SpectroscopicEllipsometryPrinciples and Applications

Hiroyuki FujiwaraNational Institute of Advanced Industrial Science and Technology, Ibaraki, Japan

Spectroscopic Ellipsometry

SpectroscopicEllipsometryPrinciples and Applications

Hiroyuki FujiwaraNational Institute of Advanced Industrial Science and Technology, Ibaraki, Japan

Japanese Edition, Copyright 2003, Hiroyuki Fujiwara, ISBN 4 621 07253 6Published by Maruzen Co. Ltd, Tokyo, Japan

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Library of Congress Cataloging in Publication Data

Fujiwara, Hiroyuki.Spectroscopic ellipsometry : principles and applications / Hiroyuki Fujiwara.

p. cm.Includes bibliographical references and index.ISBN-13: 978-0-470-01608-4 (cloth : alk. paper)ISBN-10: 0-470-01608-6 (cloth : alk. paper)1. Ellipsometry. 2. Spectrum analysis. 3. Materials—Optical properties. I. Title.QC443.F85 2007620.1′1295—dc22

2006030741

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 978-0-470-01608-4

Typeset in 10/12pt Times by Integra Software Services Pvt. Ltd, Pondicherry, IndiaPrinted and bound in Great Britain by TJ International, Padstow, CornwallThis book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two treesare planted for each one used for paper production.

Dedicated to my father, Sadao Fujiwara

Contents

Foreword xiii

Preface xv

Acknowledgments xvii

1 Introduction to Spectroscopic Ellipsometry 11.1 Features of Spectroscopic Ellipsometry 11.2 Applications of Spectroscopic Ellipsometry 31.3 Data Analysis 51.4 History of Development 71.5 Future Prospects 9References 10

2 Principles of Optics 132.1 Propagation of Light 13

2.1.1 Propagation of One-Dimensional Waves 132.1.2 Electromagnetic Waves 182.1.3 Refractive Index 19

2.2 Dielectrics 242.2.1 Dielectric Polarization 242.2.2 Dielectric Constant 252.2.3 Dielectric Function 29

2.3 Reflection and Transmission of Light 322.3.1 Refraction of Light 322.3.2 p- and s-Polarized Light Waves 332.3.3 Reflectance and Transmittance 392.3.4 Brewster Angle 402.3.5 Total Reflection 42

2.4 Optical Interference 432.4.1 Optical Interference in Thin Films 432.4.2 Multilayers 46

References 48

3 Polarization of Light 493.1 Representation of Polarized Light 49

3.1.1 Phase of Light 493.1.2 Polarization States of Light Waves 50

viii Spectroscopic Ellipsometry

3.2 Optical Elements 523.2.1 Polarizer (Analyzer) 533.2.2 Compensator (Retarder) 573.2.3 Photoelastic Modulator 583.2.4 Depolarizer 59

3.3 Jones Matrix 603.3.1 Jones Vector 603.3.2 Transformation of Coordinate Systems 623.3.3 Jones Matrices of Optical Elements 663.3.4 Representation of Optical Measurement by Jones

Matrices 683.4 Stokes Parameters 70

3.4.1 Definition of Stokes Parameters 703.4.2 Poincaré Sphere 723.4.3 Partially Polarized Light 753.4.4 Mueller Matrix 77

References 78

4 Principles of Spectroscopic Ellipsometry 814.1 Principles of Ellipsometry Measurement 81

4.1.1 Measured Values in Ellipsometry 814.1.2 Coordinate System in Ellipsometry 844.1.3 Jones and Mueller Matrices of Samples 86

4.2 Ellipsometry Measurement 874.2.1 Measurement Methods of Ellipsometry 874.2.2 Rotating-Analyzer Ellipsometry (RAE) 934.2.3 Rotating-Analyzer Ellipsometry with Compensator 974.2.4 Rotating-Compensator Ellipsometry (RCE) 994.2.5 Phase-Modulation Ellipsometry (PME) 1044.2.6 Infrared Spectroscopic Ellipsometry 1064.2.7 Mueller Matrix Ellipsometry 1114.2.8 Null Ellipsometry and Imaging Ellipsometry 113

4.3 Instrumentation for Ellipsometry 1174.3.1 Installation of Ellipsometry System 1174.3.2 Fourier Analysis 1204.3.3 Calibration of Optical Elements 1224.3.4 Correction of Measurement Errors 127

4.4 Precision and Error of Measurement 1304.4.1 Variation of Precision and Error with Measurement

Method 1314.4.2 Precision of ����� 1354.4.3 Precision of Film Thickness and Absorption Coefficient 1374.4.4 Depolarization Effect of Samples 139

References 141

Contents ix

5 Data Analysis 1475.1 Interpretation of ����� 147

5.1.1 Variations of ����� with Optical Constants 1475.1.2 Variations of ����� in Transparent Films 1505.1.3 Variations of ����� in Absorbing Films 155

5.2 Dielectric Function Models 1585.2.1 Lorentz Model 1605.2.2 Interpretation of the Lorentz Model 1625.2.3 Sellmeier and Cauchy Models 1705.2.4 Tauc–Lorentz Model 1705.2.5 Drude Model 1735.2.6 Kramers–Kronig Relations 176

5.3 Effective Medium Approximation 1775.3.1 Effective Medium Theories 1775.3.2 Modeling of Surface Roughness 1815.3.3 Limitations of Effective Medium Theories 184

5.4 Optical Models 1875.4.1 Construction of Optical Models 1875.4.2 Pseudo-Dielectric Function 1895.4.3 Optimization of Sample Structures 1915.4.4 Optical Models for Depolarizing Samples 191

5.5 Data Analysis Procedure 1965.5.1 Linear Regression Analysis 1965.5.2 Fitting Error Function 1995.5.3 Mathematical Inversion 200

References 204

6 Ellipsometry of Anisotropic Materials 2096.1 Reflection and Transmission of Light by Anisotropic Materials 209

6.1.1 Light Propagation in Anisotropic Media 2096.1.2 Index Ellipsoid 2136.1.3 Dielectric Tensor 2156.1.4 Jones Matrix of Anisotropic Samples 217

