handbook of ellipsometry (2005)

HANDBOOK OF ELLIPSOMETRY

Edited by

Harland G. TompkinsThin Films Materials Science Consultant

Chandler, Arizona

and

Eugene A. IreneDepartment of Chemistry, University of North Carolina

Chapel Hill, North Carolina

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Copyright 2005 by William Andrew, Inc. No part of this book may be reproduced or utilized in any form or by any means,electronic or mechanical, including photocopying, recording or by any informationstorage and retrieval system, without permission in writing from the Publisher.

Cover art 2005 by Brent Beckley / William Andrew, Inc.ISBN: 0-8155-1499-9 (William Andrew, Inc.)ISBN: 3-540-22293-6 (Springer-Verlag GmbH & Co. KG)Library or Congress Catalog Card Number: Library of Congress Cataloging-in-Publication DataHandbook of ellipsometry / Harland G. Tompkins and Eugene A. Irene (eds.).

p. cm.Includes bibliographical references and index.ISBN 0-8155-1499-91. Ellipsometry. I. Tompkins, Harland G. II. Irene, Eugene A. QC443.H26 2005535.52dc22

2004014676

Printed in the United States of AmericaThis book is printed on acid-free paper.10 9 8 7 6 5 4 3 2 1

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Contributors

Ilsin AnDepartment of Physics and Materials Research InstituteThe Pennsylvania State UniversityUniversity Park, PA 16802 USA

Hans ArwinLaboratory of Applied OpticsDepartment of Physics and Measurement TechnologyLinkping UniversitySE-58183 Linkping, Sweden

Chi ChenDepartment of Physics and Materials Research InstituteThe Pennsylvania State UniversityUniversity Park, PA 16802 USA

Robert W. CollinsProfessor and NEG Endowed Chair of Silicate and Materials ScienceThe University of ToledoDepartment of Physics and AstronomyMail Stop 111Toledo, OH 43606-3390, USA

Andre S. FerlautoMaterials Research LaboratoryThe Pennsylvania State UniversityUniversity Park, PA 16802, USA

James N. Hilfiker J. A. Woollam Co., Inc.645 M Street, Suite 102Lincoln, NE 68508, USA

Josef HumlcekInstitute of Condensed Matter Physics, Faculty of ScienceMasaryk University BrnoKotlrsk 2, CZ-61137 Brno, Czech Republic

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Eugene A. IreneDepartment of ChemistryUniversity of North CarolinaChapel Hill, NC 27599, USA

Gerald E. Jellison, Jr.Solid State DivisionOak Ridge National LaboratoryOak Ridge, TN 37830, USA

Joungchel LeeDepartment of Physics and Materials Research InstituteThe Pennsylvania State UniversityUniversity Park, PA 16802, USA

Frank A. ModineSolid State DivisionOak Ridge National LaboratoryOak Ridge, TN 37831-6030, USA

Arnulf RselerInstitut fr Spektrochemie und Angewandte SpektroskopieD-12489 Berlin, Albert-Einstein-Strasse 9, Germany

Mathias SchubertFakultt fr Physik und Geowissenschaften Institut fr Experimentelle Physik IIUniversitt Leipzig, 04103, Germany

Harland G. TompkinsThin Films Materials Science ConsultantChandler, AZ, 85224, USA

Juan A. ZapienDepartment of Physics and Materials Research InstituteThe Pennsylvania State UniversityUniversity Park, PA 16802, USA

xiv CONTRIBUTORS

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Preface

Light has been used for thickness measurements for a long time.Scientists in the 1700 time frame such as Boyle, Hooke, and Newton,looking at thin transparent layers, observed colors that were the result ofinterference. Brewster1 notes that these observations are of extensive useand may be regarded as presenting us with a micrometer for measuringminute thicknesses of transparent bodies by their colors, when all othermethods would be inapplicable. This phenomenon uses the intensityof the light and is the basis for a current metrology technique calledreflectometry, which excels when measuring reproducible films thathave thicknesses that are a few thousand angstroms thick. In the early1800s, the concept of polarized light was developed. In the late 1800s,Drude used the phase shift induced between mutually perpendicular com-ponents of polarized light to measure film thicknesses down to a few tensand ones of angstroms. When the mutually perpendicular components ofpolarized light are out of phase, the light is said to be elliptically polarized;and hence the technique that evolved from Drudes early measurementscame to be called ellipsometry. With further development of this technique,the lower limit of thickness has been reduced such that submonolayer cov-erage can be measured.

For random scientific measurements to evolve into an analyticaltechnique, there must be a killer application. This came in the 1960swith the transition from making single transistors out of large chunks ofsilicon or germanium to integrated circuits that used planerization methodswith silicon wafers and thin oxide and nitride films. A silicon dioxide filmon a polished silicon wafer is the ideal sample for ellipsometry because ofthe large contrast in the index of refraction of the two materials. Thisapplication area is the subject for Chapter 8. A second important develop-ment that led directly to the elevation of ellipsometry to an importantmodern surface and thin film technique is the availability of digital com-puters, also in the same era of the 1960s. In fact, a modern personal com-puter is adequate for virtually all the typical ellipsometry calculations andsimulations.

For the first three quarters of the 1900s, most ellipsometry was donewith only a single wavelength. During the last quarter of the 1900s, spec-troscopic ellipsometry has evolved significantly, and single-wavelengthellipsometry has now been relegated to tasks such as routine metrologyand fast real-time studies on well-understood samples.

Several international conferences have been held on the subject ofellipsometry, and the technical literature on the topic is voluminous. In

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xvi PREFACE

1977, R. M. A. Azzam and N. M. Bashara authored a book2 titledEllipsometry and Polarized Light, and this work has been the key sourceto be cited in most technical writing on the subject. Ellipsometry andPolarized Light is a scholarly book and not particularly suited to begin-ners. Editor H. G. Tompkins published books in 1993 and 1999 that wereintended for users who were specialists in other fields (e.g., process engi-neering) but who needed to use ellipsometry.

Considerable progress has been made in the field since Ellipsometryand Polarized Light was published, and it appears that a more currentscholarly book is needed. Accordingly, a group of authors from the inter-national scientific community were asked to make contributions to theHandbook of Ellipsometry. This book is divided into four sections.Chapters 1, 2, and 3 explain the theory of ellipsometry. Chapters 4 through7 discuss instrumentation. Chapters 8 and 9 are critical reviews of someapplications in the field. The last three chapters 10, 11, and 12, deal withemerging areas in ellipsometry.

Finally, the editors would like to acknowledge the enormous effort ofthe authors in writing the various chapters; of their students and co-workersfor contributions, reviews, and other assistance; and of the employers andfunding agencies that supported these efforts. The encouragement of theellipsometry scientific community is also appreciated.

1. D. Brewster, Treatise on Optics, Longman, Rees, Orme, Brown, Green, andTaylor, London, (1831) p. 108.

2. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North-Holland Publishing Co., Amsterdam (1977).

H. G. Tompkins and E. A. Irene, November 2004

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Contents

Contributors ..................................................................... xiii

Preface ............................................................................ xv

Part I. Theory of Ellipsometry 1. Polarized Light and Ellipsometry ...................................... 3

1.1 A Quick Guide to Ellipsometry ............................... 4 1.1.1 Light Waves and Photons ......................... 4 1.1.2 Polarization of Light .................................. 6 1.1.3 Ellipsometric Configurations ..................... 9 1.1.4 Null Ellipsometry ....................................... 12 1.1.5 Photometric Ellipsometry and

Polarimetry ................................................ 13 1.2 Maxwell and Wave Equations ................................ 19

1.2.1 Linear Local Response ............................. 20 1.2.2 Linear NonLocal Response ..................... 22 1.2.3 Dipole Moment, Susceptibility and

Inductions .................................................. 23 1.2.4 Relationships between Optical

Constants .................................................. 24 1.2.5 Wave Equation for Monochromatic

Fields ........................................................ 26 1.2.6 Plane Waves in Isotropic Medium ............. 29

1.3 Representations of Polarization ............................. 31 1.3.1 Representation by Ellipsometric Angles ... 32 1.3.2 Special Cases: Linear and Circular

Polarization ............................................... 35

vi Contents

This page has been reformatted by Knovel to provide easier navigation.

