in Figure F-5 results in the partial output listing of Figure F-7.

;./cpost mcsi.cmd

;*** Original SPICE File ;Ca0 net1 0 1e-15f;Cab net2 0 2e-15f;Cb0 net1 net2 4e-15f;*** Command list;INPUT cpost1.spice;GROUND net1;END

;==> SPICE Models for Reduced Capacitance Matrix [Farads]

C_1_0 net2 0 6e-15

;./cpost mcsi.cmd

;*** Original SPICE File ;Ca0 net1 0 1e-15f;Cab net2 0 2e-15f;Cb0 net1 net2 4e-15f;*** Command list;FLOAT net2;END

;==> SPICE Models for Reduced Capacitance Matrix [Farads]

C_1_0 net1 0 2.33333e-15

Figure F-3 Partial output listing from command file mcsi.cmd

Figure F-4 Circuit model for example SPICE file cpost2.spice

net1 net217 fF

8 fF2 fF

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The resulting simplified capacitance matrices, with net1 of file cpost1.spice grounded in the first case and net2 of file cpost2.spice floated in the second, result in capacitances of 6 and 7.44 femtofarads. These results are as expected from basic circuit theory.

Example 3: Interactive Mode and Inductance

This example illustrates both the interactive mode and use of the INDUCTANCE command. The capacitance data are derived from an RC2 simulation using the same structure as in example raexc22 with the dielectric elements removed. An edited version of the RC2 output appears in Figure F-6. To conform to SPICE for-mat specifications, the comment character ‘;’ must be inserted in front of all lines not describing capacitance values.

Cpost is invoked in this case by giving only the SPICE file name. That is,

cpost cpost3.spice

After the program title and other licensing information, it shows:

The inversion of the capacitance matrix is then carried out by the commands:

cpost> UNIT f/um, h/um

cpost> INDUCTANCE

cpost> end

input cpost1.spiceground net1endinput cpost2.spicefloat net2end

Figure F-5 Cpost command file mcmi.cmd

*** ORIGINAL MATRIX [* 10^-18]

m1 m2 m3 m1 79.7165 -38.7261 -1.56784m2 -38.7261 108.209 -28.7765m3 -1.56784 -28.7765 62.6138

Interactive mode.Enter commands:Use END command to processQUIT to exit.cpost>

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It gives rise to:

The above agrees with the results from RC2 (see Figure 2-11, p. 26).

*** REDUCED MATRIX [* 10^-18]

m1 m2 m3 m1 79.7165 -38.7261 -1.56784m2 -38.7261 108.209 -28.7765m3 -1.56784 -28.7765 62.6138

==> SPICE Models for Reduced Matrix [Dimensionless Units]

C_1_0 m1 0 3.94225e-17C_1_2 m1 m2 3.87261e-17C_1_3 m1 m3 1.56784e-18C_2_0 m2 0 4.07062e-17C_2_3 m2 m3 2.87765e-17C_3_0 m3 0 3.22694e-17

Calculating inductance matrix...*** REDUCED MATRIX [ H / uM]

m1 m2 m3 m1 1.75788e-13 7.30048e-14 3.79538e-14m2 7.30048e-14 1.4746e-13 6.9599e-14m3 3.79538e-14 6.9599e-14 2.10638e-13

==> SPICE Models for Reduced Inductance Matrix [Henries Per uM]

L_1 m1 m1_y 1.75788e-13K_1_2 L_1 L_2 0.45344K_1_3 L_1 L_3 0.197239L_2 m2 m2_y 1.4746e-13K_2_3 L_2 L_3 0.394909L_3 m3 m3_y 2.10638e-13

; ==> SPICE Models for Entire Capacitance Matrix [Farad / (1e-06*m)]

C_1_2 m1 m2 3.872614e-17 C_1_3 m1 m3 1.567842e-18 C_1_0 m1 GROUND_RC2 3.942252e-17 C_2_3 m2 m3 2.877654e-17 C_2_0 m2 GROUND_RC2 4.070618e-17 C_3_0 m3 GROUND_RC2 3.226940e-17;; *** INDUCTANCE CALCULATION [Henry / (1e-06*m)]; m1 m2 m3 ; m1 1.757885e-13 7.300512e-14 3.795376e-14 ; m2 7.300499e-14 1.474608e-13 6.959907e-14 ; m3 3.795418e-14 6.959942e-14 2.106384e-13 ;; ==> SPICE Models for Inductance Matrix [Henry / (1e-06*m)]

; L_1 m1 m1_y 1.757885e-13; K_1_2 L_1 L_2 4.534396e-01; K_1_3 L_1 L_3 1.972392e-01; L_2 m2 m2_y 1.474608e-13; K_2_3 L_2 L_3 3.949094e-01; L_3 m3 m3_y 2.106384e-13

Figure F-6 File cpost3.spice: RC2 free-space capacitance simulation data

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;./cpost mcmi.cmd

;*** Original SPICE File ;Ca0 net1 0 1e-15f;Cab net2 0 2e-15f;Cb0 net1 net2 4e-15f;*** Command list;INPUT cpost1.spice;GROUND net1;END ;==> SPICE Models for Reduced Capacitance Matrix [Farads]

C_1_0 net2 0 6e-15

;./cpost mcmi.cmd

;*** Original SPICE File ;Ca0 net1 0 2e-15f;Cab net2 0 8e-15f;Cb0 net1 net2 1.7e-14f;*** Command list;INPUT cpost2.spice;FLOAT net2;END ;==> SPICE Models for Reduced Capacitance Matrix [Farads]

C_1_0 net1 0 7.44e-15

Figure F-7 Partial output listing from command file mcmi.cmd

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APPENDIX G

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Frequently Asked Questions (FAQs)G

RC2 and RC31. What boundary conditions are assumed on the window bound-

aries?