6.2 Fresnel Equations for Anisotropic Materials 2226.2.1 Anisotropic Substrate 2226.2.2 Anisotropic Thin Film on Isotropic Substrate 224

6.3 4×4 Matrix Method 2266.3.1 Principles of the 4×4 Matrix Method 2266.3.2 Calculation Method of Partial Transfer Matrix 2326.3.3 Calculation Methods of Incident and Exit Matrices 2336.3.4 Calculation Procedure of the 4×4 Matrix Method 236

6.4 Interpretation of ����� for Anisotropic Materials 2376.4.1 Variations of ����� in Anisotropic Substrates 2376.4.2 Variations of ����� in Anisotropic Thin Films 241

x Spectroscopic Ellipsometry

6.5 Measurement and Data Analysis of Anisotropic Materials 2436.5.1 Measurement Methods 2436.5.2 Data Analysis Methods 245

References 246

7 Data Analysis Examples 2497.1 Insulators 249

7.1.1 Analysis Examples 2497.1.2 Advanced Analysis 252

7.2 Semiconductors 2567.2.1 Optical Transitions in Semiconductors 2567.2.2 Modeling of Dielectric Functions 2587.2.3 Analysis Examples 2627.2.4 Analysis of Dielectric Functions 268

7.3 Metals/Semiconductors 2767.3.1 Dielectric Function of Metals 2767.3.2 Analysis of Free-Carrier Absorption 2817.3.3 Advanced Analysis 286

7.4 Organic Materials/Biomaterials 2877.4.1 Analysis of Organic Materials 2877.4.2 Analysis of Biomaterials 292

7.5 Anisotropic Materials 2947.5.1 Analysis of Anisotropic Insulators 2957.5.2 Analysis of Anisotropic Semiconductors 2967.5.3 Analysis of Anisotropic Organic Materials 299

References 303

8 Real-Time Monitoring by Spectroscopic Ellipsometry 3118.1 Data Analysis in Real-Time Monitoring 311

8.1.1 Procedures for Real-Time Data Analysis 3128.1.2 Linear Regression Analysis (LRA) 3138.1.3 Global Error Minimization (GEM) 3178.1.4 Virtual Substrate Approximation (VSA) 323

8.2 Observation of Thin-Film Growth by Real-Time Monitoring 3288.2.1 Analysis Examples 3288.2.2 Advanced Analysis 331

8.3 Process Control by Real-Time Monitoring 3338.3.1 Data Analysis in Process Control 3348.3.2 Process Control by Linear Regression Analysis (LRA) 3348.3.3 Process Control by Virtual Substrate Approximation

(VSA) 340References 342

Contents xi

Appendices

1 Trigonometric Functions 345

2 Definitions of Optical Constants 347

3 Maxwell’s Equations for Conductors 349

4 Jones–Mueller Matrix Conversion 353

5 Kramers–Kronig Relations 357

Index 361

Foreword

It is a pleasure and an honor to comment on this outstanding book, SpectroscopicEllipsometry: Principles and Applications by Dr H. Fujiwara. It is a tutorialintroduction, yet offers considerable depth into advanced topics such as generalizedellipsometry and advanced dispersion and oscillator models for analysis of complexmaterials systems. Each chapter is extremely well referenced, with over 400literature citations in total, providing the reader rapid access to considerablepublished literature from fundamentals to recent advances. It is also well illustrated,with over 200 figures, making this an excellent possible textbook for teachingellipsometry at both the beginning and intermediate to advanced levels. The bookwill be appropriate as a text in an educational institution. Equally it will be excellentto help educate and train researchers in institutes and industrial laboratories to learnpractical applications of the technique.

For decades the book, Ellipsometry and Polarized Light, by R. M. A. Azzamand N. M. Bashara, (North-Holland, New York, 1977), has probably been the mostwidely cited general reference on ellipsometry. However, this book is now 30 yearsold, and out of print. Fujiwara-san’s book offers the reader a modern, up-to-date,clear discussion of many of the same topics: fundamentals of optics, polarization,ellipsometry and instrumentation, in the first few chapters. This follows naturallyinto more advanced and well referenced chapters on data analysis, anisotropy,experimental examples, and in situ ellipsometry.

A perspective of the role of Fujiwara-san’s book in the context of existingliterature on ellipsometry might be helpful. Often cited references on ellipsometryare:

• Infrared Spectroscopic Ellipsometry, by A. Röseler, (Akademie-Verlag, Berlin1990);

• Selected Papers on Ellipsometry, R. M. A. Azzam, Ed., SPIE Milestone Series,MS 27, (SPIE, Bellingham 1990);

• R. Muller, Ellipsometry as an in situ probe for the study of electrode processes,in Techniques of Characterization of Electrodes and Electrochemical Processes,R. Varma and J. Selman, Eds, (John Wiley & Sons, Inc., New York, 1991);

• H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry andReflectometry: A User’s Guide, (John Wiley & Sons, Inc., New York, 1999).

There are also recent books for specialists in ellipsometry. These include M.Schubert, Infrared Ellipsometry on Semiconductor Layer Structures: Phonons,Plasmons and Polaritons (Springer, Berlin, 2004); and Handbook of Ellipsometry,

xiv Spectroscopic Ellipsometry

H. G. Tompkins and E. A. Irene, Eds, (Andrew, Norwich, 2005). One can also findcontributed papers and brief reviews in proceedings of the International Conferenceson Spectroscopic Ellipsometry. However, Fujiwara-san’s book covers the topic inunique and valuable ways, subsequently allowing advanced literature, such as foundin the conference proceedings and books named above, to be comfortably read andunderstood.

Dr Fujiwara’s Spectroscopic Ellipsometry: Principles and Applications offers awelcome new contribution as both a tutorial text and an introduction to advancedtopics and applications. This book will become a ‘must have’ for every new user anduniversity student, as well as a specialist wishing for a greater depth of understandingof this technique. It also contains complete and up-to-date references to a wealth ofpublished information on spectroscopic ellipsometry and its applications.