1.3.3 Orthogonal Polarization States ................. 37 1.3.4 Representation by Complex Numbers ...... 37 1.3.5 Light Intensity, Detection of Polarization

State .......................................................... 40 1.4 Propagation of Polarized Light ............................... 45

1.4.1 Jones Vectors ........................................... 45 1.4.2 Jones Matrices .......................................... 48 1.4.3 Quantum Mechanical Description, Partial

Polarization ............................................... 53 1.4.4 Stokes Vectors .......................................... 56 1.4.5 Mueller Matrices ........................................ 59

1.5 Reflection and Transmission of Polarized Light at Planar Interfaces .................................................... 67 1.5.1 Matching Plane Waves at a Planar

Interface .................................................... 67 1.5.2 Fresnel Coefficients .................................. 72 1.5.3 Special Values of the Angle of

Incidence ................................................... 74 1.5.4 Ratio of Amplitude Reflectivities ............... 76 1.5.5 Propagation Matrices, Stratified

Structures .................................................. 80 1.5.6 SubstrateFilmAmbient System .............. 85

1.6 References ............................................................. 90 2. Optical Physics of Materials ............................................. 93

2.1 Introduction ............................................................ 93 2.2 Propagation of Light in Solids ................................ 102

2.2.1 Optically Isotropic Solids and the Complex Dielectric Function ..................... 102

2.2.2 Optically Anisotropic Solids and the Dielectric Tensor ....................................... 110

2.2.3 Dispersion Relationships .......................... 124

Contents vii


2.3 Classical Theories of the Optical Properties of Solids ..................................................................... 125 2.3.1 Semiconductors and Insulators: the

Lorentz Oscillator Model ........................... 125 2.3.2 Metals: The Drude Free Electron Model ... 129 2.3.3 Plasmons .................................................. 132 2.3.4 Optical Sum Rules .................................... 136

2.4 Quantum Mechanical Theories of the Optical Properties of Solids ................................................ 137 2.4.1 Quantum Theory of Absorption and

Dispersion ................................................. 138 2.4.2 Direct Interband Transitions in Solids ....... 146 2.4.3 Band Structure and Critical Points in

Solids ........................................................ 150 2.4.4 Indirect Interband Transitions in Solids ..... 153 2.4.5 Intraband Transitions in Metals ................. 157

2.5 Modeling the Optical Properties of Solids .............. 159 2.5.1 Classical Lorentz Oscillator Models .......... 159 2.5.2 Classical Drude Models ............................ 172 2.5.3 Generalized Quantum Mechanical

Models ...................................................... 178 2.5.4 Specialized Quantum Mechanical

Models ...................................................... 207 2.6 Overview and Concluding Remarks ....................... 227

Acknowledgments .................................................. 230 2.7 References and Bibliography ................................. 230

2.7.1 Numbered References .............................. 230 2.7.2 Bibliography .............................................. 233

3. Data Analysis for Spectroscopic Ellipsometry .................. 237 3.1 Introduction ............................................................ 237 3.2 Ellipsometry Parameters ........................................ 239

3.2.1 Calculated Parameters: Jones Matrices ... 240

viii Contents


3.2.2 Measured Parameters: Mueller Matrices .. 241 3.2.3 Mueller-Jones Matrices ............................. 242

3.3 Calculation of Complex Reflection Coefficients ..... 246 3.3.1 Isotropic, Homogeneous Systems ............ 246 3.3.2 Anisotropic Systems ................................. 248 3.3.3 Inhomogeneous Layers ............................ 251

3.4 Models for Dielectric Functions .............................. 252 3.4.1 Tabulated Data Sets ................................. 253 3.4.2 Lorentz Oscillator Model ........................... 254 3.4.3 Optical Functions of Amorphous

Materials ................................................... 255 3.4.4 Models for Crystalline Materials ................ 258 3.4.5 Effective Medium Theories ....................... 260

3.5 Fitting Models to Data ............................................ 262 3.5.1 Figures of Merit ......................................... 263 3.5.2 Errors in Spectroscopic Ellipsometry ........ 265 3.5.3 Convergence Routines ............................. 268 3.5.4 An Example: (a-SixNy:H) ........................... 271

3.6 Determination of Optical Functions from Spectroscopic Ellipsometry Data ........................... 276 3.6.1 Optical Functions from Parameterization .. 278 3.6.2 Newton-Raphson Algorithm ...................... 280 3.6.3 Optical Functions of Bulk Isotropic

Semiconductors and Insulators ................. 282 3.6.4 Optical Functions of Anisotropic

Materials ................................................... 285 3.6.5 Optical Functions of Thin Films ................. 286

3.7 Depolarization ........................................................ 289 Acknowledgements ................................................ 293

3.8 Further Reading and References ........................... 293 Optics and Ellipsometry ......................................... 293

Contents ix


Data Reduction ...................................................... 294 Numbered References ........................................... 294

Part II. Instrumentation 4. Optical Components and the Simple PCSA (Polarizer,

Compensator, Sample, Analyzer) Ellipsometer ................ 299 4.1 General .................................................................. 299 4.2 The Components .................................................... 301

4.2.1 Methods of Obtaining Polarized Light ....... 301 4.2.2 Double Refraction ..................................... 302 4.2.3 Calcite Crystals ......................................... 303 4.2.4 Polarizers and Analyzers .......................... 305 4.2.5 Wollaston Prisms ...................................... 307 4.2.6 Compensators, Quarter-Wave Plates,

and Retarders ........................................... 308 4.2.7 Photoelastic Modulators ............................ 316 4.2.8 Monochromators ....................................... 317 4.2.9 Goniometers ............................................. 321

4.3 Ellipsometer Component Configurations ................ 322 4.3.1 Early Null Ellipsometer Configurations ...... 322 4.3.2 Photometric Ellipsometer Configurations .. 323 4.3.3 Spectroscopic Ellipsometers ..................... 324 4.3.4 Other Configurations ................................. 326

4.4 References ............................................................. 327 5. Rotating Polarizer and Analyzer Ellipsometry .................. 329

5.1 Introduction ............................................................ 329 5.2 Comparison of Ellipsometers ................................. 333 5.3 Instrumentation Issues ........................................... 343

5.3.1 Optical Configuration ................................ 343 5.3.2 Optical Components and Spectral

Range ....................................................... 345 5.3.3 Alignment .................................................. 351

x Contents


5.3.4 Electronic Design and Components .......... 356 5.4 Data Reduction for the Rotating Polarizer and

Analyzer Ellipsometers ........................................... 364 5.4.1 Ideal PXSAr Configuration ......................... 364 5.4.2 Errors in the PXSAr Configuration ............. 371 5.4.3 PrXSA Configuration ................................. 378

5.5 Precision Considerations ....................................... 386 5.6 Calibration Procedures ........................................... 392

5.6.1 Ideal Rotating Polarizer and Analyzer Ellipsometers ............................................ 394

5.6.2 Detecting and Correcting Errors in Calibration ................................................. 407

5.6.3 Detecting and Correcting Compensator Errors ........................................................ 423

5.7 Summary: Recent and Future Directions ............... 425 5.8 References ............................................................. 429

6. Polarization Modulation Ellipsometry ................................ 433 6.1 Introduction ............................................................ 433 6.2 The Photoelastic Modulator (PEM) ........................ 436

6.2.1 General Description and Historical Perspective ............................................... 436

6.2.2 Mathematical Description of a PEM .......... 440 6.2.3 Stokes Vector Descriptions of the PSG

and PSA .................................................... 442 6.3 Experimental Configurations of Polarization

Modulation Ellipsometers ....................................... 446 6.3.1 Polarization Modulation Ellipsometry

(PME) with Analog Data Acquisition ......... 446 6.3.2 Phase Modulated Ellipsometry (PME)

with Digital Data Acquisition ...................... 447 6.3.3 Two-Channel Spectroscopic Polarization

Modulation Ellipsometer (2-C SPME) ....... 449

Contents xi


6.3.4 Two-Modulator Generalized Ellipsometer (2-MGE) .................................................... 450

6.4 Light Intensity through a Polarization Modulation Ellipsometer ........................................................... 452 6.4.1 Mueller Matrices for Various Samples ...... 452 6.4.2 Intensity for a Standard PME .................... 455

6.4.3 Intensity for the 2-Modulator Generalized Ellipsometer (2-MGE) .................................. 457

6.5 Waveform Analysis ................................................ 461 6.5.1 Basis Function .......................................... 463 6.5.2 Phase-Sensitive Detection ........................ 465 6.5.3 Digital Waveform Analysis ........................ 466 6.5.4 Two-Modulator Systems ........................... 467

6.6 Calibration Procedures ........................................... 469 6.6.1 One-Modulator PMEs ............................... 470 6.6.2 Two-Modulator PMEs ............................... 472

6.7 Errors ..................................................................... 474 6.7.1 General Discussion ................................... 474 6.7.2 Systematic Errors of PMEs ....................... 475

6.8 Further Reading and References ........................... 479 6.8.1 Further Reading ........................................ 479 6.8.2 Numbered References .............................. 479

7. Multichannel Ellipsometry ................................................. 481 7.1 Introduction ............................................................ 481 7.2 Overview of Instrumentation .................................. 483

7.2.1 Self-Compensating Designs ..................... 483 7.2.2 Rotating-Element Designs ........................ 487 7.2.3 Phase-Modulation Designs ....................... 491 7.2.4 Design Comparisons ................................. 493 7.2.5 Errors Unique to Multichannel Detection

Systems .................................................... 497 7.3 Rotating-Element Designs ..................................... 502

xii Contents


7.3.1 Rotating Polarizer ..................................... 502 7.3.2 Single Rotating Compensator ................... 523 7.3.3 Dual Rotating Compensator ...................... 546

7.4 Concluding Remarks .............................................. 562 Acknowledgements ................................................ 564

7.5 References ............................................................. 564

Part III. Critical Reviews of Some Applications 8. SiO2 Films ......................................................................... 569

8.1 Introduction ............................................................ 569 8.1.1 Preeminence of SiO2 in Microelectronics:

the Ellipsometry Connection ..................... 569 8.1.2 Electronic Passivation ............................... 570 8.1.3 Properties of SiO2 Films ............................ 571

8.2 Historical Perspective Prior to 1970 .................... 578 8.3 Modern Studies Since 1970 ................................ 585

