Semiconductors and semimetals: Indentification of defects in semiconductors

Identification of Defects in Semiconductors SEMICONDUCTORS AND SEMIMETALS Volume 51B Semiconductors and Semimetals A ...

Author: Michael Stavola | Robert K. Willardson | Eicke R. Weber

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Identification of Defects in Semiconductors SEMICONDUCTORS AND SEMIMETALS Volume 51B

Semiconductors and Semimetals A Treatise

Edited by R. K. Willurdson CONSULTING PHYSICIST

Eicke R. Weber DEPARTMENT OF MATERIALS SCIENCE

SPOKANE, WASHINGTON AND MINERAL ENGINEERING UNIVERSITYOF CALIFORNIA AT BERKELEY

Identification of Defects in Semiconductors SEMICONDUCTORS AND SEMIMETALS Volume 51B Volume Editor

MICHAEL STAVOLA DEPARTMENT OF PHYSICS

LEHIGH UNIVERSITY BETHLEHEM, PENNSYLVANIA

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Contents

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LISTOF CONTRIBUTORS. . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix xiii

Chapter 1 Optical Measurements of Point Defects

Gordon Davies 1. SCOPEOF THE CHAPTER . . . . . . . . . . . . . . . . . . . . . . . . 11. OVERVIEW OF THE TECHNIQUES . . . . . . . . . . . . . . . . . . . . . 1. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Optical absorption . . . . . . . . . . . . . . . . . . . . . . . . . 3. Photoluminescence generated by above-hand-gap excitation . . . . . . . . 4. Cuthodoluminescence . . . . . . . . . . . . . . . . . . . . . . . . 5. Photoluminescence generated by below-band-gap excitation . . . . . . . . 6. Absolute luminescence . . . . . . . . . . . . . . . . . . . . . . . I11 . CHEMICAL IDENTIFICATION . . . . . . . . . . . . . . . . . . . . . . . 1. Chemical incorporation . . . . . . . . . . . . . . . . . . . . . . . 2. Stable isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Unstable isotopes . . . . . . . . . . . . . . . . . . . . . . . . . 4. Photoluminescence excitation by x-ruq’s . . . . . . . . . . . . . . . . IV . STRUCTURAL IDENTIFICATION . . . . . . . . . . . . . . . . . . . . . . 1. Uniaxial stress perturbutions . . . . . . . . . . . . . . . . . . . . 2. Electric Jeld perturbations . . . . . . . . . . . . . . . . . . . . . 3. The Zeemuneffect . . . . . . . . . . . . . . . . . . . . . . . . . 4. Polarized luminescence measurements . . . . . . . . . . . . . . . . . 5. Reorientation effects and multistahility . . . . . . . . . . . . . . . . V . RADIATIVELIFETIME EFFECTS . . . . . . . . . . . . . . . . . . . . . . 1. Magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Temperature dependence . . . . . . . . . . . . . . . . . . . . . . 3. Delayed luminescence detection . . . . . . . . . . . . . . . . . . . . 4. Excitation power effects . . . . . . . . . . . . . . . . . . . . . . . VI . SPATIAL LOCALIZATION . . . . . . . . . . . . . . . . . . . . . . . . VII . ROLEOF VIBRATIONS I N “ELECTRONIC” TRANSITIONS . . . . . . . . . . . . 1. Single modes of vibration (local modes) . . . . . . . . . . . . . . . . V

2 3 3 3 8 10

12 13 14 14

16 20 22 23 28 36 38 39 41

45 45

49 51

53 54 56 51

vi

CONTENTS

Rand modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Khronic coupling . . . . . . . . . . . . . . . . . . . . . . . . . Isotope &crs on uihriitionul sirlehrrntis . . . . . . . . . . . . . . . . Zero-plionon isotope effects . . . . . . . . . . . . . . . . . . . . . VIII . E L t C I R O N l C EXCITEIISTATkS . . . . . . . . . . . . . . . . . . . . . . 1. Electron effective mass states . . . . . . . . . . . . . . . . . . . . 2. Hole-ejiectice muss stutes . . . . . . . . . . . . . . . . . . . . . . 3 . Applicubility qf the modeling . . . . . . . . . . . . . . . . . . . . . 1x. ZERO-PHONON LINESHAPES. . . . . . . . . . . . . . . . . . . . . . . X . CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 3. 4. 5.

63 66 68 70 74 74 81 82 84 87 89

Chapter 2 Defect Identification Using Capacitance Spectroscopy P . M . Moonry I . INTKODUC'TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I . CHARACTERISTICS OF DEEPLEVELS . . . . . . . . . . . . . . . . . . . . 111. C A P A C l l ANI'f: SI'E('TR0SCOPY METHODS . . . . . . . . . . . . . . . . . . 1. Tronsicn/ Crrpuci/cincr Mutrsurements . . . . . . . 2. Decl, Lrvcl Trawicnt Slxctroscupy . . . . . . . . 3. Instrutnmtii/ion . . . . . . . . . . . . . . . . . 1V. Dtt.~(.iIDENTIFICATION. . . . . . . . . . . . . . . 1. De/>c/ Conrp1e.x-csin Si . . . . . . . . . . . . . . 2. D X Cenrers in I l l - V C'oinpound Serriicondnctor Alloys V . CONCLUSIONS . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . .

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93 94 98 99 107 115 119 120 131 146 146

Chapter 3 Vibrational Spectroscopy of Light Element Impurities in Semiconductors

Michurl Stavola 1. INTKOI)IJCIION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. LOCALVIBRATIONAL M O D E S A N D VIURA I IONAL SPECTROSCOPY . . . . . . . . 1 . Model.t/i)r Irnzpirritj, Vibrutions . . . . . . . . . . . . . . . . . . . . 2. It?frured Absorption Sprctroscopy . . . . . . . . . . . . . . . . . . 3. Free C(irriers iind Clirirge Strrtr Efects in D o p d St.micor~ductors . . . . . 4 . Raniun Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 111. ISOTOPESHIFTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. O.x,vgm ii7 Si und Ge . . . . . . . . . . . . . . . . . . . . . . . . . 2 . lrnpuriti~sin Compound Semiconrhrctors . . . . . . . . . . . . . . . . 3 . De/ic t Coinple.rr~ . . . . . . . . . . . . . . . . . . . . . . . . . . IV. ANHARMONICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . O t ~ ~ - ~ i i ~ ~ e i ~Models. . s i u n a lf o r K~rlroReti-rontuiningComp1ere.s . . . . . . . 2. 1748 c i t ~ Combination Mode ~ f i n t r r s t i t i dOvygrn in Si . . . . . . . . 3 . Ferrni Rosoncrnc~e . . . . . . . . . . . . . . . . . . . . . . . . . . v . SYMMI-TKY STRESS ALIGNMENT.A N D REORIENTATION KINETICS FROM UNIAXIAL STRESS DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Stress-inclirceil Splitting of Vibro/ional Lines . . . . . . . . . . . . . . 2 . Kinctics o/ Defect Motion . . . . . . . . . . . . . . . . . . . . . . .

153 154 154

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160

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163 165 167 168 173 183 193 195 201 203

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206 207 211

vii

CONTENTS 3. Groundstore Energy Shift

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Chapter 4 Defect Processes in Semiconductors Studied at the Atomic Level by Transmission Electron Microscopy P . Schwander. W.-D. Ruu. C. Kisielowski. M . Grihelyuk and A . Ourmazd 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . I1. INFORMATION FROM HRTEM LATTICEIMAGES . . . . . 1. Approaches to Qucintitntive HRTEM . . . . . . . . 111. QUANTITATIVE ANALYSISBY REAL-SPACEM~THODS. . . 1 . Principle . . . . . . . . . . . . . . . . . . . . . 2. Implrmentut ion . . . . . . . . . . . . . . . . . . 3. Guidelines . . . . . . . . . . . . . . . . . . . . I V . APPLICATIONS. . . . . . . . . . . . . . . . . . . . 1. Difu.sion in AIGuAs . . . . . . . . . . . . . . . . 2. Difusion in GeSi . . . . . . . . . . . . . . . . . 3. SilSiO, Interjuciul Roughness . . . . . . . . . . . . V . CONCLUSIONS . . . . . . . . . . . . . . . . . . . . REWRENCES. . . . . . . . . . . . . . . . . . . . .

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226 228 229 231 232 233 244 248 248 253 255 251 258

I . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Cross-sectionul Scanning Tunneling Microscopy ( X S T M ) . . . . . . . . 2. Requirements and Limitations qf’XSTM . . . . . . . . . . . . . . . . 3. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. IMAGING AN11 S P t C T R O S C O P l C TECHNIQUES. INSTRUMENTATION . . . . . . . 1. Scunning Tunneling Microscopy Basics . . . . . . . . . . . . . . . . 2. Scanning Tunneling Spectroscopy . . . . . . . . . . . . . . . . . . 3. Cross-sectionulSranning Tunncling Microscopy ( X S T M ) . . . . . . . . I11. THEGaAs(110) SURFACE. . . . . . . . . . . . . . . . . . . . . . . . 1V . SHALLOW DEFECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. p-Dopunts in GuAs . . . . . . . . . . . . . . . . . . . . . . . . . . 2. S i n GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Dopant Projfling . . . . . . . . . . . . . . . . . . . . . . . . . . V . DEEPLEVEL DEFECTS. . . . . . . . . . . . . . . . . . . . . . . . . . 1. Arsenic Antisite in GaAs . . . . . . . . . . . . . . . . . . . . . . . 2 . Vucuncies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . COMPLEX DEFECTS. . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Zn- Vacancy Defect Complexes in InP . . . . . . . . . . . . . . . . . 2. Arsenic Clusters in LT-GaAs . . . . . . . . . . . . . . . . . . . . . 3. Compunsution Mechani.sms in Si-Doped GuAs . . . . . . . . . . . . . V I I . CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

262 262 . 263 264 . 265 . 265 . 261 . 267 270 212 272 213 271 282 282 283 288 . 288 290 . 291 293 294

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Chapter 5 Scanning Tunneling Microscopy of Defects in Semiconductors

Nikos D . Juger und Eicke R . Weher .

viii

CONTENTS

Chapter 6 Perturbed Angular Correlation Studies of Defects

Thomus Wichert 1.

I N TROI>UCTION

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1 . Anulyticul Tecliniques and Srw~ifivitirs . . . . . . . . . . . . . . . . .

2. Rutliouciive I . ~ ~ t o p e.s . . . . . . . . . . . . . . . . . . . . . . . . 11. EXPERIMENTAL METHOD . . . . . . . . . . . . . . . . . . . . . . . . 1. Hjycrjitie Inteructions . . . . . . . . . . . . . . . . . . . . . . . . 2. The Perturhed y-;~ Angulirr Correlution Tc.clniique . . . . . . . . . . . . .

3. Rtrtliouctive Probe Atotns . . . . . . . . . . . . . . . . . . . . . . . 4 . E,xperinientul A.specis . . . . . . . . . . . . . . . . . . . . . . . 111. SUBSTITUTIONAL DOPANTS . . . . . . . . . . . . . . . . . . . . . . . . 1. Donor- Accepior Pairing . . . . . . . . . . . . . . . . . . . . . . . 2. Lurtice Siies ~If’lsoluietlProhe A i o n l s . . . . . . . . . . . . . . . . . 3. Clitsiering of’DfJpunts . . . . . . . . . . . . . . . . . . . . . . . . 4. Effect qf Chemical Trunsmuiution . . . . . . . . . . . . . . . . . . Iv . LIGHTELEMENTS A N D TRANSITION METALS. . . . . . . . . . . . . . . . I . Hydrogen in Elenientul Setnicontluciors . . . . . . . . . . . . . . . . 2. Hjdrogon in III- V Setniconductors . . . . . . . . . . . . . . . . . . 3. Trunsition Met& . . . . . . . . . . . . . . . . . . . . . . . . . . V . INTRINSICDEFECTS. . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Irrudiution und Quenrhing . . . . . . . . . . . . . . . . . . . . . . 2. Single Recoil P roccss . . . . . . . . . . . . . . . . . . . . . . . . . 3. O.-stoichiotiietrj, . . . . . . . . . . . . . . . . . . . . . . . . . . V I . SUMMARY AND O U T L O O K . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGE MEN.^ . . . . . . . . . . . . . . . . . . . . . . . . . . REFER~. NCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INDEX CONTENTS OF VOLUMES IN

THIS

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297 298 299 300 302 305 316 320 322 324 337 340 342 351 352 365 372 382 382 388 393 400 401 401

407 419

Preface This is the second of two volumes on the identification of defects in semiconductors. The purpose of these volumes is to bring together chapters by leading researchers on methods used to determine defect properties. Chapter authors have prepared contributions that explain the fundamentals of the various measurement techniques and their interpretation at a level that should enable a reader to understand the current literature. The application of each technique is also illustrated by several modern examples. Thus, the many advances in the methods used to study defects in semiconductors are explained by expert practitioners, and current problems are discussed from several perspectives. It is hoped that these volumes will be a valuable reference for anyone who is interested in the characterization of semiconductor materials. In Chapter 1, Davies discusses absorption and luminescence spectroscopies and their application to semiconductor defect studies. The atoms in a defect can be identified from isotope shifts of the optical transitions. Additional perturbations, such as uniaxial stress, electric fields, or magnetic fields, can provide information about defect symmetry. An introduction is provided to the models commonly used to understand the electronic transitions of defects and the interaction of the electronic states with the lattice. In Chapter 2, Mooney discusses capacitance spectroscopy, an important method for determining the electrical properties of deep level defects. When correlated with the results of other identification methods, or combined with additional perturbations such as uniaxial stress, capacitance spectroscopy provides a powerful method to associate electrical properties with specific defect structures. Defect reactions can be conveniently followed, with the added benefit that the defect charge state can be controlled easily. The vibrational spectroscopy of the local modes of light element impurities is surveyed by the editor in Chapter 3. The frequency shifts of the vibrational lines that result from isotopic substitutions provide a means to ix

X

PREI~ACL

identify the vibrating impurity, its nearest neighbors, and, in favorable cases, to determine local symmetry. Vibrational spectroscopy has the advantage that the defects being studied need not be electrically active or paramagnetic, making it complementary to other methods. Chapters 4 and 5 introduce the use of high resolution microscopies to probe defects and defect processes. In Chapter 4, Schwander and others survey the recent advances in transmission electron microscopy that allow quantitative concentration and composition information to be derived from image contrast. Interface roughness and diffusion can be probed with near-atomic resolution. In Chapter 5, Jiger and Weber discuss scanning tunneling microscopy (STM) as a probe of defect properties. The STM measurements made on semiconductor surfaces created by cleaving in ultrahigh vacuum permit individual point defects to be imaged in real space. This exciting area is in its infancy, but STM has already demonstrated its potential to provide a new kind of information about semiconductor defects. In Chapter 6, Wichert discusses perturbed angular correlation spectroscopy (PAC), a new method in semiconductor defect studies based upon nuclear physics. In PAC, one follows the nuclear spin precession of an unstable nuclear isotope that has been introduced into the semiconductor sample. A surprisingly large number of defects can be formed in which one of the components is a radioactive probe nucleus. The electric field gradient at the probe nucleus is measured in the PAC experiment to provide a fingerprint of the defect and structural information. The information derived from several experimental and theoretical methods is often necessary to develop a complete picture of a defect’s microscopic properties (providing one of the motivations for assembling these volumes). The same defect is sometimes discussed in several chapters, but from the different vantage points offered by the measurement technique being used. Thus, it becomes important to confirm that different experimental methods do indeed probe the same defect. The use of uniaxial stress in conjunction with the various spectroscopic methods to determine defect properties appears as a common theme in several chapters and provides a means to correlate the spectroscopic signatures observed by different techniques. Another recent approach discussed in this volume is the use of unstable isotopes in conjunction with other techniques such as optical spectroscopy or capacitance spectroscopy. In this case, the disappearance of a defect that contains an unstable isotope and the appearance of its decay products, on a timescale determined by the half-life of the unstable nucleus, provides an identification of a defect atom and a means to correlate results obtained by different experimental methods. In spite of this field’s long history, the study of defects and their properties continues to be important and exciting. The demands of advanced proces-

PREFACE

xi

sing, new materials, and new applications inevitably lead to challenging defect problems and advances in the methods used to solve these problems. I thank the authors who have conveyed, with their excellent chapters, the excitement of the research performed on semiconductor defects.

MICHAEL STAVOLA

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List of Contributors

Numbers in parenthesis indicate the pages on which the authors’ contribution begins.

GORDONDAVIES ( l), Physics Department, King’s College London, Strand, London, WC2R 2LS. M. GRIBELYUK (93, 2251, Corporate Reseurch Laboratories, North Central Expressway 13588, Texas Instruments Inc., Dallas, T X 75243.

NIKOSD. JAGER(153, 261), Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., Berkeley, CA 94720. C. KISIELOWSKI (225), Lawrence Berkeley National Laboratory, One Cyclotron Road, Bldg. 621203, Berkeley, CA 94720. P. M. MOONEY (931, IBM T. J. Watson Research Center, Yorktown Heights, NY10598.

A. OURMAZD (225, 297), Institute ,for Semiconductor Physics, Walter-Korsing-Str. 2, 15230 Frankfurt (Oder), Germany and Technical University of Brandenburg- Cottbus, 030 13 Cottbus, Germany. W. D. RAU(225), Institute for Semiconductor Physics, Walter-Korsing-Str. 2, 15230 Frankfurt (Oder) Germany.

P. SCHWANDER (225), Institute jor Semiconductor Physics, Walter-KorsingStr. 2, 15230 Frankjurt (Oder) Germany.

MICHAELSTAVOLA (153), Department of Physics, Lehigh University, 16 Memorial Drive, East, Bethlehem, PA 1801.5. xiii

xiv

LISTOF

CONTRIBUTORS

EICKER. WEBER(26 l), Muterials Science Depurtment, University qf Calijorniu n f Berkeley, Berkeky, C A 94720.

THOMAS WICHERT(297), Technische Physik Geh 38, 0-66041 Saarbrucken, Fe&rul Repuhlic of’ Germnny.

SEMICONDUCTORS A N D SEMIMETALS. VOL 5 1 8

CHAPTER 1

Optical Measurements of Point Defects Gordon Davies PHYSICS I h I ‘ A K l M k K T

KING’SCOLIC(;L LONDON STKANIJ.

LOhlXJh

U N I T 1 1) KIN(IIXJM

I . SCOPEOF THE CHAPTER . . . . . . . . . . . . . . . . . . . . . . . . 11. OVkRVIEW OF THE TECHNIQUES . . . . . . . . . . . . . . . . . . . . .

111.

IV .

V.

VI . VI1.

VIII .

1. Nonienclarure . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Optirul ahsorption . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Photc~luminescenceyeneruted by uboae-batzd-giip excitation . . . . . . . . 4 . Cathodoluminescence . . . . . . . . . . . . . . . . . . . . . . . . 5. Below-Band-Gap Photoexcitation . . . . . . . . . . . . . . . . . . . 6. Ahsolute luminescence . . . . . . . . . . . . . . . . . . . . . . . CHEMICAL IIXNTIFICATION. . . . . . . . . . . . . . . . . . . . . . . I . Chemicul incorparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Stuhle isotopes 3 . Unstuhle isotopes . . . . . . . . . . . . . . . . . . . . . . . . . 4. Photoluminescence excitation hy x-ruys . . . . . . . . . . . . . . . . STRUCTURAL INFORMATION . . . . . . . . . . . . . . . . . . . . . . 1 . Uniuxial stress perturhutions . . . . . . . . . . . . . . . . . . . . 2. Electric field perturhutions . . . . . . . . . . . . . . . . . . . . . 3. The Zeeaiun gffect . . . . . . . . . . . . . . . . . . . . . . . . . 4. Polarized luminesrenre meu.surrments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Reorientation $errs and multistability RADIATIVELIFETIME EFFECTS . . . . . . . . . . . . . . . . . . . . . . 1. Muynitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Temperature dependence . . . . . . . . . . . . . . . . . . . . . . 3. Drluyed luminescence detection . . . . . . . . . . . . . . . . . . . . 4. Excituiiori power 6ffecf.s . . . . . . . . . . . . . . . . . . . . . . . SPATIALLOCALIZATION. . . . . . . . . . . . . . . . . . . . . . . . ROLEOF VIBRATIONS I N “ELECTRONIC” TRANSITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Single tnodes of vibrarion (Iocu1 modes) 2. Bandmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Mhronic coupling . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Isotope ejects on ~.ibrutionulsidehands . . . . . . . . . . . . . . . . 5. Zero-phonon isotope ejrects . . . . . . . . . . . . . . . . . . . . . ELECTRONIC EXCITEDSTATES . . . . . . . . . . . . . . . . . . . . . .

2 3 3 3 8 10 12 13 14 14 16 20 22 23 28 36 38 39 41 45 45 49 51 53 54 56 57 63 66 68 70 74

1 Copyright C. 1998 by Academic Press All rights of reproduction in any form reserved ISBN 0-12-752165-8 ISSN 0080-8784 $30.00

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GORDONDAVIES I . Elecrrori effectiiie rnuss states 2. Hole-q‘jectiue maxs stutes . 3. Applicubility OJ the niodelirig IX. ZERO-PHONON LINESHAPES . . X. CONCLUSION. . . . . . . . Refcrerlccs . . . . . . . . .

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87 89

I. Scope of the chapter Luminescence and optical absorption provide powerful methods of obtaining microscopic data on defects in semiconductor materials when they work. This qualifier is inserted to make it clear from the outset that in some cases optical techniques are incapable of providing data. One example is when the defect one wishes to investigate does not luminescence under the conditions being used, perhaps because the temperature of the crystal is either too high or too low. However, in favorable cases optical techniques can give unambiguous information on the chemistry of the defect, place very strong constraints on the structure of the defect and, in contrast to most techniques discussed in this book, easily provide very detailed information on the properties of the excited electronic states of the defect. This chapter presents a survey of the ideas involved when optical methods are used to probe optical centers, so that someone new to the field can pick up some knowledge of both the techniques and the language. It is not intended to go into great detail on the theoretical aspects, but rather to illustrate the common techniques and properties by examples. The chapter begins with a general overview of the techniques. In Section 111 we examine how they are applied to chemical identification of the optical centers. (From this point we will use “optical center” or “center” to refer to the point defect we are interested in studying, to distinguish it from the other defects that may be present in the crystal). In Section IV we examine the use of optical methods to derive structural information on the center, omitting the magnetic resonance techniques, which are discussed in detail in dedicated chapters in Volume 51A of this series. One key parameter in studying an optical center is the lifetime of its luminescence, as discussed in Section V. Another important characterization of a center is to determine its spatial location in the crystal, and the use of optical techniques for this purpose are described in Section VI. To understand the spectra produced by the centers we need to look into the role of vibrations in the luminescence spectra, Section VII, and in Section VIII we will also consider some aspects of the excited states that are accessible by experiment. Finally, we see that very useful information can be derived from the lineshapes of the zero-phonon lines of luminescence spectra, Section IX. ~

1

OPTICAL

MEASUREMENTS OF POINT DEFECTS

3

The examples used in this chapter will be taken from the two group I V semiconductors whose luminescence centers are best understood: silicon and diamond. This chapter will concentrate on the properties of isolated centers rather than on processes involving more than one center, such as donoracceptor pair spectra (see, e.g., Schmid ~t al, 1983, for pair spectra in silicon). 11. Overview of the techniques

1. NOMENCLATURE

Before outlining the available standard techniques of optical studies of centers it is useful to list the techniques. To generate luminescence requires an input of energy; the method of providing that energy determines the prefix to the name of the luminescence technique. Thus, triboluminescence uses friction to input the energy, electroluminescence utilizes injection of charges, cathodoluminescence uses an electron beam to excite the crystal, and photoluminescence employs a beam of photons. In scientific studies of optical centers, the primary tools are photo- and cathodoluminescence. Triboluminescence and the luminescence emitted by breaking crystals tend to be transitory and thus are of little interest here. (Recent work on breaking silicon can be traced through Busch et al, 1996). Thermoluminescence (light emitted on heating a sample by the ionization of traps) is of little current scientific interest for semiconductor research, although of considerable importance in archeological dating. Phosphorescence (very long-lived light emission), which can similarly arise from the slow ionization of centers after excitation, is not discussed here, although there is current work ( e g , Wantanabe et ul., 1997). Cathodoluminescence is especially useful for wide-band-gap semiconductors; electrons of a few tens of kiloelectronvolts are easily handled and can generate electron-hole pairs across any size of energy gap met within a semiconductor. Photoluminescence requires simpler equipment, with fewer requirements for vacuum spaces. Photoluminescence has the considerable advantage that the exciting light can be polarized, and can be tuned in energy (for example, to lie above or below the band gap of the host lattice). Both cathodo- and photoluminescence allow for easy time-dependence studies of the luminescence (Section V). 2. OPTICAL ABSORPTION

Luminescence has the very important advantage that it can be an extremely sensitive technique, allowing some centers to be observed at a concentration of less than one in many billions of host atoms. However,

4

GORDON DAVIES

optical absorption by the center is the more fundamental technique: it is not affected, for example, by nonradiative processes at the centers. In absorption, a beam of light is focused on one face of the sample, and the beam transmitted through the sample is examined spectroscopically. The situation is shown in Fig. la, where I, is the transmitted intensity of light at a photon frequency v, and I , is the incident intensity at the same frequency. In the case of a gaseous sample of thickness r the absorption coefficient p is simply determined by

Note that the logarithm is the natural log to the base e. The software of many spectrometers allows the “absorbance” A of the sample, defined as

to be calculated, where the logarithm is to base 10, and there is no allowance for the thickness of the sample. When using a new spectrometer or new software. be warned.

(b)

(1 - R)A0exp(-pt/2)exp(-iknt) (1 - R)RA, exp(-3pt/2) exp(-3iknt) (1 - R)R2A,exp( - 5 4 2 ) exp( -5iknt)

t FIG. 1. (a) Transmission through a norireHecting medium allows the absorption to be defined as / I = l r ~ ( / ~ ~ / / , ) / r . With reflections, the transmitted amplitude is the sum of phaseshifted waves (b) as in Eq. ( 3 ) .

1

OPTICAL MEASUREMENTS OF POINT DEFECTS

5

This simple equation is in any case wrong for a solid sample. There may be significant reflections occurring at each face, so that a fraction r of the intensity is reflected at the normal incidence of the light. The detected signal is built up from individual multiply reflected beams, each of which is phase-shifted by the different optical path-length in the solid Fig. lb. The total transmitted intensity is

where k is the wavenumber of the light in a vacuum and n is the refractive index of the crystal. The cosine term describes the interference oscillation in the signal, which can obscure absorption lines. The fringes can be reduced by polishing the sample into a wedge shape or by tilting it so that it is not normal to the incident light. Equation (3) reduces to Eq. (1) if the absorption is so high that the reflected beams are rapidly attenuated in a double pass through the crystal, apart from the multiplying factor (1 - r)’, which simply produces an offset in the absorption spectrum because then p (l/t) ln(Io/lc) ln(1 - r)’. Otherwise reflections should be allowed for (and rarely are), using the frequency-dependent reflection coefficient. Note that I , enters Eq. (2) in the denominator, and so must be known accurately. If the absorption is so strong that Zz/lodecreases below 0.1, then we must expect errors in the absorption coefficient. The reason why we want to know the absorption coefficient accurately is that the absorption (defined appropriately) is proportional to the concentration of the centers in the crystal, subject to constraints such as the centers being homogeneously distributed in the crystal. By “the absorption defined appropriately” it is meant that the optical spectra we are concerned with in this chapter are always produced by transitions between electronic states. In a monatomic gas, the absorption spectrum would take the form of a series of very sharp lines. The lines would have a finite width, produced by the intrinsic lifetime broadening of the state, or collision broadening, or whatever. Even in this simple example of a monatomic gas, the absorption lines have a finite spectral width, and so the appropriate measure of the absorption is to integrate the absorption coefficient across the extent of the line. For an optical center in a crystal the electronic transition is usually spread out by the vibronic coupling (Section VII). Again, a suitable measure of the total absorption is to integrate the absorption coefficient across the entire optical band that is being produced by the electronic transition. The most useful expression relating the integrated absorption to the concentration N of the centers is derived by detailed balance for the optical absorption and luminescence processes. In SI units, assuming the Lorentz

-

+

GORDON DAVIES

6

correction for the effective electric field, the equation is

This form appears to be accurate to within a factor of 2 for the localized centers found in diamond (Davies et ul., 1997), although an effective field of EeIf 1.3E is better for the more diffuse states usually found at centers in silicon (Bohnert e l ul., 1993). In Eq. (4)g i and g, are the degeneracies of the initial and final states of the luminescence transition, and z is the radiative lifetime of the transition, in the absence of any nonradiative processes. t will be discussed in Section V. However, even though z may not be known, the equation shows that the integrated absorption is strictly proportional to the concentration of the centers in a homogeneous crystal, and this fact allows useful quantitative work to be done, for example, as discussed in Section 111.1 on the production of centers. We will see in the next subsections that the quantitative attribute of absorption measurements contrasts strongly with luminescence measurements. We have noted that an absorption band may have a considerable width in a crystal as a result of “vibronic processes.” To get a quantitative measure of the absorption it may be better to use some appropriate part of the band. In particular it may be useful to limit measurements only to the “zerophonon line.” This term will be defined in Section VII, but can be regarded as being produced by an electronic transition between two electronic states, see Figure 2. At low temperature, the shapes of zero-phonon lines are frequently given not by Gaussian or Lorentzian shapes but by a profile midway between them

-

I ( v )= I/( 1

+ 4(2’Ip- I)

(yyr

r is the full width at half height. This lineshape is sometimes refered to as the “Pearson VII” function. If the parameter p has the value 1 then the lineshape is Lorentzian, and as p + 03 the lineshape tends to Gaussian. A fit to experimental data for the 969 meV zero-phonon line in silicon is shown in Fig. 2, with p = 2.5 (Davies and do Carmo, 1981). A particularly convenient form of Eq. (5) is obtained when p = 2, giving I ( v ) = u/(b

+ (vo -

v)2)2

(6)

where the line peaks at frequency vo, and has a peak height of Z(v,) = a/b2,a full width at half height of r = 2 J m ) , and an integrated area of

MEASUREMENTS OF POINTDEFECTS 1 OPTICAL

r-1 0.2

7

I

-

-

i 0

969.25

969.5

969.75

Photon energy (mev) FIG. 2. (a) The absorption spectrum of the zero-phonon line of an optical center in silicon measured at 2 K. The center contains two C atoms and one unique Si atom. The line shows the measured absorption and the dots are the spectrum calculated assuming that the lineshape is as Eq. (5) with p = 2.5, and that the Si atom may be a 28Si atom (component “l”), or a 29Si atom (component “2”) or a 3”Si atom (component “3”). (b) The spectrum measured in photoluminescence is distorted, with the main peak reduced in intensity relative to the weaker components, because the luminescence can be resonantly absorbed by the centers. After Davies and do Carmo (1981).

1.9O1(v0)T.This lineshape, with p = 2, is numerically extremely close to the Holtzmark lineshape, obtained from inverse Fourier transform of exp( - l / ~ ) ~ ’ that ~ ) , is, from the Levy distribution with exponent 3/2. If we want to obtain the area of the line by integrating it then it is very important that the zero level of absorption in the line is correctly defined. For example, note that 28% of the total area of a Lorentzian lies below 10% of its total height, so that errors in the zero level must be minimized. It may be better to multiply the height by the width to get a measure of the area, especially if we are confident that the lineshape is the same in the different samples of interest. What determines the lineshapes at low temperatures will be discussed in Section IX. With increasing temperature the lineshapes are usually broadened by a Lorentzian component, so that the observed lineshape is the convolution of Eq. ( 5 ) with the Lorentzian thermal component. We may want to know the

8

GORDONDAVIES

L /

Lorentzian width FIG. 3. Thermal broadening of zero-phonon lines is of Lorentzian lineshape [i.e., p = 1 in Eq. (5). The width of the Lorentzian thermal broadening is given on the horizontal axis so that an infinitely sharp line would be broadened to the width shown by the line W. The convolution of the thermal broadening with a Lorentzian of width unity results in another LorcntLian line whose width is simply the sum of the two Lorentzians (upper straight line). The convolution of R Lorentzian with the commonly observed lineshape of Eq. ( 6 ) results in a spectral feature whose width is given by the thicker line.

magnitude of that thermal component. The total width is not the sum of the two linewidths but can be derived from the convolution of the two lineshapes and is given in Fig. 3.

3. PHOTOLUMINESCENCE GENERATED BY ABOVE-BAND-GAP EXCITATION

Photoluminescence may be generated by excitation with photons whose energy is above or below the energy gap of the host lattice. The results are significantly different. We take first above-band-gap excitation. The absorption coefficient for above-band-gap light is typically high, so

9

1 OPTICAL MEASUREMENTS OF POINT DEFECTS

that the energy is deposited near the surface of the crystal. For example, the absorption coefficient for silicon is shown in Fig. 4 over a wide range of energies for room temperature. (Data for lower photon energies derived from luminescence measurements are given by Daub and Wurfel, 1995.) If silicon is excited by an Ar' laser operating at 2.41 eV, the excitation decreases exponentially into the crystal [Eq. (l)] with a decay length of l/,u = 1 pm. Using a Kr' laser at 1.84 eV increases 1/11 to 5 pm, and changing to the near ultraviolet (UV) only reduces the length to about 0.5 pm. The absorption of above-band-gap light is by the host lattice, within a few micrometers of the surface. We will return to the spatial implications in Section VI. Here we follow the energetics of the process. The immediate end-point of the excitation is the creation of free electrons

1

2

3

4

5

Photon energy (eV) FIG. 4. The absorption coefficient of pure crystalline silicon measured at 10 and 300 K. After Jellison and Modine (1982).

10

GOKDON DAVIES

and holes in the lattice. They may partially recombine to form excitons, as long as the temperature is low enough that the excitons are not ionized, because in silicon, for example, the electron and hole are bound together by only 14.3 f 0.5 meV (Shaklee and Nahory, 1970). As silicon is an indirect gap semiconductor, the exciton lifetime can be considerable, with values over 6 0 p s being reported (Merle et a/., 1978). In that time the excitons can diffuse far enough that they may be captured by an optical center, so that the energy of the exciton is inserted into the center-in other words the center has been put into an excited electronic state. We expect that the energy levels of molecular systems depend on the particular elements making up the molecule, and so (in the great majority of cases) the energy of the luminescence emitted as the center deexcites is unambiguously characteristic of the particular species of optical center. It is this specificity that gives optical measurements their power in studies of defects in semiconductors -if we observe an optical transition of a certain energy, it will almost always indicate that there is one particular species of optical center in the crystal. From this outline we identify the following problems:

a. Luminescence, especially in an indirect gap semiconductor, can occur a substantial distance from the point of excitation. b. The input energy may be captured by many species of defect, not simply by the center of interest, or by no defect at all (when free-exciton emission will be observed). Consequently, the luminescence intensity emitted by a center is not, in general, directly proportional to the concentration of the center, in contrast to the absorption technique (Section 11.1). c. When the energy has been captured by a center, the luminescence intensity is affected by the presence of any nonradiative processes. Such processes will be discussed in Section V.