John A. WoollamJ.A. Woollam Company (Founder and President)

University of Nebraska (George Holmes University Professor)10 July 2006

Preface

Historically, the development of a new measurement technique implies advance inscience. Unless clear scientific evidence is presented, scientific facts are often treatedas merely experimental knowledge. In other words, advances in scientific fields canbe viewed as a consequence of various measurements used to confirm scientificsignificance. In the last 50 years alone, a variety of characterization techniques havebeen established, and some scientific fields owe their progress to the innovation ofsuch measurement techniques. The development of scanning tunneling microscope(STM), for example, has revolutionized surface science and contributed greatly tothe rapid progress of surface science.

Ironically, basic principles of ellipsometry were established more than 100 yearsago, but ellipsometry had been perceived as an ‘unproductive instrument’ untilrecently. During the 1990s, however, this situation changed drastically due torapid advances in computer technology that allowed the automation of ellipsometryinstruments as well as ellipsometry data analyses. With the commercialization ofsuch spectroscopic ellipsometry instruments in the mid-1990s, the ellipsometrytechnique became quite popular, and now is applied to wide research areasfrom semiconductors to organic materials. Recent developments in spectroscopicellipsometry have further allowed the real-time characterization of film growthand evaluation of optical anisotropy. Consequently, spectroscopic ellipsometry hasestablished its position as a high precision optical characterization technique, andmore researchers in universities and companies have started using this technique.Nevertheless, principles of ellipsometry are often said to be difficult, partly dueto a lack of proper knowledge of polarized light used as a probe in ellipsometry.Besides, the meaning of ����� obtained from ellipsometry measurements is notstraightforward, and procedures of ellipsometry data analysis are rather unique.

The key objective of this book is to provide a fundamental understanding forspectroscopic ellipsometry particularly for researchers who are not familiar with theellipsometry technique. Although some aspects are complicated, the understandingof the ellipsometry technique is not essentially difficult, if one comprehends theprinciples in order. Based on this point of view, this book provides generaldescriptions for measurement and data analysis methods employed widely inspectroscopic ellipsometry. Since ellipsometry is quite a geometrical measurementmethod, various illustrations are included to help readers. To simplify descriptions,unnecessary equations for electromagnetics and quantum mechanics have beeneliminated. Instead, the derivations of important formulae used in spectroscopicellipsometry are shown in this book.

xvi Spectroscopic Ellipsometry

In order to comprehend spectroscopic ellipsometry, however, a fundamentalknowledge of optics is required. In the book, therefore, ‘Principles of optics’ and‘Polarization of light’ are described in Chapter 2 and Chapter 3, respectively. Fromthese two chapters, ‘Principles of spectroscopic ellipsometry’ (Chapter 4) can beunderstood more easily. We focus on data analysis of spectroscopic ellipsometryin Chapters 5–8. In particular, principles and physical backgrounds of ellipsometryanalysis are discussed in detail in Chapter 5. Since there is growing interestin optical anisotropy, the data analysis of anisotropic materials is explained inChapter 6. In ‘Data analysis examples’ (Chapter 7), examples of ellipsometryanalyses for various materials used in different fields are described. In ‘Real-time monitoring by spectroscopic ellipsometry’ (Chapter 8), the applications ofspectroscopic ellipsometry for growth monitoring and feedback control of processingare addressed.

Most of the content in this book is a translation from the Japanese bookSpectroscopic Ellipsometry, published in 2003 by Maruzen. In this English edition,the overall content is expanded and the description for anisotropic materials(Chapter 6) has been added. For the English edition, I am especially grateful toProf. John A. Woollam (University of Nebraska, Lincoln and J. A. Woollam Co.)who kindly reviewed this book and gave me very thoughtful comments. The authorgratefully acknowledges Prof. Isamu Shimizu for his continued support. I wouldalso like to thank Dr Michio Kondo (AIST), Dr Akihisa Matsuda (AIST), and Prof.Christopher R. Wronski (Pennsylvania State University) for their kind advice. I amgrateful to Mr Michio Suzuki (J. A. Woollam Co., Japan), Mr Teruaki Kuwahara(Maruzen) and Miss Jenny Cossham (John Wiley & Sons, Ltd) who have supportedthe publication of this book. Finally, I wish to express my sincere gratitude to Prof.Robert W. Collins (University of Toledo) who has taught me everything concerningreal-time spectroscopic ellipsometry.

Hiroyuki Fujiwara

Acknowledgments

The author wishes to thank the authors and publishers for permission to reproducethe following figures and tables used in this book:

Fig. 1.4 and Table 1.3, K. Vedam, Thin Solid Films, 313–314 (1998) 1. Fig. 1.5,I. An, Y. M. Li, H. V. Nguyen, and R. W. Collins, Rev. Sci. Instrum., 63 (1992) 3842.Fig. 1.6, D. E. Aspnes, Thin Solid Films, 455–456 (2004) 3. Fig. 2.12, S. Adachi,J. Appl. Phys., 53 (1982) 8775. Fig. 3.9(a), S. N. Jasperson and S. E. Schnatterly,Rev. Sci. Instrum., 40 (1969) 761. Fig. 3.9(b), J. C. Canit and J. Badoz, Appl.Opt., 22 (1983) 592. Fig. 4.11, W. M. Duncan and S. A. Henck, Appl. Surf. Sci.,63 (1993) 9. Fig. 4.13, A. Röseler, Thin Solid Films, 234 (1993) 307. Fig. 4.14,P. S. Hauge, Surf. Sci., 96 (1980) 108. Figs. 4.16(a) and 4.16(c), G. Jin, R. Jansson,and H. Arwin, Rev. Sci. Instrum., 67 (1996) 2930. Fig. 4.22(a), D. E. Aspnes,J. Opt. Soc. Am., 64 (1974) 812. Fig. 4.23, B. Johs, Thin Solid Films, 234 (1993)395. Fig. 4.24, J. Lee, P. I. Rovira, I. An, and R. W. Collins, J. Opt. Soc. Am. A,18 (2001) 1980. Fig. 4.26, S. Kawabata, OYO BUTURI, 57 (1988) 1868. Fig. 4.27,J. Lee, P. I. Rovira, I. An, and R. W. Collins, Rev. Sci. Instrum., 69 (1998) 1800.Fig. 4.28, D. E. Aspnes and A. A. Studna, Appl. Opt., 14 (1975) 220. Fig. 4.29,R. W. Collins and K. Vedam, ‘Optical properties of solids,’ in Encyclopedia ofApplied Physics, Vol. 12, Wiley-VCH (1995) 285. Fig. 5.6(b), R. J. Archer andG. W. Gobeli, J. Phys. Chem. Solids, 26 (1965) 343. Figs. 5.15(b), 5.19 and 5.20(a),K. Kobayashi, Physics of Light: Why Light Refracts, Reflects and Transmits, inJapanese, Tokyo University Publisher (2002). Fig. 5.21(b) G. E. Jellison, Jr andF. A. Modine, Appl. Phys. Lett., 69 (1996) 371. Fig. 5.28, H. Fujiwara, J. Koh,P. I. Rovira, and R. W. Collins, Phys. Rev. B, 61 (2000) 10832. Fig. 5.30, J. Koh,Y. Lu, C. R. Wronski, Y. Kuang, R. W. Collins, T. T. Tsong, and Y. E. Strausser,Appl. Phys. Lett., 69 (1996) 1297. Fig. 5.31(b), R. H. Muller and J. C. Farmer, Surf.Sci., 135 (1983) 521. Fig. 5.37, M. Kildemo, R. Ossikovski, and M. Stchakovsky,Thin Solid Films, 313–314 (1998) 108. Fig. 5.38, G. E. Jellison, Jr andJ. W. McCamy, Appl. Phys. Lett., 61 (1992) 512. Figs. 5.43 and 7.17, D. E. Aspnes,A. A. Studna, and E. Kinsbron, Phys. Rev. B, 29 (1984) 768. Fig. 7.5,C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, and W. Paulson,J. Appl. Phys., 83 (1998) 3323. Fig. 7.7(a), J. R. Chelikowsky, and M. L. Cohen,Phys. Rev. B, 14 (1976) 556. Fig. 7.7(b), U. Schmid, N. E. Christensen, andM. Cardona, Phys. Rev. B, 41 (1990) 5919. Figs. 7.10(b) and 7.11, T. Yang, S.Goto, M. Kawata, K. Uchida, A. Niwa, and J. Gotoh, Jpn. J. Appl. Phys., 37 (1998)L1105-1108. Fig. 7.12(b), P. Petrik, M. Fried, T. Lohner, R. Berger, L. P. Bíro, C.Schneider, J. Gyulai, H. Ryssel, Thin Solid Films, 313–314 (1998) 259. Figs. 7.13(a),

xviii Spectroscopic Ellipsometry

7.14 and Table 7.3, F. L. Terry, Jr, J. Appl. Phys., 70 (1991) 409. Fig. 7.13(b),D. E. Aspnes, S. M. Kelso, R. A. Logan and R. Bhat, J. Appl. Phys., 60 (1986)754. Figs. 7.15 and 7.16, P. Lautenschlager, M. Garriga, L. Viña, and M. Cardona,Phys. Rev. B, 36 (1987) 4821. Fig. 7.18(a), C. Pickering and R. T. Carline, J. Appl.Phys., 75 (1994) 4642. Fig. 7.18(b), R. T. Carline, C. Pickering, D. J. Robbins,W. Y. Leong, A. D. Pitt, and A. G. Cullis, Appl. Phys. Lett., 64 (1994) 1114.Fig. 7.19(b), S. Boultadakis, S. Logothetidis, S. Ves, and J. Kircher, J. Appl. Phys.,73 (1993) 914. Fig. 7.20(a), H. Ehrenreich and H. R. Philipp, Phys. Rev., 128 (1962)1622. Fig. 7.20(b), Frederick Wooten, Optical Properties of Solids, Academic Press(1972). Figs. 7.22(b), 7.23, 7.24 and 7.25, H. Fujiwara and M. Kondo, Phys. Rev.B, 71 (2005) 075109. Fig. 7.26, T. E. Tiwald, D. W. Thompson, J. A. Woollam,W. Paulson, and R. Hance, Thin Solid Films, 313–314 (1998) 661. Fig. 7.27,Y.-T. Kim, D. L. Allara, R. W. Collins, K. Vedam, Thin Solid Films, 193/194(1990) 350. Fig. 7.28, D. Tsankov, K. Hinrichs, A. Röseler, and E. H. Korte,Phys. Stat. Sol. A, 188 (2001) 1319. Fig. 7.29, K. Postava, T. Yamaguchi, andM. Horie, Appl. Phys. Lett., 79 (2001) 2231. Fig. 7.30(b), A. C. Zeppenfeld,S. L. Fiddler, W. K. Ham, B. J. Klopfenstein, and C. J. Page, J. Am. Chem. Soc.,116 (1994) 9158. Fig. 7.31, H. Arwin, Thin Solid Films, 313–314 (1998) 764.Fig. 7.32, U. Jönsson, M. Malmqvist, and I. Rönnberg, J. Colloid. Interface Sci., 103(1985) 360. Fig. 7.33, D. E. Gray, S. C. Case-Green, T. S. Fell, P. J. Dobson, andE. M. Southern, Langmuir, 13 (1997) 2833. Figs. 7.34(b) and 7.35, J. Humlícek,and A. Röseler, Thin Solid Films, 234 (1993) 332. Figs. 7.37 and 7.38, M. Schubert,B. Rheinländer, J. A. Woollam, B. Johs and C. M. Herzinger, J. Opt. Soc. Am. A,13 (1996) 875. Fig. 7.39, T. Wagner, J. N. Hilfiker, T. E. Tiwald, C. L. Bungay,and S. Zollner, Phys. Stat. Sol. A, 188 (2001) 1553-1562. Figs. 7.41 and 7.42,C. M. Ramsdale and N. C. Greenham, Adv. Mater., 14 (2002) 212. Fig. 8.2,H. Z. Massoud, J. D. Plummer, and E. A. Irene, J. Electrochem. Soc., 132 (1985)2685. Fig. 8.4, M. Wakagi, H. Fujiwara, and R. W. Collins, Thin Solid Films, 313–314 (1998) 464. Fig. 8.10, H. Fujiwara, Y. Toyoshima, M. Kondo, and A. Matsuda,Phys. Rev. B, 60 (1999) 13598. Fig. 8.14, H. Fujiwara, J. Koh, C. R. Wronski, andR. W. Collins, Appl. Phys. Lett., 70 (1997) 2150. Fig. 8.15, H. Fujiwara, J. Koh,C. R. Wronski, R. W. Collins, and J. S. Burnham, Appl. Phys. Lett., 72 (1998) 2993.Fig. 8.16, Y. M. Li, I. An, H. V. Nguyen, C. R. Wronski, and R. W. Collins, J. Non-Cryst. Solids, 137&138 (1991) 787. Figs. 8.17 and 8.18, H. Fujiwara, M. Kondo,and A. Matsuda, Phys. Rev. B, 63 (2001) 115306. Figs. 8.19 and 8.20, H. Fujiwara,M. Kondo, and A. Matsuda, J. Appl. Phys., 93 (2003) 2400. Fig. 8.21, E. A. Irene,Thin Solid Films, 233 (1993) 96. Fig. 8.22, H. Fujiwara, and M. Kondo, Appl. Phys.Lett., 86 (2005) 032112. Figs. 8.23, 8.24, and 8.25, H. L. Maynard, N. Layadi,J. T. C. Lee, Thin Solid Films, 313–314 (1998) 398. Fig. 8.26, D. E. Aspnes,W. E. Quinn, M. C. Tamargo, M. A. A. Pudensi, S. A. Schwarz, M. J. S. P. Brasil,R. E. Nahory, and S. Gregory, Appl. Phys. Lett., 60 (1992) 1244. Fig. 8.27,B. Johs, D. Doerr, S. Pittal, I. B. Bhat, and S. Dakshinamurthy, Thin Solid Films,233 (1993) 293.