8.3.1 Thick SiO2 Films ........................................ 585 8.3.2 Thin SiO2 Films ......................................... 599 8.3.3 Recent Results on Ultra Thin SiO2 Films

and the Si-SiO2 Interface ........................... 619 8.4 Conclusions ............................................................ 632

Acknowledgements ................................................ 633 8.5 References ............................................................. 633

9. Theory and Application of Generalized Ellipsometry ........ 637 9.1 Introduction ............................................................ 637 9.2 The Generalized Ellipsometry Concept .................. 638

9.2.1 Comments on Notations in GE ................. 638 9.2.2 The Optical Jones Matrix .......................... 640 9.2.3 The Generalized Ellipsometry

Parameters ............................................... 643 9.2.4 Generalized Ellipsometry Acquisition

Techniques ............................................... 647

Contents xiii


9.3 Theory of Generalized Ellipsometry ....................... 650 9.3.1 Birefringence in Stratified Media ............... 650 9.3.2 4 x 4 Maxwells Equations in Matrix

Form .......................................................... 652 9.3.3 Transmission and Reflection GE .............. 656

9.4 Special Generalized Ellipsometry Solutions ........... 657 9.4.1 Biaxial Films (Symmetrically Dielectric

Materials) .................................................. 657 9.4.2 Bi-Biaxial or Magneto-Optical Films (Non-

Symmetrically Dielectric Materials) ........... 661 9.4.3 Chiral Biaxial Films (Axially Twisted

Symmetrically Dielectric Materials) ........... 663 9.4.4 Isotropic Dielectric Films ........................... 669 9.4.5 Further Solutions: [1 1 1] Superlattice

Ordering in III-V Compounds (CuPt-Ordering) ................................................... 671

9.5 Strategies in Generalized Ellipsometry .................. 675 9.5.1 Data Acquisition Strategies for

Anisotropic Samples ................................. 676 9.5.2 Strategies for Treatment of Sample

Backside Effects ....................................... 679 9.5.3 Model Strategies ....................................... 682

9.6 Generalized Ellipsometry Applications ................... 683 9.6.1 Anisotropic Bulk Materials ......................... 684 9.6.2 Anisotropic Films ....................................... 693

9.7 Conclusions ............................................................ 710 Acknowledgements ................................................ 710

9.8 Further Reading and References ........................... 711 9.8.1 General Reading ....................................... 711 9.8.2 Numbered References .............................. 712

xiv Contents


Part IV. Emerging Areas in Ellipsometry 10. VUV Ellipsometry ............................................................. 721

10.1 Introduction ............................................................ 721 10.2 Historical Review of Short Wavelength

Ellipsometry ............................................................ 722 10.2.1 BESSY Ellipsometer ................................. 722 10.2.2 EUV Ellipsometer ...................................... 724

10.3 VUV Ellipsometry Today ........................................ 726 10.3.1 Current VUV Instrumentation .................... 726

10.4 Importance of VUV Ellipsometry ............................ 732 10.5 Survey of Applications ............................................ 737

10.5.1 Lithography ............................................... 740 10.5.2 Gate Dielectrics ......................................... 748 10.5.3 High-Energy Optical Constants ................. 749

10.6 Future of VUV Ellipsometry .................................... 757 10.7 Acknowledgments .................................................. 757 10.8 References ............................................................. 757

11. Spectroscopic Infrared Ellipsometry ................................. 763 11.1 Experimental Tools ................................................ 763

11.1.1 Two Kinds of Instruments ......................... 763 11.1.2 Optical Equipment for the Infrared-

Ellipsometry .............................................. 768 11.1.3 The Degree of Polarization ....................... 771 11.1.4 Linearity of the Detection System ............. 775 11.1.5 Infrared Synchrotron Radiation ................. 775

11.2 Applications ............................................................ 776 11.2.1 Optics of Absorbing Media ........................ 776 11.2.2 Vibration Modes the Concept of Weak

and Strong Oscillators ............................... 778 11.2.3 Inversion of Infrared Ellipsometric

Measurements .......................................... 781

Contents xv


11.2.4 Anisotropy Features in the Infrared Ellipsometric Spectra ................................ 786

11.3 References ............................................................. 797 12. Ellipsometry in Life Sciences ............................................ 799

Poem and Dedication ....................................................... 799 12.1 Introduction ............................................................ 800 12.2 Historical Background ............................................ 802 12.3 The Interfaces under Study .................................... 802 12.4 From Optics to Biology ........................................... 804

12.4.1 The Unique Possibilities ............................ 804 12.4.2 Verification of Ellipsometric Results .......... 805

12.5 Methodology for Data Evaluation from and to Biologically Related Parameters ........................ 806 12.5.1 A Thin Biolayer on a Flat Ideal

Substrate ................................................... 806 12.5.2 A Thick Biolayer on a Flat Ideal

Substrate ................................................... 817 12.5.3 Adsorption of Biomolecules into Porous

Structures .................................................. 817 12.5.4 Surface Roughness .................................. 819 12.5.5 Use of Dispersion Models ......................... 820 12.5.6 Anisotropy ................................................. 820

12.6 Methodology Experimental .................................. 821 12.6.1 Instrumentation ......................................... 821 12.6.2 Cell Designs .............................................. 822 12.6.3 In situ Considerations for Biological

Interfaces .................................................. 824 12.6.4 Some Model Surfaces ............................... 825 12.6.5 Studies on Real Biological Surfaces ......... 827 12.6.6 Complementary and Independent

Information ................................................ 828 12.6.7 Experimental Design ................................. 828

xvi Contents


12.7 Applications ............................................................ 829 12.7.1 Introduction ............................................... 829 12.7.2 Adsorption of Biomolecules to Model

Surfaces .................................................... 830 12.7.3 Spectroscopy ............................................ 839 12.7.4 Imaging ..................................................... 841 12.7.5 Biological Surfaces ................................... 843 12.7.6 Biosensors Based on Ellipsometric

Readout .................................................... 844 12.7.7 Engineering Applications .......................... 845

12.8 Outlook ................................................................... 846 Acknowledgements ................................................ 847

12.9 References ............................................................. 847

Index ............................................................................... 857

Theory of Ellipsometry

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1 Polarized Light and EllipsometryJosef Humlcek

Institute of Condensed Matter Physics, Faculty of Science Masaryk University Brno, Czech Republic

Polarization is a fundamental property of light. It has been treatedcomprehensively in many texts such as Born and Wolf[1], Brosseau[2],Chipman[3], and Saleh and Teich[4]. In this chapter, we summarize thebasic theoretical knowledge on the properties of polarized light on twolevels of complexity.

First, we collect in Section 1.1 information needed by plug-and-play users of commercial equipment, who are interested mostly in someof the applications of ellipsometry. As the precision and accuracy of state-of-the-art ellipsometers increase, detection of partially polarized lightbecomes increasingly important; we have included a short discussion ofits basics in the quick-guide section.

Second, selected topics are treated in more detail in the following sec-tions. The phenomenological background, ranging from Maxwell equa-tions to plane electromagnetic waves, is covered in Section 1.2. It alsoprovides the link of field amplitudes to measurable quantities (intensitiesof light waves), which is essential in any ellipsometric setup. The micro-scopic origin of optical response is discussed by Collins and Ferlauto inChapter 2. The properties of general elliptic polarization are describedusing several representations in Section 1.3. Each of the representations isfrequently encountered in ellipsometric literature; we discuss their useand provide mutual links for reference. In addition, we explain principalways of measuring the parameters of polarized light in photometric ellip-someters. Photometric setups are dealt with in more detail by Tompkins,Jellison, and Roeseler elsewhere in this volume. Section 1.4 is devoted toa systematic description of the propagation of purely and partially polar-ized light through optical systems. We aim at selected optical componentsand try to explain the ways of predicting the polarization phenomena. Anumber of additional items is discussed by Jellison and Modine inChapters 3 and 6. The closing section of this chapter, 1.5, deals with thesimplest isotropic surfaces and homogeneous and isotropic film systems.We provide a detailed mathematical description and several illustrations ofa typical behavior of polarized light at planar interfaces. The more complexcase of anisotropic materials is treated by Schubert in Chapter 9.

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4 THEORY OF ELLIPSOMETRY

1.1 A Quick Guide to Ellipsometry1.1.1 Light Waves and Photons

Light is a transverse electromagnetic wave with frequencies rangingfrom about 3 1011 to 3 1016 Hz. The corresponding wavelengths in vac-uum are from about 1 mm to 0.01 m. In the range of optical frequencies,as well as below it (microwaves and radio waves), matter behaves as acontinuum. Above the optical range, starting with soft X-rays, the discreteatomic structure of matter becomes important even for the phenomenol-ogy of radiationmatter interaction, since it leads to diffraction.