4. CATHODOLUMINESCENCE

Cathodoluminescence is typically generated using a beam of electrons of energy of a few tens of kiloelectronvolts. When the electrons enter the crystal they can interact with the nuclei and with the electrons of the crystal; both types of particle have similar magnitudes of charges so that similar impulses p are transferred to either type from the incident electron. The kinetic energy of the recoiling particle p2/2rn is considerably greater for the electrons (of small rn) than for the nuclei, so that the energy loss is primarily through excitation of the electrons of the crystal. The rate of energy loss per unit length by a nonrelativistic electron of energy E = rnu2/2 is

1

dE

-_dX

OPTICAL


ne4 [In(mu2) - ln(21) - i l n 2 4lrE,2mv2

-~

+ 31

11

(7)

Here I is the mean excitation potential of the atoms and n is the density of electrons in the lattice. For example, numerical integration of Eq. (7) for diamond ( n = 1.06 x lo3" m-3 and I = 81 eV) results in a total pathlength L (pm) of an electron of incident energy V(keV) of

L - 0.018V'.825pm

(8)

to 0.1 pm accuracy for 5 < V < 70 keV. For example, a 10 keV electron travels 1.4 pm and, for V = 50 keV, L = 23 pm. The distance L is the total distance traveled, but the path is convoluted by the scattering processes. The majority of the energy is deposited at depths between 0.4 L and 0.8 L and in a roughly spherical volume, giving an excitation volume of about 10pm in diameter for V = 50 keV (Yacobi and Holt, 1986). This limits the spatial resolution of the technique, and a further loss of resolution comes from migration of the excitons, which are produced following the creation of the electrons and holes. The sequence of exciton capture and the production of luminescence follows as in the previous subsection. In a head-on collision an electron of kinetic energy E and mass m can transfer an energy

T

= 2(E

+ mc2)E/Mc2

(9)

to a lattice atom of mass M , and for electrons of a few tens of kiloelectronvolts the energy transfer is below the threshold for damage; for example, for Si the threshold is E 150 keV (Canham, 1988). However, damage can be caused by a cathodoluminescence beam, as a result of ions being accelerated to the sample. In the case of Si this effect has been investigated by Canham and Lightowlers (1986). They found that an electron beam of energy 20 to 60 keV can create di-carbon centers, which are known to be produced when a carbon interstitial atom Ci migrates to a substitutional carbon atom C,. Consequently, this low-energy electron beam was producing Ci atoms. When they used a sample doped with 13Cand 12C in the ratio (4.5 0.5) : 1, they found that, especially at low beam energies, the Ci atoms were primarily I2C atoms, identifying their source as outside the crystal. (See Fig. 5.) Ions can be carried down the electron beam, and implanted into the sample. To avoid this source of damage, the electron beam can be bent onto the sample, so that the more massive ions travel straight past it. Surface contaminants can also be driven into a sample during high-energy electron bombardment, allowing, for example, SiGe alloys to be constructed

-

12

GORDONDAVIES

5

0

ItI 0

Bulk mediated

I

I

1

20

1

I

40

I

60

Electron beam energy (keV) Fici. 5. Ratio of the intensities of the I3C to ”C local mode absorption of a two-carbon atom center in silicon for subthreshold radiation damage by electrons of 20 60 keV in a sample doped with [13C]/[12C] = 4.5. The local mode involves motion of predominantly only one C atom in the center. Conscqucntly, if the damage occurred entirely inside the crystal the ratio would be 4.5 (“bulk mediated”). and if one C atom was introduced from outside the crystal thc ratio would be 0.7 (“surface mediated”). After Canham and Lightowlers (1986).

by knocking Ge atoms from a layer of germanium on a silicon wafer (Wada ct at., 1997).

5. BELOW-BAND-GAP PHOTOEXCITATION

Photons of energy less than the energy gap are not absorbed by a pure crystal, but optical centers in the crystal may absorb the light giving direct excitation of the centers. Direct excitation has the following advantages relative to above-band-gap excitation: a. There is no intermediate process between the photon being absorbed and the center being excited, so that the polarization of the exciting light can be exploited as discussed in Section IV.4. b. By tuning the energy of the photons used for excitation, in favorable cases the spectrum for exciting the center (i.e., its absorption spectrum)

1

OPTICAL


13

can be measured; the technique is then referred to as photoluminescence excitation. The disadvantage of using below-band-gap excitation is that frequently the center may produce negligible absorption of the radiation, either because its concentration, or its oscillator strength, is too small. There is then too small a deposition of energy in the centers to create measurable luminescence.

6. ABSOLUTELUMINESCENCE An absorption spectrum is calculated from the ratio of the transmitted and incident intensities at each frequency; See Eq. (1). Consequently, the wavelength-dependent intensity of the light source cancels out, as does the wavelength-dependent sensitivity of the equipment. In a luminescence spectrum we simply measure the emission from a sample. Consequently, we may need to correct for the wavelength-dependent response of the equipment, and we may want to obtain an absolute measure of the luminescence efficiency. To correct for the wavelength-dependent sensitivity we need to measure the spectrum from an object with a known spectrum. A tungsten lamp, placed at the site of the sample and using the same optics, can be used as a quasi-blackbody emitter. Its effective temperature may be measured by an optical pyrometer and the spectrum it emits can be calculated using the known emissivity of tungsten (de Vos, 1954). Consequently, the wavelengthdependent factor can be calculated, which converts the measured spectrum into a spectrum proportional to the number of photons emitted per unit time per frequency interval. We note that the correction factor will normally be different for different polarizations of light passing through the spectrometer, as the reflectivity of each mirror in the spectrometer depends on the plane of polarization relative to the reflecting plane; further, in a grating monochromator there are large changes in the reflectivity near the blaze angle of the grating. To calculate the absolute quantum efficiency of the center, the traditional technique is to use an integrating sphere to capture all the light emitted by the crystal, and compare the intensity with the emission of a standard emitter at the same wavelength region. This is easily visualized for work at room temperature; at lower temperatures it is necessary to compare with the emission from a standard source such as a calibrated blackbody (e.g., Brown and Hall, 1986).

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GORDON DAVIES

111. Chemical identification We are usually interested in the concentrations of impurities in semiconductors at the level of the order of one part per billion atoms. It is very likely that there will be contamination of the sample by unknown impurities at this and higher levels. The problem of identifying an optical feature with a particular impurity is therefore severe.

INCORPORATION 1. CHEMICAL In some favorable cases it is possible to control the concentration of one impurity and correlate the optical signal with the measured concentration. For example, carbon can be grown into silicon as an isolated substitutional impurity, and its concentration measured (e.g., by doping with I4C and using its radioactivity). Good correlations have been reported between the absorption at 607 cm- and the concentration of 12C (Newman and Willis, 1965). In luminescence there is the additional problem that the signal is not absolute. At the very least some form of internal calibration of the intensity is required, for example, to overcome the effects of not having different samples in exactly the same place in the equipment. In some cases the intensity of the Raman line from the sample may be a good internal calibrant (Clark et al., 1995). We will follow in detail the measurement of phosphorus in silicon. Dean et al. (1967) first linked the luminescence of Fig. 6 with the decay of excitons trapped on the P donors in Si, by using samples of known major dopant and resistivity. Figure 7 shows the strength of luminescence from the excitons as a function of the P concentration (Colley and Lightowlers, 1987). There is an accurate linear relationship over 3 orders of magnitude. In this case the internal calibrant is the intensity of luminescence from the free excitons. As the concentration of P increases, the probability that an exciton e P-emission will be captured at a P atom increases, and so the ratio l p / l f of to free-exciton emission will increase. The concentration of P can also be assessed from its electrical activity, if one allows for the presence of the other donors and acceptors that are revealed by the optical spectra. It is important to point out that all shallow donors and acceptors are seen simultaneously in the luminescence measurements; compensation of the minority acceptors by the majority donors will occur when the crystal is in equilibrium, but in a photoluminescence experiment the excitation places the sample in a nonequilibrium state, allowing all donors and acceptors to be populated. The data of Fig. 7 look surprisingly simple to use. However, for reproducibility the spectra have to be taken under conditions of saturating the P

’

1 OPTICAL MEASUREMENTS OF POINTDEFECTS

1,

a,(LO) 1146

1148

1150

15

1152

yT0)

1090

1130

1110

1I50

Photon cncrey (mcV)

FIG. 6 . Photoluminescence from silicon doped with [PI = 3.2 x 10'4cm-3. The inset shows an expanded view of the zero-phonon line. After Colley and Lightowlers (1987).

9

10-6

L

10-5

1 0 - ~

1

*

I

&

10-2

NP line-area/FE peak-height ratio (eV) FIG. 7. The concentration of boron, aluminum and phosphorus in silicon as a function of the ratio of the integrated luminescence in the zero-phonon line of the bound excitons to the peak height of the free exciton. These data are accurate only for the particular experimental setup used, but any equipment could be calibrated to give similar plots. After Colley and Lightowlers (1987).

16

GORDONDAVIES

donors -the excitation intensity is increased until luminescence is observed from multibound excitons on the P. We have noted in Section 111 that the excitons may drift considerable distances in silicon, so that the excitation density varies with depth into the crystal. Consequently, the degree of saturating the P signal varies with depth into the crystal, and a different value of l p / l f ewould be recorded by sampling different depths into the crystal. The orientation of the sample relative to the optics collecting the luminescence is then important -different l p / l f eare recorded if the luminescence is observed from the excited face or from a perpendicular face. In general, each experimental rig must be precalibrated against a standard set of samples. In this example, all the P atoms form the same sort of optical center, that is, simply one substitutional P atom. But often an optical center involves only a small fraction of the total amount of an impurity in the crystal. One example is in silicon, where a small fraction (perhaps 10%) of the carbon impurity may combine with a small fraction (say, 1%) of the oxygen impurity to form an optical center known as the “ C center. How can we verify that carbon and oxygen are involved? One possibility is to model the production of the center. The carbon-oxygen C center in silicon is created when a carbon atom is placed into an interstitial site by radiation damage. The carbon atom Ci is mobile at room temperature. If Ci is trapped at an 0 atom the “C” luminescence band is produced with its zero-phonon line at 789 meV. This center also produces local vibrational mode absorption at 865 cm-’, and Fig. 8a shows the production of the absorption (and hence the concentration of the center) as a function of radiation dose. At low doses the concentration is expected to be proportional to the dose, but this limit is not readily observed (Fig. 8a) because the Ci - 0 centers can capture a Sii atom and convert to so-called “C4” centers with a local mode absorption at 1020 tin- *. The C4 absorption is shown in Fig. 8b for the same samples as used in Fig. 8a. The lines on the plots are calculated taking into account all the competing reactions that occur during the irradiation (Davies et al., 1987a). The fit strongly suggests that the centers are being created by the chemical reactions used in the model, and in particular establish that the 789 meV band is formed from one Ci moving to one 0 atom, a result subsequently confirmed in detail (Trombetta and Watkins, 1987).

2. STABLE ISOTOPES

A simple technique that can produce unambiguous identification of the chemistry of an optical center is to exploit the effects of different isotopes of a suspected impurity. The isotopes may be present naturally, as in the case

1

OPTICAL


17

A

0

5

10

5

10

Electron dose 1017 cm-2 FIG.8. Modelling of the production of an optical center may allow its chemical composition to be determined. Here optical absorption (filled circles) produced by a carbon-interstitial -oxygen-interstitial pair is modeled (solid lines) using the chemical reactions triggered by the radiation damage. The slowdown in production of the centers is a result of their capturing a silicon-interstitial atom, producing the "C4" centers (open circles, modeled by broken lines). After Davies et al. ( 1987a).

of Ni in diamond. Nickel enters diamonds when they are synthesized by high-temperature, high-pressure conversion of graphite into diamond, using Ni as a solvent-catalyst (Kanda and Sekine, 1994). Diamonds can now be synthesized with sufficient perfection that fine structure can be observed in the two zero-phonon lines near 1.4 eV, one of which is shown in Fig. 9a. Figure 9b is a reconstruction of the observed lineshape assuming the lineshape of Eq. (5) for each component, and a relative intensity of the components equal to the ratio of the natural abundances of the Ni isotopes

58Ni:"oNi:61Ni:62Ni:64Ni = 67.76:26.16: 1.25:3.65: 1.16

(10)

The quality of the fit implies that Ni is involved in the center, and that the effects are produced by one Ni atom (see Section VII.5 this chapter; Nazare et al., 1991). In other cases we cannot rely on naturally abundant isotopes and have to introduce different isotopes into the crystal. Figure 10 shows the effect on

GORDONDAVIES

0

0.6

1.2

Photon energy -1402.8 (mev) FIG. Y. (a) Absorption at 1 1 K of the 1.404 eV zero-phonon line in a synthetic diamond. The line contains substructure produced by the effects of one nickel atom. A reconstruction of the spectrum is shown by curve (b), where each component has a relative intensity given by the ratio of the abundances of the isotopes of Ni, and the lineshape is Eq. ( 5 ) with p = 2. The computed lineshape matches closely the measured spectrum, upper curve. After Nazarb et ul. (199 I).

the 1150.9 meV line in Si of doping a crystal with both H and D in equal amounts, establishing that H is involved in the center (Safonov and Lightowiers, 1996). We will discuss the origin of the effect in Section VII.5, and here will deal with some of the practical issues. First, the number of components observed in the splitting tells us how many impurity atoms are involved in producing the isotope effect -this is then a lower limit to the number of those atoms in the center. That is, there could be other atoms at the center that do not produce an isotope effect. Taking the H example of Fig. 10, the presence of three lines with equal concentrations of H and D implies that two atoms are active at the center and that they occupy equivalent sites in the center. Assuming random assignments of the atoms, if the crystal has been doped with both H and D atoms, in the ratio p / q with p + q = 1, the probabilities of two H, one H and one D, and two D atoms are given by successive terms in the expansion of ( p q)2. If the hydrogen

+

1 OPTICAL MEASUREMENTS OF POINT DEFECTS I

1

19

I

2H

HD

D A

1.1505

1.151

1.1515

Photon energy (eV) FIG. 10. Spectra of the B,, center in silicon doped with hydrogen, deuterium or an H, D mixture. The pure-isotope spectra consist of a lower energy line from a triplet state and a higher energy line from a singlet state (see Section VIII.l). In the mixed-isotope sample, one additional line appears midway between each of the singlet and triplet states, indicating that two closely equivalent hydrogen/deuterium atoms are involved in the center. After Safonov et. a1 (1997b).

atoms occupied inequivalent sites, then the configurations H - D and D - H would be different, and four lines of equal intensity would be observed for p = q = 0.5. Similarly, for the Ni example of Fig. 9, the fact that the intensity ratio of the fine structure equals the natural abundance ratio implies that only one Ni atom is active in producing the isotope effect. This argument relies on the assumption that different isotopes can be incorporated into an optical center with equal probabilities. This postulate

20

GORDON DAVIES

is not necessarily true -- the diffusion coefficient of an impurity depends on its mass. It is also possible that the capture of an impurity to form a new optical center will also be mass-dependent. This is especially likely when the migrating impurity is H or D, as they differ by a factor of two in their in their zero-point energies, although in practice masses, and hence by only small deviations from equal probabilities of capture are observed The replacement of 12C by I 3 C in silicon typically leads to changes in zero-phonon energies of less than 0.1 meV (Davies et al., 1987). Because the isotope splittings in zero-phonon lines are small, it is important to be sure that the observed structure is produced by isotopes and not some other effect in the crystal. One technique is to note that the different relative abundances of the isotopes occur at spatially separated optical centers. In the Ni example, one center may be made of '*Ni and another of 60Ni-we do not expect energy to be transferred between these two centers and so the energy states of different energy cannot come into thermal equilibrium with each other. Similarly, the same intensity ratio should be measured in absorption and luminescence measurements of components ascribed to 58Ni and to 60Ni, as is observed (Nazare et al., 1991). If an electronic state was being split at the center by spurious strains, then we would expect intensity transfer between the two states. It is essential that the host crystals used for the different isotope-doped samples are very similar.

4

3. UNSTABLE ISOTOPES If an unstable isotope is present in an optical center in a crystal, the optical signal should evolve as the isotope decays, because the optical spectra are highly characteristic of the chemical elements present. Figure 11 shows a test example, in which luminescence from the well-known phosphorus donor in Si (Section 111.1) has been monitored; the P in this case was unstable 31P.The luminescence from the P has been normalised with respect to the luminescence from the stable B acceptors, to provide the necessary internal calibration. The line on Fig. 11 is an exponential decay with the lifetime of 25 days of 31P.In this case the 3 1 Pdecays by beta emission, with an energy of only 0.25 meV, and so little radiation damage will result. Changes in the luminescence efficiency of the crystal are allowed for by the boron calibrant. In some cases there may be substantial concern that the crystal is changing significantly during the period of observation, which needs careful consideration (Magerle et al., 1995). Typically the unstable isotope will be implanted into the crystal, and so there may be annealing of the implantation damage as well as new damage being caused by the unstable nuclei. One solution that has been adopted (Henry et al., 1996) is

21


I

I

m

0.5

-

0.0

-

1

1

I

I

I

I

I

Time (days) F~G.11. Luminescence measured from the phosphorus donor (Fig. 6 ) in silicon implanted with "P atoms. The points show experimental data and the line is an exponential decay calculated with the decay time of the 3 1 Patoms. Cower (1996) private communication.

to observe the decay of a luminescence signal as an unstable element decays and in a separate experiment to observe the growth of the same signal as the element is created. For example, they identified luminescence from silicon at 777 meV as being caused by Pt by observing its decay as '"Pt(2.9 days) + Ir, and by its growth through the reaction '95Au(183 days) -+ Pt. The unambiguous link overrules a previous assignment (based on the chemical preparation) that the center was caused by Ag. This example demonstrates the danger in assuming that the dominant impurity introduced is the cause of a photoluminescence spectrum : luminescence can be caused by very low concentrations of trace impurities. Figure 12 shows the growth of the 777 meV band in p-type silicon that has been implanted with 193Au,the growth following a saturating exponential, [l - exp( - t / z ) ] , with z = 32 _+ 12 h, equal to the decay time (z = 25 h) for lY3Au 192Pt(Knopf et al., 1998). In n-type silicon, the decay of the 735 meV band (z = 21 L- 4 h) establishes that band as gold-related. Note that the luminescence probe in silicon penetrates only of the order of a micrometer, which is similar to the implantation depths, so that a good matching is achieved as long as there is no significant long-range migration of the excitons; see Section 11.3. -+

22

GORDONDAVIES

Photon energy(meV) 775 750 725

775 750 725 64 49

35

f

2

1

A

20

O

r 12

13 ---J.--Ih____ I

js50

l600 1650 1700 1750

1550

~

, . , , , . 1600 1650 7700 9750

,

Wavelength(nrn) ~ at 60keV FIG. 12. Spectral changes in silicon implanted with I012cm-Z 1 9 3 A atoms ~ to Pt with a decay time of 25 h. followed by annedling at 950°C and quenching. 1 9 3 Adecays In p-type Si (left) the 777meV band increases with a saturating exponential with a decay time of 32 +_ 12h, consistent with the band being produced by Pt. In n-type Si the 735 meV band decays with T = 21 4 hours, consistent with that hand being gold-related. After Knopfet nl., 1998).

4.

PHOTOLUMINESCENCE EXCITATION BY

X-RAYS

One problem in identifying the chemical origin of an optical center is that the outer electronic states of the impurity atoms are changed in a nontrivial way by their environment. However, sufficiently localized electronic states will still retain their chemical signature. If an atom is excited by exciting an inner electron, the energy required will be a characteristic signal of the atom. A synchrotron X-ray source, scanned through the characteristic atomic energy levels, can therefore act as a chemical probe, with luminescence used as the detected signal. Figure 13 shows that the intensity of X-ray excited optical luminescence (XEOL) from porous silicon, which has been freshly etched, follows the silicon X-ray edge at 1844 eV, while with progressive oxidation it shifts to the 1851 eV edge of silicon oxide (Pettifer et al., 1995).


1840

1850

23

1860

Photon energy (eV) FIG. 13. X-ray excited luminescence spectra recorded from silicon (a) and silicon-oxide (b) -(d) in various states of oxidation, as a function of the X-ray excitation energy. After Pettifer et al. (1995).

IV. Structural information Suppose that an optical center consists of only one defect atom. That one defect can still take up a variety of sites : a substitutional site with tetrahedral symmetry, as for a vacancy in diamond; a substitutional site with lower symmetry, as for a vacancy in silicon; a trigonal symmetry, as in the example in Section 111.2 of Ni in diamond; a trigonal bond-centered site, as postulated for H in Si (Bech Nielsen, 1988), and so on. These different symmetries can be distinguished by various simple extensions of optical techniques. Other techniques are discussed in separate chapters of this volume and volume 51A of this series. Before looking at the techniques, let us see why we want to have symmetry information on the centers. Apart from helping to define the structure of a center, knowledge of its symmetry places severe constraints on its expected properties. There are many texts that provide an introduction to the use of symmetry arguments and group theory in quantum mechanics (e.g., Tinkham, 1964; Wherrett, 1986). As a very brief outline,

24

GORDON DAVIES

consider a center such as an F-center in an alkali halide (Fig. 14). One halide atom (e.g., CI in NaCI) is missing, and the nearest neighbors are the 6 N a atoms along the positive and negative x, y , z axes. Suppose that we are interested in the properties of hydrogen-like p orbitals, which are directed along these same x, y , z axes. What happens if the atoms along the x axis move inward‘? In the new configuration it is obvious (see Fig. 14b) that the p, orbitals will have a different energy now from the p, and p, orbitals-the Na ions along the x axis have been moved in, changing the Coulomb interaction with the p, orbitals while not greatly affecting their interaction with the p, and p, orbitals. Further, the py orbital will be affected in precisely the same way as the p, orbital, because both are perpendicular to the main motion (along the x axis). In quantum theory, the perturbation of the states depends on integrals such as (p,16,Qxlp,) where O,Q, is the change in the potential of the electrons as a result of the movement by Q, of the ions and 0, is the corresponding operator. Thus,

and just from the symmetry of the center we have derived useful information about the effect of perturbations on electronic states. We have not used the detailed form of the wavefunctions, merely the fact that they are identical in form except for their different orientations. All the related combinations of perturbations and states have been tabulated for all point groups of interest (1963). by Koster er d., The same ideas can be applied to optical transitions, because the intensity of a transition between two states and $ 2 depends on the square of an Irl$2), where r is the direction of the electric field of the integral such as radiation. For example, at an F-center, with the electric vector of the light polarized along the x axis, transitions from an s-like state can only occur to p, because (s I x I p,) = (s I x I p,) = 0: this follows simply from the odd parity of p,, (and p,) and of x, but the even parity of s. In contrast, parity arguments show that (slxlp,) # 0. These selection rules have been derived purely from the shapes of the orbitals and of the environment, and are not dependent on the details of the wavefunctions. For example, if all the orbitals were made twice as large, the same selection rules would be followed. Symmetrybased arguments are trivial to apply to perturbations whether they are time-independent (e.g., stresses) or time-dependent (e.g., optical transitions). To report some of the important results we note that a spatial function is “totally symmetric” in the point group of the center when it is not changed by any symmetry operation at the center. For example, an s state is spherically symmetric and so is not changed by any rotation or reflection at thc F center; it is a totally symmetric function in the 0, point group.

1

OPTICAL


25

l z

X

I

t t FIG. 14. An F center in an alkali halide is a missing halide ion. In the position of least vibrational potential energy, the center has the 0, point group with equal distances to the 6 nearest neighbor alkali atoms (a). Moving the two x-oriented ions inwards (b) will perturb the p, orbitals in a different way to the p, and p, orbitals. In terms of symmetry, lowering the symmetry to C,, symmetry has partially lifted the orbital degeneracy.

26

GORDON DAVIES

Similarly, a six-pointed star, with each point directed along x, y or z, is unchanged by the symmetry operations. Orbitally degenerate states such as px, py. p, have the property that the symmetry operations of the F center transform one into another, or, in general, into a linear combination of the three states; for example, a rotation through 4 2 transforms p, into p,. Two important results from the use of group theory are that: a. A nondegenerate state is perturbed in first order only by a perturbation that is totally symmetric in the point group of the center. Consequently, an s state in the F center (Fig. 14) may be perturbed by a radial movement of all the ions but not by a symmetry-lowering distortion. This is a powerful selection rule to determine which perturbations are effective in a low-symmetry center, when all states are non-degenerate. b. Group theory can tell us when an integral must be zero. However, an integral may be effectively zero simply from the particular functions concerned. Thus, group theory is excellent for determining if a term is zero, but not at determining how large a nonzero term is going to be. These simple applications of group theory in quantum mechanics demonstrate how essential a technique it is. The results will be used without further justification in much of the remainder of this chapter. The symmetries of some centers have the full symmetry of an atomic site in the crystal (tetrahedral symmetry in semiconductors, T, point group). Others have lower symmetry, such as an off-center impurity, or a combination of several impurities. Lower symmetry centers possess “orientational degeneracy.” For example, a trigonal center such as the N-V center in diamond, oriented along the (1 11) axes of the crystal, can have its major axis oriented along any of the [lll], [lTl] . . . [TTT] directions, giving 8 possibilities. In general, the number n of different orientations is given in terms of the number G of symmetry elements in the 0, point group of the diamond structure (G = 48) and the number g of symmetry elements in the point group of the center, as n = G/g. For a trigonal C , , point group there are 6 symmetry elements that leave the center unchanged : 3 reflections, 2 rotations by 21113, and one identity operation (Fig. lS), giving the n = 4816 = 8 possible orientations. In an ideal crystal we would expect that all these orientations would be equally populated. However, if the crystal has different growth rates in different directions, as occurs with epitaxial growth, different orientations may not have equal populations. Similarly, there may be different rates of including impurities when a bulk crystal grows at different rates in different crystallographic directions, so that different parts of the crystal have unequal populations of optical centers (see Collins, 1989). We will assume in this chapter that the centers are equally distributed.

1

OPTICAL MEASUREMENTS OF POINTDEFECTS

27

FIG. 15. Representation of a trigonal center viewed along its main axis. The center can be rotated by +2n/3 producing an equivalent configuration of the atoms. Similarly, reflection in any of the three planes containing a filled circle and an open circle leaves the center unchanged.

The trigonal center of Fig. 15 is oriented along the (1 11) axes and so will luminesce differently with respect to that major axis. It might be expected that the orientation of the centers could be determined by simply measuring the intensity of luminescence in different planes of polarization. However, in a cubic crystal, the luminescence averages over the whole crystal to a value that is independent of the symmetry of the center. The simplest example is a tetragonal optical center with its electric dipole oriented along its main (001) axis. Consider luminescence emitted by the [001]-oriented centers, Fig. 16. The intensity of the luminescence polarized with electric vector oriented along [pqr] will have a strength proportional to r 2 , the square of the projected electric vector on the dipole axis. Consequently, the luminescence is strongest when r = 2 1, providing at first sight a method of identifying the (001) orientation of the center. However, centers oriented along [loo] and [OlO] emit with strengths proportional to p 2 and q2 respectively, and the total luminescence of p 2 q2 r2 = 1 is independent of the plane of polarization of the light. The anisotropy of the centers is thus made “latent” by the “orientational degeneracy” of the centers. The same can be shown to be true for any orientation of dipole. To reveal the symmetry axis of a center we can make the different orientations inequivalent to each other, so that it is possible to detect the properties of individual orientations of center. The simplest conceptual way of achieving this is to compress the crystal along one axis. In the example,

+ +

28

GORDON DAVIES

Y 2

/

FIG.16. Centers with their primary axcs along the (001) directions will, usually, have equal populations along the [loo], [OlO] and [OOl] axes. If the centers absorb light whose electric vector is polarized along the main axis of each center, then the absorption by each center is proportional to the square of the electric vector projected on the axes of the centers, respectively pz, y’ and r’, which always sum to unity.

a [OOl] compression will make the [OOl] centers absorb differently from the [loo] and [OlO] centers, revealing the anisotropy. In addition to stresses, electric fields and magnetic fields can be used (Section. 1V.2 and 3). By polarizing the exciting light and looking at the polarization of the emitted light we can also overcome the latent anisotropy (Section IV.4).

1. UNIAXIAL STRESS PERTURBATIONS

It is easy to place a crystal under a variable stress in a low temperature cryostat. Strains of up to 0.1 YOare adequate for deriving much information about the centers. Note that these strains are, however, much smaller than the typical zero-point oscillations of an atom. For a carbon atom vibrating with a quantum of Zzo = 70 meV in an optical center in silicon the root mean square (rms) displacement r in the zero point state is r = Jj;/2mcrl= 5 x nm, or 2 % of the bond length. The easily achieved strains are, therefore, small compared to the natural motions of the atoms, and this places a limit on the amount of information we can derive on the “electron-phonon interactions” described in Section VII In the following outline it is assumed that the measurements are made in absorption. In cathodo- or photoluminescence exactly the same arguments


29

apply, as long as the photoexcitation is above the band gap so that the polarization of the exciting light is lost (Section 11.3). When the excitation of the centers is by below-band-gap light, the absorption per center depends on the polarization of the exciting light relative to the axes of the center. Results with this additional complication have been tabulated by Mohammed et al., (1982). In a cubic semiconducting crystal 6 “stress tensor” components sij are required to fully define the stress. To calculate the tensor components we need to know the magnitude of the applied stress, its direction of application, and the orientation of the area on which we are considering the stress. For an applied stress of magnitude s, the tensor components are defined by

s,,= s cos(s, i) cos(s, j )

(12)

where (s, i ) is the angle between the directions of s and the ith cubic axis of the crystal (i = x,y , z). The stress tensor component s,, has the physical meaning of being the force directed along the x cubic axis of the crystal acting on a face perpendicular to the x axis, that is, a compression along the x axis. The component S, describes a force directed along the x cubic axis of the crystal acting on a face perpendicular to the y axis, that is, a shear of the y faces. It is useful to note that the stress tensor component sytransforms under the symmetry operations in the same way that the product i j transforms. To obtain a complete set of stress data on an optical center requires stresses to be used that involve all 6 stress tensor components s,, sYy,s,, syz,s,, sxy.Compressions are used along the (OOl), (1 11) and (1 10) axes. The (001) compression generates stresses in the cubic axes of the crystals of the form s,, and (111) compressions create shear stresses like sxy. Compression along (1 10) generates both types of stresses and so gives no new perturbation data, but it does provide a powerful self-consistency check and gives information about the type of dipole moment. The optical spectra for [l 101 stress are different for viewing along [lTO] and [OOl]; if the spectra are the same for light polarized parallel to the [ l l O ] stress axis then the transition is electric-dipole allowed, and if they are the same for light polarized perpendicular to the stress axis the dipole is magnetic (Kaplyanskii, 1964a, 1964b). The way that a uniaxial stress perturbs the electronic states is that the stresses move the atoms and so the electron-nuclear interactions. We can estimate the size of the perturbation by approximating the electrons at an optical center as being trapped in a potential like a cubic box. The energy levels for such a system scale with the inverse square of its linear dimensions. For example, for a particle in an infinitely deep one-dimensional square-well

30

GORDON DAVIES

potential the energy levels are E = h2n2/8ma2,where a is the width of the box. Consequently, we expect the change in energy to be dE/E

= - 2da/u

(13)

implying a fractional change in energy of 0.2% under 0.1% strain. This is easily resolved, being x2 meV for a typical transition in silicon, over ten times larger than the widths of typical zero-phonon lines (Section IX). The stress required to achieve this amount of strain near the optical center depends on the value of the elastic constants near the center, which will be different from those in the bulk crystal as a result of the change in the chemistry at the center. A rough rule is that for localized centers, larger changes in energy per unit stress are observed at vacancy-related centers than at interstitial-related centers, because the vacancies soften the crystal (see, e.g., Table I in Zaitsev et d.,1996). Note that the strains interact with the orbital components of the electronic states, and that there is no first-order effect on the spins. Consequently, the uniaxial stress technique gives no information on the spin states, in contrast to magnetic field effects (Section 1V.3). There are extensive compilations of the effects of uniaxial stresses on electronic transitions observed in optical absorption, or by photoluminescence above the band gap, or cathodoluminescence. For centers with low symmetries see Kaplyanskii (1964) and Davies et al. (1988); for trigonal symmetry see Hughes and Runciman (1967) and Davies and Nazare (1980); for tetrahedral symmetry see Kaplyanskii (1964b). These compilations are for cases of “isolated” electronic transitions, that is, those where the ground and excited electronic states of the transitions do not interact, under the stress perturbation, with other electronic states. This situation is always true for small enough stresses, and allows the symmetry class of the center to be determined (for example, whether it is a trigonal center, but not whether it is C,, or D,, within that class; for this refinement electric fields can be used, Section IV.2). An example where these tabulations apply is the splitting of the 3942 cm-’ zero-phonon line for a defect in silicon, as shown in Fig. 17 (Davies et al., 1987b). We note here that the energies of the stress-split components are all linear in the applied stress: there is no interaction with other electronic states. Under (1 10) stress the same pattern is observed with the electric vector of the absorbed light parallel to the stress, but the observation of component “g” with electric vector perpendicular to stress is direction-dependent, indicating that the transition is an electric dipole transition. The intensities of the stress-split components were found to be independent of temperature and of the magnitude of the splittings as the stress is increased, indicating that the split states cannot come into thermal


31

K s II till>

h

0

0.2

0.4

0

0.2

0.4

? 3900 0

0.2

0.4 Stress

IGPa)

FIG. 17. The effect of uniaxial compression along the (OOl), (111) and (110) crystallographic axes on a zero-phonon line in silicon. For [I lo] stress component 'g' appears when the electric vector E is parallel to [IiO] but not with E )I [OOl], while the spectra with E parallel to the stress are identical for both viewing directions. This identifies the transition as electric-dipole allowed. The number of stress-split components and their polarizations are consistent with a monoclinic I center. For this type of center, the perturbations can be described using 4 parameters [ A , to A , of Eqs. (14, 15)], and the lines show the best fit to the measured data (points); the 9 shift rates are very well described by the 4 parameters. After Davies et al. (1987b).

equilibrium with each other. This implies that the splitting is unlikely to occur at electronically degenerate ground states in these absorption measurements. Likewise, in a luminescence experiment, if there is no thermal equilibrium between the excited states then we expect to see no variations in intensity with stress. These results suggest that there is no electronic degeneracy at the center, and so the splitting is caused only by the lifting of orientational degeneracy. The number of stress-split components for each direction of stress, and their

32

GORDON DAVIES

relative intensities in the different polarizations identify this center as having the monoclinic I symmetry class. In silicon this implies a center that is left invariant under only two operations, the identity operation and a reflection operation in a plane perpendicular to a (110) direction (Fig. 18). For this type of center the symmetry is so low that there can be only nondegenerate orbital states, and so the first-order perturbation of each state by the stress is given by those combinations of the stress that are totally symmetric in the symmetry operations. It is useful to define a set of axes for each orientation of the center. For the center oriented with the perpendicular to the reflection plane along the [llO] axis, let the local Z axis be that [ l l O ] diection (Fig. 18), and let X be parallel to [OOl] and Y along [lTO]. Stress tensor

FIG. 18. A monoclinic I center in silicon has one reflection plane oriented perpendicular t o a ( 1 lo) direction. One particular orientation of center has its reflection plane perpendicular to [llO]. This primary axis is denoted the Z axis. Cartesian coordinates local to each center can be defined by the right-handed coordinates X , Y and Z as shown. However, note that while Z is uniquely defined, because there is only one normal to the reflection plane, X and Y can lie at any orientation in the reflection plane. Here they are arbitrarily defined so that X 11 [OOI] and Yll [lTO].