1 Introduction to SpectroscopicEllipsometry

Because of recent advances in computer technology, the spectroscopic ellipsometrytechnique has developed rapidly. As a result, the application area of spectroscopicellipsometry has also expanded drastically. In spectroscopic ellipsometry, processdiagnosis including thin-film growth can be performed in real time by employinglight as a measurement probe. More recently, ‘feedback control,’ in whichcomplicated device structure is controlled in real time, has been carried out usingspectroscopic ellipsometry. In this chapter, we review the features and applicationsof spectroscopic ellipsometry. This chapter will provide an overview of measurementtechniques and data analysis procedures in spectroscopic ellipsometry.

1.1 FEATURES OF SPECTROSCOPIC ELLIPSOMETRY

Ellipsometry is an optical measurement technique that characterizes light reflection(or transmission) from samples [1–4]. The key feature of ellipsometry is thatit measures the change in polarized light upon light reflection on a sample (orlight transmission by a sample). The name ‘ellipsometry’ comes from the factthat polarized light often becomes ‘elliptical’ upon light reflection. As shownin Table 1.1, ellipsometry measures the two values �����. These represent theamplitude ratio � and phase difference � between light waves known as p- ands-polarized light waves (see Fig. 4.1). In spectroscopic ellipsometry, ����� spectraare measured by changing the wavelength of light. In general, the spectroscopicellipsometry measurement is carried out in the ultraviolet /visible region, butmeasurement in the infrared region has also been performed widely.

The application area of spectroscopic ellipsometry is quite wide (Chapter 7). Forreal-time monitoring, not only characterization of thin-film growth but also processdiagnoses including etching and thermal oxidation can be performed (Chapter 8). Inparticular, spectroscopic ellipsometry allows characterization of thin films formed in

Spectroscopic Ellipsometry: Principles and Applications H. Fujiwara© 2007 John Wiley & Sons, Ltd

2 Spectroscopic Ellipsometry

Table 1.1 Features of spectroscopic ellipsometry

Measurement probe: LightMeasurement value: �����

Amplitude ratio � and phase difference � between p- and s-polarizedlight waves

Measurement region: Mainly in the infrared–visible/ultraviolet regionApplication area:

Semiconductor Substrates, thin films, gate dielectrics, lithography filmsChemistry Polymer films, self-assembled monolayers, proteins, DNADisplay TFT films, transparent conductive oxides, organic LEDOptical coating High and low dielectrics for anti-reflection coatingData storage Phase change media for CD and DVD, magneto-optic layers

Real-time monitoring: Chemical vapor deposition (CVD), molecular beam epitaxy (MBE),etching, oxidation, thermal annealing, liquid phase processing etc.

General restrictions: i) Surface roughness of samples has to be smallii) Measurement has to be performed at oblique incidence

solution (Section 7.4), because light is employed as the probe. However, there aretwo general restrictions on the ellipsometry measurement; specifically: (1) surfaceroughness of samples has to be rather small, and (2) the measurement must beperformed at oblique incidence. When light scattering by surface roughness reducesthe reflected light intensity severely, the ellipsometry measurement becomes difficultas ellipsometry determines a polarization state from its light intensity. If the sizeof surface roughness exceeds ∼30 % of a measurement wavelength, measurementerrors generally increase, although this effect depends completely on the type ofinstrument (Section 4.4).

In ellipsometry, an incidence angle is chosen so that the sensitivity for themeasurement is maximized. The choice of the incidence angle, however, variesaccording to the optical constants of samples. For semiconductor characterization,the incidence angle is typically 70–80� (Section 2.3.4). It should be noted that, atnormal incidence, the ellipsometry measurement becomes impossible, since p- ands-polarizations cannot be distinguished anymore at this angle (Section 2.3.2). Oneexception is the characterization of in-plane optical anisotropy. In this case, theellipsometry measurement is often performed at normal incidence to determine thevariation of optical constants with the rotation of a sample (Chapter 6).