The light wave in a medium is linked to the induced motion of theelectric charges of electrons and atomic nuclei, constituting macroscopicelectric currents. On the other hand, it is mostly safe to neglect theinduced magnetization, since the motion of magnetic moments of elec-trons and nuclei is too slow to follow the rapid optical oscillations. Thus,the most important quantity describing the light wave is the vector of itselectricfield intensity, E. The existence of preferential directions of theaction that the electric field in the wave exerts on electric charges in mat-ter is the reason for the importance of its polarization. Early experimentswith the optical behavior of calcite crystals led to the discovery of thepolarization of light by Malus in 1808, well before the classical electro-magnetic theory had been established. The spatial and temporal depend-ence of the electric field E(r,t) in a uniform, isotropic medium of thecomplex permittivity e, is described by the wave equation[1,5]32 4 E(r, t) 0, (1.1)where w is the angular frequency of the light wave, and c is the lightvelocity in vacuum. A useful solution of the wave equation is the mono-chromatic plane wave propagating along the zaxis of an orthogonal coor-dinate system,

E(z,t) Re53 4ei(kzzwt)6. (1.2)

Here Ex and Ey are the complex amplitudes of E

along the x and yaxes;they can be arranged conveniently in the 2 1 column vector. The sym-bols Re{f} and Im{f} mean real and imaginary parts of a complex quan-tity f, respectively; for example, the x component of the field intensity isEx(z,t) Re{Ex}cos(kzz wt) Im{Ex}sin(kzz wt). We use the com-plex form of most of the equations, since they are simple and transparent.

ExEy

2t2

e(w)

c2

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POLARIZED LIGHT AND ELLIPSOMETRY, HUMLCEK 5

However, care should be taken in adopting a number of different possibleconventions[6].

In order to satisfy the wave equation, Eq. (1.1), the nonvanishingcomponent of the propagation vector of the plane wave of Eq. (1.2)assumes the values given by the dispersion equation

kz e N, (1.3)

where N n ik e is the complex refractive index of the medium.By choosing the planewave solution, Eq. (1.2), with the time dependence ofexp(iwt), we are adopting the standard physics convention, where theimaginary parts of e and N are positive. Another convention, in which thetwo imaginary parts are negative (usually preferred in optics) is easilyrecognizable, since all expressions containing e and N become complexconjugate.

The intensity of a light wave is its energy crossing a unit area perpen-dicular to the direction of propagation per unit time. It is given by the mag-nitude of the time average of the Poynting vector, E(t) H(t), where His the intensity of the magnetic field. In SI units, the intensity of the planewave of Eq. (1.2) in free space is

I E (t) H(t) (Ex2 Ey2), (1.4)

where e0 is the permittivity of vacuum. The signals measured by detectorsin ellipsometric setups are proportional to the intensity, i.e., to the squaredmodulus of the complex amplitude of the electric intensity. Since ce0 2.654 103 AV, a wave with the electric field amplitude of 1 V/m hasthe intensity of about 1.3 mW/m2. At low light intensities used in ellipso-metric measurements, the response of materials to the light wave is linear.

The understanding of the lightmatter interaction requires, as a rule,quantum mechanical description. A monochromatic wave of the angularfrequency w carries energy in the quanta of hw, where h 1.05457266 1034 Js is the Planck constant. To convert the wavelength l to the photonenergy hw in the practical units of electronvolts, we use the followingrelation,

hw[eV] . (1.5)

The photon energy ranges from about 1 meV in the farinfrared to about100 eV in the vacuum ultraviolet. For example, the green light of a mercurydischarge lamp has the vacuum wavelength of 0.5461 m, corresponding

1.23985l [m]

ce02

wc

wc

Ch_01.qxd 1/1/04 1:31 AM Page 5

to the photon energy of 2.270 eV; a collimated light beam with the inten-sity of 1.3 mW/m2 (mentioned above as having the electric field intensityof 1 V/m in the classical picture) consists of the flux of 3.6 1015 photonsper second per m2. A striking consequence of the quantized nature of thelight wave is the onset of sensitivity of photoconductive or photovoltaicdetectors. For example, siliconbased detectors provide signals only forphoton energies above the bandgap (about 1.1 eV); a light beam of lowerphoton energy is not absorbed, irrespective of its intensity.

The second fingerprint of the quantum nature of a light wave (i.e., theflux of individual photons) is the superposition of possible polarizationstates of the photons, described by the quantummechanical probabilityamplitudes. This is formally identical with coherent superposition of thecomplex amplitudes of plane waves in the classical picture. The advan-tage of the quantum mechanical description becomes obvious especiallyin the treatment of unpolarized or partially polarized light.

1.1.2 Polarization of LightThe most general polarization of a monochromatic light wave, Eq. (1.2),

is elliptic. Since the wave is transverse, the endpoint of the electricfieldintensity vector precesses along an elliptic trajectory in any plane per-pendicular to the direction of propagation. One revolution is achieved inthe very short time interval of 2w. The time evolution can be viewedas a superposition of harmonic vibrations along two perpendicular axes.If the two vibrations are shifted in phase, the resulting trajectory is ellip-tic. An example of the polarization ellipse is shown in Fig. 1.1. Thewave is assumed to propagate along the z axis of the righthanded carte-sian coordinate system xyz. The amplitudes of the electric field in thex and y directions are denoted by X and Y, respectively; both of them arereal, nonnegative quantities. The time dependence of the vector E(t) ofEq. (1.2) in the plane z 0 can be written down conveniently using thefollowing complex form,

E(t) 3 4 Re53 4eiw(tto)6. (1.6)

At the initial time t t0, the y component is at its maximum as indicatedby the dashed arrow in Fig. 1.1; the maximum value of the x componentis reached after the time interval of w (dotted arrow). Note that theangle between the dashed and dotted vectors in Fig. 1.1 is related to therelative phase of the vibrations along x and ydirections. For positive val-ues of , the sense of precession is clockwise, and the polarization is

XeiY

Ex(t)Ey(t)


Ch_01.qxd 1/1/04 1:31 AM Page 6


called righthanded. For negative values of , the polarization ellipse isdescribed counterclockwise; it is called lefthanded polarization. Thevalues of are usually limited to the interval from zero to 2, or from to . The nomenclature regarding the sense of rotation is also notunique[6].

Besides the phase shift , the state of elliptic polarization is deter-mined by the amplitudes X and Y. More precisely, only relative amplitudeXY is relevant in ellipsometric measurements; multiplying both X and Yby a common constant changes merely the light intensity. The relativeamplitude can be expressed with the help of the angle shown in Fig. 1.1;it is given by tany XY, varying from zero to 2. Consequently, theelliptic polarization of Eq. (1.6) can be represented by the Jones vector3 4 , (1.7)which is determined by the two real angles y and . An important notionin dealing with polarization is that of the orthogonality of polarizationstates. Two states are called orthogonal, when their Jones vectors are

sinyeicosy

Figure 1.1 Polarization ellipse, described by the ellipsometric angles y and . Thepolarized wave propagates in the positive sense of the z axis, which pointstowards the reader.

Ch_01.qxd 1/1/04 1:31 AM Page 7

orthogonal in the usual sense of vector algebra. The polarization ortho-gonal to that of Eq. (1.7) therefore is3 4 3 4 . (1.8)Simple special cases of a general elliptic polarization are:

Linear polarization, for which 0 or . This is equivalentto the vibrations along x and y either inphase, or with theopposite phase. The polarization ellipse collapses into thelinear segment connecting the upperright and lowerleftcorners of the rectangle in Fig. 1.1 for 0, or theupperleft and lowerright corners for . The anglebetween this direction and the x axis is called azimuth; itvaries between zero and and defines uniquely the state oflinear polarization. Two linear polarizations with theirazimuths at the right angle are orthogonal. Linearly polar-ized light is produced by linear polarizers; these are devicesthat reject (either absorb or deflect)[7] the unwanted orthog-onal linear polarization. The angular setting of the polarizerdetermines the azimuth of the resulting light polarization.

Circular polarization, for which y 4, and 2 or2; the light is called right or leftcircularly polarized,respectively. The x and ycomponents have the same magni-tude and are 2 off phase. The polarization ellipse becomesa circle, with the two possibilities of the sense of precession.Right and leftcircular polarizations are mutually orthogonal.Circular polarization can easily be produced by a linearpolarizer followed by a retarder. A quarterwave retardershifts the phase of the light polarized linearly along its fastaxis by 2 with respect to the orthogonal linear polarizationalong its slow axis. If a linear polarizer is set to 4 withrespect to the fast axis of a quarterwave retarder, the twocircularly polarized states result.

In the quantummechanical picture, the general state of elliptic polar-ization of Eqs. (1.6) and (1.7) is a superposition of the state of linear polar-ization along x with the complex amplitude sinyexpi, and the state oflinear polarization along y with the complex amplitude cosy. The squaredmoduli of the two amplitudes, sin2y and cos2 y, are the probabilities of thephoton being transmitted through ideal linear polarizers oriented along xand y, respectively. At the same time, it is a superposition of any other

sin(p2 y)ei(p)cos(p2 y)

cosyeisiny


Ch_01.qxd 10/15/04 1:08 PM Page 8


pair of orthogonal polarizations, e.g., the right and leftcircular ones.Such a state is called pure; the predictions of the behavior of a light beamcomposed of identical photons are probabilistic, governed by the laws ofquantum mechanics. The unavoidable fluctuations become observablewhenever the flux is weak enough. An additional level of uncertaintyarises when the polarization state of any photon in a light beam resultsfrom random processes. The best known case is the completely randompolarization of the light emitted from thermal sources; ideally, any possi-ble pure state of polarization is found with the same probability of onehalf. In other words, the intensity of the completely unpolarized lightis halved by any lossless ideal polarizer (linear, circular, or elliptic), irre-spective of the polarizer settings.