It is assumed that the normal component of the E-field at the window bound-aries is zero (i.e., ∂V/∂n=0). It is called the Neumann, or reflective boundary condition. The term magnetic ground plane or even mode is also used. Both the structures and potential distribution can be considered as if they were peri-odic.

2. How do I compare RC2 and RC3 results with those using the boundary-element method?

A program based on the boundary element method typically assumes an open space and does not require window boundaries. The RC2 and RC3 results are similar to those of a boundary element based program, if the window size is large enough to make the actual E-field at the window boundaries negligible. In general, the open-space condition can be approximated by assigning the window boundaries at 5 times the maximum distance between any two con-ductor surfaces away from the structures themselves. R.E. Collin, in his text-book, Field Theory of Guided Waves (p. 282), shows that for an object located between two ground planes of separation d, the field intensity decays to 1% of its original value at a distance 1.5d away.

Both RC2 and RC3 use a local conductor as reference. While a 2D boundary-element program assumes the same local reference, a 3D boundary-element program typically uses the reference at infinity. It is necessary to convert the 3D results from a boundary-element program to use a local reference when comparison with RC3 is to be made. The conversion is simple because the infinity node can be treated as if it is a floating node, and RC3-BEM has this conversion built-in.

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3. Should I always use a large window?

It depends on the application. If an isolated structure (in open space) is to be simulated, then a large window is needed. In simulating the chip wiring, how-ever, the open-space assumption may be invalid, because of the surrounding structures.

4. How many grid points should I use?

Beginning in Raphael v3.3, the default number of grid points is dynamically adjusted so that a larger number is automatically assigned to a more compli-cated structure. This default number guarantees that the simulation runs suc-cessfully (if not limited by the hardware storage) with reasonable accuracy. You can experiment with more or fewer grid points for better accuracy or effi-ciency.

5. Why is the C-matrix asymmetric?

The regridding and the iterative matrix solver that causes RC2 and RC3 to give asymmetric matrices. Most programs average the off-diagonal terms, so they appear to have a symmetric C-matrix. Keeping the original numbers in RC2 and RC3 provides some indication of how large the numerical errors might be. In case of severe asymmetry, you can set max_regrid=0 and/or reduce iter_tol.

RI36. Why does RI3 not require the specification of a window?

RI3 is not a finite-difference program. It is based on a volume-integral approach to compute inductances. Starting in Raphael v3.3, a current-sheet formulation is used for parallel conductors. For nonparallel conductors, a cur-rent-filament formulation is used.

RPD7. How can I change:

a. The window boundaries?

b. The number of grid points?

c. The conductor’s cross-sections for the RC2 and RC3 simulations?

Beginning in Raphael v3.3, you can change the option settings or redefine the geometry by customizing the simulation template files.

For additional details see the Raphael Tutorial.

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GLOSSARY

Glossary

This glossary contains terms frequently used in the Raphael Reference Manual. A list of acronyms is included as the last section in the Glossary. For references to more informa-tion about a term, see the Index.

Aanisotropic dielectric A dielectric material with unequal dielectric constants in the x, y, or z directions.

BBoundary Element

Method (BEM)Numerical technique for solving electromagnetic problems based on integral equations. It can be more efficient than the finite different method (see FD) for isolated structures with large open regions.

Ccapacitance matrix Matrix of capacitance values representing the couplings between individual members of a

set of electrodes and ground.

capacitance table Table containing various capacitance values with respect to design parameters (width and spacing).

Cpost Post-processing utility program in the Raphael package to allow manipulation of capaci-tance matrix data.

Ffield simulation The use of numerical techniques to solve electromagnetic problems for quantities such as

electric fields, voltages, current densities, etc.

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field solver templates Template file for the input file of a field solver containing geometry information of generic structures.

Finite-Difference (FD)method

Numerical technique for solving electromagnetic problems based on the finite difference approximation to differential equations.

floating conductor A conductor upon which the total amount of electrical charge is fixed, typically at zero.

fringing fields Fields generated by discontinuities in interconnects.

GGDS II Popular binary file format for integrated circuit layouts.

generic structures Predefined structures used to generate capacitance models.

GUI Graphical User Interface. A visual interface for controlling a computer program typically using a mouse or other pointing device.

Iinterconnect library Library containing parasitics data for common interconnect parasitic structures such as

bends, vias, and crossovers.