Table 1.2 summarizes the advantages and disadvantages of the spectroscopicellipsometry technique. One of the remarkable features of spectroscopic ellipsometryis the high precision of the measurement, and very high thickness sensitivity�∼0�1 Å� can be obtained even for conventional instruments (Section 4.4.3).As we will see in the next section, spectroscopic ellipsometry allows variouscharacterizations including optical constants and thin-film structures. Moreover, asthe ellipsometry measurement takes only a few seconds, real-time observation andfeedback control of processing can be performed relatively easily (Chapter 8).

The one inherent drawback of the ellipsometry technique is the indirect natureof this characterization method. Specifically, ellipsometry data analysis requires an

Introduction to Spectroscopic Ellipsometry 3

Table 1.2 Advantages and disadvantages of spectroscopic ellipsometry

Advantages: High precision (thickness sensitivity: ∼0�1 Å)Nondestructive measurementFast measurementWide application areaVarious characterizations including optical constants and film thicknesses arepossibleReal-time monitoring (feedback control) is possible

Disadvantages: Necessity of an optical model in data analysis (indirect characterization)Data analysis tends to be complicatedLow spatial resolution (spot size: several mm)Difficulty in the characterization of low absorption coefficients �� < 100 cm−1�

optical model defined by the optical constants and layer thicknesses of a sample(see Fig. 5.39). In an extreme case, one has to construct an optical model evenwhen the sample structure is not clear at all. In addition, this ellipsometry analysisusing an optical model tends to become complicated, which can be consideredas another disadvantage of the technique. The spot size of a light beam used forspectroscopic ellipsometry is typically several millimeters, leading to the low spatialresolution of the measurement. However, it is possible to determine the surface arearatio of different materials that cover the sample surface (see Fig. 5.31). Recently,in order to improve spatial resolution, imaging ellipsometry has been developed(Section 4.2.8). As shown in Table 1.2, in ellipsometry, characterization of smallabsorption coefficients ��<100 cm−1� is rather difficult (Section 4.4.3).

1.2 APPLICATIONS OF SPECTROSCOPIC ELLIPSOMETRY

Spectroscopic ellipsometry has been applied to evaluate optical constants andthin-film thicknesses of samples. However, the application area of spectroscopicellipsometry has been expanded recently, as it allows process diagnosis on the atomicscale from real-time observation. Figure 1.1 shows various physical propertiesthat can be determined from spectroscopic ellipsometry. In particular, this figuresummarizes the characterization by ex situ measurement. Here, ex situ measurementmeans a measurement performed after finishing sample preparation (processing).

As shown in Fig. 1.1, spectroscopic ellipsometry measures ����� spectra forphoton energy h or wavelength . In general, the interpretation of measurementresults is rather difficult from the absolute values of �����. Thus, construction ofan optical model is required for data analysis. From this data analysis, physicalproperties including the optical constants and film thicknesses of the sample canbe extracted. Unlike reflectance/transmittance measurement, ellipsometry allowsthe direct measurement of the refractive index n and extinction coefficient k,which are also referred to as optical constants. From the two values �n� k�,the complex refractive index defined by N ≡ n − ik�i = √−1� is determined.

4 Spectroscopic Ellipsometry

Measuredvalues:

ψ (hν)Δ (hν)

Constructionof opticalmodel

Optical constants:

Complex refractive index n(hν), k(hν): N = n – ikComplex dielectric constant ε1(hν), ε2(hν): ε = ε1 – iε2Absorption coefficient α = 4πk/λ

Reflectance: RTransmittance: T

Film thickness(structure):Surface roughness layerBulk layerMultilayer

UV/visibleregion

Infraredregion

Band structure:Bandgap (Eg) Direct transition: α = A(hv – Eg)1/2

Indirect transition: α = A(hv–Eg)2

Surface temperature (ºC)Alloy composition (at.%)Phase structure(crystal, amorphous, void)Grain size (Å)

Free-carrier absorption:Carrier concentration (cm–3)Carrier mobility (cm2⁄ Vs)Conductivity (S/cm)Infrared absorption:LO and TO phononsLocal structures (Si–H, –OH)

Figure 1.1 Characterization of physical properties by spectroscopic ellipsometry.

The complex dielectric constant � and absorption coefficient � can also be obtainedfrom the simple relations expressed by � = N 2 and � = 4�k/, respectively(Chapter 2). Moreover, from optical constants and film thicknesses obtained, thereflectance R and transmittance T at a different angle of incidence can be calculated.

From the measurements in the ultraviolet /visible region, interband transitions(band structures) are characterized. In particular, the bandgap Eg can be deducedfrom the variation of � with h (Section 7.2.1). Since band structure generallyvaries according to surface temperature, alloy composition, phase structure, andcrystal grain size, these properties can also be determined from the spectral analysisof optical constants (Section 7.2.4). In the infrared region, on the other hand,there exists free carrier absorption induced by free electrons (or holes) in solids.When carrier concentration is high enough �>1018 cm−3�, electrical propertiesincluding carrier mobility, carrier concentration, and conductivity can be obtained(Section 7.3.2). Moreover, in the infrared region, lattice vibration modes (LO andTO phonons) as well as local atomic structures, such as Si–H and –OH, can alsobe studied (Sections 7.5.1 and 7.4).

In real-time spectroscopic ellipsometry, ����� spectra are measured continuouslyduring processing. This technique further allows a number of characterizationsillustrated in Fig. 1.2 (Chapter 8). From real-time monitoring, for example, initialgrowth processes or interface structures can be studied in detail (Section 8.2). Ina compositionally modulated layer in which alloy composition varies continuouslyin the growth direction, the alloy compositions of each layer are determined. Inparticular, the real-time measurement enables us to characterize reaction rate during

Introduction to Spectroscopic Ellipsometry 5

ψ (hν,t) Δ (hν,t)

Measuredvalues: Construction

of opticalmodel

Thin-film structure:

Volmer–Weber

Substrate

Substrate Substrate

Stranski–Krastanov

Initial growthprocess

Interfacestructure

Interface

Bulk layer AxB1–x layer

Substrate

Compositionallymodulated layer Reaction rate

time t

Start processing

A(t)

Optical constants:n(hν,t), k(hν,t)ε1(hν,t), ε2(hν,t)

Real-time structural control(Feedback control)

Figure 1.2 Characterization of thin film structures by real-time spectroscopic ellipsometry.

processing. Real-time spectroscopic ellipsometry can be applied further to performprocess control. From real-time observation, the feedback control of semiconductoralloy composition has already been performed (Section 8.3.3). Accordingly, theability of spectroscopic ellipsometry has opened up a new way for more advancedprocess control.