A state of partial polarization is called mixed in quantum mechan-ics[8]. It results from a probabilistic preference of a certain pure (elliptic,in general) polarization state. This preference is most naturally quanti-fied by the probability wc of finding the specified pure state labeled c.Thus, instead of the pair (y,) of two real quantities defining any purestate, the mixture is given by the three real numbers (y,,wc). Themixed state becomes pure for wc 1 or wc 0, i.e., when an observa-tion of the polarization leads with certainty to the specific pure state orto the pure state orthogonal to it, respectively. Another convenient meas-ure of the preference towards a certain pure state is the degree of polar-ization Pc, related to the probability wc by

Pc 2wc 1, wc . (1.9)

It is usually introduced as the measure of incoherent addition of classicalwaves. The intensity of a partially polarized beam is decomposed into thefraction Pc of the totally polarized light and the fraction 1 Pc of thetotally unpolarized light[1]. Except for the laser sources, the light enteringan ellipsometer is usually almost totally unpolarized. After passing thefirst polarizer, the degree of polarization is typically fairly high, unlesssome of the optical elements or the measured sample exercise a depolar-izing action. The properties of partial polarization states are discussed inmore detail in Section 1.4.

1.1.3 Ellipsometric ConfigurationsEllipsometric technique is based on a suitable manipulation of the polar-

ization state by auxiliary polarizing elements and measured sample. The basicPCSA configuration of an ellipsometer is shown in Fig. 1.2, consisting of a

1 Pc

2

Ch_01.qxd 1/1/04 1:31 AM Page 9

light source, linear polarizer (P), retarder (called also compensator, C),sample (S), linear polarizer (called analyzer, A), and detector. The arm withthe source, polarizer and retarder prepares a known polarization state oflight incident on the sample. The arm with the analyzer and detector is usedto detect the change of polarization produced by the sample. An alternativePSCA scheme results from moving the compensator between the sampleand analyzer.

A planar sample is assumed in Fig. 1.2, with the angle of incidencedenoted by j. The beams incident on, and reflected from, the sample, lie inthe plane of incidence, which contains also the normal to the sample sur-face. The oblique incidence (j 0) is typical, since it leads to pro-nounced changes of the polarization of incident light upon reflection fromthe sample. The reason is a pronounced difference in the behavior of thetwo basis linear polarizations, parallel to the plane of incidence (ppolar-ization) and perpendicular to it (spolarization; from senkrecht, Germanfor perpendicular). In isotropic materials, currents induced by the lightwave flow in the direction of the electric field. The currents are differentwhen driven along the sample surface or interfaces within a layered sam-ple for spolarization, or partly perpendicular to the surface (ppolariza-tion). For anisotropic samples, measurable changes of polarization statecan occur even at normal incidence, j 0, since the induced currents neednot be parallel to the electric field.

In the scheme of Fig. 1.2, the directions of linearly polarized lighttransmitted by the polarizer and analyzer are defined by the angles P andA, respectively. The azimuth of the fast axis of the compensator is C. All ofthese angles are measured from the local xaxes lying in the plane of inci-dence, with the positive values for the clockwise rotations when looking in


Figure 1.2 Polarizercompensatorsampleanalyzer (PCSA) configuration of anellipsometer.

Ch_01.qxd 1/1/04 1:31 AM Page 10


the positive zdirection (i.e., using the righthand rule). The dependence ofthe field amplitude seen by the detector on the relevant parameters can beeasily calculated by performing the following steps:

(i) Assume the complex amplitude EP of the linearly polarizedwave transmitted by the polarizer. Note that this amplitudewill be independent of the azimuth P for an unpolarized orcircularly polarized wave incident on the polarizer.

(ii) The wave incident on the compensator is a superposition ofthe component aligned along the fast axis, having the com-plex amplitude EPcos(P C), and the orthogonal compo-nent along the slow axis, having the amplitude EPsin(P C).The transmitted wave is modified by multiplying the twocomponents by the complex transmittances tfast and tslow,respectively. Consequently, the field amplitudes transmittedby the compensator along its fast and slow axes are

Efast tfast EPcos(P C), Eslow tslowEPsin(P C). (1.10)(iii) The wave incident on the sample is a superposition of the

component parallel and perpendicular to the plane ofincidence, i.e., p and spolarizations. The correspon-ding complex amplitudes are EfastcosC Eslow sinC, andEfast sinC EslowcosC; the reflected wave is modified bymultiplying the two components by the complex reflectivi-ties rp and rs, respectively. Consequently, the amplitudes ofp and spolarized components reflected from the sample are

Erp rP(Efast cos C Eslow sin C),Ers rs(Efast sin C Eslow cos C). (1.11)

(iv) Finally, the wave transmitted by the analyzer results fromthe addition of incident, p and spolarized components,projected onto the direction of the azimuth A.Substituting the intermediate amplitudes of Eqs. (1.10)and (1.11), the result isEA Erp cosA Erssin A

EP{rp cos A[tfast cos(P C)cosC tslow sin(P C)sinC] rs sin A[tslow sin(P C)cosC tfast cos(P C)sinC]}.

(1.12)According to Eq. (1.4), the light intensity measured by the detector isproportional to |EA|2.

Ch_01.qxd 1/1/04 1:31 AM Page 11

A simpler ellipsometric scheme is the polarizersampleanalyzer(PSA) configuration, resulting by removing the compensator fromPCSA. Note that the compensator is effectively removed from the sys-tem by aligning either of its axes parallel to the azimuth of the polarizer.The field amplitude on the detector results from Eq. (1.12) by insertingtfast tslow 1 as

EA EP(rp cosPcos A rs sinPsin A) EPrs(r cosPcos A sin Psin A). (1.13)

We have denoted by r the complex reflectance ratio,

r tanyei, (1.14)

which is usually expressed in terms of the angles y and defined here.The field amplitudes are usually calculated using Jones vectors and

matrices in properly chosen coordinate systems. For example, the calcu-lation for the PSA configuration can be summarized as follows,

EA3 4 3 4 3 4 3 4 3 4EP. (1.15)Here, the column vector on the righthand side is the normalized Jonesvector of the linearly polarized light transmitted by the polarizer, in thexy coordinate system of the source arm. This vector is multiplied fromthe left by three matrices. The first of them is the (diagonal) Jonesmatrix of the sample, consisting of the amplitude reflectivities for the pand spolarization. The next matrix multiplication rotates the xy coor-dinate system of the detection arm by the angle A. The first coordinateis then aligned with the transmission axis of the analyzer, allowing us touse the Jones matrix of the analyzer in its eigenpolarization system. Therelation of the scalar field amplitudes EA and EP resulting from Eq. (1.15)is, of course, the same as that of Eq. (1.13). The advantages of the matrixdescription of the propagation of polarized light become even moreobvious for imperfect (partially polarizing) optical elements. A conven-ient scheme uses Stokes vectors and matrices.

1.1.4 Null EllipsometryThe PCSA configuration can be used to determine fairly simply the

two ellipsometric parameters of the sample. The procedure consists offinding component settings for extinguishing the light at the detector. Theazimuths of the polarizer, Po, and compensator, Co, are adjusted so as the

cos Psin P

rp 00 rs

cos A sin Asin A cos A

1 00 0

10

rprs


Ch_01.qxd 1/1/04 1:31 AM Page 12


resulting elliptically polarized light is reflected from the sample as lin-early polarized. The reflected beam is then extinguished by a suitablyadjusted analyzer azimuth, Ao. Setting EA 0 in Eq. (1.12), we obtain alinear equation for the complex reflectance ratio. Its solution reads

r tan A0. (1.16)

We have denoted by tc the complex transmittance ratio of the compen-sator,

tc tan ycei c, (1.17)

which can be expressed in terms of the angles yc and c.A significant simplification results for a quarterwave (tc i) com-

pensator set at the azimuth of C 4. Namely, the output intensity ofEq. (1.12) is extinguished for two settings P1, A1 and P2, A2 related by P2 P1, A2 A1. In addition, these azimuths are straightforwardlyrelated to the measured ellipsometric angles

y A1, 2P1 p2, (1.18)

for A1 0. Using this very simple scheme with a laser source, highqualitypolarizing elements and mountings, highprecision and accuracy meas-urements are achievable even with a manual operation of the instrument.However, the operation is slow and tedious. The usual way of calibratingthe instrument consists of performing multiplezone measurements[9],which determines and corrects the unavoidable errors in the zero positionsof P, C and A and the retardation angle of the compensator. Essentialingredients of this procedure are measurements performed at the secondsetting of the compensator at C 4.

1.1.5 Photometric Ellipsometry and PolarimetryPhotometric ellipsometry is based on measurements of intensity for a

number of suitably chosen settings of the optical components influencingthe polarization state of light. Unlike the nulling variant, it is well suitedfor spectroscopic studies in broad spectral intervals, increasing tremen-dously the amount of information obtainable from the measurements.Advances in instrumentation transformed ellipsometry from a cumbersomelaboratory tool to a widespread, efficient technique for both basic research

tslowtfast

tc tan(P0 C0) tanC0tan(P0 C0) tan C0 1

Ch_01.qxd 10/15/04 1:08 PM Page 13

and applications [10,11]. Of course, a very important issue in any photomet-ric system is the sensitivity, linearity, response time, and polarizationdependence of light detectors.