LLayout Parameter

ExtractorCAD tool to extract circuit netlist (circuit simulator input file) from layouts. These employ empirical rule-based methods for estimating electrical characteristics.

lateral couplingcapacitance

Capacitance between two lateral metals that do not overlap vertically.

Nnonplanar technology Process technology that results in nonplanar structures.

Ooverlap capacitance Capacitance between two vertically overlapping metals, Overlap capacitance is the sum of

the area and perimeter capacitances. (See perimeter capacitance)

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Pparasitics Unintentional electrical effects in circuits due to such things as interconnect capacitances,

dielectric charge leakage, etc. These are typically modeled by adding nondevice compo-nents to SPICE models.

perimeter capacitance Fringing field component of overlap capacitance. (See overlap capacitance)

Poisson’s equation Partial differential equation stating the relationship between voltage and charge distribu-tion.

RRaphael Synopsys tool for electrical and thermal analysis of two- or 3D structures such as on-chip

interconnects, packages, circuit boards, etc.

Raphael-NES Raphael Net Extraction System. High-speed tool from Synopsys for calculation of on-chip interconnect capacitances.

RC2 2D field solver to compute interconnect resistance and capacitance.

RC3 3D field solver to compute interconnect resistance and capacitance.

RI3 3D tool to compute interconnect resistance and inductance with skin effect.

RIL Raphael Interconnect Library. A database program that generates and stores electrical model parameters for interconnect elements.

SSakurai’s model Empirical capacitance model developed by T. Sakurai and K. Tamaru.

SPICE Popular format for representing electrical circuit models.

TTCAD Technology Computer-Aided Design. Generally refers to physics-based simulation soft-

ware that solves partial differential equations to model semiconductor processing, device characteristics, and interconnects.

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AcronymsBEM Boundary Element Method

FD Finite-Difference (Method)

LPE Layout Parameter Extractor

RIL Raphael Interconnect Library

SPICE Semiconductor Parameter Integrated Circuit Extraction

TCAD Technology Computer-Aided Design

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INDEX

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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Index
AADD 7-5
add, RIL command 7-5, 8-12

analytic formulae C-10

aniso.rc2 example file 2-37

aniso.rc3 example file 4-38, 4-39

anisotropic dielectric materials, RC2 example 2-36

anisotropic dielectric materials, RC3 example 4-37

APIS Program Commands 8-18

array above substrate below dense array Fig. A-21

ASSIGN 8-20

automatic node assignment by RIL Fig. 7-11

Bbatch mode, Cpost example F-7

BEM solvers, boundary conditions B-1

BLOCK 4-5

block

RC3 input file statement 4-5type geometric element (direction and perp

are arbitrarily oriented) Fig. 4-7BOOLEAN 8-21

boundary conditions

BEM solvers B-1Dirichlet B-3Neumann B-2open-space B-1, E-2two magnetic planes facing B-3

Boundary-Element Method (BEM) 3-1, C-7

box

RC2 input file statement 2-5type geometric element Fig. 2-6

CCAPACITANCE 4-19

CAPACITANCE

RC2 input file statement 2-14RC3 input file statement 4-19

capacitance and potential analysis of a crossover structure, RC3 4-29

capacitance matrix C-4

capacitance postprocessing, Cpost F-1

capacitance values calculated by RC3-Taurus-Topography interface Fig. E-12

capacitance, RC3 example 4-29

capacitances for level array between 2 ground planes Fig. 7-24

CHECK 7-5

check, RIL command 7-5

circ1


circ2


circuit models

after floating the poly Fig. 4-37before floating the poly Fig. 4-37example SPICE file cpost1.spice Fig. F-7example SPICE file cpost2.spice Fig. F-8

circular ring, RI3 example 6-23

command conventions 1-6

command editor 1-7

command line syntax E-3

commands

RC2 2-1RC2 options 2-2RC2-BEM 3-3RC2-BEM options 3-3, E-4

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A B C D E F G H I J K L M N O P Q R S T U V W X Y ZRC3 4-1RC3 options 4-2RC3-BEM 5-2RC3-BEM options 5-2RI3 6-4RI3 options 6-4RIL 7-3, 8-3

comparisons

RC2 and RC2-BEM for equidistant three-microstrip line structure Fig. 3-8

RC2 and RC2-BEM for Figure 3-6 Fig. 3-12RC2 and RC2-BEM for the resistance computation

Fig. 3-14RC2-BEM and RC2 3-5RC3-BEM and RC3 5-4

conductors

next to electric plane and corresponding equivalent structure Fig. B-3

next to magnetic plane and corresponding equivalent structure Fig. B-2

CONFORMAL 8-26

Conformal Dielectric 8-11

conformal dielectric layer

metal layer on top of parallel lines Fig. A-27metal layer on top of parallel lines with sidewall

spacers Fig. A-30on top of parallel lines Fig. A-25on top of parallel lines with sidewall spacers Fig. A-

29CONNECTING 8-33

conventions

command 1-6naming for examples 1-7RIL 7-10typographical 1-xi

convergence of capacitance value versus grid size for single plate structure in Figure 5-1 Fig. 5-7