1.3 DATA ANALYSIS

Figure 1.3 shows (a) optical model consisting of an air/thin film/substrate structureand (b) (�, �) spectra obtained from a hydrogenated amorphous silicon (a-Si:H)thin film formed on a crystalline Si (c-Si) substrate. As mentioned earlier, an opticalmodel is represented by the complex refractive index and layer thickness of eachlayer. In Fig. 1.3(a), N0, N1 and N2 denote the complex refractive indices of air,thin film, and substrate, respectively. The transmission angles ( 1 and 2) can becalculated from the angle of incidence 0 by applying Snell’s law (Section 2.3.1). Asshown in Fig. 1.3(a), when light absorption in a thin film is small, optical interferenceoccurs by multiple light reflections within the thin film. In particular, this figureillustrates the optical interference in which each optical wave is superimposeddestructively. Of course, the total intensity of the reflected light becomes smaller inthis case.

In ellipsometry, the two ellipsometry parameters (�, �) are defined by� ≡ tan � exp�i�� (Section 4.1.1). In the optical model shown in Fig. 1.3(a), � isexpressed by the following equation (Section 5.1):

tan � exp�i�� = ��N0�N1�N2�d� 0� �1.1�

6 Spectroscopic Ellipsometry

Air

(a)

N0

N1

N2

Thin film

Substrate

θ0

θ1

θ2

d

(b)40

30

20

10

2 3 4100

120

140

160

180

200

220

Photon energy (eV)

Incidence angle: θ0 = 70°

ψ (d

eg)

Δ (d

eg)

a-Si:H (2500 Å)/c-Si substrate

Optical interference(layer thickness)

Optical properties andsurface roughness

Figure 1.3 (a) Optical model consisting of an air/thin film/substrate structure and (b) �����

spectra obtained from an a-Si:H thin film (2500 Å) formed on a c-Si substrate.

Notice that the above equation shows only variables used in the calculation. Thecomplex refractive index of air is given by N0 = 1, and the values of N2 and 0 are usually known in advance. In the ����� spectra shown in Fig. 1.3(b), theoptical interference effect appears in the energy region where optical light absorptionis relatively small �<2�5 eV�. From the analysis of this interference pattern, thethin-film thickness d can be estimated. If d is determined from this analysis, theunknown parameters in Eq. (1.1) are only N1 = n1 − ik1. In this condition, thesetwo values (n1� k1) can be obtained directly from the two measured values �����(Section 5.5.3). In spectroscopic ellipsometry, the optical constants and thickness ofthe thin film are determined in this manner. In the high-energy region, on the otherhand, light absorption in samples generally increases and penetration depth of lightbecomes smaller. Thus, optical interference is negligible in this region. From theanalysis of this energy region, band structure and effect of surface roughness canbe studied. In spectroscopic ellipsometry, therefore, from ����� spectra measuredin a wide energy range, characterization of various physical properties becomespossible.

Figure 1.4 shows the data analysis example of a multilayer structure byspectroscopic ellipsometry [5]. In this figure, (a) the cross-sectional image obtained

Introduction to Spectroscopic Ellipsometry 7

Pene

trat

ion

dept

h of

ligh

t

(a)

TEMSiO2

c-Si + a-Si

25 Å

120 ± 20 Å

550 ± 50 Å

c-Si

a-Si

c-Si

250 ± 50 Å

SESiO2 24 ± 3 Å

c-Si0.82 + a-Si0.18 ± 0.03

c-Si0.21 + a-Si0.79 ± 0.03

119 ± 19 Å

c-Si1.03 ± 0.03

511 ± 21 Å

270 ± 30 Å

c-Si

NOT direct techniquebut nondestructive,quantitative, andinexpensive

(c)(b)

Direct techniquebut NOT nondestructive

Figure 1.4 (a) Cross-sectional TEM image of a Si(100) wafer implanted with Si ions, (b)structure obtained from TEM, and (c) structure estimated from spectroscopic ellipsometry (SE).Reprinted from Thin Solid Films, 313–314, K. Vedam, Spectroscopic ellipsometry: a historicaloverview, 1–9. Copyright (1998), with permission from Elsevier.

from transmission electron microscope (TEM), (b) the structure obtained fromTEM, and (c) the structure estimated from spectroscopic ellipsometry (SE) areshown. The sample is a Si(100) wafer implanted with Si ions and, by this Siion implantation, a partial phase change from c-Si to a-Si occurs. As confirmedfrom Fig. 1.4, the results obtained from TEM and spectroscopic ellipsometryshow excellent agreement. Nevertheless, spectroscopic ellipsometry further allowsthe characterization of the volume fractions for the c-Si and a-Si components.As shown in Fig. 1.4(a), structural characterization by TEM is very reliablesince TEM is a direct measurement technique. In TEM, however, difficulties insample preparation as well as measurement itself generally limit the number ofsamples for the measurement. In contrast, although spectroscopic ellipsometry isan indirect measurement technique, highly quantitative results can be obtained.Moreover, spectroscopic ellipsometry provides fast and easy measurement, whichpermits characterization of many samples. Accordingly, for samples that allowproper data analysis (see Fig. 5.32), spectroscopic ellipsometry is a quite effectivecharacterization tool.