The PSA configuration provides basic possibilities for photometricmeasurements. The light intensities are measured for several properlychosen azimuths of the polarizer and analyzer. Since the analyzed stateof polarization is independent of absolute intensities, one of them canbe used as a reference for the measurement of relative values.Consequently, at least three independent intensities are required todetermine the two real ellipsometric parameters of Eq. (1.14). Equation(1.13) gives, for a fixed azimuth P, the intensity transmitted by theanalyzer as

I(A) I(P)|rs|2cos2 P(tan2y cos2 A tan2Psin2 A 2tany cos tan P cosA sinA). (1.19)

Taking the intensity for A 2 as the reference, the ellipsometric angley is obtained from the relative intensity measured for A 0 as

tany tan PI(0)I(p2). (1.20)

The third intensity can be measured for A 4; used in Eq. (1.19), itprovides the following explicit result for the ellipsometric angle ,

cos sgn(P) [2I(p4) I(0) I(p2)][2I(0)I(p2) ]. (1.21)The obvious limitation here is the impossibility of distinguishing thesign of , which is confined to an interval of the length of . Theazimuth P of the polarizer is the disposable quantity in this scheme; itshould be chosen properly to minimize measurement errors. Equation(1.20) suggests the favorable setting of P y, for which I(0) I(2).In any case, the values of P close to either zero or 2 should beavoided, since they lead to a rapid loss of sensitivity. It should beemphasized that a possible polarization sensitivity of the detector hasbeen neglected; if present, it can be measured and accounted for using asample with known parameters, or straightthrough operation with nosample.

The number of measured intensities has to be increased when themeasurement should also correct the results for unknown zero positions Poand Ao of the polarizer and analyzer, respectively. The two values aredefined as the readings Pr and Ar of the angular scales for the polarizationexactly parallel to the plane of incidence, i.e., for the zero actual values Pand A in Pr P Po and Ar A Ao. Alternatively, Po and Ao represent


Ch_01.qxd 1/1/04 1:31 AM Page 14


minus the actual azimuths for zero readings of the two scales. Note that atilt of the sample in Fig. 1.2 along its axis lying in the plane of incidencechanges the zero positions. The three intensities of Eqs. (1.20) and (1.21)are actually equal to I(Ao), I(Ao 4) and I(Ao 2). Averagingof the results obtained for the intensities measured with the readings P Po and (P Po) at the polarizer scale provides the correction to thefirst order in small Po and Ao. In addition, it specifies the zero positions ofboth polarizer and analyzer.

The PSA scheme with the fixed polarizer is usually operated with alarger number of intensity measurements at different analyzer azimuths.The most common way is to use equidistantly spaced points covering ahalfrotation (rotatinganalyzer-ellipsometer, RAE)[12]. The intensity ofEq. (1.19) can be rewritten as

I(A) (1 a cos 2A b sin2A), (1.22)

where

a , b . (1.23)

The a and b are cosine and sine Fourier coefficients of I(2A), normal-ized to the constant background. They result from the discrete Fouriertransform of the measured intensities. Once computed, they are used inthe following relations equivalent, in principle, with Eqs. (1.20) and(1.21)

tan y tan P11

aa

, cos sgn(P) . (1.24)

The use of a larger set of analyzer positions reduces noise and allows thedetection of spurious effects, such as nonlinearity of detection system.Note that both Fourier coefficients vanish for P y and 2;these are the conditions for the circularly polarized light in front of theanalyzer.

The photometric data can also be collected for a fixed position ofthe analyzer and several azimuths of the polarizer (rotatingpolarizer-ellipsometry, RPE). This scheme suppresses possible polarization sensi-tivity of the detector. However, the source has to be either unpolarized orcircularly polarized for a simple treatment identical to the RAE caseabove; note that the amplitude of Eq. (1.13) is symmetric with respect to

b1 a2

2tan P tany cos

tan2y tan2 Ptan2y tan2Ptan2y tan2P

I(P)|rs|2 cos2P2(tan2y tan2P)

Ch_01.qxd 1/1/04 1:31 AM Page 15

the azimuths P and A for the field EP independent of P. The RPE modeshould be preferred to RAE for a weaker polarization of the source com-pared with the polarization sensitivity of the detector.

Although the PSA scheme is very efficient in many circumstances,the inclusion of a compensator in the PCSA or PSCA configurationsoffers substantial advantages. First, the polarization dependence of bothsource and detector can be suppressed by fixing P and A; the necessarynumber of intensity readings can be obtained by changing the azimuthand/or phase shift of the compensator. Second, the ellipsometric phaseshift can be measured with improved precision and accuracy, in theinterval from zero to 2. Third, partial polarization can be detected andquantified.

Assume a sample illuminated by a linearly polarized light from anideal polarizer at the azimuth P in the PSA scheme. With no depolariza-tion effects, the reflected light is in a pure elliptic polarization state cshowing up in the Fourier coefficients of Eqs. (1.23). A partly polarizedstate results when the state c appears in the reflected beam with the prob-ability wc 1, and the orthogonal pure state c with the complementaryprobability of 1 wc. This can occur due to either imperfections of thesample, such as a spread of its complex reflectance ratio across the illu-minated surface, or the imperfections of the incoming beam other than itspolarization state, such as a spread of the angles of incidence or wave-lengths. The total flux behind the analyzer at the azimuth A is composedof two parts,

I(A) wc(1 acos2A bsin2A) (1 wc)(1 acos2A bsin2A). (1.25)

Here a and b are the Fourier coefficients that would be observed if thereflected light had the polarization , i.e., for wc 0; it can be easily veri-fied that a a and b b. Consequently, the measured intensity

I(A) 1 (2wc 1)a cos2A (2wc 1)bsin2A 1 aa~ cos2A b~sin2A (1.26)

is of the same form as for the pure elliptic polarization in Eq. (1.22).However, the cosine and sine Fourier coefficients are reduced in magni-tude by multiplying with the degree of polarization Pc 2wc 1, seeEq. (1.9),

a Pc , b Pc . (1.27)2 tanP tany cos

tan2 y tan2Ptan2y tan2Ptan2y tan2P


Ch_01.qxd 10/15/04 1:08 PM Page 16


Both Fourier coefficients vanish for the totally unpolarized lightreflected from the sample, when wc 12, Pc 0. A rotating ideal linearpolarizer cannot distinguish between unpolarized and circularly polar-ized light.

The two real Fourier coefficients obtained from the intensities meas-ured in the PSA configuration do not allow obtaining the three real param-eters of a partial polarization. Using a compensator in the PCSA or PSCAconfiguration can solve the problem. A simple possibility consists ofadding the measurements of the Fourier coefficients with rotating ana-lyzer for the axis of the compensator parallel or perpendicular to the planeof incidence. Using the notation of Eqs. (1.14) and (1.17), the intensitiescalculated above are modified by simply taking the product

rtc tany tantc ei(c) (1.28)in place of the complex reflectance ratio r of the sample. Accordingly, theFourier coefficients

ac Pc , bc Pc

(1.29)are measured with the properly aligned compensator, in addition to thoseof Eqs. (1.27). With given values of the relative attenuation of the com-pensator, tan yc, and its phase shift, c, Eqs. (1.27) and (1.29) can besolved for the parameters of the sample, y and , and the degree of polar-ization, Pc. Techniques used to measure partially polarized light are some-times called polarimetry.

The desired quantities are very simply extracted for an ideal quarter-wave retarder (tanyc 1, c 2). Since Eqs. (1.27) and Eqs. (1.29)simplify as

a ac Pc , b Pc ,

bc Pc ,

(1.30)

we obtain the following explicit results for the sought quantities:

cos sgn(P) , sin sgn(P) , tan ,(1.31)

bcb

b cb 2 bc2

bb 2 bc2

2tanP tany sin

tan2y tan2P

2tanP tany cos

tan2 y tan2 Ptan2y tan2Ptan2y tan2P

2tan P tany tan2 yc cos( c)

tan2y tan2yc tan2 Ptan2y tan2 yc tan2Ptan2y tan2 yc tan2P

Ch_01.qxd 1/1/04 1:31 AM Page 17

tany |tan P| , (1.32)

Pc . (1.33)

Note that Eqs. (1.31) determine the angle in the zero to 2 range, andimprove the precision for the values close to zero or . It should be empha-sized that the degree of polarization depends on both the sample (whichprovides the incoherent sum of polarization states upon reflection), and thepolarizer azimuth P. For P 0 or 2 the reflected light is pure, linearlypolarized p or s wave, respectively, for any isotropic sample; the sineFourier coefficients vanish, b bC 0; the cosine coefficients are unity,a aC 1, resulting in the unit degree of polarization, Pc 1. Of course,these two eigendirections are unusable for measuring the sample proper-ties. At the preferable setting of |P| y, we have a aC 0, leading to

Pc . (1.34)

In this case, we can easily follow the consequences of limiting the meas-urement to the part without the quarterwave retarder, and assuming nodepolarization:

the reported value of y obtained from Eq. (1.24) is correct,y |P|, since the cosine Fourier coefficient a is zero;

the reported value r obtained from Eq. (1.24) is in error,since the sine Fourier coefficient b is modified by the degreeof polarization in the second of Eqs. (1.30); its relation to thetrue value of is given by

cos r Pc cos; (1.35)

the decrease of Pc from unity to zero is misinterpreted as thetendency towards circular polarization, p2.