COPY, RC2 input file statement 2-11

COPY3D 4-15

COPY3D, RC3 input file statement 4-15

corresponding schematic for the capacitance simulation Fig. 2-14

Cpostbatch mode F-7command file mcmi.cmd Fig. F-9command file mcsi.cmd Fig. F-7command input files F-2examples F-7input files F-1invoking F-1

options F-2SPICE input files F-2

Cpost examples

interactive mode and inductance F-9multiple input files F-8

Cpost input file statement

DEFAULT FLOAT F-4DEFAULT GROUND F-4END F-6FLOAT F-3GROUND F-4INDUCTANCE F-4INPUT F-3OUTPUT F-3QUIT F-6SIGNAL F-4UNIT F-5

CPU time vs. grid size for single plate structure in Figure 5-1 Fig. 5-7

crossover capacitance, RC3-BEM 5-8

crossover of two traces above ground plane Fig. A-14

CURRENT


current density

3D via structure for Example raexc31 Fig. 4-28and resistance, RC3 example 4-25RC2 example 2-29

current filaments D-4, D-5

CYLINDER 4-9

cylinder

geometric element Fig. 4-10RC3 input file statement 4-9

Ddata flow

RI3 6-4RI3 Fig. 6-4

DEFAULT FLOAT, Cpost input file statement F-4

DEFAULT GROUND, Cpost input file statement F-4

device structures

Example raexc23, with left and right dark gray areas representing electrical contacts Fig. 2-31

Example raexi33 Fig. 6-25Example raexi34 which simulates four bond leads

Fig. 6-30DIELECTRIC 8-24

Dielectric Stack 8-10

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A B C D E F G H I J K L M N O P Q R S T U V W X Y ZDirichlet boundary condition (electric ground plane) B-3

discretization of cross-section of multi_bar Fig. 6-12

discretization of plane and suffixes defined by RI3 Fig. 6-8

DPLOTexamples, introduction to plots 4-24file dpraexc32 for Example raexc32 Fig. 4-33graphical output generated by Line 15 of file

dpraexc32 Fig. 4-34input file dpraexc22 for Example raexc22 Fig. 2-27input file dpraexc23 for Example raexc23 Fig. 2-30input file dpraexc31 for Example raexc31 Fig. 4-27input file dpraexi33 for Example raexi33 Fig. 6-24input file dpraexi34 for Example raexi34 Fig. 6-30output showing potential contours for Example

raexc22 Fig. 2-28output showing structure and grid for Example

raexc22 Fig. 2-28visualization tool 1-5

dpraexc22 example file 2-27

dpraexc23 example file 2-30

dpraexc31 example file 4-26, 4-27

dpraexc32 example file 4-30 to ??dpraexi33 example file 6-24

dpraexi34 example file 6-30

Eeditor, command 1-7

effective inductance D-3

electric ground plane (Dirichlet boundary condition) B-3

electrostatic analysis 2-3

electrostatic and resistance analyses, incompatibility 2-3

END, Cpost input file statement F-6

environment variables, RIL 7-9

equal width/spacing traces

between two ground planes Fig. A-20one ground plane Fig. A-19without ground plane Fig. A-18

equations

Maxwell’s C-1, D-1Poisson’s C-2, D-2

equivalent circuit

array above substrate below dense array Fig. A-21conformal dielectric layer and metal layer on top of

parallel lines Fig. A-27conformal dielectric layer and metal layer on top of

parallel lines with sidewall spacers Fig. A-30

conformal dielectric layer on top of parallel lines Fig. A-25

conformal dielectric layer on top of parallel lines with sidewall spacers Fig. A-29

crossover of two traces above ground plane Fig. A-14

equal width/spacing traces between two ground planes Fig. A-20

equal width/spacing traces with one ground plane Fig. A-19

equal width/spacing traces without ground plane Fig. A-18

level 1 array under parallel level 2 array above substrate Fig. A-32


overlap conductor above metal trace and ground plane Fig. A-31

pad above plane Fig. A-13RI3 6-16three level crossover above ground plane Fig. A-16three parallel bonding wires Fig. A-17trace bend above plane Fig. A-7, A-8trace narrowing above plane Fig. A-11trace narrowing between 2 planes Fig. A-12trace widening between 2 planes Fig. A-9, A-10two conformal dielectric layers and metal layer on

top of parallel lines Fig. A-28two conformal dielectric layers on top of parallel

lines Fig. A-26two level crossover above ground plane Fig. A-15unequal width/spacing traces between two ground

planes Fig. A-24unequal width/spacing traces with one ground plane

Fig. A-23unequal width/spacing traces without ground plane

Fig. A-22equivalent circuit for 2 Pins/Vias

pads above plane with antipads Fig. A-6pads above solid plane Fig. A-5pads in antipad holes of plane Fig. A-4

equivalent representation of conductor between two magnetic planes Fig. B-4

example files

aniso.rc2 2-37aniso.rc3 4-38, 4-39dpraexc22 2-27dpraexc23 2-30dpraexc31 4-26, 4-27dpraexc32 4-30 to ??dpraexi33 6-24