1.4 HISTORY OF DEVELOPMENT

Table 1.3 summarizes the history of development for ellipsometry instruments(ellipsometers) [5].AsshowninTable1.3,ellipsometrywasdevelopedfirstbyDrude in1887. He also derived the equations of ellipsometry, which are used even today. Drudeis well known from ‘the Drude model’ which expresses the optical properties of metals

8 Spectroscopic Ellipsometry

Table 1.3 History of ellipsometry development

Year Techniquea Parametersdeterminedb

Numberof data

Timetaken (s)

Precision(deg)

Author andreference

1887 E ��� 2 Theory and first experiment Drude [6]1945 E ��� 2 3600 � = 0�02 Rothen [7]

� = 0�011971 E ���� R 3 3600 � = 0�02 Paik,

� = 0�01 Bockris [8]1975 SE ����� 200 3600 � = 0�001 Aspnes,

� = 0�0005 Studna [9]1984 RTSE �������t 80 000 3–600 � = 0�02 Muller,

� = 0�01 Farmer [10]1990 RTSE (PDA)c �������t 2×105d

0.8–600 � = 0�02 Kim,� = 0�01 Collins,

Vedam [11]1994 RTSE (PDA)c ������ R��t 3×105d

0.8–600 � = 0�007 An, Collins� = 0�003 et al.[12]

a ellipsometry (E), spectroscopic ellipsometry (SE), real-time spectroscopic ellipsometry (RTSE)b reflectance (R), wavelength ��, time (t)c photodiode array (PDA),d maximum capacity. Reprinted from Thin Solid Films, 313–314, K. Vedam, Spectroscopic ellipsometry: ahistorical overview, 1–9. Copyright (1998), with permission from Elsevier.

(Section 5.2.5). Until the early 1970s, most ellipsometers were operated manually andthe ellipsometry measurement was very time consuming. In 1975, however, Aspneset al. realized the complete automation of spectroscopic ellipsometry measurements[9] (Section 4.2). As shown in Table 1.3, the development of this instrument improvednot only the measurement time but also the measurement precision significantly.A spectroscopic ellipsometry instrument for real-time monitoring was reported firstby Muller and Farmer in 1984 [10], and this instrument increased the number ofmeasurement data drastically. In 1990, a group from the Pennsylvania State Universitydeveloped a real-time instrument that has been used widely up to now [11]. Inparticular, this instrument unitizes a photodiode array (PDA) detector that allowsthe simultaneous measurement of light intensities at multiwavelengths (Section 4.2).Figure 1.5 shows real-time spectra obtained from this instrument [13]. In this figure,��1� and ��2� represent pseudo-dielectric function that can be calculated from (�, �)spectra (Section 5.4.2). In this measurement, the total of 250 spectra were measured in16 seconds with a repetition time of 64 ms during the a-Si:H growth on a c-Si substrate.From analysis of the real-time data set, the initial growth process of the thin film canbe characterized on the atomic scale (Section 8.2).

Up to now, spectroscopic ellipsometry instruments have been improvedcontinuously and four different types of instruments are mainly used. Nevertheless,ranges and errors for the ����� measurement vary significantly depending on thetype of instrument (see Tables 4.2 and 4.3). In order to perform accurate dataanalysis, therefore, understanding of the ellipsometry measurement is necessary.

Introduction to Spectroscopic Ellipsometry 9

Figure 1.5 Real-time spectra obtained from the spectroscopic ellipsometry measurementperformed during the a-Si:H growth. Reprinted with permission from Review of ScientificInstruments, 63, I. An, Y. M. Li, H. V. Nguyen, and R. W. Collins, Spectroscopic ellipsometryon the millisecond time scale for real-time investigations of thin-film and surface phenomena,3842–3848 (1992). Copyright 1992, American Institute of Physics.

1.5 FUTURE PROSPECTS

Recently, optically anisotropic materials have been studied extensively by applyingMueller matrix ellipsometry that allows the complete characterization of opticalbehavior in anisotropic materials (Section 4.2.7). For the characterization ofconventional isotropic samples, current spectroscopic ellipsometry instruments arehighly satisfactory. Thus, most of recent ellipsometry studies have been made onmaterial characterization, rather than the development of ellipsometry instruments.

Figure 1.6 shows the number of papers published each year with ‘ellipsometry’in the title [14]. The two large peaks at 1993 and 1997 are due to publicationsof the ellipsometry conference proceedings [15–17]. Since the early 1990s,research that applies spectroscopic ellipsometry has increased drastically due tothe commercialization of spectroscopic ellipsometry instruments. During the 1990s,spectroscopic ellipsometry was mainly employed to characterize semiconductormaterials. Now, from advances in instruments as well as data analysis methods,

10 Spectroscopic Ellipsometry

Figure 1.6 Number of papers published with ‘ellipsometry’ in the title versus year. Reprintedfrom Thin Solid Films, 455–456, D. E. Aspnes, Expanding horizons: new developments inellipsometry and polarimetry, 3–13. Copyright (2004), with permission from Elsevier.

the application of the spectroscopic ellipsometry technique has become quitecommon in wider scientific fields from semiconductors to biomaterials (Chapters 7and 8). Moreover, some characterizations including the feedback control of alloycomposition can be performed only using spectroscopic ellipsometry. Therefore,the application of spectroscopic ellipsometry is expected to expand further inthe future. For some materials, however, no optical data is available. Thus, theconstruction of a larger optical database has been required in this field. As mentionedearlier, ellipsometry data analysis requires the construction of an optical model.In Chapters 5–8, we will see examples that will explain how data analyses areperformed using various optical models and when data analyses are difficult.

REFERENCES

[1] R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North-Holland,Amsterdam (1977).

[2] H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry:A User’s Guide, John Wiley & Sons, Inc., New York (1999).

[3] H. G. Tompkins and E. A. Irene, Eds, Handbook of Ellipsometry, William Andrew,New York (2005).

[4] M. Schubert, Infrared Ellipsometry on Semiconductor Layer Structures: Phonons,Plasmons, and Polaritons, Springer, Heidelberg (2004).

[5] For a review, see K. Vedam, Spectroscopic ellipsometry: a historical overview, ThinSolid Films, 313–314 (1998) 1–9.

[6] P. Drude, Ann. Phys., 32 (1887) 584; Ann. Phys., 34 (1888) 489.[7] A. Rothen, The ellipsometer, an apparatus to measure thicknesses of thin surface films,

Rev. Sci. Instrum., 16 (1945) 26–30.[8] W. Paik and J. O’M. Bockris, Exact ellipsometric measurement of thickness and optical

properties of a thin light-absorbing film without auxiliary measurements, Surf. Sci., 28(1971) 61–68.