A finite spectral resolution and/or an imperfectly collimated beam canproduce depolarization effects for perfect samples, since their responseto the polarization state of the light beam depends on the photon energyand angle of incidence. On the other hand, a nonuniformity in composi-tion across the illuminated sample surface is a potential source of depo-larization effects, since it can lead to the incoherent addition of reflected

b 2 b c2b 2 bc2

a 2 b 2 b c2 a a2 b 2 bc2

aa 2 b 2 bc2

a a2 b2 bc2b 2 bc2


Ch_01.qxd 1/1/04 1:31 AM Page 18


waves for an incident ideal monochromatic plane wave. Another obvi-ous source of depolarization is a spread of film thicknesses in layeredsamples. As the precision of the ellipsometric technique increases, itbecomes necessary to treat properly the possible partial polarization(see, e.g., the recent work[13]). We deal with partially polarized light inmore detail in Section 1.4.

1.2 Maxwell and Wave EquationsPhenomenological description of the optical behavior of any material

involves a few characteristics, called response functions. They are intro-duced conveniently within the framework of Maxwell equations andmaterial relations; the latter are sometimes called constitutive relations.

The macroscopic electromagnetic field is described by the intensitiesE

and H

of the electric and magnetic components, respectively. In anonmagnetic medium, they obey Maxwells equations (in SI units)

H

e0 j

,

EE

m0 , (1.36)

E

,

H

0, (1.37)

satisfied at any position r and time t. Here r and j are the macroscopiccharge and current densities; they are usually divided into external andinduced parts, r rext rind, j

jext j

ind. In Eqs. (1.36) and (1.37), e0is permittivity of vacuum, m0 is permeability of vacuum:

e0 Fm1, m0 4p107 Hm1, e0m0 , (1.38)

where c is the velocity of light in vacuum.The macroscopic picture involves mean values of quantities varying

rather strongly on the atomic scale. For example, we are dealing with thespatial average of the microscopic electric field intensity, e, over somemacroscopic volume V:

EE(r,t)

V

e(r,t) dr. (1.39)

V is centered around r; it has to be large enough to smooth out the micro-scopic variations of e, but small compared to the wavelength of light. Forcrystals, the volume for averaging is just one unit cell.

1V

1c2

1074pc2

re0

HH

t

E

t

Ch_01.qxd 1/1/04 1:31 AM Page 19

1.2.1 Linear Local ResponseWe assume no external charges and currents, i.e., the densities in Eqs.

(1.36) and (1.37) are due to the electromagnetic field itself. Moreover, themacroscopic average of the charge density rind vanishes in transverse fields.Consequently, the only relevant material property is that relating the cur-rent density, j jind, to the field intensities. We shall assume a lineardependence, which is valid for weak fields. This is a direct generalizationof the simplest material equations for quasistationary fields, which are usu-ally written down separately for conductors and insulators, respectively:

j(r,t) s0 E(r,t), (1.40)

j(r,t) (e0 1) e0 . (1.41)

In the case of a conductor containing free charge carriers, s0 in Eq. (1.40)is the real tensor of static (i.e., dc) conductivity. The case of an insula-tor with no free charges is slightly more complex. The induced currentdensity of Eq. (1.41) is due to the motion of bound charges, vanishing inthe dc limit. This dielectric displacement is proportional to the rate of tem-poral change of EE

; e0 in Eq. (1.41) is the real tensor of (dimensionless)

static permittivity. The bound charges in a conductor also contribute to theinduced current; it is impossible to distinguish it from the contribution offree carriers on the macroscopic level at optical frequencies.

The needed generalization of the material equations (1.40) and (1.41)retains their linearity. Moreover, for most substances, their local charactercan also be preserved. The substantial change, however, consists in replac-ing the synchronous response by the weaker requirement of causality theinduced current depends on the electric field only before the time of obser-vation. The linear and local material equation generalizing Eq. (1.40) istherefore

j(r,t) t

f(t t) E(r,t)dt 0

f(t) E(r,t t)dt. (1.42)

Consequently, the optical response of a homogeneous material isdescribed by a real tensor function f(t) with the dimension of conductiv-ity/time, defined for t 0. It represents the current density produced by afunction pulse of the electric field. The usual approximation of optics isthe monochromatic field of circular frequency w,

E

w(r,t) E

w(rr)exp(iwt). (1.43)

EE(r,t)

t


Ch_01.qxd 10/15/04 1:08 PM Page 20


Inserting it into Eq. (1.42), we obtain the sought generalizations of Eqs.(1.40) and (1.41) for conductor and insulator, respectively:

jw(r,t) s0

f(t) exp(iwt)dttEw (r,t), (1.44)jw (r,t) s

0f(t) exp(iwt)dtteo . (1.45)

For the monochromatic field, these material equations are of the sameform as Eqs. (1.40) and (1.41), with the complex, frequencydependentconductivity tensor

s (w) 0

f(t) exp(iwt)dt, (1.46)

and the complex, frequencydependent tensor of permittivity (usuallycalled dielectric function, or sometimes also dielectric constant)

e(w) 1 0

f(t) exp(iwt)dt 1 s(w). (1.47)

The dependence of the conductivity and permittivity on frequency iscalled dispersion.

The real and imaginary parts of s(w) are the cosine and sine Fouriertransforms of the real tensor function f(t), respectively; the real and imag-inary parts of e(w) are very simply related to them. The following sym-metry property

e(w) e*(w), s(w) s*(w), (1.48)

where the asterisk denotes the complex conjugation, follows directly fromthe definitions of Eqs. (1.46) and (1.47). We can use interchangeably eachof s(w) and e(w) to characterize the linear response. They differ, however,rather substantially in the behavior at low frequencies. For a conductor,the complex conductivity reaches the real dc value of s0 at w 0, whilethe complex permittivity diverges as is0(we0) for w 0. For aninsulator, the complex permittivity reaches the real static limit of e0 atw 0, while the complex conductivity goes to zero as iwe0(e0 1) forw 0. Thus, the static dielectric behavior is described by the slope of thefrequency dependence of s(w).

iweo

iweo

EE

w (r,t)

ti

weo

Ch_01.qxd 1/1/04 1:31 AM Page 21

1.2.2 Linear NonLocal ResponseIn a further generalization of the starting material equations (1.40) and

(1.41), the assumption of the induced current density, j(r), to be propor-tional to the field strength at the same point r is relaxed. This case shouldoccur when the average path traveled by the charge carriers is not negli-gible with respect to the spatial variations of the optical field. In additionto the harmonic timedependence of Eq. (1.43), we assume also the har-monic spatial distribution of the electric field intensity:

E

wk(r,t) EE

wk ,0 exp(iwt ik

r). (1.49)

For a real wavevector k

of the magnitude k, the field pattern is repeated atthe distance of the wavelength, l 2k. When the average path of thecharge carriers is not small compared with l, the induced current is influ-enced by the fields at different locations. In analogy with Eq. (1.42), thelinear response becomes

j(r,t) dtdr f(t, r)E(r rr, t t). (1.50)The spacetime extent of the integration has to warrant both causality andfinite speed of the electromagnetic interaction. When inserting the mate-rial equation (1.50) into Eq. (1.49) we obtain

jwk(rr,t) dtdrf (t, r) exp(iwt ik rr) Ewk( r,t). (1.51)Thus, we arrive at the w and k

dependent complex tensor of

conductivity

s(w,k) dtd rf(t,r) exp(iwt ik r). (1.52)By a similar reasoning as in the previous subsection, we obtain also thew and k

dependent complex tensor of permittivity, e(w,k). The

dependence of conductivity and permittivity on the wavevector iscalled spatial dispersion. The local response is recovered when thefunction f (t,r) in Eq. (1.50) becomes factorized and its spatialdependence is like, f (t,r) f(t t) d(r r). This is equiva-lent to taking the k

0 values of s and e, which is also called

longwavelength limit. The nonnegligible spatial dispersion occursrather rarely; the usual way of reporting response function includes


Ch_01.qxd 1/1/04 1:31 AM Page 22


only their frequency dependence. However, the spectacular phenomenonof optical activity (i.e., of rotating the direction of polarization of a lin-early polarized wave traversing the material) can be linked to thenonlocal nature of the optical response[17].

1.2.3 Dipole Moment, Susceptibility and Inductions

Macroscopic polarization P

is the spatial average of the induced dipolemoment per unit volume. The linear response of Eq. (1.42) or (1.50) impliesa linear dependence of P

on the macroscopic field strength; with the har-

monic temporal and spatial dependence exp(iwt ik rr) of P and E, thisrelation is

P

wk c (w,k) e0 E

wk. (1.53)

The dimensionless tensor function c (w,k) is called electric susceptibility.The electric induction, D

, is also a linear function of the intensity E

,

D

e0E

P

(1 c)e0E

ee0 E

, (1.54)

where e 1 c is the (relative) permittivity. The polarization andinduced current are proportional in monochromatic fields,

P

wk ij

wkw, j

wk iwP

wk. (1.55)

Macroscopic magnetization M

is the spatial average of the induced mag-netic moment per unit volume. The assumption of nonmagnetic materialsmeans vanishing M

, and magnetic induction preserves its vacuum value,

B

m0H

M

m0H

.