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A B C D E F G H I J K L M N O P Q R S T U V W X Y Zdpraexi34 6-30raexc21 2-23raexc22 2-26 to 2-28raexc22.pot 2-27raexc23 2-30, 2-31raexc24 2-33, 2-34raexc25 2-35raexc26 3-7raexc27 3-11raexc28 3-14raexc31 4-25 to 4-28raexc31.pot 4-26raexc32 4-30 to ??raexc32.pot 4-30raexc33 4-36raexc34 5-6raexc35 5-9raexc36 5-11raexc37 5-13raexi31 6-17raexi31.out 6-18, 6-19raexi32 6-20raexi32.out 6-20, 6-21raexi33 6-23 to 6-25raexi33.mat 6-23 to 6-26raexi33p 6-26raexi34 6-27 to 6-31raexi34.mat 6-28

examples, naming conventions 1-7

EXT, RI3 input file statement 6-14

external nodes, RI3, see EXT 6-14

EXTRACT 4-20

EXTRACT


Ffile cpost3.spice, RC2 free-space capacitance simulation

data Fig. F-10

final structure after Taurus-Topography simulation Fig. E-11

Finite-Difference method (FD) 3-1, C-8

Finite-Element Method (FEM) 3-1, C-8

FLOAT, Cpost input file statement F-3

floating conductors

RC2 2-20RC2 example 2-34

floating-gate transistor

Fig. 4-35RC3 example 4-35

flow of Taurus-Topography to Raphael interface Fig. E-2

FREQUENCY, RI3 input file statement 6-15

GGDS II interface 1-5

GDS II stream format interface 1-3

GENERATE 7-5

generate, RIL command 7-5

graphical output showing current density distribution for Example raexc23 Fig. 2-31

graphical representation of the spice subcircuit created by Example raexc24 Fig. 2-34

graphical user interface 1-3

Green’s function 3-2, C-2

GROUND Cpost, input file statement F-4

ground planes C-3

HHEADER 8-18

Iillustration of the automatic node assignment by RC2

when spice subcircuit is created Fig. 2-18

INDUCTANCE

Cpost input file statement F-4RC2 input file statement 2-16

inductance

calculation of two parallel microstrips Fig. 6-17microchips, RI3 example 6-17RC2 example 2-22wires, RI3 example 6-27

inhomogeneous cases C-4

inhomogeneous dielectric layers, RC2-BEM 3-9

INPUT CHECK / PLOTS 7-6

input check/plots, RIL command 7-6

input file for Taurus-Topography Fig. E-10

input files

RC2 2-3RC3 4-3RI3 6-5RIL 7-8

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A B C D E F G H I J K L M N O P Q R S T U V W X Y ZINPUT, Cpost input file statement F-3

interactive mode and inductance, Cpost example F-9

interfaces

GDS II 1-5graphical user 1-3

internal and external inductances D-4

introduction 1-xi to ??, 1-1 to ??plots, RC3 example 4-24RC3-BEM 5-1RI3 6-1RI3 examples 6-17typographical conventions 1-xi

LLabeling File Format 8-34

layername parameter E-10

length units

RC2 2-4RC3 4-4



LIST 7-6

list, RIL command 7-6

loop inductances Fig. 6-2

Mmagnetic energy D-2

magnetic groundplane (Neumann boundary condition) B-2

main components and information flow of Raphael Interconnect Library (RIL) Fig. 7-2, 8-2, 8-3, 8-4, 8-5, 8-6, 8-7, 8-8, 8-9, 8-10, 8-11, 8-12, 8-13, 8-27

mask file in TL2 format, sample.tl2 Fig. E-8

MATRIX, RI3 input file statement 6-13

Maxwell’s equations C-1, D-1

MERGE 4-16

MERGE


MERGE3I, RI3 input file statement 6-14

Method of Moments (MOM) C-6

methods

Boundary-Element (BEM) 3-1comparison 3-3Finite-Difference (FD) 3-1Finite-Element (FEM) 3-1

MULTI_BAR, RI3 input file statement 6-10

Nnaming conventions for examples 1-7

Net Constructing 8-9

Neumann boundary condition (magnetic ground plane) B-2

NEW 7-6

new, RIL command 7-6

NODE, RI3 input file statement 6-6

notes

RC2-BEM 3-4RC3-BEM 5-3

Oopen-space boundary condition B-1, E-2

OPTIONS 4-20

OPTIONS

RC2 input file statement 2-19RC3 input file statement 4-20, E-6

options, Cpost F-2

OPTIONS3I, RI3 input file statement 6-13

output file for Example raexi33 Fig. 6-24

output file generated by RC2

Example aniso.rc2 Fig. 2-37Example raexc24 Fig. 2-33Example raexc25 Fig. 2-35

output from RI3 after post-processing file raexi33.mat with file raexi33p Fig. 6-26