33

components that transform as X 2 , Yz, 2’ and X Y are all invariant under the symmetry operations of the center. Transforming these axes to the Cartesian axes i,j, k of the crystal shows that the perturbation can be written as

where the ai are operators acting on the electronic states (Kaplyanskii, 1964a). This expression applies to both the ground and the excited state of the transition, and, consequently, to the difference between the excited and ground states. We define four parameters

as the difference in the matrix elements of the operators ai acting on the excited and ground states. The lines on Fig. 17 are calculated using this form of perturbation, applied to the 24 different orientations of the monoclinic I center. The agreement between the fit, using the four parameters A , to A , as adjustables, and the 9 measured perturbations of the components confirms the assignment to a monoclinic I center. Note that because the optical measurement monitors the difference between the excited and ground states, we only get information on the diference in the perturbations of the ground and excited states. In contrast, paramagnetic resonance, for example, gives information on the orbital state in which the resonance is observed. The example given here is a simple case. One common complication is that the excited states of centers are multiplets of nearly degenerate states, which interact under stress. Some quantitative details will be given in Section VITI, but a simple picture of an excited state of many centers in semiconductors is that an electron has been excited from a relatively localized state, leaving a localized hole. The electron is still attracted to the center through its Coulomb attraction to the hole, but the Coulomb attraction is reduced by the large relative permittivity of the semiconductor (c = 1 1.7 for Si) allowing the electron to orbit in effective mass states, which are closely separated in energy. The lowest energy electron states are likely to be split by the low symmetry field of the central cell of the optical center. Consequently the electron is likely to be perturbed in a similar way to the conduction band states, and the hole as in the Kaplyanskii theory. Thonke et ul. (1985) first used this approach for a carbon-oxygen optical center in Si, as shown in Fig. 19. The symmetry of the optical center is again monoclinic I (Foy, 1982) but is sufficiently close to tetragonal that the data can be fitted using this higher symmetry. The lines on Fig. 19 have been

GORDON DAVIES

34

(001)

k ---------------------

L

0

0.1 0.2 0.3

0

0.1 0.2 0.3

0

0.1 0.2 0.3

Stress (GPa)

FIG. 19. Perturbations of the electron effective-mass states of the interstitial-carbon, interstitial-oxygen center in silicon. Points show measured data by Foy (1982) and lines calculated by Thonke rt ul. (1985). The calculation is described in Sections IV.l and VIII.l with the lines computed using Eq. (41).

calculated by assuming that an electron is loosely trapped on a tetragonal center, which provides the initial (zero-stress) splitting of the six conduction band minima. In this case the ground (or hole) state of the center is negligibly perturbed by the stress and so the observed splitting is caused primarily by the perturbation of only the effective-mass electron. An example of a center where both the hole and electron are perturbed is given by the Cu S center in Si, where again the excited state can be treated as an electron in an effective mass state, with a hole in a tightly bound state, see Fig. 20 (Jeyanathan et ul., 1994). The same arguments can be used for the centers where the transition involves a tightly bound electron and an effective mass hole, as for another monoclinic I center, the carbon-oxygenhydrogen “ T center in Si: See Section V11.2 and Fig. 45 in this chapter. The existence of two particles in the excited states of the centers raises the question of their spin coupling. Several cases exist depending on the amount of orbital angular momentum of the particles, and the possibilities have been reviewed by Dean and Herbert (1979). Centers such as the Si: Cu + S center are particularly simple in that all orbital angular momentum of the hole and the electron is quenched by the low (monoclinic I) symmetry of the center, so that only the spin angular momentum survives. The spins (s = 1/2) of the hole and the electron combine to produce a spin triplet ( S = 1) and a spin singlet ( S = 0), which are separated in energy by an amount depending on the spatial localization of the particles, so that, in general, the

+

1

35


(110) stress

(111) stress

(001) stress

g7F t:.

960

950

0.0

0.5

940

0.0

0.5

0.0

0.5

Applied compressive stress (GPa)

FIG. 20. Effect of uniaxial compressive stresses on the spin-triplet of the A configuration of the Si:S + Cu system (Section VIII.1). The spin-singlet state lies 11 meV to higher energy, but its different spin state prohibits stress-induced interaction with the singlet. The triplet states are, however, repelled from higher lying triplet states. The lines are calculated assuming an effective-mass-likeexcited state and a tightly bound hole ground state. Data from Jeyanathan (1995).

singlet-triplet splitting increases with the binding energies of the particles. Because the uniaxial stresses do not couple to the spins of the particles, the singlet and triplet states do not interact appreciably under stress, although they do interact (respectively) with higher lying singlet and triplet states of the higher energy effective mass states of the electron, producing the curved perturbation plots for the triplet states of the Si:Cu S center shown in Fig 20. We noted that one useful observation is whether the stress-split components are in thermal equilibrium with each other; if they are in equilibrium the initial state of the transition is probably orbitally degenerate and has split under the stress. However, the degeneracy of electronic states can be lifted without thermal equilibrium being reached between the split components. For example, the conduction band minima in silicon are located at different k values, and are not able to come into thermal equilibrium at low temperatures until the splitting is about 16 meV, large enough for a k-conserving transverse acoustic phonon to be emitted as the electron makes the transition between the different minima (Kulakovskii et al., 1978). The

+

36

GORDON DAVIES

same effect can carry over to electrons loosely trapped on optical centers 1984) although the k-based selection rules are inevi(Ciechanowska rt d., tably weaker in view of the localization of the electron on the center. Conversely, the observation of thermalizing lines does not necessarily imply that electronic degeneracy is being lifted. Thermal equilibrium can sometimes be reached between different orientations of centers which have different energies as a result of a uniaxial stress. For example, a monoclinic I center can be derived, formally, from a trigonal (C3J center by lowering its symmetry through the removal of the three-fold axis to leave only one reflection plane: we could create the monoclinic I center by placing one new impurity atom on one of the reflection planes of a trigonal center. There are three ways the symmetry can be lowered depending on which reflection plane contains the new atom. An applied stress may then increase the energy of one of the monoclinic I orientations relative to the other two, and the new atom may be able to flip into one of the other reflection planes. This easy re-orientation between the monoclinic I configurations derived from one trigonal orientation is seen, for example, in the 969 meV luminescence of the two-carbon atom center in silicon (Foy et ul., 1981). As always, a simple rule (“thermalisation = electronic degeneracy”) can have exceptions. When the effects of uniaxial stresses are known on an optical transition, they can be used to monitor the state of the crystal. For example, the linewidth may indicate the general background strains in the crystal (Section IX), and if a zero-phonon line is observed to be split in an as-grown crystal, the strains in the crystal may be estimated (Pickard et al., 1997). We will consider other uses of uniaxial stresses in Section IV.5.

2. ELECTRIC FIELDPERTURBATIONS We have seen that the uniaxial stress technique produces large perturbations relative to the zero-phonon linewidth so that the symmetry group can be determined. However, a uniaxial stress requires equal and opposite forces to be applied to the crystal, if the crystal is to be static, and so the perturbation has even parity under inversion. As a result, the technique cannot distinguish between those centers with inversion symmetry (e.g., a trigonal center with the D,, point group) from those which do not (e.g., trigonal with C3J. To distinguish them requires a perturbation that is axial but is directed, and an electric field is suitable. Large electric fields can only be applied to crystals that are insulating. Even then the perturbation is small. The potential energy of an electron is changed by only meV between two points separated by 1 p\ (0.1 nm) along a field of 1000 V/cm, and so it is immediately apparent that lock-in

1

OPTICAL


37

techniques will be required with modulated electric fields if any effect is to be seen. Under these conditions an optical center with inversion symmetry will show no first-order perturbation of an electronic state, but the perturbation by the squared field will result in a lineshape given by the first derivative of the lineshape. A center with no inversion symmetry can have a first-order perturbation of its electronic states, resulting in a lineshape of second derivative form (Kaplyanskii et ul., 1971; Kaplyanskii et al., 1970). Figure 21 shows the absorption spectrum and its field-modulated derivative for a nitrogen-vacancy center in diamond (Davies and Manson, 1980). The symmetry of the center is known, from stress measurements and from the formation of the center, to be Clh.The lack of inversion symmetry allows a first-order Stark effect, and this is observed (Fig. 21b). The size of the Photon energy (mev) -10

(a)

0

10

Transmission

Electric field squared (volts/mm)'

FIG. 21. Transmission (a) through a diamond, showing absorption at 2.417eV, compared to the differential signal (b) with niodulatcd electric field. The second derivative lineshape indicates that the center does not possess inversion symmetry; from uniaxial stress measurements it is known to have monoclinic 1 (inversion-free) symmetry. The energy scale is centered on the zero-phonon line. In (c), the size of the differential signal is shown to be proportional to the square of the electric field. After Davies and Manson (1980).

38

GORDON DAVIES

V

-10

0

10

Photon energy (mev)

FIG. 22. The differential signal for the zero-phonon line of the neutral tetrahedral vacancy in diamond is of the correct second-derivative shape in sample (a). In sample (b), with a slightly asymmetric absorption lineshape, the signal has a complex shape when measured in unpolarized light, and is of first derivative shape, (c), when viewed in one polarization (electric vector perpendicular to the electric field). The expected shape is second derivative. The energy scale is centered on the zero-phonon line. After Davies and Manson (1980).

differential signal increases with the square of the applied electric field (Fig. 21c). The small perturbations can, however, produce misleading results if the optical center is perturbed by its environment. The vacancy in diamond is tetrahedral but can be perturbed by nitrogen impurity atoms close to it, producing an asymmetric lineshape. Figure 22 shows the Stark effect on unperturbed vacancies, giving the expected second derivative shape, in contrast to the Stark effect on an asymmetric line. The technique has to be used carefully; lines of irregular shape can produce misleading results. 3. THEZEEMAN EFFECT

Data on the spins of states cannot be derived from uniaxial stress or electric field perturbations but can be obtained from magnetic perturba-


39

tions. We have seen in Section IV.l that states with a spin of S = 1 and g = 2 can occur at centers in semiconductors as a result of the coupling of the spins of an electron and a hole. Assuming g = 2, a field of only 5 tesla

will split the M , = + 1 and 0 states by 0.58 meV, which is usually an easily resolved splitting at a zero-phonon line, as seen in Fig. 23 for the Si :Cu S center (Singh et d.,1989). However, in contrast to paramagnetic resonance measurements, hyperfine splittings cannot be resolved in these measurements. Consequently, no chemical data can be obtained directly from Zeeman experiments. A further limitation on Zeeman spectroscopy is that in a pure spin system there is no coupling of the spin to the orbital motion, and so the Zeeman data for a singlet-triplet system do not contain any information regarding the symmetry of the center. In this case the Zeeman splitting is isotropic (Fig. 23). A particle with orbital motion has the motion influenced by the local symmetry of the center, and so can give symmetry information. Figure 24 shows data for the T center in Si, a complex of two C atoms and one H atom (Safanov et a/., 1995). The analysis is in terms of recombination of an exciton bound to a center with C,, symmetry. The electron has isotropic spin angular momentum only (with g = 1.95) but the hole is derived from the valence band maximum and so has orbital as well as spin angular momentum, combining to give a total angular momentum with components J i on the cubic axes of the crystal. The Hamiltonian for the hole can be written as

+

where the fit in Fig. 24 uses g1 = 1.3 and y, expected for a shallow trapped hole in Si.

= -0.1,

close to the values

LUMINESCENCE MEASUREMENTS 4. POLARIZED We take again the simple example of Section IV (in the Introduction on page 23) where a center may absorb light when the electric vector of the light has a component along the (001) axis of the center. Consequently, light polarized along [OOl] is absorbed only by the [OOl] centers. If the energy remains on that center and if it is emitted as one photon, the intensity of the luminescence will be polarized along the same [OOl] axis. Rotating a polarizer to analyse the emission will reveal an intensity varying as cos2 8

40

GORDON DAVIES

967.6

0

I

I

I

2

I 3

I

1

4

5

1

2

3

4

5

812.4

811.6

0

Magnetic field (T)

I--

R11 =5Tcsln

'I'=16K

Photon cncrgy (mcV)

FIG. 23. (a) Splitting of the triplet slates of the 'A' and ' B configurations of the Si:Cu + S system under a magnetic field. The field is actually directed along the (001) axis, but could be along any crystallographic axis. because at the resolution of a photoluminescence spectrum, the splitting is isotropic; (b) The appearance of the spectrum for the 'B' configuration, measured at 5 Tesla and 16 K. At this temperature the excited singlet state is readily observable, and is not split by the field. The triplet states are populated according to a Boltzmann distribution. After Singh et ul. (1989).


41

0

p 30

h

u)

a,

z!, a,

EI 60

2

90

I

934.5

935.5 Photon energy (mev)

FIG. 24. Magnetic perturbations of the ‘T’line, a carbon-hydrogen related system in silicon. One line is observed at zero field. At 5 T, the observed splittings are a function of the direction of the magnetic field, and can be fitted using Eq. (16). After Safonov er a/. (1995).

where 8 is the angle between the transmission axis of the polarizer and the [OOl] axis. From an experimental point of view, the variation in signal for this crystal geometry would characterize the transition as being produced by a [OOl] dipole at a [OOl] oriented center. The technique requires sufficiently strong absorption directly by the center of interest, and so requires an easily tuned excitation source. Consequently it is limited in its use. The technique has been widely used in studies of centers in diamond and alkali halides where it is easy to obtain resonant excitation of the strongly absorbing centers. Examples are given by Clark and Norris (1970, 1971).

5.

REORIENTATION EFFECTS AND MULTISTABILITY

A center with symmetry lower than T, has several equivalent orientations in the crystal (Section TV, see the Introduction on page 000). For example, a trigonal center may be oriented along the four axes [lll], [lil], [I111 and [lli] plus their four inversions. A species of center present in a semiconductor at a concentration of 5 x 10l6~ r n has - ~ a fractional concentration of about 1 in lo7 lattice atoms, and so we expect there will be about 200 host atoms between each optical center : the optical centers; are typically well separated and isolated from each other. Now examine Fig. 25,

42

GORDON DAVIES

which shows the effects of [lll] uniaxial stresses on a trigonal center in Si. One of the excited states (labelled ‘a’), which is derived from a [111]oriented center interacts with a state (‘b’) from the other orientations. It follows that the electronic wavefunctions for the different orientations of this center have an appreciable overlap with each other, which is unreasonable if the centers are separated by hundreds of Si atoms. The explanation is that at this center the trigonal symmetry is the result of a distortion from T, symmetry, and so any one center may distort into any of the trigonal orientations; all the orientations can occur at the same lattice site (Davies, 1994). The origin of this effect is discussed in Section VIII.1. A center that distorts spontaneously into a lower symmetry is unusual; it is far more usual that a center has a low symmetry simply because that is its molecular structure. We have noted that differently oriented centers are well separated in the crystal, and so we do not expect them to interact with

1050

h

5

v

h

8

bo

1045

8

Y

42

c4

m ‘

1040 0.0

0.1

0.0

a

0.1

Applied stress (GPa) FIG. 25. Effect of uniaxial stresses on the transitions at the ‘Q’ four-lithium atom complex in silicon. At zero stress, two lower energy lines are observed at 1044 and 1045meV, from triplet and singlet states of the Q center (cf. Fig. 23b). Under (001) stress all the trigonal centers are equally perturbed, and the splitting is a result of the stress lowering the symmetry of the trigonal centers. Under ( 1 11) stress the electronic degeneracy is not lifted at this center. However, centers oriented parallel to the stress (labeled a) are perturbed differently from those nonparallel (h). The triplet and the singlet states undergo an anti-crossing, showing that the different orientations can all occur at each optical center. After Davies (1994).

1


43

each other. However, the energy barrier between the different orientations may be small enough for a uniaxial stress to produce a reorientation, even at cryogenic temperatures, as we saw in Section IV.l. A uniaxial stress would then align the centers along one orientation. If the center has a simple structure, reorientation may lead to migration through the lattice. For example, the interstitial oxygen atom in Si can be rotated to specific lattice directions by applying stress : this orientation requires an atomic jump between equivalent sites and this jump is one step in the diffusion of the impurity. Consequently, reorientation of individual centers allows the individual diffusion step to be monitored. See, for example, the chapter by Stavola (pages 153 to 219) in this volume. All these orientations leave the center in an equivalent environment, and so its optical transitions have the same energy for each orientation. A different class of restructuring occurs when the atoms comprising the center are stable (or are metastable) in distinguishable configurations. Typically, one atom of the complex is an interstitial atom. We may picture this atom as migrating towards a trapping point in a periodic potential, with the periodicity determined by the distance between the interstitial sites, plus an attractive potential: See Fig. 26. Movement over the energy barriers will involve relaxation of the atoms at the saddle point, and so the energy barrier will be affected by the different chemistry near the trapping point. Near the

B

Distance through crystal FIG. 26. Schematic diagram of the potential of an impurity as it migrates towards a trapping point at point ‘0’.The potential is periodic in the perfect lattice distant from the trap. The potential minimum nearest the trap may have a low potential (point ‘A’) but with an increased barrier (point ‘B’) before the final trapping point. The increased barrier may be caused, for example, by additional difficulties in moving the lattice atoms apart, to move through a saddle-point, near the trapping point.

44

GORDON DAVIES

defect, relatively stable sites may be formed. An example where this hindered motion is seen in an optical experiment is when a carbon interstitial approaches a substitutional carbon atom in silicon (Davies ef al., 1988) : the optical signal of the Ci atom disappears as the Ci atom approaches C,, but the Ci must then rebond with C, before the Ci - C, pair is observable (Fig. 27). The classic example of more than one stable configuration is the Fe-AI pair in silicon. Here the Al is substitutional and the Fe interstitial. The pair can exist as a nearest-neighbor pair or as a second nearest pair (Chantre

Oe2

r

0

500

1000

1500

Time (mins) FIG. 27. Migration in silicon of a carbon interstitial atom Ci to a trapping point, a substitutional carbon atom C,, produces the two-carbon ‘ G center. The optical absorption (circles) produced by the C , atoms decrcases exponentially with annealing time. However, the C, musi overcome a barrier of 938meV in order to combine finally with the C , atom; schematically, this harrier is the energy difference between points B and A on Fig. 26. Consequently there is a delay in the growth of the G absorption. The delay is clearer in (b) for annealing at 0°C than in (a) for annealing at 2O’C; the line labeled “exp. grth.” (which stands for “saturating exponential growth”) would he the growth curve of the G centers if they were produced without delay. Thc harrier height for motion of the C, at a large distance from the C, atom is 862 _+ 10 meV. After Davies ct (I/. (1988).

OF POINT DEFECTS 1 OPTICAL MEASUREMENTS

45

FIG. 28. (a) Configurational coordinate sketch of the energy levels of the C,-P center in its negative charge state, the neutral charge state when it has lost one electron, and the repulsive state after gaining two holes. (b) Similar sketch for the fifth configuration of the C,-P center. See Fig. 26 for a simple schematic form of the figure. After Giirer et (I/. (1992).

and Bois, 1985). The different configurations are identifiable by the different symmetries of the centers producing paramagnetic resonance [see Sakauchi et al., (1996)l. A more complicated version of this behavior occurs in silicon for the substitutional-phosphorus interstitial-carbon center, where the Ci atom has relatively stable positions at several different sites relative to the P, atom, see Fig. 28 (Gurer et al., 1992). In terms of optical centers the Cu-S pair in Si is currently unique in showing two different optical bands, associated with different configurations (Fig. 29). Cooling a sample in the dark to 4.2 K results in the center’s being trapped in the “A” configuration, producing zero-phonon luminescence near 912 meV. Excitation (to create the luminescence) converts the center into the “B” configuration, with zero-phonon line near 812 meV. The B configuration is then stable until the sample is heated sufficiently for the defect to overcome a barrier of about 100 meV, taking it back into the A structure (Singh et al., 1993; Jeyanathan et al., 1994; Chen et al., 1994).

V. Radiative lifetime effects

1. MAGNITUDES We have seen that the radiative lifetime T of a transition is useful in calibrating the strength of the absorption in a band to the concentration of

46

GORDONDAVIES

Construction

Measurement One-phonon part

700

800

900

1000

Photon energy (mev) FIG. 29. The spectrum measured from the two Si:S + Cu centers is shown by the noisy line at the bottom. The spectrum has been corrected for the response of the system as described in Section 11.3. The upper spectra are calculated using the vibronic theory of Section VII.3, with the contributions from the one-phonon processes as shown by the insets to the measured spectra.

the center (Section 11.2). The z is defined through the exponential decay of the intensity I of the luminescence after the excitation has ceased

I = I, exp( - t / z )

(17)

This equation assumes that the excitation of the center stops abruptly at t = 0. In practice it may be possible for energy to excite the center from another defect, so that energy stored in the crystal continues to be transferred to the center after the primary excitation has ceased, making the decay time of luminescence from the center longer than the intrinsic value of 7 . Equation (17) also assumes that all the decay is by the emission of photons in the luminescence transition. Frequently there are also nonradiative processes, in which case the measured decay time zd is less than the radiative lifetime z,. The total decay time zd for independent processes is


47

given by the sum of the reciprocals

where zni are the decay times in the nonradiative channels. It may be easier to regard Eq. (18) as saying that the reciprocals of the decay times are the probabilities per unit time of the decay, and the total probability is the sum of each probability. The decay times are especially likely to be affected by competing processes in an indirect gap semiconductor, such as Si, where the radiative lifetimes may be long. The Wannier exciton has a well-separated electron-hole pair, so that there is very little overlap of the two particles, reducing their recombination probability. Also, in silicon the electron and hole have very different wavevectors as a result of the indirect bandgap, and so recombination does not just involve the electron and hole but necessitates one or more phonons to conserve wavevector. These factors result in the free exciton in Si having a long lifetime, with measurements of at least 60 pi in pure Si being reported at low temperature (Merle et a/., 1978). When the exciton is captured by a shallow donor or acceptor its decay time is significantly reduced. For example, for the interstitial Li donor the lifetime at 4.2 K is 1.1 ,us (Steiner and Thewalt, 1984). The lifetime is reduced by two factors. First, localization in real space on the donor results in an increase in overlap between the electron and hole, and it also produces a spread in wavevector, increasing the possibility of the electron and hole recombining without the need to involve a phonon. Second, the donor is in the neutral charge state, which means that in the unexcited state of the center there is one electron orbiting it in an effective-mass orbital. When an exciton is bound at the donor, nonradiative decay is possible through the Auger emission of an electron : the energy of the exciton may be used to ionize an electron off the center rather than to produce a photon. With increasing binding of the exciton on the donor or acceptor, both the localization and the probability of emitting the Auger particle increase, resulting in a rapid decrease in decay times. For donors in silicon the binding energies and exciton decay times (measured at 4.2 K) are given in Table I, and the corresponding values for acceptors in Si are given in Table 11. The luminescence efficiency, defined as the probability of emitting a photon after excitation, is zd/z,, and so rapidly decreases down these progressions of donors and acceptors. Extrapolating this situation to “deeper” centers in silicon, where the exciton is bound to the center with a substantially larger binding energy, it is not surprising that the luminescence efficiency can become very low if there is a third particle (electron or hole) on the center that can be kicked off by the decaying exciton in an Auger

48

GORDON DAVIES TABLE 1

EXCHON DECAYTIMESAT 4.2 K Donors Lifetimc Energy

Li 115011s 3.6 meV

AND

EXCITON BINIIINGENERGIES OF SOME DONORSIN SI

P 272 ns 4.7nieV

Bi 8.6 ns 7.7 meV

As

183 ns 5.4meV

Data are from Vouk and Lightowlers (1977a, 1977b); Thewalt (1977); Dean et al. (1967); Steiner and Thewalt (1984).

process. Consequently, in silicon, those deep centers observed in luminescence very rarely contain a third charge in their excited states; the excited state can be described as one exciton, and the ground state is isovalent with the lattice. The ground state of the center is then expected to have no spin and so will not be observable by paramagnetic resonance. We have “derived” here a rule of thumb that in silicon the optical centers are usually not observable by paramagnetic resonance until their charge state is modified. This rule is very well (but not universally) valid; the “T” center of Fig. 24 is expected to be one exception, although paramagnetic resonance has not yet been observed in the ground state (Newton, 1997). The measured decay time of luminescence may be affected by interactions with other defects in the crystal, in which case the measured lifetime may be specimen-dependent. We note two possibilities: energy transfer between centers and charge transfer. An example of energy transfer occurs when the luminescence from a center occurs at the same photon energies as the absorption band of another defect. Instead of the center emitting a photon as luminescence, it may transfer the energy to the other defect in what we can picture as the emission and absorption of a virtual photon (Dexter, 1953). The ‘“3” center, a triangle of three nitrogen atoms next to a vacancy in diamond, has a radiative decay time which has been observed to vary between 41 and 18 ns in different samples (Thomaz and Davies, 1978). The reduction in decay time correlates with the increasing concentration of pairs of nitrogen atoms in the samples, and can be fitted by assuming that the electric dipole of the EXCITONDECAYTIMESAT 4.2 K Acceptors Lifetime Energy

B 105511s 3.8 meV

AND

TABLE 11 EXCITONBINDING ENERGIES OF

Al 76 ns 5.1 meV

Ga

77 ns 5.7 meV

SOME

In 2.7 ns 14.0meV

ACCEPTORS IN SI TI 0.27 ns 43.8meV

Data are from Vouk and Lightowlers (1977a, 1977b); Thewalt (1977); Steiner and Thewalt ( 1984).

1

OPTICAL

MEASUREMENTS OF P O l N T DEFECTS

49

N3 centers couple to an electric quadrupole at the nitrogen-pairs. The probability per unit time of energy being transferred by this mechanism decreases rapidly with the distance r between the luminescing center and the absorbing defect, and is proportional to l/r*. Consequently, the energy transfer becomes increasingly important as the concentration of the defects increases. An example of charge transfer is provided by the transfer of an electron from the excited state of a C - 0 center to the V - 0 “A” center in Si (V = vacancy), giving

(C-O)*

+ (V-0)

-+GO)+

+ (v-0)-

where the star indicates an excited state. This process is detectable by the paramagnetic signals of the C-0’ and V - 0 - centers (Frens e t al., 1994). The transfer time was estimated in some samples as 50 ns, much smaller than the radiative decay time of 14 ps for the G O * center (Bohnert et a)., 1993). Consequently, the charge transfer process does not affect the measurements of the radiative decay time, but would reduce the measured luminescence efficiency of the center in a sample-dependent way.

2. TEMPERATURE DEPENDENCE

Tables I and I1 show that excitons may be bound to acceptors and donors by only a few millielectronvolts. With increasing temperature the excitons are thermally ionized from the centers before they recombine; for example, in silicon the luminescence from these shallow centers decreases rapidly above about 8 K. The excitons emitted from shallow traps become available for capture by other optical centers, so that the intensity of the luminescence from deep centers often increases with increasing temperature (see Fig. 30). At higher temperatures the excitons are in turn ionized from the deep centers, and the deep luminescence also decreases. This is the central problem in making successful light-emitting diodes for room-temperature operation from crystalline silicon. These ionization processes are shown in Fig. 30 by data for the ‘ P carbon-oxygen center in silicon, which is observed after heating oxygencontaining silicon at about 450°C. The data are readily fitted by an equation of the form

Here the growth in luminescence as the shallow traps are ionized is

50

GORDON DAVIES

0

80

40 Temperature

(K)

FIG. 30. Temperature dependence of the 767 meV ‘ P zero-phonon line in silicon. The P centers have been generated by heating oxygen-rich silicon for 48 h at 450°C. The line is calculated using Eq. (19). The increase in intensity at low temperatures corresponds to release of excitons from traps whose exciton binding energy is 3 meV. After Davies (1989).

+

described by the t e r m f = u/[l bT3’’exp( - E , / k T ) ] , where E, is the trap ionisation energy. The term p = gT3’2exp( - E / k T ) describes the loss of intensity from the deep center by its ionization (ionization energy E ) , and a, b, g are constants. Typically, the shallow traps have a binding energy of 4 meV, and their origin is currently unknown. The increase in intensity by a few orders of magnitude between 4 and 30 K has severe implications for the use of photoluminescence as a diagnostic tool-optical centers may exist but not be observable under the convenient experimental conditions of immersing the sample in liquid helium. . Corresponding to the decrease in luminescence under steady-state excitation at the higher temperatures, the measured decay time also decreases, as seen in Fig. 31 (Thonke er ul., 1985). The multiplet structure of many excited states of centers also causes intensity and lifetime changes. For example, under steady-state excitation there may be a significant increase in intensity with increasing temperature if transitions from the lowest excited state are relatively forbidden compared with transitions from a higher lying state. Figure 32 shows data for the A configuration of the Si: S + Cu band. The lowest excited state of this center is a spin triplet, while the ground state has

-

1

OPTICAL


51

Temperature (K) I

20

10

5

I

I

I

1

200 n

9

50

W

10

1

0

0.1

0.2

1/T (K-l) FIG. 31. The main plot shows the measured radiative decay time of the 789meV band produced by a Ci-Oi complex in silicon. The inset shows the decay of the intensity as a function of time for temperatures (from top to bottom) of 10, 7 and 4.2K. At T < 10K an initial fast decay is observed as shown by the squares on the main figure. After Thonke et al. (1985).

S = 0. Consequently, the rules of atomic physics suggest that this transition is forbidden. In a crystal, the transition is allowed (see Section VIII.l) although it has a transition probability some 600 times smaller than transitions from the S = 0 excited state which, as shown on Fig. 20, lies about 11 meV above the S = 1 state. With increasing temperature this S = 0 state becomes populated, and the luminescence transition probability increases substantially.

3. DELAYED LUMINESCENCE DETECTION The different decay times of optical transitions can be used to simplify spectra that contain more than one different luminescence band. A longlived luminescence band may be enhanced relative to a quickly decaying transition by exciting the crystal using a pulsed source. The detector is then gated so that it is active after the short-lived luminescence has died away, when the long-lived luminescence will dominate the spectrum. An example

52

GORDON DAVIES

-5’

o

-5

0

‘

I

20

20

’

I

40

40

‘

I

60

60

‘

I

80

80

’

I

loo

100

100O/T K-’ FIG. 32. The Si:S + Cu bands (Fig. 29) have spin-singlet and spin-triplet excited states (See also Fig. 20). Transitions from the higher-lying singlet are considerably more allowed than from the lower energy triplet, with the two sets of states in thermal equilibrium over the measurable ranges. The projections to infinite temperature give intensity ratios of 570:l for the A band and 13001 for the B band.

is shown in Fig. 33, where photoluminescence from Co-doped diamond is dominated, under steady-state conditions, by emission from nitrogen-related centers (‘H3’ with a radiative decay time of about 16 ns and “3’ with 40 ns). A 1 p s delay after excitation discriminates against the emission from these centers, and allows studies to be made of the Co-related luminescence (Lawson et ul., 1996).

1


53

14x103 -

12 10 86-

42-

01,

I

I

2.0

2.2

2.4

2.6 2.8 Photon Energy (eV)

3.0

2.0

2.2

2.4 2.6 2.8 Photon Energy (eV)

3.0

I

I

I

FIG.33. Top spectrum shows photoluminescence from a diamond containing zero phonon emission at 2.156eV (labeled 575nm), at 2.463eV (labeled H3) and at 2.985eV (labeled N3). Luminescence from all these bands decays with a lifetime of a few tens of nanoseconds. The lower spectrum has been recorded with a l p s gate delay, so that the short-lived bands dominant in the upper spectrum have decayed and the longer-lived components can be observed clearly. After Lawson et al (1996).

4.

EXCITATION POWEREFFECTS

Finally, note that the relative intensities of different contributions to the luminescence spectrum of a crystal may vary with the excitation power. For example, if the excited state of a center has a long lifetime, then it will be easy to saturate the luminescence from that center by exciting it with enough power. A center in the same crystal with a short-lived excited state cannot

54

GORDONDAVIES

be as easily saturated, and under sufficiently high-power excitation it will dominate the luminescence. It follows that the spectrum may change depending on the geometry being used in the experiment; if the luminescence is collected from the excited face then the observation is of some average over the different excitations into the crystal, while if the luminescence is detected perpendicular to the excitation then the specific excitation level depends on the section of the crystal being monitored.

VI. Spatial localization We noted in Section I1 that the volume excited in a luminescence experiment may be on the scale of micrometers, whether the excitation is by an electron beam or by a beam of light. However, further degradation of the spatial resolution can occur if the excitation results in the creation of excitons that can diffuse through the crystal before they are trapped at the optical centers. In silicon, where we have noted that the exciton lifetime may be tens of ps, considerable diffusion may occur. In Si doped with l O I 3 cm-3 B atoms, it has been estimated that 10% of the excitons can diffuse through a thickness of 150 pm at liquid helium temperature (Gregson, 1987), decreasing to about 1% at l0l4 ~ m - With ~ . increasing concentration of traps for the excitons, the diffusion length decreases. One example is shown in Fig. 34 where the luminescence from a carbon center is being detected as the beam in an electron microscope is scanned across a multigrain sample of silicon (Davies et al., 1993). At the grain boundaries, the luminescence decreases; the line is calculated assuming a diffusion length of 6 pm for the excitons, significantly larger than the 3 pm diameter of the excitation sphere created by the 15 keV electron beam (Yacobi and Holt, 1986). Variation of the concentration of centers may occur in a thick film of epitaxial material if the growth conditions vary during the growth. Figure 35 shows data for an epitaxial layer of Si, 11 pm thick, grown on Si. The surface of the sample has been progressively etched away, and the luminescence from each optical center is shown as a function of the etch depth (de Mello, 1990). To allow for any variation in the equipment it is necessary to normalise the measured intensity against a standard, and in these experiments the standard was provided by one part of the surface of the layer, which was always protected from the etchant. The calculations allow for the 1 pm exponential decay of the 514.5 nm Ar' ion laser light into the sample, and they assume that the excitons can diffuse from their point of creation with a Gaussian distribution. To fit the data a diffusion length of 3.5 pm had to be used. The remaining variable is the concentration profile of the optical

1

I

0

OPTICAL

I

20

I

55


I

40

I

I

60

I

I

80

I

I

I

100

Distance (pm) FIG.34. Cathodoluminescence intensity of the two-carbon atom ‘ G complex as measured by cathodoluminescence topography using a 15 keV electron beam. The line through the points is calculated for an exciton diffusion length of 6pm and assuming a partial reduction in the luminescence at the grain boundaries, whose positions are shown by the vertical lines. The stight linear increase in signal across the scan is an artifact and has been allowed for in the fit. After Davies et al. (1993).

center, and the lines fitting the measured distributions have been calculated with the profiles shown in the right-hand figures. Unfortunately, there are no independent data on the profiles to check against the calculations. Excitons do not necessarily diffuse randomly. In a strain field the valence band and the conduction band of a crystal are likely to be split, so that some of the split exciton states may be lowered in energy. The exciton is then attracted into that strained region so that it can lower its energy. It has long been known that the electron-hole “liquid” created by the strong excitation of germanium can be concentrated into part of the crystal by straining that part. Similarly, excitons that enter a region of reduced bandgap are incapable of escaping from it. For example, excitons are spatially localized by a quantum wire and so may be captured by the individual optical centers

GORDON DAVIES

56

I

d-line PL and flt

0

4

8

12

0

b -

4

I

g-line PL and f i t

I

L flttlna profila

8

12

fitting profile

I

.-

0

4

8

12

Etch depth pm

0

4

8

12

Depth in layer pm

FIG. 35. Photoluminescence intensities of two zero-phonon lines, arbitrarily labeled ‘d’ and ‘g’, observed in an epitaxial layer of silicoii. The layer was 11 Ltm thick. O n the diagrams in thc left column, measurements of the intensities after successive etching of the surface are shown by the points, as functions of the thickness removed by the etching. The lines are calculated using thc dopant profiles shown in the diagrams in the right column, allowing for the penetration of the exciting light and an assumed 3.5pm diffusion length of the excitons. After dc Mello (1990).

in the wire, giving a way of observing individual centers in the wire. This technique has been used to detect cathodoluminescence from single carbon acceptor atoms in GaAs quantum wires in AlCaAs (Samuelson and Gustafsson, 1995). An alternative high-resolution excitation process is by carrier injection from a scanning tunneling microscope (Pfister et al., 1994).

V11. Role of vibrations in “electronic” transitions

Optical absorption and luminescence transitions involve changes in the electronic state of an optical center. However, the vibrations of the center are extremely important in determining many of the observed properties, such as the shape of the spectrum and its isotope dependence. Since the ideas are

1

OPTICAL


57

so fundamental we will only briefly review here the “vibronic” (combined electronic and vibrational) properties of the center. A. Single modes of vibration (local modes) In Section IV.l we saw that an electronic state is perturbed when the crystal is strained, and the change AE in energy of the state is linear in the strain e as long as the state is sufficiently different in energy from other electronic states; that is, A E = A e where we expect A to have a magnitude of about one eV per unit strain, Eq. (13). The strain has to transform as the A, representation in the point group of the optical center, if the electronic states are non-degenerate. Suppose now that we have no strain in the crystal, and that we freeze the positions of the atoms in their equilibrium positions in the ground electronic state of the center (at Q = 0 on Fig. 36). Now excite the electron to energy E,. The energy of the electronic state could be reduced by allowing the atoms to move so that they create a strain - e at the defect, giving an energy reduction of -Ae. This strain would correspond to a movement of the nearest neighbor atoms to the center through a distance Q = -u,e, where a, is the interatomic spacing. Increasing the magnitude of the strain would allow the electronic state continuously to decrease in energy. However, the

I

0

Q-

FIG. 36. A two-dimensional sketch of an optical center in a crystal. The energy of an electronic state of the center can be lowered, according to Eq. (13), by moving the atoms outwards through a distance Q, creating a local strain -Q/ao at the center. However, the chemical bonds will be distorted, increasing the elastic energy.