For monochromatic transverse fields in the longwavelength limit,Maxwell equations (1.36) and (1.37) and the material equation (1.47)combine into the following set

B

w (r) iw E

w (r),

E

w (r) iwB

w (r), (1.56)

E

w (r) 0,

B

w (r) 0, (1.57)

providing the spatial dependence of the field quantities in a homogeneousmaterial of the dielectric function e.

e(w)

c2

Ch_01.qxd 1/1/04 1:31 AM Page 23

1.2.4 Relationships Between Optical ConstantsIt is customary to introduce the complex refractive index N as the square

root of the (scalar) dielectric function, i.e., e N2. The relationshipsbetween optical constants valid at any frequency w are listed in Table 1-1,using the most common notation for their real (s1, e1, n) and imaginary(s2, e2, k) parts. We have added the line with negative inverse of thedielectric function; its imaginary part is called loss function, since it isrelated to the energy loss of a longitudinal wave in the material. It can bedirectly measured using the electron energy loss spectroscopy (EELS),and compared with the data obtained from optical measurements.

Causality of the linear material equations implies the validity of anumber of so called KramersKronig relations. They can be derivedrigorously from the analytic behavior of s(w) and its functions, like e(w),N(w), and others, in the upper halfplane of the complex values of .Using w ig, g 0, in Eq. (1.46), the complex conductivitybecomes

s() s(w,g) 0

f(t) exp(iwt) exp(g t)dt. (1.58)

This complexvalued function has obviously no poles in the upperhalfplane of for any physically reasonable response function f, being


Optical Constant(symbol) Real part Imaginary partconductivity(s s1 is2) s1 we0e2 s2 we0(e1 1)dielectric function e1 1 s2(we0) e2 s1(we0)(e e1 ie2)

e1 n2 k2 e2 2nk

refractive index n (e1 e21 e22)2 k (e1 e21 e22)2(N n ik)

n e2(2k) k e2(2n)negative inverse of dielectric function e1(e21 e22) e2(e21 e22)(e1)

Table 1-1. Relationship between Optical Constants.

Ch_01.qxd 1/1/04 1:31 AM Page 24


limited and continuous including its derivative. Moreover, s()( w)decays fast enough for || to ensure the vanishing value of the con-tour integral over the semicircle at infinity. This is essentially due to thedominance of inertial effects at high frequencies over the restoring and/ordissipative forces[15]. Consequently, the use of the Cauchy integral for-mula for a suitable closed contour in the complex plane[16] leads to thefollowing integral relations linking the real and imaginary parts of thecomplex conductivity:

s1(w)

d 0

d, (1.59)

s2(w)

d 0

d, (1.60)

Since the integrands diverge for w , the Cauchy principal value has tobe taken. This is indicated by the integral sign with a line through it.

The equivalent integral relation for the dielectric function is

e1(w) 1 0

d, e2(w) 0

d.

(1.61)

Here s0 is the dc conductivity, which vanishes for insulators. For therefractive index, the KramersKronig relations become

n(w) 1 0

d, k(w) 0

d. (1.62)

Note that there is no pole for metals at the origin w 0 in the refractiveindex. Instead, the lowfrequency behavior is described by a squarerootsingularity

N(w) (1 i) . (1.63)The KramersKronig relations are linked with a number of sum rules.

Using the first of Eqs. (1.61) for w gives the asymptotic behavior

limw

e1(w) 1 , (1.64)w2pw2

1w

so2eo

isoweo

n() 12 w2

2w

pk()2 w2

2p

e1() 12 w2

2wp

s0we0

e2()2 w2

2p

s1()2 w2

2w

ps1() w

1

p

s2()2 w2

2p

s2() w

1p

Ch_01.qxd 1/1/04 1:31 AM Page 25

where

w2p 0

e2 ()d 0

s 1()d. (1.65)

Here wp denotes the plasma frequency 4pNe2m, where N is the totalelectron density, e and m are electron charge and mass, respectively (in CGSunits). Equation (1.65) is the wellknown f sum rule. A necessary conditionfor the integrals in Eqs. (1.65) to exist is that e2 and s1 fall off faster thanw2 and w1, respectively. Consequently, the highfrequency limit of thesecond of Eqs. (1.61) gives the asymptotic behavior

limw

e2(w) 0

[e1() 1]d , (1.66)

where

s0 0

[e1() 1]d, (1.67)

since the term proportional to w1 has to vanish. For any insulator (i.e., fors0 0), the deviation of e1 from its vacuum value of 1 averaged over allfrequencies is zero. This is a sum rule concerning dispersive processes,accompanying the f sum rule of Eq. (1.65) for absorptive processes.Equation (1.67) can also be taken to be a direct consequence of theKramersKronig relation for s1, Eq. (1.59), since s0 s1(0) ands2() e0[e1() 1]. The basic sum rules involving the complexrefractive index are

w2p 0

k()d, 0

[n() 1]d 0, (1.68)

as obtained from the highfrequency expression N(w) 1 w2p(2w2) .

1.2.5 Wave Equation for Monochromatic FieldsFor a monochromatic optical field of angular frequency w, the spatial

distribution of the intensities, E

and H

, can be found from the wave equa-tion, supplemented by boundary conditions. Maxwell equations (1.56) and(1.57), valid in the longwavelength limit, can be transformed into a singledifferential equation of the second order in the following way. Applying the

4p

2e0p

2pw

s0we0

2peo

2p


Ch_01.qxd 1/1/04 1:31 AM Page 26


curl operation to the second of Eqs. (1.56) we find

[ Ew(r)]

[ Ew(r)] (

)Ew(r) iw

B

w(r).(1.69)

Since the divergence of E

w vanishes, and the righthand side of Eq. (1.69)is given by the first of Eqs. (1.56), we have the wave equation for the elec-tric field intensity:

3 e(w)4Ew(r) 0. (1.70)Here is the Laplace operator,

2x2 2y2 2z2.

Elementary solutions of the wave equations can be sought in the formof the plane waves

E

w (r) E

0 exp(ik

r) E0 exp[i(kxx kyy kzz)], (1.71)

and a general solution as a superposition of these plane waves. The ampli-tude E

0 has cartesian components E0x, E0y, and E0z. The wave equation (1.70)transforms into a set of three homogeneous linear equations for the threecomplex components of the amplitude,

3k k e(w)4 s t 0. (1.72)In terms of the cartesian components of the tensor e, the wave equationcan be written as

s t s t 0, (1.73)where

D (kx2 ky2 kz2). (1.74)

A symmetric tensor e is assumed; we do not include possible opticalactivity of the material, which can be described phenomenologically byintroducing an antisymmetric imaginary part of the matrix e [17]. The sym-metry of the dielectric tensor of a crystal with no optical activity results

c2w2

E0xE0yE0z

exx D exy exzexy eyy D eyzexz eyz ezz D

E0xE0yE0z

w2c2

w2c2

Ch_01.qxd 1/1/04 1:31 AM Page 27

from energyflow considerations [18]. A nonzero solution of Eqs. (1.73)exists if the determinant of the matrix of the system vanishes.

Consider, without loss of generality, a wave of Eq. (1.71) propagat-ing in the z direction, i.e., kx ky 0. Since k

E

0 0, the wave ispolarized in the xy plane, E0z 0. The wave equation (1.73) reduces tothe following pair of homogeneous linear equations for the nonzeroamplitudes,

s t 0. (1.75)Nonzero solutions exist for the vanishing determinant of its matrix,which is a quadratic equation for kz2 with the roots

k2z(1,2) . (1.76)

Thus, two linearly polarized waves can propagate along an arbitrary direc-tion, with the magnitude of their wavevectors given by the last equation.This leads to the phenomenon of birefringence in anisotropic materials. Thedirections of the linear polarizations can easily be found from Eq. (1.75).

When the two roots of Eq. (1.76) coincide, the wave equation(1.75) is satisfied by any amplitudes E0x and E0y. Consequently, wavesof any polarization can propagate along the zdirection, which is calledoptical axis. The multiple root of (1.76) occurs for exx eyy, exy 0 inour coordinate system. This has a simple geometrical interpretation interms of the ellipsoid associated with the tensor e,

[x y z] s t s t exxx2 eyyy2 ezzz2 2exyxy 2exzxz 2eyzyz 1.

(1.77)

Namely, its crosssection perpendicular to z is circular for exx eyy, exy 0. A light wave polarized in the xy plane sees the same dielectricfunction irrespective of the direction of its vibrations.

In general, either two, one, or any, direction of propagation has theproperty of optical axis for a given (symmetric) dielectric tensor e. Thisleads to the following classification of the optical anisotropy:

xyz

exx exy exzexy eyy eyzexz eyz ezz

exx eyy (exx eyy)2 4e2xy

2w2c2

E0xE0y

exx wc2

2kz2 exy

exy eyy wc2

2kz2


Ch_01.qxd 1/1/04 1:31 AM Page 28


BiaxialTwo optical axes exist for the dielectric tensor with mutuallydifferent values along the principal axes of the ellipsoid ofEq. (1.77). Occurs for crystals of triclinic, monoclinic, andorthorhombic structure. The directions of optical axes mayvary with frequency w.

UniaxialA single optical axis exists for the dielectric tensor havingrotational symmetry of its ellipsoid of Eq. (1.77). Occurs forcrystals of rhombohedral, tetragonal, and hexagonal structure.The optical axis coincides with the highsymmetry axis of thecrystal.

IsotropicThe ellipsoid of Eq. (1.77) is a sphere. all of the directions ofpropagation are equivalent. Occurs for cubic crystals or amor-phous mat

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