output generated by Example

aniso.rc3 Fig. 4-38raexc31 Fig. 4-25raexc32 Fig. 4-32raexc33 Fig. 4-36

OUTPUT, Cpost input file statement F-3

overlap conductor above metal trace and ground plane Fig. A-31

overlapping rule

RC2 2-4RC3 4-4

RA 2006.03 Index-5


A B C D E F G H I J K L M N O P Q R S T U V W X Y ZPpad above a plane Fig. A-13

PARAM 4-4

PARAM

RC2 input file statement 2-4RC3 input file statement 4-4RI3 input file statement 6-6supported functions 2-4, 4-4

parameters, RC2

ANG 2-5, 2-9ANG1 2-7ANG2 2-7CHRG 2-6 to 2-13COLOR 2-6 to 2-10COORD 2-10CX 2-5 to 2-8CY 2-5 to 2-8DIEL 2-5 to 2-13DX 2-11DY 2-11FAC_REGRID 2-19FILE 2-17FLOAT 2-5 to 2-11FROM 2-11GND 2-17GRID_SLIP 2-19H 2-5ITER_TOL 2-19LENGTH 2-17MAX_ITER 2-19MAX_REGRID 2-19NAME 2-5 to 2-10PX 2-8PY 2-9R 2-7REGRID_TOL 2-19RHO 2-6 to 2-17SET_GRID 2-19TO 2-11UNIT 2-19VOLT 2-5 to 2-11W 2-5X1 2-12X2 2-12Y1 2-12Y2 2-13

parameters, RC3

CENTER 4-11CHRG 4-6 to 4-18, E-5

COLOR 4-7 to 4-14COORD 4-13CTC 4-6 to 4-18DIEL 4-6 to 4-18, E-5DIRECTION 4-5 to 4-16, 8-18 to ??, 8-18 to ??,

8-21 to ??FAC_REGRID 4-21FLOAT 4-6 to 4-16, E-5FROM 4-15GRID_SLIP 4-20HEAT 4-7 to 4-18HEIGHT 4-5 to 4-13, 8-18 to ??, 8-18 to ??, 8-

18 to ??, 8-21 to ??ITER_TOL 4-21, E-6LENGTH 4-6MAX_ITER 4-21, E-6MAX_REGRID 4-21NAME 4-5 to 4-13, 8-18 to ??, 8-21 to ??, 8-21

to ??, 8-22 to ??, 8-23 to ??, 8-26 to ??, 8-30 to ??, 8-31 to ??, 8-32 to ??, 8-33 to ??, 8-34 to ??, 8-34 to ??, E-5

PERP 4-6, 4-13RADIUS 4-9, 4-11REGRID_TOL 4-21RHO 4-6 to 4-18, E-5SET_GRID 4-20TEMP 4-6 to 4-16TO 4-15UNIT 4-21V1 4-5 to 4-17, 8-18 to ??, 8-21 to ??, 8-21 to

??, 8-22 to ??, 8-24 to ??, 8-25 to ??V2 4-6 to 4-17VOLT 4-6 to 4-16, E-5WIDTH 4-6

parameters, RI3

BASE_NODE 6-12CENTER 6-8CRIT_ANGLE 6-13DIRECTION 8-18 to ??FILAMENT 6-13H 6-9 to 6-12L1 6-8L2 6-8N1 6-8N2 6-8NAME 6-7 to 6-12NH 6-11NODE1 6-9, 6-10NODE2 6-9, 6-10NORMAL 6-7NW 6-10

Index-6 RA 2006.03


A B C D E F G H I J K L M N O P Q R S T U V W X Y ZPLANE 6-7POSITION 6-7RHO 6-9 to 6-12SEGMENT 6-13UNIT 6-13W 6-9, 6-10

Parametric Library Fig. 7-2

parametric structure to calculate capacitances of central trace between two ground planes Fig. 7-21

parasitics extraction flow process 1-8

part of RC3 output file

and capacitance values Fig. E-14and resistance values Fig. E-15

partial inductances Fig. 6-2

partial listing

Example raexc21 Fig. 2-23output file raexc34 for RC3-BEM Fig. 5-6

partial listing of RI3 output file

raexi31.out showing example of inductance calculation Fig. 6-19

raexi32.out Fig. 6-21partial output

Example raexc22 Fig. 2-26file of raexc23 computes current density in bent line

Fig. 2-30partial output listing

command file mcmi.cmd Fig. F-11command file mcsi.cmd Fig. F-8raexc28 simulates the resistances among 3 vias Fig.

3-14PLANE, RI3 input file statement 6-12

PLANE_NODE, RI3 input file statement 6-7

Poisson’s equation C-2, D-2

poly


POLY3D 4-12

poly3d

RC3 input file statement 4-12type element with 4 vertices in polygon base Fig. 4-