58

GORDON DAVIES

strain distorts the lattice, introducing an elastic energy kQ2, which will constrain the size of the deformation. The equilibrium distortion occurs when kQ2 - UQ is minimized, where for convenience we write u = A/u,. The total energy of the center in its excited state is then V, = E,

+ kQZ

-

UQ = E , + k(Q - uj2k)' - a2/4k

(20)

V, is shown on Fig. 37; it is parabolic in Q and results in harmonic motion of the atoms about an equilibrium point at Q , = a/2k. At Q,,, the center has a total energy a2/4k less than the energy E , at Q = 0; this reduction in energy can be regarded as driving the deformation. In general, all optical centers have atomic vibrations about different equilibrium positions in different electronic states, which can be interpreted as simply saying that the chemical bonding in a crystal is changed when the electronic states are changed. This coupling of the electronic states to the way the center vibrates is referred to as the electron-phonon coupling. Figure 37 is referred to as a configurational coordinate diagram, and relates the total energy of a center to a distortion coordinate Q. This particular figure shows that the center in its round state would vibrate harmonically at an angular frequency n) = 2k/m about the equilibrium point Q = 0. In the excited state the vibrations would occur with the same frequency but about Q, = u / m o 2 . Here m is the mass of the mode, which depends on the type of motion and the definition of Q . For a simple picture we assume that the change in electron density when the transition occurs is confined to within an atomic spacing of the optical center. We could then take Q to be the movement of one nearest neighbor to the center, and m its mass. The Born-Oppenheimer approximation states that the total wavefunction for the center can be factorized into an electronic part d(r), which primarily depends on the electronic coordinates r, and a vibrational part x(Q). In this approximation we write the total wavefunctions in the ground g and excited e electronic states as

,;"

where n, N are the quantum states of the vibration in the ground and excited electronic states. For an electric-dipole transition between these states the transition probability depends on the square of the integral


59

400 200

0 -0.025

0

0.025

Coordinate (nm) Fiti, 37. The single-coordinate configurational coordinate diagram drawn to scale for a vibrational quantum of 30meV, using the mass of one silicon (''Si) atom. Excitation of the electron drives a relaxation shown here as corresponding to a Huang-Rhys factor S = 2.0. For reference, the nearest-neighbour interatomic spacing in silicon is 0.234 nm. The vertical lines show those luminescence transitions nearest the zero-phonon line, which are observed at low temperature. See also Fig. 38.

which can be factorized into an electronic part and a vibrational part

The first integral, which depends on the details of the electronic states, is usually not known; however, it will have some particular value that we will regard as an unknown constant and in the following we will concentrate on the second, vibrational, integral. This second integral simply involves harmonic oscillator states, which are fully understood and so can be readily

60

GORDON DAVIES

evaluated. From the orthogonality of states with different energies we expect that when Q, = 0, r

where bN,”= 1 if n = N and is zero otherwise. The orthogonality breaks down when there is a relative displacement of the equilibrium points of the oscillators, that is, when Qo # 0. We are particularly interested in transitions measured at low temperature, when only the zero-point vibrational states are occupied. For the case of absorption transitions, which start from n = 0, the squared overlap integral is

where the “Huang-Rhys factor” S = a 2 / 2 m h ~ ~has 3 , the physical interpretain units of the quantum of the tion of being the relaxation energy u2/2m~02 vibration k o . Note that the transition probabilities in Eq. (25) have the property that summing them over all the phonons gives

The physical meaning of this equation is that the optical transition is driven by the coupling of the electronic states to the radiation field, and not by the vibrational states. The total transition probability is not affected by this form of vibronic sidebands; it only spreads out the absorption or luminescence over a finite range of photon energies. The appropriate measure of the absorption for use in detailed balance arguments (Section 11.1) is therefore the absorption integrated across the absorption band. Using Fig. 37, the absorption transitions from the n = 0 level in the ground electronic state to the N = 0 level of the excited electronic state occurs at the energy hv, = E , - u2/2mw2;this transition involves no change in phonon number and so is the “zero-phonon” line. From Eq. (25) the intensity of the zero-phonon line is

I, = I, exp( - S)

(26)

1

OPTICAL


61

where I , is the total intensity of the entire absorption band. The one-phonon sideband occurs at an energy tio higher in energy, with an intensity S times larger, from Eq. (25), and the relative intensity of the higher phonon sidebands is given by successive terms in Eq. (25). Consequently, we can use the absorption strength of each feature (e.g., the zero-phonon line) as a measure of the total absorption, and usually it is more accurate to use one well-defined feature rather than attempt to integrate over the extent of the band. In luminescence the same spectrum would be obtained, but with the phonon sidebands displaced to lower energy than in absorption (Fig. 38). The spectra of Fig. 38 have been drawn for S = 3. To estimate the typical values expected we can use the relationship of Eq. (25), S = a2/2mho3,in conjunction with the expected value a 1 eV per unit strain, Eq. (13). Then, with the mass of a Si atom used for m, and tio = 30meV, as is typical for silicon, we obtain S 4. Typical values are about 2 for optical centers in silicon (e.g., Davies et ul., 1987). This simple model describes the spectra when only one vibrational mode is involved. Spectra of this form are sometimes observed (see e.g., Fig. 39). For optical centers in crystals, local vibrational modes fit the description. Local modes are created by a vibration with a frequency higher than the cut-off frequency of the lattice, so that the vibration is not able to propagate into the crystal and is localized in the vicinity of the optical center. The vibrational frequency is well-defined with a width that is usually determined by its lifetime; for example, the local mode may decay by breaking into two lower energy vibrations that can travel into the crystal. In Si, local modes with hru = 75 meV are often observed in luminescence spectra, and the width (in excess of the zero-phonon width) is about 0.5 meV. To identify a local mode sideband of a vibronic sideband we can therefore use the rules

-

-

Luminescence

t:l,

Absorption

I 0

1 5c

L 10

150

Photon energy (mev) FIG. 38. The Poisson distributions of absorption and luminescence lines predicted by the single-coordinate diagram for the same parameters as in Fig. 37.

62

1000

GORDONDAVIES

1010

1030

1020

1040

Photon energy (mev) FIG. 39. An approximation to the single-mode spectrum is observed at a lithium-related center produced by radiation damage in silicon. The transitions are not exactly equi-spaced in energy, as shown by the energy separations at the top of the spectrum. The lines are labelled 0 for the zero-phonon line to 5 for the fifth-phonon sideband. After Canham et al. (1985).

~

980

1000

~~

1020

1040

Photon energy (mev) FIG. 40. The spectra show the one-phonon part of the vibronic sideband of the “Q” photoluminescence band in silicon, whose zero-phonon line is at 1045 meV. Peaks A, B, and C are resonances of the center, superimposed on the one-phonon lattice continuum. The peaks to the left of C are the sidebands involving optic mode phonons. The two spectra are for samples doped with ’Li and 6Li in the ratios as shown, and in each resonance the components move to lower photon energy, that is, higher phonon energy when ‘Li replaces ’Li.The spikes show the predicted distributions in mode A if only two Li atoms contribute to the motion, and in mode B if all 4Li atoms contribute equally (as in a breathing mode). After Canham et al. (1983).

1

OPTICAL


63

that:

a, its intensity ratio relative to the intensity of the zero phonon line is fixed (at the value S ) in all samples regardless of preparation; b. its width must be at least the width of the zero-phonon line; c. higher-order sidebands will be in the intensity ratio given by Eq. (25), and at, or close to, multiples of the local mode frequency from the zero-phonon line; d. for nondegenerate electronic states, the vibrational mode has to transform as the A, representation in the point group of the optical center to produce a first-order perturbation of the electronic states (Section IV); and e. following from (d) its response to perturbations (e.g., uniaxial stresses) must be qualitatively (and, in practice, quantitatively) like the zerophonon line. For a local mode, the frequency must also exceed the cut-off of the host lattice. However, a sharply defined mode of lower energy, a resonance mode, will obey the same rules; an example is shown in Fig. 40 for Li atoms in Si. Local modes are very important when isotopes are used to identify impurities, as we will see in Section VII.4.

2. BANDMODES

So far, we have considered the case where only one mode of vibration interacts with the electronic states. Movements of the atoms near an optical center can also be constructed by a linear superposition of band modes, so that the change by Q, in the equilibrium position of the neighboring atoms can also lead to vibrational sidebands of lattice modes. They are typified by a continuum of modes, as in Fig. 29. If we use the approximation that the energy of the electronic states is changed linearly in the displacement, and the displacement is the sum of contributions from different modes, then the energy of the electronic state is perturbed simply by the cumulative effect of each mode, and Eq. (20) becomes

the summations being over all the modes. We could now draw V, in a multidimensional space with axes Qi and with the equilibrium position

GOKDON DAVIES

64

shifted by ui/2ki in the ith mode. The important point is that the modes are all behaving independently. This allows us to define a Huang-Rhys factor for each mode Si= uf/2rnihw’. We recall from the previous section that this type of electron-phonon coupling spreads out the transition but does not change the integrated absorption or luminescence. For convenience let us take the total absorption or luminescence intensity to be unity. The contribution to the total band from the one-phonon sideband involving the ith mode is, from Eq. (25), Si. The sum of all the one-phonon sidebands allows us to define a total Huang-Rhys factor as S = c i s i .Now consider the two-phonon processes. The two-phonon sideband has a contribution from two quanta of the ith mode, giving an intensity from Eq. (25) of S2/2. There are also two-phonon processes involving the simultaneous creation of one quantum of the ith mode (probability Si) and one quantum in another mode, say, the j t h mode (probability S j ) . The probability of creating these two modes simultaneously is Si x S j of the zero-phonon line. The total two-phonon intensity is then

The final expression is exactly what is expected from Eq. (25) for the two-phonon process using the total Huang-Rhys factor S = Si.Similarly, Eq. (25) applies to the total intensity of any phonon sideband. As the different modes are assumed to be independent of each other here (the harmonic approximation), the combinations of phonons occur statistically just as described. For these random combinations it is reasonable to use integrals rather than summations when dealing with a quasi-continuum of phonon energies. We can define I,(co)dn) to be the strength (relative to the zero-phonon line) of the one-phonon sideband involving phonon frequencies between (n) and n) dw. Then the shape I,(x) of the two-phonon sideband is the random combinations of these modes, giving I , as the convolution

+

Note that the arguments (x - w ) and w of the I , expressions sum to the final frequency x. The intensity of the two-phonon processes can be found by summing all the independent combinations, and is simply S2/2 times the zero-phonon intensity, where S = c i s iis the sum of all the mode-Huang-

1

OPTICAL

MEASUREMENTS OF

POINT

DEFECTS

65

Rhys factors. The generalization is simple: the shape of the nth phonon sideband can be generated by the convolution of the ( n - 1)th sideband and the first sideband, by

IJX)

=

s:'

dwl, l(x - rr,)ll(o) ~

with relative intensity P / n ! times the zero-phonon intensity. If we guess a trial one-phonon sideband we can calculate all the other sidebands by successive convolutions, and hence add them to obtain the predicted total vibronic bandshape. This process can, today, be done trivially using a spreadsheet. Iterative adjustment of the one-phonon sideband then allows convergence on the experimental bandshape. We recall from Section 11.6 that the bandshape as measured in a luminescence experiment is modified by the system throughput, and has to be corrected by using a luminescence source of known absolute spectral intensity. Second, a luminescence spectrum measures (or can be used to derive) the number of photons per second per unit energy range, say L(v) at photon frequency v, which is related to the transition probabilities I(v) used in this discussion by L(v)a= l(v)v3. Figure 29 shows the calculated and measured bandshapes for the Cu S optical center in Si, which exhibits two different configurations (Section IV.5). The purposes of this type of calculation are:

+

a. that it determines whether all the features of the optical band derive from these simple considerations; and b. if the calculations apply, then the relaxation energy xiuf/4ki in the excited state can be calculated. In fact it can simply be measured from the difference between the centroid of the luminescence band and the zero-phonon line. The importance of the relaxation will be discussed in Section VIII. What is happening if the calculated bandshape cannot be made to fit the measured spectrum? One reason may be that the electronic states are degenerate, in which case coupling may occur to degenerate modes of vibration, producing the effects known generally as Jahn-Teller interactions. These are not discussed further here; several reviews have been published; see, for example, works by Sturge (1967); Ham (1972); Engleman (1972); Bates (1978); and specifically the work on diamond by Davies (1981).

GORDONDAVIES

66

3. VIBRONICCOUPLING A second reason for the failure of the theory in Section VII.2 may be that the electronic states are not sufficiently separated in energy from other states, and the vibrational modes may make the states interact. This is the direct analogy of coupling between states in the stress experiments (Section IV.1). In a stress experiment, two zero-phonon states $l,o(r, Q) = 41(r)xo(Q) and $z,o(r, Q) = $2(r)xO(Q)in the notation of Eq. (21) may interact through a static deformation of the crystal, which perturbs the potential energy of the center by A V = Ale where e is the strain and A^ is an operator that couples the two electronic states

where (in this example) the overlap of the vibrational states is taken to be unity. If we now replace the static strain by a vibrational term so that A V = A^Q, the two zero-phonon states are not coupled; the interaction is now r

r

r

and the final integral is zero (as seen by considering the parity of the terms). However, the zero-phonon level of one electronic state can interact with the one-phonon level of the other, for

This interaction is shown schematically in Fig. 41b. In general, the nth vibrational state of one electronic state will interact with the (n+ 1)th state of the other because


12 K

67

B

4.2 K A

6 W

6060

6120

6180

6240

6300

6360

Photon energy (cm-’)

(b)

Excited state

Ill

a

I

b

Ground state

FIG. 41. Photoluminescence from a silver-related center in silicon. In (a), the bottom spectrum, measured at 2K, shows a zero-phonon line A and much stronger one-phonon sidebands F’ and F”, with very weak two-phonon sidebands. This sequence is not consistent with the Poisson distribution of Eq. (25). Increasing the temperature shows the rapid increase in intensity of line A relative to F‘ and F”. The mechanism is outlined in (b). The phonon sidebands F and F are vibronically induced by mixing the zero-phonon state a, from which transitions are allowed as line A, with the one-phonon level b of a state from which transitions are forbidden. After Zhu et al. (1997).

68

GORDON DAVIES

The square root term is closely related to the root-mean-square displacement r of a harmonic oscillator. In the zero-point state, r = JG. Evaluating this using the mass of a Si atom and a typical vibrational quantum for silicon, hw = 30 meV, r is 1.4% of the bond-length between the atoms in a silicon crystal. The instantaneous strains created by the vibrations are considerably larger than the strains used in uniaxial stress experiments (typically less than about 0.2%), so that vibronic mixing may occur between states for which stress mixing cannot be observed. Figure 41 shows a spectrum at 2 K from an Ag-related center in silicon. We apparently see the zero-phonon ‘A’ and the much more intense onephonon sidebands F’ and F”, but the two-phonon sidebands are almost nonexistent and are certainly too weak to satisfy the Poisson distribution of Eq. (25). Heating to 4.2 K produces a strange effect. We have not considered here temperature effects, but it is reasonable to expect that there will be no significant temperature dependence in a vibronic spectrum at low temperature, when the temperature is considerably less than hulk. (We recall that k T = 1 meV when T = 11.6 K.) What is observed is a growth in intensity I , of the “zero-phonon line” A relative to the intensity I , of the one-phonon sidebands F‘ and F”, with the relationship lo/llK exp( -E/kT) where the activation energy E = 0.27 meV (Zhu, 1997). Detailed inspection of the spectra shows a very weak zero-phonon line at this energy below the apparent zero-phonon line A . A simplified energy level structure is shown in Fig. 41b. There are two excited electronic states shown, numbered 1 and 2, with optical transitions from state 1 being allowed. They produce the zero-phonon line A . Transitions from a slightly lower energy set of levels, numbered 2, are forbidden. However, coupling of the zero-phonon level of state 1 (u) to the one-phonon level of state 2 ( h ) through Eq. (33) does allow one-phonon transitions, producing the sideband F’. In the real center, two more excited electronic states exist, producing allowed transitions with zerophonon lines B and C in Fig. 41a.

4.

ISOTOPE

EFFECTS ON VIBRATIONAL SIDEBANDS

When the isotope of an atom involved in the host lattice is changed, the only effects that can be resolved in optical spectra come from the change in nuclear mass because luminescence measurements do not have the resolution to detect nuclear-spin effects. Changing the mass of a mode of vibration changes the vibrational frequency of a harmonic oscillator, as w K 1/&. However the mass m is for the mode, not the atom. If we have one ”C atom vibrating against one 28Si atom, the reduced mass of the pair is 8.4. Increasing the mass by using the isotope 13C changes the vibrational


69

frequency to 0.973 of its original value, giving a change in frequency of only 69% of that expected for the C atom alone = 0.961). In contrast, consider one hydrogen atom vibrating against a 2sSi atom. Changing the 'H to a 2H atom (deuterium) changes the frequency by a factor 0.719, within 95% of the shift expected for the single atom case (1/$ = 0.707) simply because the isotope is partnered by a relatively heavier and so more static atom. The magnitude of the isotope shift for a vibration is always expected to be less than that predicted simply from the change to the single atom. However, isotope shifts can be larger than the square root dependence, for example if the atom is rotating rather than vibrating. For a pure rotator the energy levels are given in terms of the quantum number J by J ( J + l)h2/ (2mr') and so follows a linear reciprocal law in m. The nearly bond-centered 0 in Si is a classic example of an approximation to this (Bosomworth et al., 1970). Figure 42 shows the splitting of a local mode of vibration in the 'T' center, a C-C-H center, in Si as the C mass is changed. The mode splits into four

(J12/13

a

I

800

I

I

I

810 815 Photon energy (meV)

805

FIG. 42. Effect of carbon isotopes on a local mode of the 'T' center in silicon observed in luminescence. In (a) the sample is doped only with "C and in (b) with both 12C and I3C. The splitting into four lines shows that the vibration is produced by the motion of two inequivalent carbon atoms. The peaks move to higher photon energy in '3C-doped material because the phonon energy is reduced. After A. N. Safonov (1996) private communication.

70

GORDONDAVIES

components, which can be identified with the configurations 12C-'2C-H, l2C-l3C-H, 13C-12C-H, and 13C-13C-H. The first and last can be identified from the spectra obtained with pure isotopes. The two intermediate cases show that the C atoms occupy inequivalent sites. The relative intensities of the four lines are expected to obey random statistics as discussed in Section 111.1. We have evidence here, then, for two C atoms in the center, and it therefore contains at least two C atoms. It is possible to have a mode of vibration in which one impurity atom does not vibrate appreciably. In that case, changing the isotope of the impurity has no effect on the frequency, and that impurity is not identified in an isotope experiment. A local mode in the ' G center, a C-C center, in silicon is a very good approximation to this case. The 71.90 meV mode of Si doped with pure "C is reduced by about 1.9 meV when one "C atom is replaced by a 13C, but the reduction is by less than 0.1 meV when the other site is changed (Davies et al., 1983). Failure to observe this small change would lead one to guess that there is only one C atom in the 'G' center: again, the isotope effects alwuys give the lower limit to the number of atoms in the center. Two final comments: First, it is often assumed that the presence of a local vibrational mode involving an atom of the host lattice implies that that atom must be at an interstitial site. This is not so; cases are known of strengthened back-bonds on substitutional atoms at vacancy-related centers which produce local modes (e.g., the 'H2' center in diamond, Jones et al., 1994). Second, resonance modes (involving an impurity atom but at a frequency within the continuum of lattice states) can also produce significant isotope effects (see Fig. 40).

5. ZERO-PHONON ISOTOPEEFFECTS Because the zero-phonon line is the sharpest feature in the optical spectrum, isotope effects are most easily resolved using it. At low temperature, the zero-phonon line is a transition between the zero-point vibrational states of the ground and excited electronic states (Fig. 37). A comprehensive discussion of the effects of changing the isotopes of the defect was provided by Hughes (1966). The essential results can be obtained by considering again the single-mode model of Section VII.l. This model was derived using harmonic vibrations and assuming that the coupling between the electronic states and each vibration is linear in the displacement of the atoms. The vibrational potential in the excited electronic state is given by Eq. (20) as V = E,

+ k(Q - a/2k)'

-

a2/4k

1

OPTICAL MEASUREMENTS OF POINTDEFECTS

71

Here k = mm2/2 is independent of the isotope, as the vibrations are harmonic. Similarly, the electron-lattice coupling term a (which tells us how the electronic energy is changed by a unit displacement in the mode) has no dependence on the isotopes of the atoms. The zero-point vibrational state is at an energy E , - a2/4k + ihw, which does depend on the isotope if it changes the vibrational frequency w. However, there is an equal change in the zero-point energy hw/2 of the ground electronic state. Consequently, in this model there is no change in zero-phonon energy when the isotopes are changed. To obtain isotope effects we must consider higher-order electron-phonon coupling and anharmonic vibrational potentials. The effect of exciting the electron distribution at the center is to change the chemical bonding, resulting in the change in equilibrium positions of the atoms we have discussed. The chemical bonding also determines the “spring constants” between the atoms, and changing the electron distribution therefore changes the vibrational frequencies. Changing the vibrational frequencies makes the zero-point energy hw,/2 in the ground electronic state different from the zero-point energy hwe/2in the excited electronic state. The zero-phonon line is then at an energy hv,

= E , - a2/41c

+ $hoe - ihw,

(34)

that is dependent on the frequencies of vibration and so does depend on the atomic mass. If the effective mass of the mode is m so that changing m results in a frequency change

then the zero-phonon shift is, from Eqs. (34) and (35), hv, - hv,

= i(l - J*)(hwel

-

hwgl)

(36)

where the frequencies are specifically those for isotope 1. Intuitively we expect that the vibrational potential will be softer in the excited state (when the ‘molecule’ is nearer to dissociating), so that w e < a,.Equation (36) then implies that the zero-phonon line will move to higher energy with heavier isotope, as in Figs. 9 and 10. This sign of the shift is usually observed, but not always. When the vibrational mode is a local mode, o r a discrete resonance mode as in Fig. (39) then hoe may be measurable from the luminescence spectrum and hw, from the absorption spectrum; however, as we will see later in this Section, this approach may not give enough information.

12

GORDON DAVIES

The theory generalizes to many modes, as for band modes. With many modes of similar frequencies it is usually not possible to determine the frequency change of each of the modes. However, the temperature-dependence of the zero-phonon energy can then be used. Figure 43 illustrates the idea using a single-mode model for clarity. The vibrational potentials have been drawn so that hw, < hw,. The vertical lines show the zero-phonon luminescence transitions, defined as being transitions that conserve the number of phonons. At OK only the transition at E + 1/2ho, - 1/2hw, is observed. At higher temperature the 3/2ho, state becomes populated as well as the zero-point state, and so a new component is seen in the zero-phonon line at E + 3/2hw, - 3/2ha),. The mean quantum state occupied at tempera-

-;t

400

t

Coordinate (nm) FIG. 43. Configurational coordinate diagram as for Fig. 37 but with the curvature of the potential surface in the excited state reduced to half. The vertical lines show the zero-phonon transitions (defined as being between states of the same phonon quanta), which progressively reduce in energy as higher quantum states are involved.

73


ture T is given by the Bose-Einstein number n = 1/(1 - exp(ho/kT)), and so the mean zero-phonon line is shifted by

Ahv,(T) = n(hop- ha,)

(37)

The temperature dependence therefore contains the information about the difference h o e - iio, required for the isotope effect, as long as the zerophonon line can be observed at temperatures that are high enough for adequate population of the modes of interest. Equation (37) can be generalized to cover all modes at a center by simply summing all the contributions from the different modes. In practice, whenever the temperature dependence of the energy of a transition is considered, one must also take into account the contribution from the lattice expansion. An application of linking the temperature dependence and the isotope shift is given for a transition in 13Cdiamond and ”C diamond by Davies et a! (1997). We have seen that the energy of the zero-phonon line is dependent on the isotope as a result of the changes in vibrational frequency when the electronic state is changed. To include the effect in the vibrational potential we extend Eq. (20) by introducing a term bQ2 so that the total energy of the center in its excited state becomes

V,= E ,

+ kQ2

-

UQ

+ bQ2 = E , + ( k + b)(Q - ___ 2 ( k r h))2

-

U2 ~

4(k

+ b) (38)

The vibrational frequency is now o,= J2o/m, different from the in the ground electronic state-the difference has frequency o,= been caused by the coupling term bQ2, which is quadratic in Q-the “higher-order coupling” referred to earlier in this Section. If the electronic states are nondegenerate, Qz must transform as the A , irreducible representation in the point group (Section IV). Contrast this with the requirement that Q must transform as A, if there is to be a vibronic sideband of the mode. These two requirements do not have to be satisfied by each mode of vibration. For example, at a center with inversion symmetry Q may have odd parity but QZ may have even parity. In general, a mode may produce a contribution to the isotope shift of the zero-phonon line but need not be visible in the vibronic sideband. As a final contribution to the zero-phonon isotope dependence let us consider the effect of anharmonicity in the vibrational spectrum. The mean length of a bond is determined by the anharmonicity of the bond, as is familiar from the thermal expansion of a material. Changing the isotope to a heavier mass reduces the zero-point energy and so produces smaller excursions into the anharmonic region of the potential: the bond is shorter.

74

GORDON DAVIES

Replacing one H atom in an optical center by one deuterium atom will therefore move the atoms closer together, an effect which is equivalent to compressing the center. Because an optical transition changes in energy by a fractional amount proportional to the strain, A E / E -2Aa/a, Eq. (13), the effective compression of the center shifts the zero-phonon line. Note that this process is independent of the different vibrational frequencies in the ground and excited states of the center. In silicon, shifts of hydrogen-related zero-phonon lines of about 0.2 meV have been reported by Safonov et al. (1997a) when H is replaced by deuterium. They write the vibrational potential in the form

-

&

is the root-mean-square displacement in the zero-point where r = state. The anharmonic term shifts the centroid of the zero-point state, and the difference in the centroids when the isotopes are changed is

Here the value for o corresponds to the mass m,. Taking the value ),/hw= to lo-' (Herzberg, 1950), and assuming that the strain is AQ/a,, where a,, is the interatomic spacing in the semiconductor, the arguments given here suggest that replacing H by deuterium would give a zero-phonon shift in the range 0.5-5 meV, if hw 250 meV (typical for hydrogen vibration) and the transition energy E = 1 eV; experimental values of -0.5 meV are observed (Safonov et al., 1997b). Clearly, this mechanism may produce considerable shifts of zero-phonon lines. In reality, the electronic states probably sample a larger volume of the crystal than just one bond length, and so this predicted value is an upper limit to the observed shift.

-

-

VIII. Electronic excited states 1. ELECTRON EFFECTIVE MASSSTATES

The order of magnitude calculation at the end of the previous section could be improved by some knowledge of the spatial extent of the electronic

1

OPTICAL

MEASUREMENTS OF

POINT

DEFECTS

75

states. Optical spectra can give an indication of the nature of the electronic states. In Section 11.3 we saw that luminescence can be generated by above-band-gap excitation. An exciton is created that diffuses to an optical center, and is captured by it. The binding of the exciton to the center is determined by the chemistry of the center and is likely to be unique to that center, so that the photon emitted when the exciton recombines will have an energy characteristic of the center. Let us stay with this picture of bound excitons for a moment. If the emitted photon is near the band-gap energy, as in Fig. 6, then we conclude that the exciton is only weakly bound to the center and it is natural to expect the properties of the bound exciton to be similar to those of the free exciton. The same phonons will be emittedthose which conserve wavevector in an indirect gap semiconductor -with only minor differences. For example, the localization of the exciton will weaken the phonon selection rules, leading to more dispersion in the phonon energies (Thewalt et af., 1976). We can now expect that the properties of the bound exciton can be described accurately by the known properties of the free exciton, perhaps modified by a lower symmetry at the optical center. For example, Thewalt et al. (1979) have discussed the effects of uniaxial stress on excitons bound to the shallow donors and acceptors in silicon in terms of slightly perturbed valence and conduction band states. This picture is valid because in silicon the binding energy of an electron to a hole, to form an exciton, is 14.6 meV, and the exciton is bound by only a few millielectronvolts (Section V.l) to the donor or acceptor; both are small energies compared to the energy gap of 1170 meV. As the binding energy increases it is not apparent that the bound exciton model will remain of use; however, it often provides a framework, or a language, for discussing the excited states of the optical center (Kleverman, 1998). When an optical center produces luminescence at an energy that is well below the energy gap (a “deep” center), it is possible to use the following very simple general description. The electronic structure in the ground state can be taken as a linear combination of the atomic orbitals of the defect atoms and their immediate neighbors. This approach limits the extent of the electronic orbitals to only one or two atomic spacings from the core of the defect. If we excite an electron from that state, the higher-energy state is expected to be more diffuse, just as a 2p orbital in a hydrogen atom is larger than a 1s orbital. In most semiconductors, the permittivity is large. For example, in silicon the relative permittivity is E = 11.7. The excited center can now be pictured as a hydrogenic system in which the core that has been positively charged by the removal of an electron, and the electron orbits the center, attracted to it by a Coulomb force, which is reduced by a factor of I/& compared to the attraction of the charges in a vacuum. Consequently the electron orbits in an effective-mass-like orbital. We have

76

GORDON DAVIES

seen in Fig. 19 that the excited state of a carbon-oxygen center in silicon can indeed by described in terms of an effective-mass particle orbiting an oppositely charged core. As with all general descriptions there are exceptions, to be discussed in what follows, but first we will present some examples in which this description is useful. Continuing with the Si: C - 0 center, it is striking that there is no splitting of the states by uniaxial stresses under compressions along the (1 11) direction (Fig. 19). This suggests the involvement of effective mass states composed from the conduction band minima in silicon, which lie in the (001) directions in wavevector space and so are affected equally by a (1 11) stress. To explore this further we need to look at the way these effective mass states are affected by a point defect. When an effective-mass electron is trapped at a point defect with T, symmetry, the degeneracy of the 6-conduction band minima states is expected to be lifted, on group theory grounds, to form orbital states that transform as an A , nondegenerate state, an E doubly degenerate state, and a T, triply degenerate state. The A, state is the only state with a nonzero density at its center (like a hydrogen 1s state) and so its energy can be substantially different from the E and T, energies. If the center of the defect is attractive to electrons (as for the phosphorus substitutional donor in silicon), the A, state is lowest in energy. For Si: P, the A, state lies 11.6 meV below the E state, which is only 1.3 meV below the T, state (Aggarwal and Ramdas, 1965). If the core is repulsive, the A, state is highest [as at the Li interstitial donor in silicon; the A , state is 1.8kO.l meV above the nearly degenerate E and T, states; (Aggarwal et ul., 1965)J. If stress is applied to these bound states, the matrix describing the perturbation is (Wilson and Feher, 1961)

1

A,

E"

E,

-A(5+4

W,o, -Bo,+

qoF

Bo,

A(l-6) Bog

+

T, 0 0

T, 0

Tx 0 0

0

0

0

0

0

+

0

0

A( 1- (5)

C( - 0 0 + Jia,, A(1 8)

+ +

- C(o, J?oJ

+A(1+6)

I

+

2C0, A(l+6)

(41)

1

OPTICAL

77


In this symmetric matrix, the stress combinations

and oEare given by

r ~ ,

where the stress tensor components s,, are defined in terms of the crystal’s axes, as in Eq. (12). No allowance has been made in Eq. (41) for the hydrostatic stresses. As the basis states of Eq. (41) have been derived from the same parent states (the conduction-band minima), hydrostatic stresses simply shift all the levels without introducing any splitting or interactions, and can be included in the matrix by adding A ( s , syy szz)to each diagonal term. In Eq. (41), 6A is the energy separation of the A, and E states of the bound electron, and 26A is the separation of the E and T, states as a result of their localization on the T, center. The parent, conduction-bandminima states are located equivalently along the (001) directions and are affected in the same way by shear stresses. Consequently, shear stresses do not appear in Eq. (41). The meaning of the parameters B, C, and W, is that a unit compression along the z axis (so that szz= 1) splits the E states by 4B, separates T, from T, and T, by 6C and couples the A, and E, states by 2 W,. As these states are all derived from the same parent states it is possible to link the coefficients, and one finds (Wilson and Feher, 1961)

+ +

B = -C=

-W,/$

(43)

Consequently, there is only one stress parameter entering Eq. (41) and from the properties of a conduction-band electron its value is (Laude et al., 1971) B

= - 7.7 meV/GPa

(44)

When an electron is trapped at a P donor in silicon, the effect of stress on the A, state is described by a value of B, which is 92% of the free electron value; for %:As B is reduced to 80% of the free electron value (Tekippe et ul., 1972). Higher-lying states of the donors appear to have the same stress-response as the conduction band. These results indicate that the localized electron is likely to have a coupling to stress that may deviate by 20% from the free-electron value. Now let us apply these ideas to the problem of an optical center which is electrically neutral and for which the excited state can be described as by a quasi-exciton, consisting of a tightly bound hole (the “ground” state) and an effective-mass electron (the “excited” state). The electron will feel to some extent the local symmetry of the optical center. Formally, when the symmetry is lower than T, we expect the E and T, states to split under

78

GORDONDAVIES

stress. For example, suppose the symmetry is tetragonal, so that the defect has a major axis along one of the (001) directions. We can simulate the reduction in symmetry from T, to DZdby imagining that the only way the symmetry is lowered is by an “internal” stress, which acts along the required axial direction. This model is made respectable by describing the internal stress as an “effective” stress. Figure 19 shows the model at work. The center considered here is known to have monoclinic I symmetry (Foy, 1982), and has stress parameters [defined in Eqs. (14) and (15)] of A , = -13.1,

A,

=

-0.4,

A,

=

1.8, A,

= 0.5meV/GPa

(45)

Note that the A , parameter dominates : the transition is affected primarily by [OOl] stresses directed down the [OOl] axis of the center. We can introduce this axiality at the center by using a (001) effective internal compressive stress. For the carbon-oxygen center the magnitude is found to be 80MPa. Using this value for s,, with all other s,,= 0,together with A = 2.1 meV, h = - 1.05 and B = - 10.8 meV/GPa, Eq. (41) gives the zerostress splitting of the bound electron states in agreement with experiment. Increasing the stress by applying an external stress also gives a very close description of the data (Fig. 19; Thonke et al., 1985). The parameters correspond, in the absence of the internal stress, to an energy separation of the A, and E states of 12.6meV (close to the Si:P value quoted in the foregoing), and a splitting of the T, and E states by -4.4meV compared to + 1.3meV for Si:P. The B coefficient is also reduced to 75% of the free-electron value, within the expected range. In silicon, an effective-mass electron has a binding energy of about 50 meV to a donor. For the carbon-oxygen zero-phonon line, at 789meV, the hole state must be located at about (789 + 50) meV below the conduction band, that is, about 330meV above the valence band. This state has also been measured at 320 k 20meV by photobleaching of the paramagnetic resonance of C - 0’ (Lee et al., 1977), and by transient capacitance measurements at 350 meV (e.g., Murin, 1987). These independent measurements confirm the model for the excited states of the neutral center as being a tightly bound hole and an effective-mass electron. Because the effective-mass-like electron orbits with a radius of many atomic spacings, it samples the surrounding lattice. If there is another defect in that region, it may be energetically favorable for the electron to transfer to the other defect. This process has been observed, as discussed in Section V. 1 (Frens et ul., 1994). For the C - 0 center, we can ignore the effect of stress on the hole and simply use the conduction band states. For other centers a major part of the splitting comes from the tightly bound hole. An example is the Si:S + Cu

1


79

center discussed in Section IV.l. Uniaxial stress measurements are shown in Fig. 20 for the zero-phonon line of the “A” configuration of the center (Jeyanathan et al., 1994). The symmetry is triclinic (i.e., the point group has no symmetry elements except for the identity operation). The fit to the data has been obtained by taking the hole to be tightly bound in a state at the triclinic symmetry, and the electron in an effective-mass state in tetragonal symmetry, The A, state of the effective mass electron is lowest, and the curvature seen under stress corresponds to the E state lying about 15meV to higher energy, very similar to the carbon-oxygen center. Describing the electron by these effective mass states is noncontroversial because the electron is assumed to be distant from the core of the center. Surprisingly, we can sometimes apply the same ideas to the hole. The valence band maximum is 6-fold degenerate, in the absence of spin-orbit coupling, and can be represented by three degenerate T, orbitals. The symmetric matrix describing the effect of stress on T, states in T, symmetry is

T,

X *

T,

Again, hydrostatic terms can be added to the diagonal elements if necessary. Compared to the T, part of the conduction band matrix equation (41) we now have no restriction on including shear stresses. To describe the effect of stress on the valence band (Laude et al., 1971) requires D = 22.6 and E = 52.5meV/GPa. In reality, the valence band maximum is split by spin-orbit interaction, which in silicon is 45meV. For the Si:S Cu transition at (968 50) meV below the conduction 968meV the hole level will be band, giving a strongly localized hole. For simplicity, let us ignore the spin-orbit splitting and construct a localized state directed along the i, j , k axis by simply taking the linear combination (iTZx jTZy kT,,). The stress response of this state can be calculated from the matrix, Eq. (46), and a fit almost as good as that shown in Fig. 20 is obtained with (i, j , k ) = (0.874, 0.32,0.365). Independent optically detected magnetic resonance (ODMR) data placed one important axis of the center along (0.78, 0.353, 0.517), which is only 10” from (i, j , k ) (Jeyanathan et al., 1994). The fit may be fortuitous, but does suggest a simple description of the ODMR axis and stress data in terms of bound valence and conduction band states. In this example of Si:S Cu, the combination (iTZx jT,, kT,,) is an orbitally nondegenerate state. The only angular momentum is derived from

-

+

+

+

+

+

+

+

80

GORDON DAVIES

the spin sh = 1/2 of the hole. This spin couples with the spin s, = 1/2 of the electron to form a pair of excited states with triplet and singlet spin states, as described in Section IV.1. In a pure-spin system, optical transitions from the triplet state to the singlet (no particle) ground state are forbidden by the spin selection rule. Here they are allowed, although the ratio of transition probabilities for the triplet and singlet zero-phonon lines (their relative intensity as would be observed in an absorption spectrum) is indeed very small, 1,/13 = 1/600 (Singh et al., 1989). To understand why the triplet transition is seen at all, remember that the maximum of the valence band in silicon has angular momentum j = 3/2, split from the j = 1/2 valence band by 44MeV. A typical state from the upper valence band is the j = 3/2, inj = 1/2 state which is of the form f l ( T , , iT2y)1 - .J2/3T2, T, where t and 1 denote the two spin states. The j = 1/2, mj = 1/2 from the split-off We valence band, is f i ( T 2 x + iT2y) 1 f n T 2 =t. (Here i is can form the pure-spin state T,, T state by taking the combination m ( 3 / 2 . 1/2) - J1/3(1/2, 1/2). However, to obtain this pure-spin form requires the axiality of the center to be strong enough to overcome the spin-orbit splitting of the valence band. For any finite axiality we can calculate the mixture of the j , mj states and hence calculate the strength of the optical transitions from the “singlet” and “triplet” states, Figure 44 shows the observed transition probabilities for a set of similar, trigonal centers in silicon doped with Li (Davies, 1994). As a measure of the “axiality” of the center, the spectroscopic binding energy has been equated to the ( 1 11) perturbation of the valence band states. As the binding becomes comparable with, and then exceeds, the valence-band spin-orbit splitting, the triplet line becomes progressively quenched. What is striking here is that a very simple model, using the properties of the valence bands of perfect silicon and with no arbitrary parameters, is able to describe approximately the behavior of these real optical centres. On Fig. 44, one set of data is labelled “Q.” It originates at an optical center consisting of 4 Li atoms (Canham et ul., 1983). Tn the ground state, this complex is predicted to have T, symmetry (Tarnow, 1992), in contrast to the C,, symmetry that is observed (Davies et ul., 1984). The lower symmetry can be assigned to the effects of the degeneracy of the valence band maximum. For simplicity, let us ignore spin-orbit coupling in the valence band. The triply degenerate Tz,, T2,, T,, states can then lead to a Jahn-Teller distortion of the center in which the. T, electronic state couples to T, modes of vibration. The effect of the distortion is to relax the T, symmetry into one of the four trigonal distortions obtained by moving one apex of the tetrahedron in or out relative to the center of the tetrahedron. At this trigonal center, the orientational degeneracy (Section IV) is the result

+

fi.)


81

0

4 -1 --. i-:

v 0 ,+

bO

0

3

-2

0

1

1

I

40

80

120

Binding energy (mev) FIG. 44. Points show the ratios l , / I s of the intensities of the zero-phonon lines from the triplet and the singlet excited states in a series of lithium-related centers in silicon. The point for the four-lithium center discussed in the text is labelled “ Q . The exciton binding cnergy is the difference between the free-exciton energy and the observed zero-phonon line. The line is calculated assuming a 44 meV spin-orbit coupling in the valence bands and equating all the binding energy to the axial component. After Davies (1994).

of a T, center spontaneously distorting into an arbitrarily chosen trigonal direction. Different orientations can therefore interact with each other, leading to the stress-induced mixing of states at different orientations (Fig. 25).

2. HOLE-EFFECTIVE MASSSTATES The paradigm discussed in the last section can be reversed so that there is a tightly bound electron and a shallow hole. This situation occurs at several hydrogen-carbon complexes in silicon (Safonov et al., 1995; Safonov et al., 1996b; Gower et ul., 1997). These centers contain one hydrogen atom, leading to a convenient situation in which, in the ground electronic state, the proton is neutralized by one electron. In the excited state, the bound exciton contributes one more electron, which spin-pairs with the first to

82

GORDON DAVIES

produce a spin-zero electronic state, and the hole is left as a single excited particle orbiting the single positively charged core. The hole can be described using the upper valence band only. In the j, mj basis set (referred to the crystal axes) the matrix for stress interactions is

1 1 3 1

12’:1 2’ 2

2’

-2

1

f -Do&

2,

3

1 1 2’ 2

3

1 L 2’ 2

1 -1

2’ 2 - E(s,, - isY,)/,/3 -Do,/2 + i E s x Y / , / 3

1 -3 2’

2

0

\

0.

The symbols are as already defined, and i = Again, the local field of the center can be represented by an equivalent internal stress, which for these centers is of the order of 60 to 100 C P a (Cower et ul., 1997; Safonov et al., 1995), similar to the value for the S i c - 0 center of Section VIII.1. This internal stress splits the j = 3/2 valence band state, producing the doublet zero-phonon structure observed at the centers, and a further externally applied stress then produces an additional perturbation of the zero-phonon states that can be described by simply adding the external stress components to Eq. (47) [Fig. 451.

3. APPLICABILITY OF THE MODELING Most deep centers in silicon can be described using effective-mass excited states. Others cannot be described in this way, and both the ground and excited states appear to be strongly localized on the center. One way of verifying that the effective-mass modeling is applicable is when a Rydberg series of excited states can be observed, but this requires access to the excited states by absorption, or more likely photoluminescence excitation (Section 11.5). Another is that perturbations of the states are explicable in terms of the free-particle states. Yet another suggestion that the effective mass approach may be valid comes from the radiative lifetime z of the luminescence. Longer radiative decay times are expected for excited states in which the particles are widely separated in space (one highly localized, one diffuse) compared to centers with both particles in compact orbitals. For example, the values of z for the allowed singlet transitions of centers discussed in this chapter that have one diffuse particle are: for the Si:Cu + S center (977 meV zero-phonon line) z = 5ps, Brown and Hall (1986); for the C - 0 center (789 meV zero-phonon line) z = 14 & 2ps, Bohnert et al. (1993); for the similar


83

940 936 932

5

h

940

W

%

c a, c

936

3 0 932 c

n

r

I

940 936 932 0

80

160

Stress (M Pa) FIG. 45. A carbon-hydrogen center in silicon produces the “T” center with a pair of zero-phonon lines near 936meV. The points show the effect of compressive stress applied along (a) the (001) axis, (b) the (111) axis, and (c) the (110) axis. The lines are calculated using a matrix of the form of Eq. (47). The valence band maximum is split by the local effective strain field of the center, and the two zero-phonon lines are transitions between the shallow valence-band-like hole states and a tightly bound electron. After Safonov et al. (1995).

‘ Pcenter (767 meV zero-phonon line) z 5 p , Bohnert et al. (1993); for the N

Li-related ‘Q’ center (1045 meV zero-phonon line) T = 9.5ps, Lightowlers et al. (1984); and hydrogen-related damage centers (zero-phonon lines at 1147 to 1137meV) T = 12 to 4 p , Kaminskii et al. (1996). In contrast, some strongly luminescent centers have much shorter lifetimes, such as the family

84

GORDONDAVIES

of radiation-damage centers whose precursor, the “W” or “11” center, has its zero-phonon lines at 1018.9meV with 5 = 28 rt_ 2 ns (Bohnert et al., 1989), with states that do not respond to perturbations in the way expected for shallow states.

IX. Zero-phonon Lineshapes We noted in Section 11.2 that a zero-phonon line is broadened into a Lorentzian lineshape at high temperature. As T+ 0, the thermal broadening is frozen out, leaving a zero-phonon line with a nonzero width. In this limit both the width and the shape of the zero-phonon line provide valuable information on the environment containing the optical centers. The observed zero-phonon line is the sum of the contributions of many optical centers, where “many” is 10” for a concentration of centers of loL6 cm-3 and an effective volume determined by excitation that is perhaps lOpm deep and l m m in diameter. Each optical center is in a slightly different environment to all the others, resulting in the zero-phonon line being at a unique perturbed energy. It is the envelope of all the transitions, which is observed as the zero-phonon line (Fig. 46). The origin of the perturbations can be any of the mechanisms that perturb optical transitions, but as we saw in Section IV strains are particularly important. The strains may be derived from a bad mounting of the sample. For example, the sample may be held by a metal clamp that is gently tightened at room temperature but tightens considerably as a result of differential contraction on cooling. This source of stress is detectable by the irregular (asymmetric) lineshape it produces. However, even if this type of spurious stress is avoided there are still strains inside the crystal as a result of the presence of defects such as impurities (other than the optical centers). For a given sample, the background strains place a lower limit on the resolution possible by orthodox optical measurements. Small perturbations can be detected by lock-in techniques when the perturbation can be modulated, as for electric-field perturbations (Section IV.2). Considerably higher effective resolution can be obtained if it is possible to “burn” a “hole” in the zero-phonon line (Kharlamov et al., 1974; de Vries and Wiersma, 1976; Howley et al., 1984). In this technique, a laser is tuned to a wavelength within the range of wavelengths covered by the zero-phonon line. The laser may reduce the optical absorption in the line, for example by ionizing those optical centers that absorb at that wavelength (see Fig. 46). Ideally the width of the hole will be determined by the convolution of the laser linewidth (say, 100 MHz = 4 x meV) and the lifetime of the excited state of the center


0

20

40

60

80

85

100

Energy FIG. 46. Representation of an observed zero-phonon line (outer envelope) as thc sum of individual components with slightly different peak positions. Each component could represent an optical transition at one optical center.

meV for z 3 s). For diamond, where optical transitions ( M, and the force constants are not changed, there are no localized modes

1 MODE

FREP

m ' = 2 0 omu

1

u(m')

I

5

0.01

254.7

0.24

308

o.o

+dTI

447

127

+

205

O'O

1' 1

I. . I

24

204

25

25

.I. 1 .l. -1. .t 1' '1 1 1 'I'

47

47

LOCAL MODE

LOCAL MODE

48

48

760

a95

. .. . .

' 4 ,

,+.

.. . .... . .

FIG. 2. (a) Diatomic linear chain model containing a light element impurity that substitutes for an atom on the light atom sublattice (m'<m) along with its eigenvectors. The highest frequency mode is the local mode of the impurity. (b) Diatomic linear chain model containing a light element impurity that substitutes for an atom on the heavy atom sublattice ( M ' < M ) along with its eigenvectors. The highest frequency mode is the local mode of the impurity and also a gap mode separates from the top of the acoustic branch. (From Barker and Sievers, 1975.)

3 VIBRATIONAL SPECTROSCOPY OF LIGHTELEMENT IMPURITIES

159

IMPURITY ION

rn'

c=I-rn'/m w,oc

n-25

.I95

376.7

dm')

L-~--J

n

0.66

.

...

6 ,

i

FIG. 3. The eigenvectors of the localized mode for decreasing values of the impurity mass rn'.As m' is decreased, the mode becomes more localized. (From Barker and Sievers, 1975.)

for the one-dimensional chain. In real crystals and for other models, there are resonant modes that give rise to broad absorption features in the lattice bands. Examples of impurities that gives rise to such modes are P, Ge and As in Si (Barker and Sievers, 1975, and the references contained therein). The broad absorption features do not usually lend themselves to the observation of sharp vibrational features from which much structural information about the defect can be determined. While calculations for one-dimensional chain models have been taken much further, the important points for the purposes of this chapter have been made. If an atom with mass m or M is replaced by a lighter impurity, localized modes appear. It is the localized vibrational modes and the information they provide about a defect's structure and properties that are of primary interest in this chapter. As we will see from the examples to be discussed, the localized modes give narrow vibrational lines, allowing shifts from isotopic substitutions or additional perturbations to be resolved. Of course the one-dimensional chain model discussed here is oversimplified and the guidelines already presented are sometimes violated in real crystals, for example, when the force constants in the vicinity of the impurity are changed. Also, there are a few important examples of resonant modes that lie in frequency regions where the density of lattice modes is small, giving rise to sharp quasi-localized modes.

160

MICHAELSTAVOLA

The linear chain model with a mass defect applies most directly to an isolated substitutional impurity in a crystal. However, a great number of important impurities and defect complexes with lower symmetry have been fruitfully studied, and limiting consideration to substitutional defects is too restrictive. An impurity in a crystal has three vibrational degrees of freedom. For substitutional impurities in a tetrahedral lattice, motions along x, y and z are equivalent and the vibrational modes are triply degenerate. If the symmetry is lowered, this degeneracy is lifted and more than one vibrational line is observed. For example, a light impurity in a trigonal center has two vibrational modes, a nondegenerate mode (A,) in which the atom motions are along the three-fold axis of the center, and a doubly degenerate mode (E) with atomic motions perpendicular to the trigonal axis. For an impurity at a site with lower than axial symmetry, there are three nondegenerate modes. The linear chain models already discussed show that the vibrations of a light impurity are localized at the impurity and its nearest neighbors. Therefore, as a good approximation, the vibrations of a light element impurity and its neighbors can be viewed as the vibrational modes of a “defect molecule.” Thus, the mathematical tools and symmetry considerations used in molecular spectroscopy (Herzberg, 1945; Wilson el al., 1955) are also relevant to the vibrations of defects in semiconductors. Simple molecular models can be of great help with the interpretation of experimental data. Little group theory will be used in this chapter, although vibrational modes will occasionally be labeled by their symmetry designations.

2. INFRARED ABSORPTIONSPECTROSCOPY Fourier transform spectrometers (Griffiths and de Haseth, 1986) with sensitive, cooled infrared detectors are now widely used for local mode studies. The practical aspects of measuring local vibrational modc spectra have been discussed (Newman, 1990a; Murray and Newman, 1989a). Fourier transform instruments have much greater sensitivity than dispersive instruments and the high resolution that is possible with commercial spectrometers has allowed the shifts in impurity modes that result from a change in the isotopic mass of the nearest neighbors to be resolved. A resolution of 0.1 cm-’ is adequate for most local vibrational mode studies in semiconductors, although much higher resolutions are readily available with commercial instruments and have been used in some studies. Semiconductor samples are typically cooled to liquid N, or liquid He temperatures for local mode absorption measurements. At low temperature, a vibrational line narrows and its pcak absorption coefficient increases, assuming the strains in the crystal are small. Thus, the sensitivity of the


161

measurement is increased and smaller frequency shifts can be resolved. The sensitivity of local mode absorption measurements depends strongly on the defect being studied and on the spectral range into which the vibrational mode falls. However, as a rough guideline, local vibrational modes can be seen for concentrations of the order of 10'4cm-3 in a 1 cm thick bulk sample, or 10'Hcm-3 in a 1 pm thick epitaxial layer. For a few impurities, lower concentrations can be detected. For exampIe, the sensitivity limit is near lo" cm-3 in a 1 pm thick layer for the hydrogen vibrations of hydrogen-containing complexes because of their high frequency and the availability of sensitive detectors in the appropriate spectral region (near 2000cm- '). As another example, Newman (1990a) has quoted a sensitivity limit of 3 x 10'' c m p 3 for Si,, in a 1 pm thick GaAs layer. Not all impurity vibrational modes are infrared active. The transition probability for an electric-dipole allowed optical transition between vibrational states is proportional to the square matrix element of the electric dipole moment of the center 12.pif12. Here 2 is a unit vector in the direction of the polarization of the light and p i f is the matrix element of the electric dipole moment of the center (which is a function of the coordinates of the defect atoms) between the initial and final vibrational states. For harmonic vibrations, a mode is infrared active if the motions of the defect atom or atoms give rise to a change in the dipole moment of the center, that is, if there is a nonzero dipole-moment derivative with respect to the vibration's normal mode coordinates. Consider the simple example of a linear defect of the form X-Y-X. The symmetric vibrational mode in which the X atom is stationary and the Y atoms vibrate 180" out of phase is infrared inactive because there is no net oscillating dipole moment. The antisymmetric mode in which the Y atoms vibrate in phase with each other and opposite to the direction of X has an allowed electric dipole transition. Furthermore, the vibrational mode will only be excited by incident light that has a component of its electric vector polarized along the direction of the oscillating dipole moment. For the simple example of the antisymmetric stretching mode of a linear defect molecule, the direction of the oscillating dipole moment (i.e., the transition moment) is along the axis of the defect molecule. The integrated absorption coefficient for a defect-absorption band is well known to be proportional to the concentration of the defect. Dawber and Elliot (1963) have used the Green's function method to derive the following commonly used expression for the integrated absorption coefficient for the local modes of an oscillating defect with mass m and an effective charge q :

s

nq2N myc'

a(o)do = -

Here N is the concentration of defects in cm-3, and y is the refractive index

162

MICHAELSTAVOLA

of the host. This equation has been written in CGS units to be consistent with the absorption coefficient, which is conventionally determined in units cm-’ and the frequency 0 given in wavenumber units, ern-'. Equation (2) is similar to Smakula’s equation (Dexter, 1958) and has been used in semiconductors without a local field correction. It has been recognized that the effective charge q is not the static charge on the vibrating impurity and is not simply calculated (Leigh and Szigeti, 1967, 1968; Newman, 1973). Instead, the effective charge is related to the derivative of the dipole moment with respect to the normal coordinate of the vibrational mode, as was discussed in the preceding paragraph. This point is underscored by the empirical result that a neutral impurity can have a greater effective charge q than a charged impurity (Newman, 1973). If there is no independent determination of the defect’s concentration to calibrate the absorption strength, a rough estimate of the concentration can be made with the approximation q2 = e’, where e is the electron charge in esu. When making this approximation, it is important to recognize that the error in the estimate of N is typically a factor of about two to four but could easily be much larger. To determine a more accurate concentration for a defect from vibrational spectroscopy, there must be an independent calibration to relate concentration to absorption strength. A variety of methods have been used, including secondary ion mass spectrometry (SIMS), Hall effect, activation analysis, etc. Typically, the calibration is given in terms of the effective charge q in Eq. (I), which is quoted as a multiple of the electron charge. The effective charges for a variety of impurities in GaAs have been tabulated by Newman (1993) with values ranging from about l e to 3e. Alternatively, the factor A in the equation

N=A

s

a(cr)do

is given (in units cni-’). In addition to the measurement error associated with the calibration method, a common problem is that a technique like SIMS measures the total concentration of an impurity even though this impurity might be involved in several defect structures, only one of which contributes to the infrared absorption band being studied. The difficulty in obtaining an accurate calibration is apparent in Bullis’s review (1994) of the determination of the concentration of oxygen in Si by infrared spectroscopy, which shows that the calibration factor has fluctuated by a factor of roughly two since 1964 when the ASTM standard was first established. Thus, while calibration factors have been reported for many defects, inconsistencies between different determinations are often as large as a factor of about two.

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VIBRATIONAL SPECTROSCOPY OF LIGHTELEMENT IMPURITIES

163

IN 3. FREECARRIERS A N D CHARGE STATE EFFECTS DOPEDSEMICONDUCTORS

Semiconductors can be doped to have a wide range of free carrier concentrations, and in many cases, it is the vibrational modes of the dopant impurities themselves that are of interest. The free carier concentration and Fermi level position can have several effects on the vibrational bands, which are discussed briefly in the following. a.

Free Currier Absorption

Free carriers in doped semiconductors give rise to a broad continuum absorption that can make semiconductor samples opaque in a frequency range of interest for high free carrier densities. The absorption coefficient due to free carriers is proportional to co-2 (Moss, 1959) and thus, is especially important at low frequencies. Cooling a sample to low temperature sometimes reduces the continuum absorption by freezing out the free carriers, but in this case the electronic absorption associated with the bound carriers can be strong and can interfere with vibrational absorption measurements. A few methods have been used to reduce the free carrier concentration. (See Newman, 1973, for a review.)

1. A strategy commonly used in recent work has been to use electron irradiation to introduce compensating deep defects. This approach was introduced in an early local mode study of Si doped with high concentrations of P and B by Smith and Angress (1963). In III-V semiconductors, Spitzer et al. (1969) used electron irradiation to compensate Si doped GaAs. 2. An alternative approach is to diffuse Cu or Li into samples to reduce the free carrier absorption. This method was used by Balkanski and Nazarewicz (1964) in early studies of B-doped Si and by Lorimor and Spitzer (1966) in studies of Si-doped GaAs. 3. Another recent strategy has been to passivate dopant impurities with hydrogen. With all of these methods, the compensating impurities or defects may interact with the impurity being studied. Therefore, one must carefully determine what these interactions are to separate the effects of the compensation method from the properties of the impurity in the uncompensated sample. In many cases there are strong interactions and it is the defect

164

MICHAELSTAVOLA

interactions and the resulting complexes that are studied. Numerous examples have been reviewed elsewhere (Newman, 1973, 1993; Barker and Sievers, 1975). In other cases, there is essentially no interaction between the compensating defect and the impurity of interest, an example of which is Si in GaAs (Spitzer et a/., 1969; Newman, 1993, 1994). h. Funo Resoncmcv

When a sharp absorption line is in resonance with a broad continuum absorption, and there is coupling between these, Fano (1961) showed that the absorption line shape could be modified. This effect, first studied for atomic transitions, can also be important in semiconductors. If a local vibrational mode occurs in a region where there is also free carrier absorption, the absorption line can be shifted and its line shape modified. This effect was studied in detail by Raman spectroscopy of the B local mode in heavily doped Si (Cerdeira et ul., 1974; Chandrasekhar et a!., 1980). The effect of Fano resonance on the line shapes of local modes has also been observed in several infrared absorption experiments. Spectra of Be doped GaAs are shown in Fig. 4, which compares the line shapes observed for the

I

I

I

476 480 484 WAVE NUMBERS (em-') FIG. 4. Spectra measured at 4 . 2 K for a Be-doped GaAs epitaxial layer with [p] = 3.7 x 10"cm- ', Spectrum (a) was measured for an as-grown sample and (b) was measured for a sample irradiated with 2 MeV electrons. (From Murray et d., 198Ya.)

3

VIBRATIONAL SPECTROSCOPY OF

LIGHTELEMENT IMPURlTlES

165

Be local mode before and after the sample was compensated by electron irradiation (Murray et al., 1989a). Fano resonance will not be important in the examples to be discussed in this chapter, and is included here only to complete the discussion of the effects of free carriers. However, it is important to recognize that in heavily doped semiconductors, the vibrational lines can be shifted and broadened. And the asymmetric Fano lineshape that dips below the baseline of the spectrum can make the area of a vibrational band impossible to determine accurately. c.

The Ef'ct cf Charge State

Many of the defects of interest have levels in the semiconductor band gap and hence more than one charge state. Not surprisingly, a change in the electronic charge of a defect changes the strength of the bonds and hence the vibrational frequencies. For shallow impurities with their delocalized wavefunctions, a change in charge state has little effect on the vibrational mode frequencies. For example, there is only a small shift of the Be local mode in GaAs shown in F i g 4 upon a change in charge state, and in this case the shift is due primarily to the Fano resonance interaction. However, for deep level defects, with their more localized states, the shifts of the vibrational lines are often large, that is, tens of cm-', and are easily observed. An early example is the oxygen-vacancy complex in Si with local mode frequencies of 835 and 884cm-' for the (0-V)' and ( 0 - V ) - charge states, respectively (Newman, 1973). Thus, the local mode frequencies of a deep level defect can be modified by a change in the defect's charge state that results from a change in the Fermi level position, or thermal ionization, or photoionization. Such charge-state-related changes in the vibrational frequencies lead to an unusual method to monitor a deep defect's charge state, that is, from the defect's vibrational spectra.

4. RAMANSPECTROSCOPY Raman spectroscopy is complementary to infrared absorption spectroscopy, and in some situations, can have important advantages. A review of the use of Raman spectroscopy for the characterization of semiconductors has been written by Prevot and Wagner (1991). For Raman scattering measurements of impurity local modes, a monochromatic laser beam is incident on a sample. Typically, the scattered light is dispersed with a double or triple monochromator and detected with a multichannel detector. For

166

MICHAEL STAVOLA

absorbing semiconductor samples, backscattering geometries are used. In addition to the elastically scattered light, there is also light that has been inelastically scattered by the excitations of the crystal. Measurements are usually made at low temperatures and only Stokes scattering processes that excite impurity vibrations have appreciable intensity. In this case, the frequency of light us scattered from local mode vibrations satisfies the relationship

where uLand wLVMare the laser and local mode frequencies, respectively. Thus the local modes can be detected from their Raman shifts. In Raman scattering experiments, the scattered light with the same polarization as the incident beam, or with perpendicular polarization, can be measured. Raman scattering is typically a weak process. Two things make it possible to study the local mode vibrations of impurities. The first is the use of multichannel detectors to analyze the scattered light, which greatly increases the signal to noise ratio (S/N) of Raman measurements when compared with the use of a scanning monochromator with photon counting detection. Second, if the exciting or scattered light has the same energy as an electronic excitation of the host crystal, then the Raman scattering intensity can be resonantly enhanced. Experiments are typically performed with incident photon energies in resonance with high-energy band gaps of the host semiconductor. For GaAs (Ramsteiner et al., 1988), the vibrational modes of several acceptors show resonance enhancement for incident photon energies approaching the E, gap (1.9-2.7eV). The SiGa donor in GaAs shows resonance enhancement in a more narrow range at higher energy (3.0eV, very near the E, gap energy). Similar behavior has been observed for other III-V semiconductors (Wagner et al., 1991). Raman measurements of impurity local modes can be made in very thin layers, that is, 10 to 100nm depending on the excitation energy, and are therefore ideal for studying heavily doped epitaxial layers and delta-doped structures. For infrared absorption experiments performed on heavily doped samples, the free carrier concentration must usually be reduced by some compensation scheme, as was already discussed in Section 11.3. For Raman scattering experiments, free carrier absorption is unimportant for the small penetration depth of the incident light, although lineshapes are sometimes modified by Fano resonance effects (Section 11.3). The Raman intensity for a local mode must be calibrated for each impurity from measurements made on a sample with a known impurity concentration. For example, the vibrational bands of Si-related donor and acceptor centers in GaAs have been calibrated and infrared absorption or Raman measurements can be

3

VIBRATIONALSPECTROSCOPY OF LIGHTELEMENT IMPURITIES

167

used for quantitative measurements (Murray et al., 1989b). The detection limit for Raman scattering is typically lo1* ~ m - ~ . Raman scattering is a second-order process in perturbation theory and is subject to different selection rules than electric-dipole allowed optical transitions. The infrared absorption strength is proportional to the square matrix element of the dipole moment of the center, whereas the Raman intensity is proportional to 12,. R-2J2,where R is the second-order Raman tensor and 2, and P, are unit vectors in the polarization directions of the scattered and incident light (Prevot and Wagner, 1991). Thus, Raman spectroscopy can provide information about vibrational modes that are infrared inactive. Furthermore, both polarized and depolarized scattering can be measured in Raman experiments, and provide information about the symmetries of the vibrational modes.

111. Isotope Shifts

The effect of isotopic substitutions on the vibrational frequencies of a defect often makes it possible for the defect atoms and their neighbors to be unambiguously identified. This is one of the primary strengths of vibrational spectroscopy. The isotopic shifts also provide information from which detailed structural parameters, for example, bond angles, can be determined. Recently, the ability to engineer the isotopic constitution of the host crystal has been attained (Haller, 1995), giving unprecedented control of hostrelated isotope shifts. For a light impurity, it can be seen from Fig.2 that the motions of the impurity and its nearest neighbors are involved in the local mode vibration. The following commonly written expression provides a simple way to account for the effect that both the mass of a light impurity and the mass of its neighbors have upon the vibrational frequency (Leigh and Newman, 1988; Leigh et al., 1994; and references contained therein):

Here m is the mass of the impurity, M , is the mass of the nearest neighbors, and X is a parameter that is typically near 2 (Newman, 1973). In the following, several examples are discussed in which frequency shifts due to isotopic substitutions of the defect atoms or host atoms have been used to identify a defect, its site location, and nearest neighbors.

168

MICHAELSIAVOLA

1. OXYGEN I N Si AND Ge Oxygen in silicon is one of the very early examples of an impurity in a solid studied by vibrational spectroscopy. Impurity and host isotope shifts have played an essential role in identifying the vibrational modes and determining the atomic structure. Kaiser et al. (1956) correlated an optical absorption band at 9 pm, measured at room temperature, to the concentration of oxygen in Si grown from a melt contained in a quartz crucible. In this early work, a model was proposed in which the oxygen occupies an interstitial position, bonded to two Si neighbors, similar to the Si-0-Si unit in quartz. The 9 pm absorption band was explained as a Si-0 vibration. This model for interstitial oxygen has been confirmed in numerous subsequent studies except that the S i - 0 - 9 bond angle is larger than was originally suggested. [See the reviews by Newman (1973) and Pajot (1994 and 1996).] In this same paper (Kaiser et ul., 1956), an absorption band observed in Ge near 11.6 ,um at room temperature was assigned to the Ge-0 bond stretching vibration. It is now well known that there arc three strong absorption bands due to near liquid He interstitial oxygen at 29 cm-', 518 cm -', and 1136 cm temperature (Newman, 1973; Pajot, 1994, 1996). There are also several overtone and combination features that have been studied. This section focuses on the 1136 cm-' and 29 c m p l bands, which have been assigned to the antisymmetric strctching and rotational/bending-like vibrations of a nearly linear Si-0-Si "defect molecule," and what can be determined from the frequency shifts to which oxygen and silicon isotopic substitutions give rise. The corresponding oxygen vibrational modes in Ge will also be discussed. The effect of oxygen isotopic substitutions on the vibrational frequency of the antisymmetric stretching band was first observed by Hrostowski and Kaiser (1957). A spectrum of silicon enriched with "0 and ''0 is shown in Fig. 5. The frequency shifts observed for the different oxygen isotopes unambiguously confirm that this is a vibrational motion of oxygen. Additionally, two weaker lines are observed for each oxygen isotope to the low-frequency side of a strong absorption line. Pajot and Deltour (1967) attributed this vibrational structure to the effect of the naturally abundant isotopes of Si (2RSi,92.23%; 29Si, 4.67%; and 30Si, 3.10%) upon the vibrational frequency of the Si-0-Si unit. The frequencies of the lines associated with different isotopes of 0 and Si are given in Table I. The relative intensities of these lines are consistent with the natural isotopic abundances of Si, and with two Si atoms being involved in the vibrational mode.

3 VIBRATIONALSPECTROSCOPY OF LIGHTELEMENT IMPURITIES

1070

1110

I090

1130

169

1150

WAVENUMBER (cm-1) FIG.5. Spectrum of the antisymmetric stretching band of interstitial oxygen in Si measured at liquid He temperature. The sample was enriched with " 0 and "0. (From Pajot et a/., 1995.) The inset shows the configuration of interstitial oxygen and the motion of the atoms for the antisymmetric stretching mode.

At elevated temperatures, thermally activated vibrational bands appear to the low-frequency side of the 1136 cm-' band. These hot bands result from the thermal population of a low-frequency mode whose first excited state is at 29.3 cm-'. Bosomworth et al. (1970) performed far infrared absorption experiments (25 to 50cm-') to study this low-frequency mode as a function of temperature and deduced the essential features of the currently accepted TABLE I

FREQUENCY (cm- * ) OF THE ANTISYMMETRIC STRETCHING MODEOF INTERSTITIAL OXYGEN IN SILICON (xSi-yO-zSi) MEASURED AT LIQUID He TEMPERATURE FOR DIFFERENT 0 AND si ISOTOPES (Pajot, 1995) Si Isotopes

l " 0

1 136.4 1134.5 1132.7 1130.8 1 129.2

170

1109.5 1107.6 1105.8

*O 1085.0 1083.0 1081.2

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MICHAELSTAVOLA

structural model of interstitial oxygen. The 29.3 c m p l line is an excellent example of a quasi-localized mode that can be observed even though it lies in the band of Si host vibrations. The substitution of l80for "0 shifted the low-frequency mode to 27.2 cm - ', unambiguously confirming its assignment to oxygen. At elevated temperatures, several transitions originating from these low-lying levels were observed in the far-infrared spectrum. A model potential was fit to these data and the low-frequency lines were assigned to energy levels arising from the anharmonic rotational/bending motion of a nearly linear molecule. An important conclusion of this work is that the Si-0-Si center is nearly linear. The potential surface for this low-frequency transverse motion has a minimum displaced only 0.22 A from the Si-Si axis, which implies a Si-0-Si bond angle of 2c( = 162". A similar value for the bond angle for the interstitial oxygen center was also determined from the isotopic structure observed for the antisymmetric stretching band (Fig. 5). Pajot and Cales (1986) treated the Si-0-Si unit as a defect molecule with C,,, symmetry and applied the results for the isotopic frequency shifts of a molecule of the form Y,X derived by DeWames and Wolfram (1964). Upon an isotopic substitution, Yci)for Y or X'i) for X, the frequency shift for the antisymmetric stretching mode, from w 3 to wy) is given by

Here m, is the oxygen mass, m,, and 2u is the Si-0-Si bond angle. To account for the effect of the host, Pajot and Cales (1986) took rn:) to be an effective Si mass, m&' + M ' , where rn& is the isotopic mass of Si and M' is an "interaction mass" that was also used as a parameter to fit the data. From a fit to the frequencies of the isotopically shifted lines for the antisymmetric stretching mode alone, a bond angle of 164" was obtained, in agreement with the conclusions of Bosomworth et ul. (1970). Several groups have performed theoretical calculations for the vibrations of oxygen in Si, reviewed recently by Pajot (1994, 1996). The rich isotopic structure observed experimentally provides an important input to phenomenological models and a critical test for first principles calculations. The recent theoretical model by Yamada-Kaneta et al. (1990), takes full advantage of the isotope data that are available to refine the model of a nearly linear molecule originally proposed by Bosomworth et al. (1970). First principles calculations for interstitial oxygen by Jones et ul. (1992) have given values for the 0 and Si isotope shifts for the antisymmetric stretching mode that are in good agreement with experiment.