15portion of output generated by RI3 for Example raexi34

Fig. 6-31

POTENTIAL 4-19

POTENTIAL


potential and capacitance analysis of a crossover structure, RC3 4-29

power-plane resistance, RC2-BEM example 3-13

print, RIL command 7-6

QQUIT 7-7

QUIT

Cpost input file statement F-6RIL command 7-7

Rraexc21 example file 2-23

raexc22 example file 2-26 to 2-28

raexc22.pot example file 2-27

raexc23 example file 2-30, 2-31

raexc24 example file 2-33, 2-34

raexc25 example file 2-35




raexc31 example file 4-25 to 4-28

raexc31.pot example file 4-26

raexc32 example file 4-30 to ??raexc32.pot example file 4-30






raexi31 example file 6-17

raexi31.out example file 6-18, 6-19

raexi32 example file 6-20

raexi32.out example file 6-20, 6-21

raexi33 example file 6-23 to 6-25

raexi33.mat example file 6-23 to 6-26

raexi33p example file 6-26

raexi34 example file 6-27 to 6-31

raexi34.mat example file 6-28

Raphaeldata flow Fig. 1-2

RC2

command options 2-2command syntax 2-1floating conductors 2-20input file 2-3

RA 2006.03 Index-7


A B C D E F G H I J K L M N O P Q R S T U V W X Y Zlength units 2-4overlapping rule 2-4

RC2 and RC3 G-1

RC2 examples

anisotropic dielectric materials 2-36current density 2-29floating conductors 2-34inductance 2-22introduction 2-22SPICE model 2-32three lines 2-24

RC2 input file statements

BOX 2-5CAPACITANCE 2-14CIRC1 2-6CIRC2 2-8COPY 2-11CURRENT 2-15EXTRACT 2-18INDUCTANCE 2-16MERGE 2-13OPTIONS 2-19PARAM 2-4POLY 2-10POTENTIAL 2-14RESISTANCE 2-15SPICE 2-16WINDOW 2-12Z0 2-16

RC2 program 2-1 to ??RC2 solver 1-3

RC2-BEM 3-1 to 3-15

command options 3-3, E-4command syntax 3-3comparison of RC2 3-5introduction 3-1notes 3-4theoretical background 3-2

RC2-BEM examples

inhomogeneous dielectric layers 3-9power-plane resistance 3-13three-microstrip line structure 3-6

RC2-BEM partial output listing

raexc26 Fig. 3-7raexc27 simulates three-microstrip lines embedded

in stratified dielectric media Fig. 3-11RC2-BEM solver 1-4

RC3

command options 4-2command syntax 4-1

input file 4-3introduction 4-1length units 4-4overlapping rule 4-4simulation commands

DC resistance analysis 4-4electrostatic analysis 4-4static thermal analysis 4-4

RC3 examples

anisotropic dielectric materials 4-37capacitance 4-29current density 4-25floating-gate transistor 4-35introduction, plots 4-24resistance 4-25

RC3 input file statements

BLOCK 4-5CAPACITANCE 4-19COPY3D 4-15CURRENT 4-19CYLINDER 4-9EXT 6-14EXTRACT 4-20FREQUENCY 6-15MATRIX 6-13MERGE 4-16MERGE3I 6-14MULTI_BAR 6-10NODE 6-6OPTIONS 4-20, E-6OPTIONS3I 6-13PARAM 4-4, 6-6PLANE 6-12PLANE_NODE 6-7POLY3D 4-12POTENTIAL 4-19REF 6-15SINGLE_BAR 6-9SPHERE 4-11TEMPERATURE 4-19THERMOCAP 4-20WINDOW3D 4-17

RC3 solver 1-4, 4-1 to 4-39

RC3-BEM

command options 5-2command syntax 5-2comparison of RC3 5-4introduction 5-1notes 5-3

RC3-BEM examples

Index-8 RA 2006.03


A B C D E F G H I J K L M N O P Q R S T U V W X Y Zcrossover capacitance 5-8single plate above plane 5-4substrate resistance 5-12trapezoidal conductor between two ground planes 5-

10RC3-BEM partial listing

output file raexc35 Fig. 5-9output file raexc37 Fig. 5-13raexc36 Fig. 5-11

RC3-BEM solver 1-4, 5-1 to 5-14

REF, RI3 input file statement 6-15

related publications

TCAD Products and Utilities Installation Manual 1-xii

relationship between these capacitances and those represented in Figure 2-5 Fig. 2-15

resistance analysis 2-3

resistance and current density, RC3 example 4-25

resistance calculation C-9

resistance dependency with frequency for microstrip Fig. 6-22

resistance values calculated by RC3-Taurus-Topography interface Fig. E-13

RESISTANCE, RC2 input file statement 2-15

results obtained with RI3 compared with Walker’s results for inductance per unit length Fig. 6-20

RI3 G-2

command options 6-4command syntax 6-4data flow 6-4equivalent circuit 6-16input file Example raexi34 Fig. 6-29input files 6-5introduction 6-1theory 6-1

RI3 examples

circular ring 6-23inductance of microstrips 6-17inductance of wires 6-27introduction 6-17skin effect simulation 6-20

RI3 solver 1-4, 6-1 to 6-31

RIL

commands 7-5, 8-5, 8-18, 8-34customization 7-8directory structure 7-8environment variables 7-9files 7-8information flow 7-1, 8-1naming conventions 7-10

running 7-3, 8-3solver 1-4SPICE netlist generation 7-10start_up file 7-8start_up file Fig. 7-10utility introduction 7-1

RIL commands

add 7-5, 8-12check 7-5generate 7-5input check/plots 7-6list 7-6new 7-6print 7-6QUIT 7-7SAVE 7-7TABLE 7-7VISUALIZE 7-7