171

The behavior of oxygen in Ge is similar to that of oxygen in Si discussed in the preceding, but with a more complicated vibrational spectrum (Pajot and Clauws, 1987). A spectrum is shown in Fig.6 for the antisymmetric stretching mode; Ge has 5 isotopes with natural abundances, 70Ge, 21.23%; 72Ge, 27.66%; 73Ge, 7.73%; 74Ge, 35.94%; and 76Ge, 7.44%. The possible combinations of Ge isotopes in a Ge-0-Ge defect molecule give 11 distinct vibrational bands. (Combinations of Ge isotopes with the same average mass for the two Ge atoms have the same frequency, that is, 72Ge-0-74Ge has the same frequency as 73Ge-0-73Ge.)An analysis of the dependence of the vibrational frequencies on the Ge mass with Eq. (5) yielded a Ge-0-Ge bond angle of 2a = 140" (Pajot and Clauws, 1987). Similar to oxygen in Si, there are also thermally populated lines associated with the antisymmetric stretching mode due to coupling with a low-energy excitation. Spectral

Wave number (ern-') FIG. 6. Spectrum of the antisymmetric strctching band of interstitial oxygen in natural Ge. The vibrational bands are labeled by the average mass (in amu) of the two G e atoms in the Ge-0-Ge defect molecule. (From Mayur et al., 1994.)

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MICHAELSTAVOLA

congestion makes a detailed study of the thermally populated bands impractical in natural Ge. Samples containing nearly a single isotope of Ge, that is, 98% 74Ge, led to a great simplification of the oxygen spectrum. Khirunenko rt d.(1990) were able to study the vibrations of 74Ge2160,without interference from the other Ge isotopes. These authors were able to conclude that the thermally populated bands were similar to those observed previously for oxygen in Si and to estimate the energies of the low-frequency rotational levels of the off-axis oxygen impurity that couple to the antisymmetric stretching mode. Mayur et ul. (1994) have performed a detailed study of the antisymmetric oxygen stretching mode in monoisotopic G e crystals grown for several Ge isotopes. These authors have found a Ge-0-Ge bond angle of 2a = 111" from the frequencies of the various isotopically shifted lines and Eq. (5), in reasonable agreement with the previous result of Pajot and Clauws (1987). In the absence of the spectral congestion characteristic of natural Ge, Mayur rt ul. could study the temperature dependence of the hot band intensities in detail. Spectra of the antisymmetric stretching mode of 0 in monoisotopic Ge are shown in Fig.7. A thermally populated band is present even at T = 2.0 K because of the small energy of the oxygen rotational mode. From the temperature dependence of the areas of the thermally populated bands, the energy levels arising from the low-frequency rotational mode, shown in the inset in Fig. 7(b), were determined. In this case, the first excited state has a frequency of only 0.2 meV (1.6 cm- '). The important point illustrated by these studies of oxygcn in Ge is that the ability to engineer the desired

I

300

'

-

"

~

I

'

'

~

~

I

~

T=9.2Kt i t I I v,=l

-

I IIIIIIV

Frequency (cm-1)

FIG. 7. Spectra of the antisymmetric stretching band of interstitial oxygen in monoisotopic "Ge. The rotational energy levels that give rise to the tllermally populated lines are shown in the inset in the T = 9.2 K spectrum. (From Mayur ef d.,1994.)

~

~


173

isotopic content of a host crystal permits experiments to be done that would otherwise be impossible.

2. IMPURTIES I N COMPOUND SEMICONDUCTORS Host isotope shifts were first identified for oxygen in Si (Pajot and Deltour, 1967). The availability of comparable data for compound semiconductors (Theis et al., 1982) represented an important advance for the determination of defect structures in these materials. Since that time, the isotope shifts due to first nearest neighbors in compound semiconductors have been used to identify the site of a defect, its symmetry, and configuration. A selection of examples, some classic and a few more recent, that illustrate what can be learned from isotope shifts of the vibrational frequencies are discussed in what follows. Numerous studies of the vibrations of defects in compound semiconductor hosts have been reviewed by Newman (1993). While all of the examples to be discussed in the following will be for local modes, recent progress has been made in studies of host-isotope shifts for gap modes by high resolution spectroscopy. For gap modes, the vibrational amplitude is more delocalized than for local modes [Fig. 2(b)] and hostisotope shifts increase in importance, becoming comparable to the frequency shifts due to the impurity isotopes. The nearest-neighbor isotopic structure for gap modes assigned to As, in G a P (Grosche et al., 1995) and Se2- in ZnS (Sciacca et al., 1996) has been resolved and studied. For BG, in G a P (Robbie et ul., 1996), interactions with next-nearest neighbors ("Ga and "Ga) are shown to affect the gap-mode lineshapes, further demonstrating the greater delocalization of the gap-mode vibrations.

u. Silicon in GuAs The Si dopant has a complicated behavior in GaAs that has been fruitfully studied by vibrational spectroscopy over many years. Spitzer (1971) reviewed early work and Newman (1994) has also written a comprehensive review. At concentrations of less than roughly 1 0 * 8 c i n ~ 3Si, occupies a G a site, SiGa,in stoichiometric GaAs where it acts as a donor. At higher concentrations or in material grown under Ga-rich conditions, Si also occupies an As site, Si,,, where it acts as an acceptor. In addition to these defects, SiGa-SiAspairs and two other Si-containing defects have been identified that affect the compensation condition of the material. Sic, gives rise to a vibrational line at 384cm-' and Si,, gives rise to a band at 399cm-' (Lorimor and Spitzer, 1966). Laithwaite and Newman

MICHAELSTAVOLA

174

(1977) noted that the S i A q band had a greater linewidth than the SiGaband (1.5 cm-' compared to 0.4cni- '), which they explained by the different local environments for SiAS,and hence closely spaced vibrational lines, which would arise from different combinations of naturally abundant Ga isotopes, "Ga (60%) and 71Ga(40%), that make up the first-neighbor shell of an impurity on an As site. An impurity on a Ga site has 75As first neighbors (100% abundant) and gives rise to a single sharp line. Spectra due to SiGa and SiA, measured by Theis et al. (1982) at high resolution (0.06 cm- l ) are shown in Fig. 8. Sics gives rise to a single line whereas several closely spaced lines are partially resolved for SiAs. The observed structure convincingly proves that the 399cm-' band is due to an As-site impurity. Continuing further with Si in GaAs as an example, a vibrational spectrum (Maguire et al., 1987) is shown in Fig. 9 for a heavily Si-doped sample grown by MBE that illustrates the power of vibrational spectroscopy as a method for determining both local structure and quantitative defect concentrations. (The free carrier concentration was reduced prior to the infrared absorption measurements by 2 MeV electron irradiation.) The as-grown sample was found to have a total Si concentration of [Si] = 3.7 x 1019cm-3 by SIMS, but a free electron concentration of 8 x 10'7cm-3. In the spectrum, one sees the line due to 28SiGaand also the shifted lines due to the natural abundance of the isotopes 29SiGaand "SiGa. The 28SiAsband with its characteristic host-isotope structure is seen at 399 cm- ', and SiG,-SiA,pairs

1 ) 1 ) 1 1 1 1 1 ~ " " " " ' 381

303

305

396

390

d

0

WAVENUMBERS (em")

FIG.8. The absorption bands assigned to the local vibrational modes of Si,, (left) and Si,, (right) in GaAs. The spectra were measured at 80 K. (From Theis et al., 1982.)

3 VIBRATIONALSPECTROSCOPY OF LIGHTELEMENT IMPURITIES 200.0

I

I

175

I

Wavenumber (cm-')

FIG.9. Absorption spectrum of heavily Si-doped GaAs grown by MBE. The Si concentration determined by SIMS was 3.7 x 10'9cm-3 and the free carrier concentration was 8 x 1017 cm-'. The sample was irradiated by high energy electrons prior to the measurement to eliminate the free carrier absorption. (From Maguire et at., 1987.)

are also present. Additionally, a band labeled Si-X, due to a Si-containing acceptor center whose structure is not well understood, is observed, The calibration factors relating infrared intensity to concentration for the various Si-containing defects have been determined (Newman, 1993; Murray et al., 1989b), allowing one to quantitatively account for the total Si concentration and compensation condition of Si-doped GaAs. Raman spectroscopy has also been used to study heavily Si-doped GaAs. Raman spectra are shown in Fig. 10 for MBE grown GaAs with a total Si concentration, measured by SIMS, of 4 x 1019cm-3 and a free carrier concentration of 5.4 x 1 0 ' * ~ m -(Wagner ~ and Ramsteiner, 1989). The spectrum measured with an incident photon energy of h a , = 3.00 eV shows the line assigned to the 28SiGadonor and also the compensating acceptor centers 28SiAsand Si-X. The Raman measurements have been made with a resolution of 5 cm-' so the nearest-neighbor isotope structure for the SiAs local mode is not resolved. Advantages of Raman spectroscopy for this application are that very thin layers can be studied and that free carrier absorption does not interfere with the measurements. As was noted in Section 11.4, the intensities of the Si-related Raman bands have been calibrated so that quantitative measurements of the concentrations of the different centers can be made (Murray et al., 1989b).

176

MICHAEL

*'Site

STAVOLA

G o A s : Si 5.1 x1018~m-3

I

350

I

I

L 50 R A M A N S H I F T (cm-') LOO

E

FIG.10. Raman spectra of heavily Si-doped GaAs grown by MBE measured at 77 K with different incident laser cnergies. The Si concentration determined by SIMS was 4 x 10'"cm-3 and the frcc carrier concentration was 5.4 x 10'acn---3.A background arising from secondorder phonon scattering was subtracted. (From Wagner and Ramsteiner, 1989.)

Figure I0 also demonstrates the importance of resonance enhancement and the effect that the energy of the exciting light can have on the scattered intensity (Ramsteiner et al., 1988). For hw, = 3.00 eV, Raman scattering from the vibrational modes of both the donor and acceptor centers is resonantly enhanced, whereas for hw, = 2.71 eV, only the acceptor local modes are observed in the spectrum because scattering from the SiGadonors is not enhanced for this lower incident photon energy. The resonant enhancement of the Si-X mode for Zzs, = 2.71 eV further supports the conclusion that it is an acceptor center. b.

Carbon in GuAs

A vibrational band at 582 cm-' was assigned to 12C in GaAs by Newman et LII. (l972), who also observed an isotopically shifted band at 561.2 cmin a sample doped with carbon enriched with " C . From a combination of

'

3 VIBRATIONAL SPECTROSCOPY OF LIGHT ELEMENT IMPURITIES 177

chemical analysis, electrical measurements, and vibrational spectroscopy, it was deduced that the C defect giving rise to the 582cm-' band is an acceptor and was therefore assigned to CAs (Brozel et ul., 1978). The assignment of the 582cm-' band to C on an As site was convincingly proved by Theis rt al. (1982), who resolved the isotopic structure due to the Ga first neighbors. Spectra measured with successively higher resolutions are shown in Fig. 11, which illustrate very well the advantage of highresolution Fourier transform spectrometers. Similar results were also obtained by Shanabrook et ul. (1984). Leigh and Newman (1982) assigned the host-isotopic structure for the CAF band to specific first-neighbor configurations (see Fig. 12). The possible configurations are: (i) 69Ga4CAsand 71Ga4CAs,each of which has a sharp triply degenerate vibrational mode; (ii) 69Ga371GaCAs and 69Ga71Ga3CAs, each of which has trigonal symmetry and A, and E modes of vibration; and

Resolution: 0.5 cm-'

Resolution:O.OB cm-'

I

I

I

I

I

582 583 WAVENUMBERS (cm-')

584

FIG. 1 1 . The absorption band assigned to the local vibrational mode of "CA, in GaAs. The spectrum was measured at 80 K with three different resolutions. (From Theis er al., 1982.)

178

MICHAELSTAVOLA

(4.0) 71Ga

(0.4) 69Ga

FIG. 12. The nine components of the local vibrational mode band assigned to 12C,, in GaAs. The individual lines have been assigned to the different combinations of ""a and "Cia first neighbors that are shown. The heights of the lines are proportional to their intensities. (From Murray and Newman, 1989.)

(iii) "Ga,71Ga,C,,, with C , , symmetry and B,, B, and A, modes of vibration. Altogether, there should be nine vibrational lines. Several are very close together (Fig. 12), consistent with the five distinct lines that are resolved experimentally. The agreement of the measured and calculated line positions confirm that the CAsimpurity has four equivalent Ga neighbors. In recent work, Leigh et ul. (1994) have applied harmonic models for the host isotope structure to higher-resolution data and conclude that anharmonicity affects the line positions. Unlike the Si impurity in GaAs, no evidence has been found by vibrational spectroscopy for a CGadonor (Davidson et ul., 1993), even for the very high carbon concentrations that have been attained by epitaxial crystal growth from metallorganic precursors (Abernathy, 1996). However, carbon complexes form upon annealing heavily carbon-doped GaAs and give rise to vibrational modes that have been studied by Raman spectroscopy (Wagner et al., 1997).

c.

Substitutional Impurities in II- VI Semiconductors

The richly structured vibrational line shapes of several impurities studied in 11-VI semiconductors will be discussed as further examples of nearest-


179

neighbor isotopic structure. Early work on impurity vibrations in 11-VI hosts has been reviewed by Barker and Sievers (1975). High-resolution vibrational spectra of Mg2+,C a Z + ,S 2 - , and 3d transition metals in CdTe, ZnTe, and CdSe have been reported (Sciacca et al., 1995,1996; Mayur et a/., 1996). Sciacca et a / . (1995) cite previous work in 11-VI hosts. Spectra for CdTe doped with Ca are shown in Fig. 13. The inset shows a lower resolution spectrum in which the vibrational modes of the naturally

Wave number (em-') FIG. 13. The absorption band assigned to the 40Ca local mode in CdTe along with the simulated spectrum. The spectrum was measured at 2.0K with a resolution of 0.02cm-'. The inset shows a lower resolution spectrum (0.5cm- ') of the local modes assigned to the three Ca isotopes. (From Sciacca et a/., 1996.)

180

MICHAEL STAVOLA

abundant 44Ca (2.086%) and ,'Ca (0.647%) isotopes are shown, confirming the assignment of the strong line at 210.6cm-' to ,'Ca (96.941%). The high-resolution (0.02 cm- ') spectrum for the ,'Ca local mode shown in Fig. 13 has a complicated structure that has been attributed to the frequency shifts that result from the isotopes of the four Te nearest neighbors that surround the Ca impurity substituting on a Cd site in the zinc blende structure. The eight isotopes of Te lead to 330 distinct combinations of CaTe, "defect molecules" when the different possible combinations of isotopes for the four Te neighbors are distinguished. Sciacca et a1. (1996) have calculated the normal mode frequencies for each of the CaTe, defect molecules, and summed these, weighted by their appropriate abundances, to simulate the vibrational line shape. Their calculated result is also shown in Fig. 13, and is in excellent agreement with the experimental line shape and confirms the site assignment of the impurity. Similar results have been obtained for ,'Ti in CdTe and ',Mg in ZnTe, where both of these impurities also substitute for the group I1 host atom. High-resolution spectra also show host-isotope structure for the 25Mglocal mode in CdSe, which has the hexagonal wurtzite structure (Sciacca et al., 1996).

The determination of the structure and properties of oxygen centers in GaAs provides another good example of how much can be learned from neighbor isotope shifts. A recent review has been written by Skowronski (1992). An oxygen-related center was first identified in GaAs by Akkerman et al. (1976), who observed a vibrational band at 836 cm- (room temperature) in GaAs containing l60and a shifted band at 790cm-' in a sample containing "0. The ratio of these frequencies is close to [m('80)/m(160)]1/2, confirming that this is an oxygen vibration. Roughly ten years later, two additional bands near 730 and 715 cm- ' were discovered and labeled A and B, respectively (Song ef al., 1987). It was suggested that these bands were also due to oxygen (Zhong et al., 1988; Zhong, 1991) because of their high frequency. The bands at 730 and 715cm-' both showed a triplet fine structure, which led Zhong rt aI. (1988) to propose that oxygen occupies an As-site where it is bound to two G a neighbors in an off-center position [see Fig. 14(a)]. For oxygen bonded to two G a neighbors, there can be three combinations of nearest-neighbor isotopes, 71Ga-0-71Ga,69Ga-O-71Ca, and 6'Ga-O-h9Ga. The natural abundances of the 69Ga (60%) and 71Ga (40%) isotopes gives relative concentrations of 0.16:0.48:0.36 for the three possible combinations of the G a isotopes, in good agreement with the relative intensities of the triplet components of the 730 and 715 cm-' lines.

'

n


W

4

5

181

FIG. 14. Vibrational spectra measured at 4.2 K with 0.1 cm- resolution of (a) the off-center, substitutional oxygen complex in GaAs and (b) interstitial oxygen in GaAs. The structures of the defects are shown In the insets. (From Schneider et al., 1989.)

182

MICHAELSTAVOLA

This structure of the substitutional oxygen center in GaAs is very similar to the structure of the well-known A-center or oxygen-vacancy complex in Si (Watkins and Corbett, 1961; Corbett et al., 1961). Photo- and thermally induced changes of the relative intensities of bands A and B were also reported by Song et al. (1987). Work by Schneider et al. (1989) confirmed and completed the picture of the structures of the oxygen centers. The band seen by Akkerman et al. (1976) at 830 cm- was observed to be shifted to near 845 cm- [Fig. 14(b)] at 4.2 K. For GaAs containing "0, the corresponding band was observed at 802cm-'. The high resolution spectrum in Fig. 14(b) shows that the 845cm-I band is split into a closely spaced pair of lines with relative intensities consistent with the natural abundances of "Ga and 71Ga. This nearest-neighbor fine structure is consistent with oxygen in an interstitial position where it is bonded to a single G a atom. [See the inset in Fig. 14(b).] This structure is similar to that of interstitial oxygen in Si or Ge discussed in Section 111.1. Schneider et (11. (1989) also showed that the 715cm-' band is shifted to 679cm-' upon substitution of l80for l 6 0 , confirming the previous suggestion that this band was due to oxygen. These authors also observed the triplet structure in high-resolution spectra [Fig. 14(a)], confirming the assignment to oxygen that is bonded to two Ga neighbors, with an off-center, substitutional structure. Subscquent studies confirmed and expanded upon the photo- and thermally induced transformations of bands A and B reported by Song et al. (1987) and a new band, B', was discovered. Altogether, there are three vibrational bands due to different charge states of substitutional oxygen with frequencies near 730.6 (A), 714.2 (B'), and 714.9 (B) c n - ' , all of which show the triplet fine structure (Alt, 1989). Studies of the optical and thermal conversion of the center between its charge states led to the conclusions that these bands correspond to charge states with zero (band A), one (band B') and two bound electrons (band B), and that substitutional oxygen is a negative-U defect (Alt, 1990; Skowronski et al., 1990; Skowronski, 1992). For a negative-U defect, the second bound electron has a greater binding energy than the first, making the charge state with just one bound electron metastable (Watkins, 1984). Thus configuration B' of substitutional oxygen can only be observed following photoexcitation at low temperature. The multiple charge states of 0 in GaAs have been studied theoretically by Jones and Oberg (1992), who have performed local-density cluster calculations. , These authors assign the A, B', and B bands to 0-, 02-,and 0 3 -with 0'- being the metastable charge state. A discussion of the negative-U properties of substitutional oxygen in GaAs is beyond the scope of this chapter. However, it is important to note that the optically and thermally induced conversions from one charge state to another and the negative-U

'

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183

properties were determined entirely from the oxygen local vibrational modes and the shift in the vibrational frequency that results from a change of charge state. 3. DEFECT COMPLEXES The preceding examples illustrate how detailed structural information is provided by the impurity- and host-isotope shifts observed in local mode spectra. Still, these examples have all involved only a single impurity and its neighboring host atoms. Numerous defect complexes that include several impurities and/or native defects have been studied, and isotope shifts have played an important role in identifying these complexes and determining their structures (Newman, 1973, 1993; Barker and Sievers, 1975). Here, only a few hydrogen-containing defect complexes will be discussed as examples. The hydrogen-passivated Sics and CAsdopants in GaAs will be treated first. These complexes have been chosen because the vibrations of both the hydrogen and the dopant atom have been studied, and the structures of the complexes are typical of other hydrogenated donors and acceptors in semiconductors. The remaining centers to be discussed are H, in Si or GaAs, and Hf in Si as examples of defects that contain more than one atom of the same element. Both H, and HT show interesting isotope effects, which help lead to the identification of their vibrational modes. There have been earlier studies of isotopic substitutions for defect complexes that contain more than one atom of the same element, a good example being the work on Li complexes in GaAs performed by Levy and Spitzer (1973) [reviewed by Barker and Sievers (1975)l.

a. Dopunt-Hydrogen Complexes Hydrogen is introduced easily into semiconductors, either intentionally or accidentally, where it can form complexes with donor or acceptor impurities (Pankove and Johnson, 1991; Pearton et ul., 1992; Pearton, 1994; Chevallier, 1996). The hydrogenated dopants are often electrically inactive and nonparamagnetic, and are therefore difficult to study by many of the typical structure-sensitive techniques. Vibrational spectroscopy, which works especially well for the light hydrogen atom because the vibrational frequencies are high, has been one of the primary methods used to determine the structures of hydrogen-containing complexes in semiconductors (Stavola and Pearton, 1991, Pearton et al., 1992; Chevallier et al., 1991; Pajot et uL, 1991; Stavola, 1994).

184

MICHAEL STAVOLA

SiG,-H in GaAs. An excellent example of a hydrogenated dopant whose structure was determined from its vibrational properties is provided by the SiGd-Hcomplex in GaAs whose H-stretching spectrum is shown in Fig. 15 (Pajot et a!., 1988a). The sample was an n-type GaAs:SiG, epitaxial layer with a Si concentration of about 3 x 10l8 c m P 3 that had been exposed to a hydrogen (or deuterium) plasma. Following hydrogenation, the free electron concentration was reduced to 10'' cm- 3, that is, the Sica donors were passivated. The strong band at 1717.3cm-' has been assigned to the longitudinal hydrogen-stretching mode (A I ) of the trigonal 28SiCd-Hcomplex shown in the inset. In this model, the H sits at an antibonding site, bonded directly to the Si,, donor. The two sidebands to the low-energy side of the 1717.3 cm-' line have relative intensities consistent with the natural abundances of "Si and 30Si and been assigned to the 2YSiG,-Hand 30SiGa-H complexes. The presence of these sidebands indicates that the hydrogen atom is bonded to the SiGa donor. In addition to the hydrogenstretching mode, the doubly degenerate hydrogen-wagging mode (E) of the complex has also been observed at 896.8 cm- There are corresponding deuterium modes for both the hydrogen-stretching and hydrogen-wagging bands, shifted to lower frequency by a factor of approximately (1/2)Il2, showing unambiguously that these are hydrogen vibrations. The frequencies of the hydrogen-stretching modes are given in Table 11.

'.

I . r I

E 400 u

v

u

C

.-0

5

0

0

*C

200

.-0 Y

Q

i

a (0

0 1715.4

1716.4 1717.4 Frequency (cm -1)

1718.4

FIG. 15. Infrared absorption spectra measured at liquid H e temperature for the hydrogenstretching mode of the Si,,-H complex in GaAs. (From Pajot et ul., 1988a.)


185

TABLE I1 FREQUENCIES (cm-') OF THE HYDROGEN-STRETCHING MODESOF SiG;H (pajot et a/,, 1988a) AND C,,-H (Davidson et ul., 1993) COMPLEXES IN GaAs FOR DIFFERENT Si, C, AND HYI~ROGENISOTOPES

Complex

Frequency

"Si-H "SSi-H "Si-H "SI-D 29Si-D "C-H I3C-H "C-D "C-D

1717.25 1716.89 1716.53 1247.61 1247.08 2635.2 2628.5 1968.6 1958.3

A vibrational mode of the Si atom in the SiG,-H complex has also been observed (Pajot et ul., 1988a). Upon exposure of a SiGadoped sample to a hydrogen plasma, the intensity of the 384 cm-' band due to the isolated Sic, donor was reduced, and a new vibrational band appeared at 409.95 cm-'. Upon the substitution of D for H, this Si mode is shifted to 409.45cm-', confirming that the 409.9cm-' band is due to the SiGadonor bonded to hydrogen. The experimental vibrational frequencies have provided a benchmark for theoretical calculations that have been performed for the SiG,-H complex by several groups, all of whom found that the structure shown in the inset in Fig. 15 has lowest energy (Briddon and Jones, 1990; Pavesi and Giannozzi, 1991; Chang, 1991). These groups all calculate vibrational frequencies that are in good agreement with the experimental values. A new result from the theoretical studies is that the Si atom was found to relax along the trigonal axis of the complex toward the hydrogen atom, resulting in a weak Si-As bond.

CA,v-Hin GaAs. Numerous examples of acceptor-hydrogen complexes have been studied in elemental and compound semiconductors. In most cases, the hydrogen occupies a bond-centered position between the acceptor and one of its nearest neighbors. A particularly interesting example is the hydrogen passivation of the CAsacceptor in GaAs. Clerjaud et uI. (1987, 1991) discovered several sharp absorption lines with frequencies above 2000cm-' in as-grown, bulk GaAs, InP, and G a P

186

MICHAELSTAVOLA

crystals. These lines were all assigned to hydrogen-vibrational modes because of their high frequency. Identification of the hydrogen-containing defects was challenging because the hydrogen and the impurities to which it was attached were introduced unintentionally. Clerjaud et al. (1990, 1991) were able to assign a line at 2635.1 cm-', observed in LEC-grown p-type GaAs, to a 12CAs-Hcomplex from the observation of a weak additional line at 2628.4cm-' that they assigned to the 13CAs-Hcomplex (Fig. 16). The weaker band has an intensity consistent with the natural abundance of 13C (1.lo%)and the frequency shift is consistent with the mass of the heavier carbon isotope. The structure of the CAs-H complex, with the H atom bonded primarily to the CAsatom, is shown in the inset in Fig. 16.

I

J

p-typeLEC GaAs 1%-H

0.8, n I

f

- 0.6Q)

u c

tu

2 a

P

0.4,

0.2 1%-H

0,.

.

.

,

J

,

2630

,

,

.

,

I

I

I

2635

Frequency (cm-1) FIG. 16. Inlrared absorption spectrum measured near liquid He temperature for the C,,-H complex in GaAs. Both "C- and I3C-related modes are observed. The bond-centered structure of the complex is shown in the inset. (From Clerjaud el a/., 1990.)

3


LIGHTELEMENT IMPURITIES

187

Carbon has become an attractive p-type dopant for epitaxial GaAs and related alloys and can be incorporated at concentrations up to 1021cm-3 in epitaxial GaAs grown by metallorganic chemical vapor deposition (MOCVD) or metallorganic molecular beam epitaxy (MOMBE) (Abernathy, 1996). In some cases, roughly 50% of the CAsacceptors are unintentionally passivated during crystal growth because hydrogen is readily available in the growth environment. The vibrations of CA,-H complexes have been detected by several groups in as-grown GaAs epitaxial layers (Kozuch et al., 1990, 1993; Woodhouse et al., 1992; Wagner et al., 1992). Thus, the vibrational spectroscopy of CA,-H complexes has attracted much attention. For the 12CA,-Hcomplex, there will be four local vibrational modes that are shown in Fig. 17 (Davidson et al., 1993; Jones et al., 1994; Wagner et al., 1995). There is the high-frequency H-stretching mode already discussed here (A; in Fig. 17), a low-frequency mode in which the C and H move in phase as a unit along the trigonal axis (At), and two transverse E modes (E- and E'). All of these local vibrational modes have been observed and assigned for 12CAs-H,l3CA3-H,"CA,-D, and 1 3 C A 5 - D in a very nice series of infrared absorption and Raman spectroscopy studies, providing an especially complete picture of the vibrational modes of C-H complexes in

pca b

/

As

6

A1- (stretch) Ai+

(X)

Ga

--

-+

E- (Yz)

t

i

E+ (Yi)

. f t

FIG. 17. Structure of the CA,-H complex in GaAs shown with the complex's vibrational modes. (From Wagner er al., 1995.)

188

MICHAELSTAVOLA

GaAs (Davidson et al., 1993; Wagner et ul., 1995). The hydrogen- and deuterium-stretching frequencies for the different carbon isotopes are listed in TableII. The experimental frequencies are in good agreement with theoretical calculations by Jones et ul. (1994). The studies of CA,-H complexes in GaAs, beginning with the work of Clerjaud et al. (1990), nicely demonstrate the use of vibrational spectroscopy as a probe of unintentional hydrogen passivation of dopants in semiconductors. And the isotopic frequency shifts that result from the substitution of I3C for 12C have played an essential role in the assignment of the local vibrational modes. b. H , und HT Isolated hydrogen in semiconductors is mobile at low temperatures [for example, near 200 K in Si (Gorelkinskii and Nevinnyi, 1991; Holm et a/., 1991)] so at room temperature, H in semiconductors is bonded to other defects or present in the form of H, aggregates. The complexes of hydrogen with various impurities and defects give rise to vibrational bands that have been studied over a number of years, as was already discussed here . The smallest H, aggregates involve just two hydrogen atoms. Two possible di-hydrogen defects have been considered theoretically, but were not observed spectroscopically until recently. The first is the hydrogen molecule H, that has been predicted to lie at a tetrahedral interstitial site in Si or GaAs (Chang and Chadi, 1989b; Van de Walle et al.; 1989, Estreicher et a/., 1994; Pavesi and Giannozzi, 1992; Breuer et al., 1996). Molecular hydrogen has often been invoked as the product of reactions that release hydrogen from other defect complexes. The second is a species that has been named HZ (Bridden et ul., 1988; Chang and Chadi, 1989a). Vibrational spectra for H, and H: will be discussed in what follows. H , in Si or GaAs. An isolated H, molecule has inversion symmetry and therefore its stretching vibration is infrared inactive because it does not give rise to an oscillating dipole moment. This is the reason why H, has been thought to be difficult to observe in semiconductor hosts. However, Raman spectroscopy can be used to study the stretching vibration of an H, molecule, illustrating its use as a technique complementary to infrared absorption spectroscopy. In exciting recent work, vibrational spectra due to H, in GaAs and Si have been discovered. These direct observations of H, confirm the long standing hypothesis that molecular hydrogen is an important, and often present, species in semiconductors (Pearton et al., 1992; Pankove and Johnson, 1991).


189

H, molecules were observed in plasma-treated GaAs by Vetterhoffer et al. (1996) in Raman measurements made at liquid N, temperature. The vibrational bands of H, (3934.1 crn-') and D, (2842.6cm-') are shown in the spectra in Fig. 18. In this study, hydrogen and deuterium were also introduced together so that the stretching vibration of the H D complex was also observed (3446.5 cm- I), unambiguously confirming that two hydrogen atoms are involved in the complex. The frequency of the vibrational band is consistent with the reduced mass of the H D molecule. Furthermore, the results for H, in GaAs show evidence for the two possible nuclear spin states of a homonuclear diatomic molecule (Vetterhoffer et al., 1996). The two hydrogen atoms in H, each have nuclear spin 4. The two hydrogen atoms can have parallel nuclear spins to give a total nuclear spin of I = 1 (orthohydrogen), or antiparallel nuclear spins to give I = 0 (parahydrogen). The total nuclear wavefunction must be antisymmetric upon exchange of the identical hydrogen nuclei, therefore the symmetric nuclear spin wavefunction of orthohydrogen requires that the rotational wavefunction be antisymmetric. Thus the lowest allowed rotational state for orthohydrogen has J = I. Similarly, for parahydrogen, the lowest allowed rotational state has J = 0 because the nuclear spin wavefunction is antisymmetric. At elevated temperature, the nuclear spin states are equally populated, so the ratio of ortho- to para-hydrogen is 3:1, consistent with the

H2

-

c .c

v)

(a) HZplasma I

.

.

*

f

1

FIG. 18. Raman spcctra measured at 77 K for (a) semi-insulating GaAs treated in an H, plasma for 8 h at T = 254°C (b) low-temperature GaAs treated in a D, plasma for 3 h at T = 200 'C, and (c) low-temperature GaAs treated in a 50% H,/50% D, plasma for 8 h. Residual laser plasma lines are marked with asterisks (*). (From Vetterhoffer et a/., 1996.)

190

MICHAELSTAVOL.A

degeneracies of the nuclear spin states. A t low temperatures, the conversion of orthohydrogen to the lower energy para configuration requires a forbidden nuclear spin flip and proceeds very slowly. The H, stretching band in GaAs was observed to be split into two lines (VetterhofTer et a[., 1996). This splitting was assigned to the vibrations of ortho- and para-hydrogen which are slightly different because of rovibrational coupling, further confirming the assignment to a molecular-like H, species in the solid. The intensities of the split lines were consistent with the 3: I ratio expected for ortho- to para-configurations at room temperature. (Conversion to the lower energy parahydrogen state at the measurement temperature was presumed to be slow.) Splittings for D, (the deuterium nucleus has spin 1) were not resolved. In an independent Raman study performed by Murakami et al. (1996), a broad vibrational band was observed at 4158 cm- at room temperature for Si samples treated in a hydrogen plasma at 400°C. When Si samples were exposed to a deuterium plasma, a band at 2990cm-' was observed. These vibrational frequencies are close to those observed previously for H, and D, in gas, liquid and solid phases, and led to the assignment of the 4158 and 2990cm-' bands to H, and D, at tetrahedral interstitial sites in Si. Subsequent studies have contested the assignment of the 4158 cm-' band to isolated H, at an interstitial site in Si. Exposure of a Si sample to a hydrogen plasma can give rise to extended platelet defects (Johnson et al., 1987). Leitch et ul. (1998, 1998b) have found that the 4158cm-' band has the same annealing behavior as Raman bands near 2100 cm- assigned previously to hydrogen-platelet defects (Johnson et al., 1987; Heyman et ul., 1992) and proposed that the 4158cm-' band is due to H, gas molecules trapped in the platelets. Leitch et al. (1998b) have assigned a new Raman band at 3618 cm-' (10 K) to an isolated H, molecule that was produced in Si by plasma exposure at 150°C. These authors propose that H, molecules are immobile at 150°C and can be produced in isolated form at this temperature whereas for higher plasma exposure temperatures the H, molecules can migrate and become trapped in platelet defects. Theoretical calculations performed for H, at a tetrahedral interstitial site in Si find that the H, stretching frequency should be shifted to a value lower than the gas phase value by more than 500 crn- (Hourahine et ul., 1997, 1998; Van de Walle, 1998). These results support the assignment by Leitch et al. (1998b) of the 3618 cm-' band to isolated H, and also the assignment of the 4158 cm-' band seen by Murakami et al. (1996) to H, molecules that have aggregated in extended defects created by the hydrogen plasma exposure. Surprisingly, infrared absorption lines have also been observed independently by Pritchard et a/. (1997) for interstitial H, and D, in Si. For these

'

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3



191

experiments the hydrogen was introduced by annealing Si at elevated temperatures (1000 < T < 1300°C) in H, gas. Two lines at 3789 and 3731 cm-' were assigned to H, molecules trapped in the vicinity of interstitial 0 impurities and a third line at 3618cm-' was also reported. The 3618cm-' line was subsequently assigned to H, at an interstitial site (Pritchard et al., 1998) in agreement with the Raman study of Leitch et al. (1998b). A line due to HD was also observed at 3264.8 cm-' t o confirm the presence of two hydrogen atoms in the defect (Pritchard et at., 1997). The vibrational absorption bands associated with H, are roughly 100 times smaller than would be expected for an allowed local vibrational mode. The presence of a weak vibrational absorption line for the H, defect has led to the conclusion that the H, lies along a lattice direction that removes the inversion symmetry of the molecule (Pritchard et ul., 1998; Hourahine et al., 1998). In the Raman and infrared absorption studies of H, in Si, no splittings of the vibrational lines due to ortho- and para-hydrogen were observed leading to the conclusion that the H, molecule does not freely rotate in Si at low temperature (Hourahine et ul., 1998). HT in Si. An alternative configuration for a di-hydrogen defect named

H f has been examined in several theoretical calculations (Chang and Chadi, 1989a, 1989b; Van de Walle, 1994; Estreicher et al., 1994). The H f defect in Si has been predicted to consist of one hydrogen atom at a bond-centered site between two Si neighbors and a second hydrogen atom bonded to one of these Si neighbors at an antibonding site (Fig. 19).

FIG. 19. Structure of the H f defect in Si. (From Holbech rr a/., 1993.)

192

MICHAEL STAVOLA

It has been known since 1975 (Stein, 1975) that implanting protons into Si gives rise to numerous hydrogen-related vibrational absorption bands. Recently, several of these bands have been assigned to the H: defect (Holbech et a/., 1993). A n infrared absorption spectrum, of proton implanted Si is shown in Fig.20. From a comparison of the experimental frequencies to a theoretical calculation for HT, the bands at 2062 and 1838 cm-' have been assigned to the stretching modes of hydrogen atoms at the bond-centered and antibonding sites, respectively. The line at 817 cm-' has been assigned to the wagging mode of the hydrogen atom at the antibonding site and the 1599cm-' band has been assigned to be the overtone of the 817 cm-l mode. The vibrations of the two hydrogen atoms in the HT center are weakly coupled. The spectrum of a sample implanted with both H and D (Fig. 20) shows new H-stretching bands at 2058 and 1852cm-' assigned to the HD* center (Holbech et a/., 1993). An HD* center can have H at the bondcentered site and D at the antibonding site, or vice versa. The shifts of the frequencies of the vibrational modes compared to those of H2: occur because the large difference in the H and D masses causes the vibrations of the H and D atoms to be decoupled. Similar considerations apply to a comparison of the D-stretching modes of DZ and HD*. The frequencies of HD* are also in agreement with theoretical calculations for this center. Similar results to those described in the preceding for H: in Si have also been obtained for an H: center in Ce, showing that it has a similar structure and vibrational properties (Budde et a/., 1996). The results of vibrational spectroscopy establish that it is the weakly coupled vibrational modes of a center that contain two inequivalent hydrogen atoms that are observed for HT. For a complex with two equivalent hydrogen sites, there would have been only one new hydrogen mode for a

817 1838

I

I

A

n

SI: H+D

SI:H

-

1860 2050

207C

frequency (cm-')

FIG. 20. Vibrational bands assigned to hydrogen modes of Hf (lower spcctrd) and HD* (upper spectra) in Si implanted with H + and/or D'. (From Holbech et ul., 1993.)


193

complex containing H and D. [This is the case for the PtH, center in Si where the two hydrogen atoms are equivalent (Uftring et al., 1995).]

V. Anharmonicity

Up to this point in this chapter, the anharmonicity of the local mode vibrations has been neglected. Anharmonicity has a number of important effects on local mode spectra: (i) overtone and combination modes are made weakly allowed; (ii) the frequencies of the vibrational modes are modified; and (iii) near-lying vibrational modes can interact. Here, anharmonicity will be discussed in an elementary fashion and a few examples will be given along with references to treatments in the literature. Elliot et a/. (1965) considered the effect of anharmonic terms in the potential in which a light impurity vibrates to explain the fundamental and higher harmonic frequencies of H- and D- ions in the alkaline earth fluorides. For a site with tetrahedral symmetry, the potential becomes V ( T J = +k(x2

+ y 2 + 2)+ B(xyz) + D,(x4 + y" + z") + D,(y2z2 + z2x2+ xZyz)+ - .

(6)

when cubic and quartic terms are included. Here k is the force constant for the harmonic term in the potential and B, D , and D , are parameters. To determine the vibrational energy levels, the effect of the cubic term was included to second order in perturbation theory and the quartic terms were included to first order. The energy levels and allowed optical transitions for several low-lying states are shown in Fig. 21. The energy-level structure and selection rules provide information about the symmetry of the defect site. For example, Elliot et al. (1965) determined that H - occupies a site with & symmetry and not 0, symmetry in the alkaline earth fluorides from fundamental and overtone absorption data. The potentials, including anharmonic terms through quartic, have been determined for various site symmetries and the energy levels have been calculated with perturbation theory (Newman, 1973). For semiconductor hosts, there are a few examples where the fundamental and overtone absorption lines for local vibrational modes have been observed and detailed analyses of the data have been performed (Hayes and Spray, 1969; Manabe et al., 1973; Sciacca et al., 1995; McCluskey et al., 1996; Alt, 1995; Leigh and Sangster, 1996). It is known that the intensities of the overtone absorption lines cannot be reliably predicted from the anharmonicity parameters determined from the frequencies of the transitions (Elliot et a/., 1965; Newman, 1973; Fowler et

194

MICHAELSTAVOLA

2

1

0

rl

Fici. 21. The lower vibrational energy levels of a substitutional impurity at a site with T, symmetry. The vibrational quantum numbers and symmetries of the levels are shown on the left.

ul., 1991). Electrical anharmonicity is the commonly given explanation, that is, the electric dipole moment of the defect is not a linear function of the mode displacement coordinate as is typically assumed to determine the intensity of vibrational transitions. In the following, the effects of anharmonicity on the vibrational modes of a few defects will be discussed. In Section 1V.1, one-dimensional models that have been used to treat the anharmonicity of the hydrogen vibrations of hydrogen-containing complexes in semiconductors will be reviewed. In Section IV.2, the discussion of the vibrational modes of oxygen in Si begun in Section 111.1 is continued. Recent studies of a combination mode provides information about the symmetric stretching mode of oxygen, for which the fundamental has not been observed. In Section IV.3, the anharmonic interaction between near-lying vibrational modes (Fermi resonance) is used to explain the observation of an “extra” hydrogen mode of the donor-H complexes in Si.


195

1. ONE-DIMENSIONAL MODELS FOR HYDROGEN-CONTAINING COMPLEXES In cases where the vibrational modes are well separated, a mode may be treated as that of a one-dimensional anharmonic oscillator. One-dimensional models have been applied to hydrogen-containing defects in semiconductors where the anharmonicity is expected to be important because the vibrational amplitude of the light hydrogen atom is large. u.

Perturbation Theory Treatment

Newman (1990b, 1991) has written the one-dimensional potential in which an impurity vibrates as V ( Z )= &kz2+ flz3

+ yz4

(7)

The effects of the cubic and quartic terms were calculated with second- and first-order perturbation theory, respectively. The fundamental frequency w is given by

where w,, is the harmonic frequency, that is, the frequency of vibration if there were no anharmonicity. Here m is the mass of the vibrating atom in amu, and B is a parameter. If the second harmonic transition is also observed, then the anharmonic shift B/m can be determined from the equation 2w

-

wZnd= B/m

(9)

[Perturbation theory gives a second harmonic frequency of WZnd = 2w, 3B/m2 for the anharmonic potential, leading to Eq. (9) (Newman, 1990).] Newman (1990, 1991) noticed that the isotope shift observed for the hydrogen-stretching vibrations of dopant-H complexes leads to inconsistent results and attributed this inconsistency to the neglect of anharmonicity for the hydrogen modes. The ratio of the vibrational frequencies for the isotopic substitutions rn"' for m or M$) for M , can be determined from Eq. (4) and written as

196

MICHAELSTAVOLA

An implicit assumption being made in Eq. (10) is that the vibrational modes are harmonic. For the 28Si,,-H complex discussed in Section 111.3, the hydrogen stretching frequency is observed at 1717.25 cm-' (Table 11). The isotopic substitution 2'Si for 28Si shifts the hydrogen-stretching frequency to 1716.89 cm-'. Similarly, the substitution 30Si for "Si shifts the hydrogenstretching frequency to 1716.53cm-'. These shifts and Eq. (10) lead to a value of X = 2.8. Equation (10) then predicts a frequency ratio for the substitution of D for H of w(H)/w(D) = 1.4053, rather than the ratio, w(H)/o>(D)= 1717.25/1247.61 = 1.3764 that is observed experimentally. Newman explained this apparent contradiction by including the effect of the anharmonicity of the hydrogen-stretching modes on the frequency ratio w(EZ)/a)(D).From Eq. (8), the ratio of the harmonic frequencies would be given by

where (11 is the experimentally observed frequency and w,, is the harmonic frequency. The observed frequency ratio will be reduced from its harmonic value because the anharmonic shift is proportional to the inverse of the oscillator mass rather than to the square root of the inverse mass. It is this ratio of harmonic frequencies, Eq. (1 I), that is determined by Eq. (10). Thus if X can be determined, in this case from the effect of Sic,, isotopic substitutions, then Eqs. (10) and (11) can be used to determine the anharmonic shift B/m for the hydrogen-vibrational mode. For SiG,-H in GaAs, Newman (1990, 1991) has found that the values X = 2.8 and B / m , = 121 cm-' lead to consistent results for the observed Si and hydrogen isotope shifts of the hydrogen-stretching mode. Data for the carbon and hydrogen isotope shifts of the H-stretching frequency have been measured for the CA,-H complex in GaAs (TableII) and the anharmonicity parameter has been estimated (Davidson et nl., 1993), similar to the SiG,-H complex just discussed. In this case a value of X = 1.1 is determined from the frequency shifts of the H - and D-stretching modes of CA,-H and CA,-D complexes that result from the isotopic substitution for "C. This value of X and Eq.(lO) lead to a harmonic frequency ratio for the hydrogen-stretching mode o,,(H)/w,(D) = 1.367, which is again larger than the experimentally observed value of w(H)/ w ( D ) = 2635.2/1968.6 = 1.339. These results and Eq. ( 1 1) lead to an anharmonic shift for the hydrogen-stretching mode of the CA,-H complex of B/m, = 176 cm (Davidson et a/., 1993). Data for isotopic substitutions of both the dopant and hydrogen atoms

'


197

are not available for most dopant-hydrogen complexes, as they are for the SiG,-H and CA,-H complexes already discussed here. Newman (1990, 1991) has estimated a lower limit for the anharmonicity parameters for the hydrogen-stretching vibrations of a variety of dopant-H complexes and finds values of B/m of the order of 100cm-’. b.

Morse Oscillator

An alternative approach is to use the Morse potential (Morse, 1929) to describe the anharmonicity of the local mode vibrations of crystal defects (Fowler et al., 1991; Darwich et al., 1993). The Morse potential is written as V ( z - z,) = De{exp[-2a(z

- ze)] - 2 exp[a(z - z,)]]

(12)

where z, is the equilibrium value of the vibrational coordinate z, and x and D , are parameters. The Schrodinger equation can be solved exactly for the Morse potential to give the vibrational frequencies

w(n) = w,(n

+ f)- w,x,(n +

+)Z

(13)

where we = .(hD,/ncp)”’ w,x, = ha2/(4ncl*)

and p is the reduced mass. In this case, the observed fundamental frequency is given by 0 = w, - 2w,x,

(15)

and the second harmonic frequency is given by

The quantity w,x, is known as the anharmonicity constant. Comparing these equations to Eqs. (8) and (9), it can be seen that 2w,x, plays the same role here as B/m does in the perturbation theory treatment already discussed here. For the Morse potential, the intensities of the second harmonic and fundamental absorption lines have a ratio of 12/11 x,. It is known for the

-

198

MICHAEL STAVOLA

vibrational modes of molecules and defects in solids that the intensities of the overtone absorption lines are not reliably predicted from the value of x, determined from the line positions. Fowler et al. (1991) have calculated the dipole moment matrix elements for vibrational transitions with Morse wavefunctions and a nonlinear dipole-moment function that could either be (i) matched to a calculated dipole-moment function or (ii) parameterized to fit experimental data. A numerical scheme based on these methods has also been developed to explore the intensities of overtone absorption lines (Woll and Fowler, 1993). These models have been applied to OH- in insulators and could be applied equally well to local vibrational modes in semiconductors. Darwich et al. (1993) have observed the second harmonic transitions of several acceptor-hydrogen complexes in GaAs and InP. For these complexes, the acceptor atom occupies a group I11 atom site and the H is believed to be bonded more strongly to one of the acceptor's As or P nearest neighbors. Thus the vibrational modes are primarily the vibrations of the As-H or P-H bonds. These authors have used the results for the Morse potential to analyze their data and determine the anharmonicity parameters. The frequencies and relative intensities of the first and second harmonic transitions are given in Table I11 along with the anharmonicity parameters 2o,x, and x,. The anharmonicity parameter 2o,x, has values close (Le., only -25% smaller) to the values of B / m estimated by Newman (1990, 1991) from the isotope dependencies of the fundamental frequencies. Further, x, is similar to the values found for As-H and P-H bonds in gas-phase molecules. The data in Table 111 show that the observed values of I J l 1 are roughly three times smaller than the value estimated for the Morse potential, 12/11 x,. The difference between the observed and predicted second harmonic intensities has been ascribed to electrical anharmonicity (Darwich

-

TABLE 111

THE:FREQUENCIES ANI) RELAIIVE INTENSITIES OF THh FIRST AND SEC'OND HARMONIC TRANSITIONS FOR SEVERAL ACCEPTOR-HYDROGTN COMPLEXES IN G a A S AND InP (Darwich et a / , 1993). THE PARAMETERS ~w,Y, AND x, ARE A MEASURF OF THE ANHARMONICITY OF THE VIBRATION

GaAs:ZnG,-H InP: Zn,,-H Zn,,-D Cd,,-H

2W,X,

(3

W2"d

(cm- I )

(cm- I )

(cm - I )

x,

I 2 l I1

2146.94 2287.70 1664.52 2332.43

4216.7 4487.80 3283.34 4580.20

77.18 87.60 45.70 84.66

0.0174 0.0184 0.0 I 34 0.0175

0.005 0.009 0.007 0.006


199

et al., 1993) and again shows that the relative intensities are difficult to

predict and cannot be relied upon to determine the anharmonicity parameters. In an alternative analysis of the anharmonicity of the SiG,-H complex in GaAs, Walck and Fowler (1995) have used a Morse potential to fit the frequencies of the hydrogen-stretching modes, treated previously by Newman (1990, 1991). An interesting feature of this work is the interpretation of the small value of the SiGaisotope shift. These authors show that a threebody interaction bet ween a host atom and the SiG,-H oscillator can explain the small Sic, isotope shift that was observed experimentally by Pajot et al. (1988a). In their model, Walck and Fowler propose that the SiG,-H bond is strengthened as the Si atom moves away from the As neighbor to which it is weakly bonded (the defect structure is shown in the inset in Fig. 15). It is this bond strength variation that leads to a heavier effective mass for the Sic, atom. The H-stretching vibration is then treated as a Morse oscillator. The fit of this model to the observed vibrational frequencies is excellent.

c.

A n ab initio Calculation of the Anharmonicity

It has become common for the vibrational frequencies to be calculated for the defect structures that are proposed in theoretical studies. The comparison of the theoretically calculated vibrational frequencies with experimentally measured frequencies can be greatly affected by the anharmonicity of the modes. This is especially true for hydrogen local mode vibrations for which the anharmonic shifts are large (i.e., of the order of 100cm-' as has been shown here). Jones et ul. (1994) have calculated the effect of anharmonicity on the hydrogen-stretching modes of C-H in the HCN molecule and in the CA,-H complex in GaAs. The CA,-H complex in GaAs was discussed in Sections 111.3 and 1V.I and is of primary interest here. Calculations were performed by local density pseudopotential methods for a large cluster (see the chapter by Jones and Briddon in Volume 51A of this series). To determine the vibrational frequencies for CA,-H in GaAs, a CGa,,As,,H,, cluster was studied. The carbon was at a central As site and the hydrogen was at a near bond-center site adjacent to the CAs.All atoms were allowed to relax in the calculation until the energy was minimized. Vibrational frequencies for CA,-H were first calculated by methods used previously. Selected atoms were displaced from equilibrium, first by 0.09 atomic units and then by -0.09 atomic units, and the resulting forces on the selected atoms were evaluated to determine force constants. Jones et al. ( 1994) emphasize that purely harmonic force constants are not determined by this method and that quartic and higher-order terms affect the results.

200

MICHAELSTAVOLA

From these force constants, a dynamical matrix was constructed and “quasiharmonic” vibrational frequencies were determined. To explore the effect of anharmonicity on the calculated frequencies, Jones et al. (1994) considered the one-dimensional hydrogen-stretching motion along the CA,-H bonding direction and neglected the coupling to other modes, which were all well separated in frequency from the hydrogenstretching mode. The energy was calculated as a function of the displacement x of the hydrogen atom from equilibrium. Here x was varied from - 1.0 to 1.0 atomic units in steps of 0.05 atomic units. The energy was then fitted to a power series m

E

=

E,

+ 1 uixi i=2

containing terms up to m = 8. The three lowest eigenvalues for this anharmonic potential were evaluated numerically to determine the fundamental and overtone frequencies for the hydrogen-stretching mode. From these calculated frequencies, anharmonicity parameters, similar to those determined from experimental data in the preceding sections, were determined. From these calculations, Jones et al. (1994) have found that:

Thc harmonic hydrogen-stretching frequency for CA,-H in GaAs, determined from the purely harmonic term in the potential, Eq. (17), is higher than the quasiharmonic frequency. The hydrogen-stretching frequency is very sensitive to the CA,-H bond length because of the anharmonicity of the potential, and that this sensitivity is the greatest source of error in the calculation of the frequency. The calculated fundamental and overtone frequencies for the different isotopes of carbon and hydrogen give parameters X = 1.23 and B/m zz 150cm-’. These results are in good agreement with the previous results of Davidson et u1. (1993) who found X = 1.1 and B/m = 176 cm-’ from their analysis of the experimentally measured frequencies for the different carbon and hydrogen isotopes. Jones et a / . (1994) also address the question of why the overtone of the hydrogen-stretching mode of the CA,-H complex has not been observed, even though the anharmonicity is large. These authors calculate the transition probability for the second harmonic transition and find that electrical anharmonicity can greatly reduce the intensity of the overtone.


20 1

These theoretical results show that the potentials calculated for hydrogen stretching modes are indeed anharmonic and that the effect of anharmonicity on the predicted frequencies can be substantial.

2.

1748 cm- ' COMBINATION MODEOF INTERSTITIAL OXYGEN IN Si

Overtone and combination modes are made weakly allowed by anharmonicity. A combination mode observed for interstitial oxygen in Si at 1748.6cm-' (Krishnan and Hill, 1981; Pajot et ul., 1985; Pajot and Cales, 1986) is an interesting example that has allowed the frequency of the symmetric-stretching mode, which is not infrared active, to be determined (Pajot et al., 1995). The overtone provides information that is not available from the infrared spectra of the fundamental modes and allows the assignments of the fundamental modes of interstitial oxygen to be completed. (See also the review by Pajot, 1996.) As was already discussed in Section 111.1, above, there are three strong absorption bands observed in oxygen-containing Si, near 29 cm518cm-', and 1136cm-' (at liquid He temperature). The 1136cm-' band is due to the antisymmetric stretching motion of a Si-0-Si defect molecule, and the 29cm-' band is due to a low energy anharmonic excitation characteristic of a nearly linear molecule (Bosomworth et al., 1970; YamadaKaneta et a/., 1990). Several weaker combination bands, also related to oxygen, have also been observed. Of particular interest here is a band at 1748.6 cm-' that is shown in Fig. 22. The 518 cm-' band has been attributed to the symmetric-stretching mode of oxygen, which might be weakly allowed for a slightly bent Si-0-Si defect, but this assignment has been less certain than for the other fundamental modes. This assignment is supported by the results of a uniaxial stress experiment, which showed that the transition moment for the 518 cm-' mode is oriented transverse to the ( I1 1 ) bonding direction of the Si-0-Si defect, as would be expected for the symmetric-stretching mode (Stavola, 1984). A frequency shift for oxygen-isotopic substitution has not been observed (Hrostowski and Kaiser, 1957; Pajot et al., 1995), but this shift would be expected to be small for the symmetric-stretching mode because the oxygen motion is small. Of more serious concern is the lack of Si isotope shifts that would result from the naturally abundant isotopes 29Si and 30Si, which have been predicted to be large (Jones et al., 1992). The 1748.6 cmband was assigned to a combination of the 1136cm-' band and phonons of the Si host crystal (Pajot et al., 1985). In this case, the observed Si isotope shifts are roughly three times larger than those observed for the 1136 cm-'

',

'

202

MICHAEL STAVOLA

1710

1730 1750 Frequency (ern-1)

1770

FIG. 22. The 1748cm-I line (Pajot et ul., 1995) assigned to the combination of the symmetric and antisymmetric stretching modes of interstitial oxygen in Si, measured at 1.5 K. Thc oxygen concentration was approximately 10" cm-3.

band (Section 111.l), which is not consistent with the involvement of host phonons in the combination mode. More appealing assignments of the 518 cm- and 1748.6 cm- bands were made by Yamada-Kaneta et al. (19901, who attributed the 518cm-' mode to defect-induced lattice absorption -that is, a transverse motion of the oxygen atom's Si neighbors, as was previously suggested by Bosomworth et al. (1970). This assignment is consistent with the transition moment direction determined previously and the lack of observed Si or oxygenisotope shifts. Yamada-Kaneta et ul. (1990) further suggested that the 1748.6cm-' band was a combination of the 1136cm-' mode and an excitation with a frequency near 613cm-' that might be assigned to the symmetric-stretching mode of oxygen. Yamada-Kaneta et al. (1990) argued that it is appropriate to consider the Si-0-Si defect molecule to have a linear, D3dstructure for which the symmetric-stretching mode would be infrared inactive, explaining its absence in the infrared spectrum. Artacho et al. (1995) have calculated the vibrational density of states and infrared absorption spectrum for oxygen in Si with a cluster-Bethe-lattice approach. The results are in excellent agreement with the assignments suggested previously by Yamada-Kaneta et al. (1990). Artacho et al. (1995) find an infrared absorption band at 517 cm-' in their calculation that is due to the transverse motion of the oxygen atom's Si neighbors. This band would correspond to the 518 cm- ' band observed experimentally. The


203

oxygen symmetric-stretching mode appears prominently in the calculated but is infrared inactive. The density of vibrational modes at 596.3 cm frequency shifts that result from Si and 0 isotopic substitutions have also been calculated. From the experimentally measured spectra of the combination mode at 1748.6cm-' and the 1136.4cm-' mode, the frequency, 612.2 cm-', can be deduced for the symmetric-stretching mode of 28Si-'602RSi(Artacho ef ul., 1995; Pajot et al., 1995), in good agreement with the theoretical result. The frequency shifts that result from Si and 0 isotopic substitutions have also been determined similarly from the experimental spectra and are in excellent agreement with the calculation of Artacho et u1. (1995). It is interesting to note the important roles played by anharmonicity and the host isotope shifts in determining the mode assignments for interstitial oxygen in Si. The mode assignments for interstitial oxygen in Si have been completed through the observation and assignment of a combination mode that was weakly allowed by anharmonicity. The previous assignment of a band at 518 cm-' was suspect because of the absence of the Si isotope shift expected for a localized mode. The deduction that the symmetric-stretching mode of oxygen is infrared inactive and lies near 612cm-1 is strongly supported by the agreement of the calculated isotope shifts with those that are determined from the observed bands near 1748 cm-' and 1136 cm-' ~

'

RESONANCE 3. FERMI If two vibrational modes of a defect lie close in frequency and have the same symmetry, they can be mixed by anharmonic interactions. The vibrational lines will be shifted in frequency and will share intensity. This effect is well known as Fermi resonance in molecular spectroscopy (Fermi, 1931; Herzberg, 1945; Wilson et al., 1955), and is particularly dramatic when a transition to one of the coupled levels would be forbidden in the absence of anharmonicity. Here, the assignment of the vibrational spectra of the donor-H complexes in Si, based upon a Fermi resonance between the second harmonic of the hydrogen-wagging mode and the stretching fundamental, will be discussed as an example (Zheng and Stavola, 1996; Zheng, 1996). The hydrogen-vibrational spectra of P-H, As-H, and Sb-H in Si (Bergman et al., 1988a) consist of three bands observed near 810,1560, and 1660cmp' for each of the complexes. There are corresponding bands at 584, 1143, and 1218 cm- for the donor-D complexes. The high-frequency bands observed for Sb-H and Sb-D are shown in Fig.23. The vibrational frequencies were found to be nearly independent of the donor, which was taken as support

'

204

MICHAELSTAVOLA (b) Sb-H

(a) Sb-D

0.012

0.03

(I)

0.008 -

-P 0

2

v)

0.004

-

0.01

0 -h

L 1120

1170

1220

1520

1620

1720

Frequency (cm -1) FIG.23. Absorption spectra measured near 4.2 K for the two high frequency bands of thc ( a ) Sb-D and (b) Sb-H complexes in Si. The positions o f the decoupled frcquencies and their separations (in c m - I ) are shown. (From Zheng and Stavola, 1996.)

for the defect structure originally proposed by Johnson et al. (1986) [Fig. 24(a)] because the hydrogen atom is attached to a Si neighbor, rather than the donor atom in the complex (Bergman et al., 1988a). The 810 and the 1560 cm-' bands were originally assigned to the H-wagging and stretching modes, respectively. The appearance of an additional H-stretching mode for a donor-H complex was unexpected and it was suggested that the 1660 cm . band might be due to some other complex. An important clue to the correct assignment of the vibrational modes came from the recognition that the second harmonic frequency of the 13-wagging mode, (02H.z 2 x 810cm-', lies between the two high-frequency bands (Zheng and Stavola, 1996). This suggests that if the second harmonic of the wagging mode and the H-stretching-mode fundamental were weakly coupled by anharmonic interactions, then vibrational transitions to these modes would share the intensily ojthe allowed transition to the H-stretching mode and give rise to both the 1560 and 1660cm-' bands. The situation is similar for the donor-D complexes. [A similar interaction between modes had previously permitted Watkins ~t al. (1990) to explain an anomalous isotope effect observed by Pajot et ul. (1988b) for the D-stretching vibrations of the "B-D and "B-D complexes in Si.] An energy-level diagram for the hydrogen modes of the donor-H complexes is shown in Fig. 24(b). The presumed hydrogen-mode frequencies are shown on the left for the assumption that there is no anharmonic coupling between the modes. The H-stretching mode transforms like the A , represen-

3



tI 1650

205

/-

=\--

16001 (H:

4p

1550 850

1

E

I

' I4111

0 1 -

-

anharmonic coupling FIG. 24. (a) Structure of the Sb-H complex in Si. (b) Energy level diagram for the donor-H complexes in Si. The positions of the decoupled frequencies are shown on the left. The effect of a weak anharmonic perturbation is shown on the right. (From Zheng and Stavola, 1996.)

tation of the trigonal point group and the H-wagging mode transforms like E; the corresponding energy levels are shown by the high-frequency line near 1620cm-' and the line near 810 cm-', respectively. The second harmonic of the wagging mode transforms like the symmetrized product E x E, or in terms of irreducible representations, A, + E. These accidentally degenerate A, and E components of the second harmonic of the wag can be split by anharmonic terms in the potential (Herzberg, 1945; Wilson et al., 1955) and are shown by the two lower-lying levels near 1620 cm-' in Fig. 24(b). The modes that transform like the same irreducible representation can be coupled by anharmonic terms in the potential, that is, the H-stretching fundamental and the A, component of the second harmonic of the wag can be mixed, as is shown on the right in Fig. 24(b). The lowest-order term of the anharmonic perturbation that can couple the wave functions for the W-stretching fundamental, Is), and the second harmonic of the wag 12w) is the cubic term H' = k , ( X 2

+ y2)z

(18)

where the displacement coordinates are defined in Fig. 24(a). The energies of the coupled modes are given by the eigenvalues E , of the following perturbation matrix and the wavefunctions are given b y t h e corresponding

MICHAEL STAVOLA

206

eigenvectors a+ls) -

+ fl+12w): -

Here E , and E,,, are the decoupled energies of the H-stretching mode and the second harmonic of the wagging mode, respectively; E,, is related to the wagging-mode frequency by E,, = 2 E , - 6, where 6 is a small frequency shift due to anharmonicity; yi is the matrix element of H’ between the . values of 6 and yi and the energies of the uncoupled states Is) and 1 2 ~ ) The uncoupled states, E *, were determined by fitting the predicted energy eigenvalues to the energies of the observed H- and D-vibrational bands (Zheng and Stavola, 1996). There is also a weak band observed at 1616.7cm-’ in Fig. 23(b) for Sb-H. This band was assigned to the transition to the E component of the second harmonic of the wag shown in Fig. 24(b). The three bands near 810, 1560 and 1660cm-’, and also the weak band at 1616.7cm- all decay together upon annealing, which confirms that they all arise from the same defect center. The intensities of transitions to the admixed states I + and I - , are determined by the fraction of the H-stretching state Is) in the corrected wavefunctions because only the transition from the ground state to Is) is allowed for harmonic vibrations and has appreciable intensity. Thus the ratio of the intensities of the coupled modes is I + , ’ - = a:/a?. The intensity ratios calculated from the eigenvectors of the matrix in Eq. (19), without additional parameters, explain the isotope dependence of the intensities of the two high-frequency stretching bands. The assignment of the vibrational spectra of the donor-H and donor-D complexes in Si, based upon a Fermi resonance between the second harmonic of the hydrogen-wagging mode and the stretching fundamental, consistently explains the unexpected presence of two high-frequency hydrogen vibrational bands and the isotope dependence of their relative intensities. Although only the Sb-H complex has been discussed here, the P-H and As-H complexes behave similarly (Zheng and Stavola, 1996; Zheng, 1996).

V. Symmetry, Stress Alignment, and Reorientation Kinetics from Uniaxial Stress Data A defect in a diamond or zinc-blende lattice can have several crystallographically equivalent orientations, consistent with its symmetry. For


207

example, a defect with trigonal symmetry can have its threefold axis along any of the ( 1 1 1 ) crystal axes. An applied uniaxial stress can partially lift this orientational degeneracy and give rise to shifts of the transition energies and ground state energy that depend on the orientation of the defect with respect to the stress direction. When used in conjunction with vibrational spectroscopy, uniaxial stress perturbations provide additional information about the symmety, reorientation kinetics and ground state energy shift of a defect. In the following, the acceptor-hydrogen complex, B-H in Si (Pearton et al., 1992; Pankove and Johnson, 1991; Stutzmann and Chevallier, 1991), will be discussed as an example of a defect whose microscopic properties have been studied with uniaxial stress techniques. A few other examples will be introduced to illustrate additional points. In stress experiments, uniaxial stresses are applied to oriented samples along the directions of high symmetry ([OOl], [llO] and [lll]). It is straightforward to construct a stress apparatus that can be used to apply a well calibrated stress and that can be used with a commercial, variable temperature, cryostat. In a typical apparatus, a stainless steel tube is used as a push rod. Force is applied to the push rod from outside of the cryostat with a pneumatic cylinder. The stress is applied along the long axis of the sample (typical dimensions 2 x 2 x 8mm3) to end faces that have been ground flat and parallel. The probing light passes through the sample along a direction perpendicular to the stress axis and can be polarized with a wire grid polarizer.

1. STRESS-INDUCED SPLITTING OF

THE

VIBRATIONALLINES

An applied stress causes a vibrational line to split into components when the vibrational transition energies of differently oriented defects are made inequivalent by the stress. In most cases, stress can be applied at sufficiently low temperature so that the defect being studied does not reorient. In Fig. 25, the bond-centered configuration of an acceptor-H complex is shown with stress applied along the [ 1lo] direction. The four (1 11) orientations of the center are divided into two inequivalent pairs by the [llO] stress that will have different vibrational frequencies. Further, through the use of polarized light in uniaxial stress experiments, transitions at centers with different orientations can be selectively excited. From the stress data, the symmetry of the defect can be determined. Kaplyanskii (1964) has considered the stress-induced splitting of the energies of transitions between nondegenerate levels of defects of the seven possible symmetry types in cubic crystals. The stress induced shift A of the transition energy of a specifically oriented defect can be written in terms of

208

MICHAELSTAVOLA

FIG. 25. An acceptor-H complex under [llO] stress. n, and nh are the populations of the orientations u and h that arc made inequivalent by the applied stress. Below, a schematic diagram is shown of the energy of the acceptor-H complex vs position of the hydrogen atom in the (101) plane that contains inequivalent orientations. The orientation-dependent shift of the transition energies between Vibrational states of a defect gives rise to the splitting of the vibrational lines. The activation energy for reorientation is the height of the barrier between different orientations and is determined from the reorientation kinetics. Jt is the oricntationdependent shift of the ground-state energy under stress that gives rise to strcss alignment when the temperature is sufficiently high for the kinetic barrier to be surmounted. The ground-state energy shift is determined from the magnitude of the alignment in equilibrium. (From Veloarisoa et d,1993.)

a symmetric piezospectroscopic tensor Ai as

Here the defect coordinate system with cubic axes x, y, and z is used. The aij are the components of the stress tensor where oij = ninjs. ni and nj are the direction cosines of the stress vector with respect to coordinate axes i and j ; s is the magnitude of the stress (defined to be positive for compression). The symmetry of a center restricts the number of inequivalent components of the piezospectroscopic tensor. For the example of a trigonal


209

center, the defect's threefold axis is taken to be along the [lll] direction. In this case, the symmetric piezospectroscopic tensor has only two independent components and is given by

A=

The transition energy shifts for each of the defect orientations can be determined from Eq. (20). The intensity of a stress-split component of a vibrational line is not given simply by the number of orientations that remain equivalent in the presence of the applied stress because the absorption of polarized light for a specific defect orientation is proportional to the square of the projection of the optical transition moment along the polarization direction. Thus the relative size of the absorption coefficient for a stress-split component is determined by summing the squares of the transition moment projections for the different defect orientations that contribute to it. The number of stress-split components, their shift rates, and their relative intensities for transitions between nondegenerate levels for the defect symmetries of interest have been tabulated by Kaplyanskii (1964). Thus a comparison of the observed splitting pattern and intensities for a few sample orientations with the tabulated results is often sufficient to determine a defect's symmetry. For degenerate vibrational levels, the applied stress can lift the level degeneracy in addition to the orientational degeneracy, thereby providing information from which the vibrational modes can be assigned. Results for the stress-induced splittings of the transitions between the degenerate states of defects with tetragonal or trigonal symmetry have been derived by Hughes and Runciman ( 1967) and tabulated. The B-H complex has a hydrogen vibrational mode at 1903cmp1 at liquid He temperature (Stavola et al., 1988a). Spectra are shown in Fig. 26 for the high symmetry stress directions and for polarizations of the exciting light that are parallel and perpendicular to the stress axis (Bergman et a[., 1988b). The positions of the bands are plotted vs the magnitude of the stress in Fig. 27. A comparison of these data with Table IV shows that the 1903 cm-' band is a longitudinal stretching mode of a defect with trigonal symmetry. In this case, the stress-induced shift rates of five lines are consistently described by two parameters: A , = 12.7 cm-'/GPa, which is determined from the shift of the center of gravity of the lines; and A , = 13.5 cmp '/GPa, which is determined from their splitting. The polarization selection rules show that the transition moment direction is along the

210

MICHAELSTAVOLA

E l a ,

,

!

,

1900

1860

o//lllol

1

,

1860

l

1900

d

I

1940 1860

1900

298MPa

O//[?lo]

,

l

l

1940 1860

L

1940

244MPa

l

1900

,

l

194(

FREQUENCY icrn-'t

FIG. 26. Spectra of the H-stretching line of the B-H complex in Si measured near 15 K with polarized light; is the applied stress with the magnitude and direction shown, and E is the electric vector of the light. (From Bergman et d.,1988b.)