RIL examples

introduction 7-12session 1 3D 7-12session 2 2D 7-16session 3 RIL and STUDIO Visualize 7-21

RPD G-2

SSAVE 7-7

SAVE, RIL command 7-7

session 1 3D, RIL example 7-12

session 2 2D, RIL example 7-16

session 3 RIL and STUDIO Visualize, RIL example 7-21

short circuit capacitance of line m2 plotted against the interwire distance Fig. 3-10

short-circuit capacitance matrix calculated by Raphael Fig. 2-15

side view of mask saved in sample.tl2 Fig. E-9

SIGNAL, Cpost input file statement F-4

simulated inductance per unit length of structure presented in Figure 2-8 Fig. 2-24

single plate above ground plane Fig. 5-5

single plate above plane, RC3-BEM 5-4

SINGLE_BAR, RI3 input file statement 6-9

SIZE 8-22

skin effect simulation, RI3 example 6-20

solvers

2D 1-1

RA 2006.03 Index-9


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z3D 1-1RC2 1-1, 1-3RC2-BEM 1-1, 1-4RC3 1-1, 1-4RC3-BEM 1-1, 1-4RI3 1-1, 1-4RIL 1-4

SPHERE 4-11

SPHERE, RC3 input file statement 4-11

SPICE model

RC2 example 2-32substrate resistance using RC3-BEM Fig. 5-14three-conductor system shown in Figure 2-22

without floating conductors Fig. 2-36SPICE netlist generation 7-10

SPICE subcircuit

associated with element 20 of Parametric Library Fig. 7-4

corresponding to structure 18 of parametric library Fig. 7-22

created by Example raexc24 Fig. 2-34SPICE, RC2 input file statement 2-16

start_up file, RIL 7-8

stripline structure with sapphire dielectric for simulation in Example aniso.rc2 Fig. 2-38

structure corresponding to element 20 of Parametric Library Fig. 7-3

structure for Example raexc31 Fig. 4-28

structure for Example raexc32 generated by line 6 of dpraexc32 Fig. 4-30

structure formed by two electrodes parallel to a ground plane Fig. 2-14

structure used to demonstrate inductance simulations Fig. 2-22

substrate resistance, RC3-BEM 5-12

TTABLE 7-7

TABLE, RIL command 7-7

Taurus Layout E-2

Taurus Layout mask E-7

Taurus Visual 1-6

Taurus-Topography 3D E-3

Taurus-Topography input file E-9

Taurus-Topography Interface E-1

TCAD Products and Utilities Installation Manual 1-xii

TEMPERATURE 4-19

TEMPERATURE, RC3 input file statement 4-19

TEXT_OPTIONS 8-29

theoretical background 3-2

theory, RI3 6-1

THERMOCAP, RC3 input file statement 4-20

THERMORES 4-20

Thomson’s theorem C-7

three contacts over two substrates Fig. 5-12

three equal-sized rectangular conductors running parallel to each other with equal spacing Fig. 2-34

three level crossover above ground plane Fig. A-16

three lines, RC2 example 2-24

three parallel bonding wires Fig. A-17

three rectangular vias on planar resistive sheet Fig. 3-13

three rectangular wires immersed in stratified dielectric media Fig. 3-10

three-dimensional solvers 1-1

three-microstrip line structure

Fig. 3-6RC2-BEM example 3-6

tools

DPLOT 1-5Taurus Visual 1-6

top views of mask saved in sample.tl2 Fig. E-9

trace bend above a plane Fig. A-7, A-8

trace narrowing

above a plane Fig. A-11between 2 planes Fig. A-12

trace widening between 2 planes Fig. A-9, A-10

trapezoidal conductor between two ground planes

Fig. 5-10RC3-BEM 5-10

two conformal dielectric layers

metal layer on top of parallel lines Fig. A-28top of parallel lines Fig. A-26

two crossover rectangular conductors above ground plane Fig. 5-8

two level crossover above ground plane Fig. A-15

two magnetic planes facing (boundary condition) B-3

two Pins/Vias with pads

above plane with antipads Fig. A-6above solid plane Fig. A-5antipad holes of plane Fig. A-4

two-dimensional solvers 1-1

two-terminal capacitance C-5

typographical conventions 1-xi

Index-10 RA 2006.03


A B C D E F G H I J K L M N O P Q R S T U V W X Y ZUunequal width/spacing traces

between two ground planes Fig. A-24with one ground plane Fig. A-23without ground plane Fig. A-22

UNIT, Cpost input file statement F-5

utilities, RIL 7-1

Vvariational method C-6

VISUALIZE 7-7

VISUALIZE, RIL command 7-7

VOLTAGE 8-32

WWINDOW, RC2 input file statement 2-12

WINDOW3D 4-17

WINDOW3D, RC3 input file statement 4-17

ZZ0, RC2 input file statement 2-16

RA 2006.03 Index-11

raphael reference manualjmbussat/physics290e/fall-2006/tcad_documentation/raphael_ref.pdfraphaeltm...

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