(1 11) symmetry axis of the defect, consistent with a longitudinal stretching mode. The donor-H complexes in Si for P, As, and Sb donors (see Section IV.3) all have a vibrational absorption line near 810cm-' (Bergman et ul., 1988a). This line was assigned to the doubly degenerate hydrogen wagging vibration

TABLE IV

A AND INTENSITY RATIOSFOR AN A , MODEOF A TRIGONAL CENTER (Kaplyankii, 1964)

STKESS-INDUCEI) SHIITS

A

Intensity and Polarization

3

VlBRATiONAL SPECTROSCOPY OF

x

0

500 0

250

211


250

E II

500 0

[rio]

250

500

Stress (MPa)

FIG.27. Stress-induced shifts of the vibrational frequency of the H-stretching line of the B-H complex in Si. (From Bergman et a[., 1988b.)

%As-H h

r

6

Y

808

0

200

400

600

0

200

400

600

Stress (MPa)

FIG. 28. Stress-induced shifts of the vibrational frequency of the doubly degenerate Hwagging line of the As-H complex in SI measured near 15 K. (From Bergman er ul., 1988.b)

of the donor-H complex. The positions of the stress-split components vs the applied stress and their polarization dependencies are shown in Fig. 28 for the As-H complex (Bergman et a/., 1988b). A comparison with the results of Hughes and Runciman (1967) shows that these data are consistent with the excitation of an E mode of a trigonal center, confirming the assignment of the 810cm-' line to the hydrogen-wagging mode. There are more components than for the 1903 cm- line of the B-H complex because the stress lifts the orientational degeneracy and the mode degeneracy. Four parameters are required to describe these stress results.

2. KINETICS OF DEFECT MOTION An applied stress gives rise to a shift of the ground state energy of a defect complex that depends on its orientation (Fig. 25). In favorable cases, there is a range of temperatures in which a defect is free to move among its

212

MICHAELSTAVOLA

possible orientations by thermally activated jumps over a barrier between sites. Stress techniques can be used to measure the kinetics of the defect motion and hence determine the height of the barrier between different orientations. In such experiments, stress is applied at a temperature that is sufficiently high for the defect to reorient. The preferential occupation of the lower-energy orientations gives rise to a net alignment of the defects. The optical transition moments for vibrational transitions have specific orientations with respect to the defect coordinates. Hence, a stress-induced alignment of the defects will be detected as an anisotropy in the polarized optical absorption. If a sample containing defects that are preferentially aligned under stress is cooled to sufficiently low temperature with the stress maintained, the alignment can usually be quenched in and will persist after the stress is removed. In subsequent annealing experiments, made in the absence of stress, the centers will redistribute randomly among their equivalent orientations. A convenient measure of the anisotropy of the polarized absorption is the dichroic ratio, defined as

where the as are the absorption coefficients for light beams polarized parallel and perpendicular to the stress direction. The kinetics for the disappearance of the stress-induced dichroism can be measured to determine the reorientation kinetics of the center. In most cases, the decay of the dichroism follows first-order kinetics. In this case

DID, = exp( - k , t ) where

k,

= v o exp( - E , / k T )

Here t is the annealing time, E , is the activation energy for reorientation of the complex, vo is an attempt frequency, T is the annealing temperature, and k, is the rate of decay of the dichroism. The rate of a jump from one specific orientation to another is proportional to k , with the constant of proportionality depending on the configuration of the defect in the lattice. Spectra measured for the B-H and B-D complexes in Si that had been aligned by a stress applied along the [l lo] direction are shown in the inset in Fig. 29 (Cheng and Stavola, 1994; Stavola and Cheng, 1995). (Both B-H and B-D had been introduced into the same sample so that an unambiguous comparison of their reorientation rates could be made.) The stress was

3



213

1415 1880 1930 FREQUENCY (cm-I)

1375

0.01

0

I

I

10000

20000

I

I

30000

40000

t (s) FIG.29. Decay of the stress-induced dichroism for the 1903 cm-' band of the B-H complex and for the 1390cm band of the B-D complex. Polarized absorption spectra for the (a) B-D and (b) B-H complexes are shown in the insets. The decay of the dichroism is sufficiently slow at the measurement temperature (65 K) that the stress-induced dichroism did not decay appreciably while the absorption measurements were made. (From Cheng and Stavola, 1994.)

applied while the sample was cooled and was then removed immediately prior to the absorption measurements made with light beams polarized parallel (Ell [110]) and perpendicular (E(ICI101)to the stress direction. The greater absorption strength for light polarized perpendicular to the stress direction shows that the orientations in the plane perpendicular to the stress direction are preferred. The decay of the dichroism is shown in Fig. 29 for B-H and B-D for two temperatures. The reorientation process corresponds to a jump of the hydrogen (or deuterium) atom from one bond-centered site to another (Fig. 30). The rate of a jump from one specific site to another, kj, is related to the rate of decay of the dichroism by (Stavola et al., 1988b) kj

= 4kd

(24)

A plot of In(kj) vs T - ' is shown in Fig. 30. If the reorientation kinetics are first-order, the value of E,/k is determined from the slope of the line fit to the data. Values for E, are given in the figure. These data have two interesting aspects. First, the rate of B-D reorientation is greater than that of B-H in this temperature range. This is a surprising result because one would expect the heavier D to jump more slowly than H. Second, the jump rate and activation energy for B-H

214

MICHAEL STAVOLA

E: 1.4

= 0.1 94 f 0.002eV

1.5 1.6 100 / T (K -1)

1.7

1.8

FIG.30. Hydrogen anc‘ deuterium jump rates vs T-’ for the B-H and B-D complexes. Activation energies that correspond to the solid lines are shown. In thc inset. the H jump from one bond-centered site to another is shown for the B-H complex. (From Cheng and Stavola,

19Y4.)

reorientation have been determined near 125 K by internal friction measurements for much faster reorientation rates (Cannelli et ul., 199 1). The internal friction data and stress dichroism data cannot be fit with a single activation energy (Cheng and Stavola, 1994). A deviation of the hydrogen jump rate from the simple Arrhenius behavior described by Eq. (23a) is well known for hydrogen diffusion in metals where tunneling plays a role in the jump process (Fukai and Sugimoto, 1985). The results for B-H in Si support a suggestion made by Stoneham (1989) that reorientation occurs by a thermally assisted tunneling mechanism rather than by hopping over a barrier. In some cases, the jump process by which a defect reorients corresponds to a diffusion jump. Measurements of the decay of the stress alignment then provide a measurement of the diffusion constant D that is related to the time constant z for a jump from one specific site to another by

where d is the jump distance. The classic example of the determination of a defect’s diffusion constant from the decay kinetics of stress dichroism is the diffusion of interstitial oxygen in Si (Corbett et ul., 1964). The reorientation of interstitial oxygen occurs by a jump from one bond-bridging site t o


215

another (Fig. 31). This reorientation process corresponds to a diffusion jump. In this case stress was applied near 400°C to align the interstitial oxygen preferentially, giving rise to a dichroism in the 1106cm-’ band (measured at room temperature) assigned to the antisymmetric stretching mode of oxygen. The decay of the dichroism was measured in subsequent annealing experiments to provide an accurate value of the diffusion constant at a much lower temperature than is accessible by mass transport measurements. In their early determination of the diffusion constant of interstitial oxygen, Corbett et al. (1964) combined their diffusion data, measured from the relaxation of dichroism, with internal friction data (Southgate, 1960; Hass, 1960) to obtain the following result:

D

= 0.23 cm2/s exp(2.561 eV/kT)

(26)

The data used by Corbett et a/. (1964), shown in Fig. 25, give a value for the diffusion constant of oxygen that is in excellent agreement with the results

YO

\llo,-o,

6?0 T (“C)

4?0

CORBETT eral.(l964) MIKKELSEN (1982) NEWMAN eraL(l983) * SOUTHGATE (1960) 0 STAVOLA etaL(1983) 0 h

io+

I

I

I

I

I

I

0.8

1.0

1.2

1.4

1.6

1031~

(~-4)

FIG. 31. Selected diffusion data plotted vs 1/T for interstitial oxygen in Si. The inset shows the configuration of interstitial oxygen in Si and a diffusion jump from one bond-bridging site to another. Attributions for the data shown in the figure are included in the References.

216

MICHAEL STAVOLA

of several subsequent studies (Newman, 1994). Through the combination of mass transport measurements at higher temperature (Mikkelsen, 1982) and stress dichroism measurements near 400°C (Corbett et al., 1964; Stavola et al., 1983; Newman et al., 1983), the oxygen-diffusion constant has now been measured for over 12 orders of magnitude. 3. GROUND-STATE ENERGY SHIFT

If the temperature is sufficiently high for the defects to reorient and there has been suficient time for equilibrium to be established under stress, then the populations of the different orientations will be determined by Boltzmann statistics. This gives rise to the preferential alignment of the defects discussed in the preceding section. For the example shown in Fig. 25, the ratio of the defect concentrations for orientations b and a would be given by nb/nu

= exp[l(E, - E b ) / k T l

(27)

where E , and E , are the ground-state energies of defects with orientations a and b, respectively. The energy difference E, - E , can be determined from the population ratio. The stress-induced shifts of the ground-state energy can be written in terms of a piezospectroscopic tensor with the same general form as that already discussed in Section V.l, but with different values for its components. (The tensor components are written with primes here to distinguish them from the tensor components for the transition energy shifts.) Only the traceless part of the tensor can be determined from stress-alignment data because the alignment is only sensitive to the difference in energy between different orientations. For the example of a defect with trigonal symmetry, only one parameter A; is required to describe the stress-induced groundstate energy difference E, - E, and stress applied along one direction is sufficient to determine this coupling to stress. For the examples shown in Fig. 25, a uniaxial stress with magnitude o 1 is applied in the [IlO] direction. For this stress direction and [OOl] optical viewing direction, the ratio of the measured absorption coefficients aL/cL,,for the hydrogen-stretching band gives the population ratio nJn, and from Eq. (27j, the groundstate energy difference, E, - E,. I n terms of the magnitude of the [ l l O ] stress, A; is given by A; = (E,

-

E,)/2o

1 I0

(28)

From Eqs. (27) and (28) one finds that the slope of a plot of In(n,/n,) vs is 2A;/kT where T is the temperature at which the defect was aligned.

cr,

3 VIBRATIONAL SPECTROSCOPY OF LIGHTELEMENTIMPURITIES

1.5 h

217

-

- Si:B-H -

E

I

0

I

100

I

I

I

I

I

200 300 STRESS (MPa)

I

400

I

500

FIG. 32. The In(ct,/a,,) vs the magnitude of the stress applied along the [ I l O ] direction at 77 K for the B-H complex in Si. (From Veloarisoa rt al., 1993.)

A plot of In(n,/n,) vs the magnitude of the stress applied along the [llo] direction is shown in Fig. 32 for the B-H complex in Si (Veloarisoa el al., 1993). Stress was applied at 77 K and maintained for 30 min prior to the measurement of the population ratio. At 77K, the reorientation is suficiently fast to ensure that equilibrium was established. From these data the value A; = 93.6 cm- '/GPa was determined (where the sign of a compressive stress is defined to be positive). Symmetry information can be obtained from the orientation dependence of the stress alignment. For a defect with trigonal symmetry, for example, the different (1 11) directions remain equivalent for stresses applied along a (001) direction and no alignment or dichroism is produced. Stress dichroism data were used to establish the trigonal symmetry of oxygen in Si by Corbett et al. (1964). And no dichroism is produced by a [OOl] stress for the B-H complex in Si, consistent with its trigonal symmetry determined from the splitting of the hydrogen-stretching line (Stavola et al., 1988).

OF MICROSCOPIC PROPERTIES 4. CORRELATION

An important aspect of uniaxial stress experiments is that stress perturbations can be combined with a variety of characterization methods. The quantitative results determined for the ground state shift or reorientation kinetics, for example, provide a means to correlate the defect spectra observed by different techniques. In the first use of stress-induced dichroism techniques, the reorientation kinetics of the A-center in Si (i.e., the oxygen-

218

MICHAEI. STAVOLA

FIG 33. The configuration of the oxygen-vacancy complex (or A center) in Si. (From Watkins and Corbett, 1961.)

vacancy complex, Fig. 33) were determined in EPR and vibrational spectroscopy experiments (Watkins and Corbett, 1961; Corbett et ul., 1961). The inset in Fig. 34 shows the stress-induced dichroism observed for a vibrational absorption band at 12pm in oxygen-rich Si that had been electron irradiated. The two data points shown in Fig.34 for the loss of stressinduced dichroism in the 12pm band were found to be in excellent agreement with the reorientation kinetics of the A-center, determined in stress experiments performed with EPR. Thus, the 12 pm band was convincingly assigned to the A-center. Since these pioneering experiments, the correlation of the results of stress experiments has often been used to prove that the spectroscopic signatures observed by different techniques are indeed due to the same defect.

ACKNOWLEDGMENTS

The author would like to thank G. D. Watkins, S. J. Pearton, W. B. Fowler, G. Davies, C. G. DeLeo, K. Bergman, D. M. Kozuch, I. A. Veloarisoa, Y. M. Cheng, J.-F. Zheng, S. J. Uftring, M. Evans, J. Zhou and M. Weinstein for enjoyable collaborations on vibrational studies of impurities. He would also like to thank R. C . Newman for the numerous


219

Reorientation spin resownca - o

-

60

t-

101

30

'f

I

I

,

,

,

7.0

I

,

,

,

0.0 1@/T

(K-1)

FIG. 34. Isothermal relaxation time 7 vs l / r for the stress-induced alignment of the A center in Si. Data shown with open circles were measured by EPR and data shown by closed circles were measured by infrared absorption. The inset shows the 12 pm vibrational absorption band assigned to the neutral charge state of the A-center. The transmission spectra were measured with light beams polarized parallel and perpendicular to the direction of an applied (100) stress of 185 MPa. The spectrum with zero applied stress is also shown. (From Corbett et al., 1961.)

discussions of local mode spectroscopy that have occurred over the years. The preparation of this chapter was supported by the National Science Foundation under Grant No. DMR-9415404 and by the US. Navy Office of Naval Research under Grant No. N00014-94-1-0117.

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Maradudin, A. A,, Montroll, E. W., Weiss, C. H., and Ipotova, I. P. (1971). Solid State Physics, H. Ehrenreich. F. Seitz, and D. Turnbull, eds., Academic, New York, Supplement 3, 2nd edition. Mayur, A. J., Sciacca, M. D., Udo, M. K., Ramdas, A. K., itoh, K., Wolk, J., and Haller, E. E. (1994). Phys. Reo. B, 49, 16293. Mayur, A. J., Sciacca, M. D., Kim, H., Miotkowski, I., Ramdas, A. K., and Rodriguez, S. (1996). Phvs. Rev. B, 53, 12884 (1996). McCluskey, M. D., Haller, E. E., Walukiewicz, W., and Becla, P. (1996). Phys. Ren. B, 53, 16297. Mikkelsen, J. C. (1982). Appl. Phys. Lett., 40, 336. Morse, P. M. (1929). Phys. Rec. 34, 57. Moss, T. S. (1959). Optical Properties qf Semiconductors. Academic Press, New York. Murakami, K., Fukata, N., Sasaki, S., Ishioka. K., Kitajima, M., Fujimura, S., Kikuchi, J., and Haneda, H. (1996). Phys. Rev. Lett., 77, 3161. Murray, R. and Newman, R. C. (1989). Landolt-Biirnstein, Vol. 22b, 0. Madelung and M. Schulz, eds., Springer-Verlag. Berlin, pp. 382--390, 505-510, 576-583. 661, 669, 708, 713. Murray, R., Newman, R. C., Leigh, R. S., Beall, R. B., Harris, J. J., Brozel, M. R., MohadesKassai, A,, and Goulding, M. (1989a). Seniicond. Sci. Technol., 4, 423. MUrrdy, R., Newman, R. C., Sangster, M. J. L., Beall, R. B., Harris, J. J., Wright, P. J., Wagner, J., and Ramsteiner, M. (1989b). f. Appl. Phys., 66, 2589. Newman, R. C., Thompson, F., Hyliands, M., and Peart, R. F. (1972). Sol. St. Commun., 10,505. Newman, R. C. (1973). Infrured Studies of Crystal Dqfects. Taylor and Francis. London. Newman, R. C., Tucker. J. H., and Livingston, F. M. (1983). J . Phys. Cc Solid St. Pkys., 16, L151. Newman, R. C. (1990~).Growth and Characterizatiun of Semiconductors, R. A. Stradling and P. C. Klipstein, eds., Adam Hilger, Bristol, p. 119. Newman, R. C. (1990b). Semirond. Sci. Techno/., 5,911. Newman, R. C. (1991). Hjidroqen in Semiconductors: Bulk and Surface Properties, M. Stutzmann and J. Chevallier, eds., North Holland, Amsterdam, p. 409. Newman, R. C. (1993). Imperfections in III/VMuferiul.s, E. Weber, ed., Academic Press, Boston, p. 117. Newman, R. C. (1994a). Oxygen in Silicon, F. Shimura, ed., Academic Press, Boston, p. 290. Newman, R. C. (1994b). Semicund. Sci. Techno/., 9, 1749. Pajot, B. and Deltour, J. P. (1967). Infrared Phys., 7, 195. Pajot, B., Stein, H. J., Cales, B., and Naud, C. (1985). .I. Electrochern. Soc., 132, 3034. Pajot, B. and Cales, B. (1986). Oxygen, Carbon, Hydrogen und Nitrogen in Crystalline Silicon, J. C. Mikkelson, S. J. Pearton, J. W. Corbett, and S. J. Pennycook, eds., MRS, Pittsburgh, p. 39. Pajot, B. and Clauws, P. (1987). Proceedings q f t h e 18th Intemutional ConfLwnce on the Physics qfSrmiconductor.s. 0. Engstrom, ed., World Scientific, Singapore, p. 91 1. Pajot, B., Newman, R. C., Murray, R., Jalil, A,, Chevallier, J., and Azoulay, R. (1988a). Phys. B, 37,4188. Pajot, R., Chari, A,, Aucouturier, M., Astier, M., and Chantre, A. (1988b). Solid State Cornmun., 67, 855. Pajot. B., Clerjaud, B., and Chevallier, J. (1991). Hydrogen in Semiconducturs: Bulk unii Surface Properties, M. Stutzmann and J. Chevallier, eds., North Holland, Amsterdam, p. 371. Pajot, B. (1994). Ovyqen in Silicon, F. Shimura, ed.. Academic Press, Boston, p. 191. Pajot, B., Artacho, E., Ammerlaan, C. A. J., and Spaeth, J.-M. (1995). J . Phyr. Condens. Matter 7, 1077. Pajot, B. (1996). Early Stages u j Oxygen Precipitation in Silicon, R. Jones, ed., Klewer, Dordrecht, The Netherlands, p. 283. Pankove, J. I. and Johnson, N. M. (1991). Hydrogen in Semiconductors. Academic, Boston. Pavesi, L. and Giannozzi, P. (1991). Phys. Rev. B, 43, 2446. ~

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SEMICONDUCTORS A N D SEMIMETALS, VOL 5 1 R

CHAPTER4

Defect Processes in Semiconductors Studied at the Atomic Level by Transmission Electron Microscopy P. Schwander and K-D. Rau I N S l l T U T b FOR SLMICONL~lI( rOR PIlYSlCS

FRAKKILIRT ( O D ~ KGERMANY ),

C. Kisielowski LAWRPW 6I.RKt ~ LEY N A I I O ~LAHORATORY A~ hKKI.I.tY, CALIbORNlh

111. Gribelyuk CORPORATL RESFARCHLABOI~ATORIFS

TI.XAS INSlRUMt'NTS,

INC..

DAIIAS. TEXAS

A. Ourmazd

1. IN~.RODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. INFORMATION FROM HRTEM LATTICE IMAGES . . . . . . . . . . . . . . . 1. Approuc1ie.s to Quuntitutive HRTEM . . . . . . . . . . . . . . . . . .

111. QUANTITATIVE ANALYSIS BY REAL-SPACE METHODS I . Principle . . . . . . . . . . . . . . . . . 2. Implenwntarion . . . . . . . . . . . . . . 3. Guidelines . . . . . . . . . . . . . . . . IV. APPLICATIONS . . . . . . . . . . . . . . . . 1. Difusion in AlGnAs . . . . . . . . . . . . 2. DrJusion in GcSi . . . . . . . . . . . . . 3. SiiSiO, Iniqfacial Roughness . . . . . . . . V. CONCLUSIONS . . . . . . . . . . . . . . . . REFEKENCES . . . . . . . . . . . . . . . . .

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225 Copyright i:, 1999 by Academic Presc All rights of reproduction in any form rescrved. ISBN 0-12-752165-8 ISSN 0080-8784 130.00

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I. Introduction The semiconductor industry roadmap for the next decade calls for the fabrication of chips consisting of up to 50 billion transistors, each 200 atoms long and 50 atoms deep. Fewer than 64 faulty transistors would be tolerated per chip. This degree of manufacturability requires unprecedented control of a variety of defect-mediated processes in near-surface regions with microscopic lateral dimensions. Figure 1 shows a high-resolution micrograph of a MOSFET in next-generation CMOS (0.1pm) technology as obtained from a modern transmission electron microscope. The length of the channel, and specifically the thickness of the gate oxide, is measured in units of atomic spacings. In order to achieve such a level of manufacturability, our fundamental understanding and control of the involved processes has to reach the atomic level. To predict and control diffusion behavior or interfacial roughness on such length scales, characterization techniques offering according resolution and sensitivity are indispensable for process development and optimization. For many of the steps associated with device fabrication a thermal budget is required. Diffusion during such annealing steps essentially relies on the reaction and behavior of point defects. To elucidate the mechanisms by

FIG. 1. High-resolution transmission electron micrograph of an n-channel MOS transistor in cross section. The inset shows a lattice image of the channel region of the device. The channel IS less than 400 atoms long (Courtesy of Y. Kim).

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which intrinsic point defect reactions occur, the thermodynamic parameters for the creation and migration of point defects must be measured. Many characterization techniques have been developed, each with its specific strengths and limitations. Each method can be classified by its spatial resolution, sensitivity, statistical reliability, and ease of use. Tradeoffs usually have to be made, depending on the application. Transmission electron microscopy (TEM) has been used for some time to image defect structures at high spatial resolution (Hirsch et ul., 1965). In contrast to most scanning probe techniques, TEM delivers information on bulk properties (projected along the beam direction). TEM images are also highly sensitive to changes in sample composition and crystalline perfection on a microscopic scale. On the other hand, extracting quantitative information from lattice images requires sophisticated techniques, not usually needed by scanning probe methods. Recently, high-resolution transmission electron microscopy (HRTEM) entered the quantitative era. Different approaches for quantitative evaluation of HRTEM lattice images have been developed for specific problem classes (Ultramicroscopy, 64, 1996; Mobus and Ruhle, 1994; Ourmazd et ul., 1989a,b; Schwander et a/., 1993). Of particular interest is the ability to measure composition profiles across specific microscopic regions of a sample with near-atomic resolution and sensitivity. We will show how quantitative diffusion studies using marker layer experiments allow thin films to be used as microscopic laboratories for the investigation of fundamental properties of point defects. This can provide valuable input for physically based diffusion models used in state-of-the-art processing simulators. This chapter is organized as follows. Section I1 provides a brief review of the capabilities and limitations of HRTEM methods. Section 111 gives a short introduction to quantitative analysis in HRTEM and describes the basic principles of real-space analysis of lattice images. The existing implementations of Chemical Mapping and QUANTITEM are introduced and the range of applicability of each method, together with practical guidelines for the use of the techniques, are discussed. Section IV describes selected applications, giving examples of (a) Investigation of diffusion in AIGaAs; (b) Interdiffusion measurements in SiGe; and (c) Measurement of the Si/SiO, interfacial roughness. We summarize and conclude in Section V. It should be noted that this chapter is neither a definitive treatment on the subject, nor an exhaustive review. The primary intention is to provide the interested reader with the conceptual basics for investigation of defect processes by quantitative HRTEM. Details can be found in the cited literature.

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11. Information from HRTEM Lattice Images

In high-resolution transmission electron microscopy, a crystalline sample is usually imaged along a low indexed zone axis. Provided there is sufficient instrumental resolution of the electron microscopic setup, the resulting lattice image can be interpreted as a fingerprint of the sample (Coulomb) potential projected along the zone axis. Figure 2 shows an example of an HRTEM micrograph of a GaAs/Al,,,Ga,,,As quantum well in cross section. The location of the quantum well, representing a random alloy structure, is easily recognized as a pattern change in the high-resolution image. The corresponding change in the projected potential can therefore be directly visualized in a qualitative way. It is important to note, however, that due to the strong interaction of electrons with solids, described by dynamical or multiple scattering, as well as due to the highly nonlinear image formation process in HRTEM that stems from the inherent aberrations of electromagnetic lenses, the quanti-

FIG.2. High-resolution lattice image of GaAs/AI,,,Ga, ,As quantum wells at two different magnifications. The area of the high-magnification lattice image is indicated in the overview picture. Note the change of the image pattern at the quantum well interface.

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tative interpretation of the contrast in lattice images is rather complicated (see Section 11.1). Nevertheless, much useful information is contained in high-resolution lattice images. Numerous HRTEM investigations of dislocations, precipitates, grain and tilt boundaries, as well as of interface structure and roughness in semiconductors have provided valuable insight into the microscopic properties of these materials (for a review see, e.g., Spence, 1988), although the images have been primarily interpreted in a qualitative way. Over the last few years, however, an increasing trend to a quuntitutive evaluation of the image contrast has emerged. In the following, we will briefly address the principles of the different approaches that have been developed. We will then focus on quantitative evaluation by the real-space techniques Chemical Mapping and QUANTITEM.

TO QUANTITATIVE HRTEM 1. APPROACHES

High-resolution transmission electron microscopy aims to image the “atomic” (Coulomb) potential of an unknown sample, projected along a symmetry direction. Quantitative HRTEM attempts to measure the atomic potential. This requires the determination of two quantities: the projected potential and the corresponding uncertainty or measurement error. Lattice images taken by HRTEM can contain local information about the sample at near-atomic resolution. In spite of their visual appeal, lattice images cannot be directly interpreted in terms of sample structure and/or composition. There are two primary reasons for this. First, electrons are multiply scattered in the sample, even after a few atomic layers (see, e.g., Spence, 1988). Second, electromagnetic lenses have severe aberrations, leading to a strong dependence of the image features on the contrast transfer function of the electron microscope (e.g., the defocus of the objective lens). For these reasons, the image depends sensitively on the sample thickness as well as the imaging conditions. Quantitative HRTEM aims to determine the structure and composition of the sample with high spatial resolution and sensitivity. There are several approaches to quantitative HRTEM. Many of them have made substantial progress in recent years. They can be roughly divided into three different classes, as described here. (1) The projected potential is determined in two steps. First, the complex electron wave at the exit surface of the crystal is determined. This requires deconvolution of the transfer characteristics of the electron microscope [Fig. 3(a)]. Second, the effect of dynamical scattering is “unfolded” to obtain the projected potential. The wave function can be obtained either by

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(8)

(b)

FIG. 3. Schematic representation of two approaches to quantitative HRTEM. In the conventional approach (a), the eflects of lens aberrations and multiple scattering are successively treated in order to retrieve the sample projected potential P(x, y ) from the image intensity I(x, y). In the “real-space’’ approach (b), the effects of lens aberration and multiple scattering are represented by a function F , which represents a nonlinear relationship between the projected potential and the image I ( x , y). The function F depends on the imaging conditions and varies from image to image. Using vector pattern recognition to quantify the content of the image, real-space approaches attempt to determine the dependence of the function F on the sample projected potential P for each image individually.

electron holography (Lichte, 1986, 1991; Orchowski et a/., 1995) or by focus variation techniques (Schiske, 1968; Kirkland, 1984; Kirkland et ul., 1985; Coene et ul., 1992, 1996; Van Dyck and O p de Beeck, 1993). To obtain the aberration-free wavefunction, the image parameters must be determined with some precision. The projected potential can be estimated from the wavefunction by an inversion of the multiple scattering process (Van Dyck et ul., 1989; Gribelyuk, 1991; Lentzen and Urban, 1996; Scheerschmidt and Knoll, 1994). These approaches require, at least in principle, no u priori information about the object under investigation. ( 2 ) The sample structure is determined by comparison of the experimental image intensity with simulated images through objective metrics. Starting with an initial guess of a model structure, successive refinements attempt to minimize the differences between the simulated and experimental images. Suitable algorithms have been developed to automate the procedure (Thust and Urban, 1992; King and Campbell, 1993; Hofmann and Ernst, 1994; Mobus and Ruhle, 1994). The image parameters need not be known

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precisely, because they may also be refined together with the model structure. These approaches usually require some a priori information about the object. (3) There are real-space techniques that attempt to extract the information without any knowledge of the imaging parameters, or comparison with simulated images, beyond what is needed to establish the initial validity of the approach [Fig. 3(b)]. These real-space techniques use pattern recognition algorithms (Ourmazd and Spence, 1987; Ourmazd et al., 1989a, b, 1990; Bode et al., 1991; Baumann et al., 1992, 1995; Schwander et al., 1993; Stenkamp and Jager, 1993; Seitz et ul., 1995; Hillebrand et al., 1995). These approaches require some a priori information about the object, for example, type of atoms and the crystalline structure type. At present, they rely on the assumption that the atoms lie entirely on coherent lattices. Therefore, at least at present, such methods cannot be applied to extended defects (i.e., they are applicable only to a subset of all possible systems). A treatment of all the approaches mentioned here would be far beyond the scope of the present chapter. In the following section, we will outline the conceptual basis and implementation of the real-space techniques Chemical Mapping and QUANTITEM, both of which belong to class (3). In spite of the limitations already mentioned, we will see in Section 1V that these techniques suffice to be applied to study a number of important materials problems.

111. Quantitative Analysis by Real-space Methods As has been outlined in Section 11, HRTEM lattice images are not simple projections of the sample structure. Traditionally, a link is established between the two by comparing lattice images with image calculations. However, this is complicated in practice because the imaging parameters are often not known and it is difficult to measure them with sufficient precision. Over the past decade, it has become clear that in some instances, information can be directly extracted from an experimental image without recourse to comparison with simulated images (Ourmazd et al., 1989a, b; De Jong and Van Dyck, 1990; Schwander et al., 1993; Walther and Gerthsen, 1993; Stenkamp and Jager, 1993). This chapter is concerned with techniques that extract information by real-space (pattern recognition) analysis of experimental images. The state of the art is such that, under appropriate conditions, the sample thickness and/or composition may be mapped at near-atomic level. Real-space analysis attempts to deduce the local properties of the sample by examining the local information content of the image.

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This approach is valid to the extent that the local information content of the image emanates from a known area in the sample of the same cross section. Taken to the extreme, real-space analysis examines individual regions containing images of single atom-columns. Usually the image is divided into small regions and the intensity distribution in each region is evaluated. In HRTEM the periodicity of the lattice can be used to divide the image into unit cells, within each of which the intensity distribution is measured. We will describe how analysis using vector pattern recognition techniques can be used to discriminate efficiently between signal and noise, allowing one to perform maximum noise reduction. The final goal of quantitative real-space analysis is the determination of two quantities at each unit cell of the lattice image: the measured projected potential; and its uncertainty or confidence level. Specifically, we are here concerned with Chemical Mapping (Ourmazd et ul., 1990) and QUANTITEM (Schwander et ul., 1993). These “direct” techniques use pattern recognition to extract the available information efficiently, producing dutu (i.e., numbers, each associated with a confidence level). For many systems of interest, an experimental image may be analyzed in a matter of minutes, to obtain detailed composition and/or thickness maps. Both Chemical Mapping and QUANTITEM are restricted to systems with atoms residing entirely on coherent lattices. 1.

PRINCIPLE

High-resolution lattice imaging is characterized by a highly nonlinear relationship between the image intensity distribution I(x, y ) and the sample projected potential P(x, y ) that may generally be described by’ I(x9 Y ) = W ( x , Y ) , Si)

where Sidenotes all the imaging parameters (defocus, accelerating voltage, etc.); F is a complicated, and in general unknown function, which relates the image intensity to the sample projected potential. In this description, changing the imaging conditions moves one to a different region of the function F . For a single image, obtained under a particular set of imaging conditions,

I(.% y )

=

WYx, yx S 3

or in other words

I(& y )

=

F”(P(x,y))

‘Strictly speaking, this relics on the projected potential approximation, in which the “vertical” position of an atom within a column is considered immaterial (see Spence, 1988).

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Real-space analysis attempts to derive the function Fa from the information contained in the image alone (Fig. 3b), thus directly relating the image intensity to the projected potential. We are concerned here with the two real-space methods, Chemical Mapping and QUANTITEM. In the case of Chemical Mapping, the nonlinear relationship between the projected potential and the intensity distribution is used to define chemically sensitive conditions. These select appropriate “operation windows” tuned to maximum sensitivity to composition changes, and minimum sensitivity to thickness changes. QUANTITEM, on the other hand, is applicable when such operation windows of enhanced chemical sensitivity are not available or are too small to be used in practice. More generally, as shown by Maurice et al. (1996), both methods may be viewed as limiting cases of a more general approach. The applicability domain for each technique must therefore be determined for each specific application.

2. IMPLEMENTATION To implement real-space approaches in practice, the lattice image is divided into unit cells, which extend over a small, integral number of crystal periods, each containing an equal number of atom-columns. The intensity distribution within each image unit cell is digitized at N pixels; N must be sufficiently large to retain the information content of the image, that is, following Shannon’s sampling theorem, the pixel frequency has to be at least twice the highest spatial frequency contained in the image (Shannon, 1948; Ourmazd et a]., 1990). For semiconductor materials, the unit cells are typically sampled at about 30 x 30 pixels. The information content of every image unit cell is then represented by a multidimensional vector with the N intensities as its components. The vectorization consists of making a list of the N ( e g , 900) gray values, which together constitute the components of an N-dimensional vector (Fig. 4). Within each unit cell, the parameters of interest (thickness, composition) are assumed to remain constant. Local changes of properties such as composition and thickness change the image, and hence “move” the corresponding unit cell vectors. The path of these vectors relates the changes in the image to the projected potential. If this link (effectively the function F“) can be quantitatively determined, the associated sample property, as well as the corresponding uncertainty, can be measured. Although the dimensionality of the vectors is generally high ( N > loo), the vectors can usually be adequately represented by linear combination of a few basis vectors, each representing an image pattern. A general image vector can then be represented by its components along these

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FIG.4. Schematic represcntation of how the information content of a lattice image can be encoded into a series of multidimensional vectors. The lattice image (a) is divided into image unit cells (b). The intensity distribution in each unit cell is digitized at N pixels (c). A multidimensional vector R is formed by taking the N intensity values as its components.

basis vectors. The path can then be represented in a space with dimensionality much smaller than N , in many cases, 2, 3, or 4 (Schwander et al., 1993; Kisielowski et a/., 1995; Schwander and Schneider, 1996; Maurice et a/., 1996). Chemical Mapping and QUANTITEM are tools to study the path traced out in multidimensional space by the vector tip of the image unit cell within the field of view. Alternatively, QUANTITEM attempts to determine the path F”, while Chemical Mapping finds conditions under which F” is particularly simple. Chemical Mapping requires the presence of chemical reflections and operation within “chemical mapping windows.” Under such circumstances, the path may be easily parameterized in terms of composition. For systems without chemical reflections, the path can often be approximated by relatively simple curves or surfaces. Indeed, for many materials and zone axes, the path is nearly an ellipse. This stems from the physics of dynamical scattering and the process of image formation. It can be understood by establishing a link between real-space techniques using vector pattern recognition and conventional theory of dynamical scattering and nonlinear image formation. The current implementation of QUANTITEM determines this ellipse, that is, the function F O , for each image, thus relating the image intensity to the sample projected potential P. QUANTITEM allows one to measure the compositional and thickness variations in the absence of chemical reflections. As an important practical point, it does not require specific imaging conditions ( e g , defocus). QUANTITEM parameterizes the path in terms of projected potential or “reduced thickness” (thickness in units of extinction distance, see Section III.2.b). This allows one to map the thickness variations when the extinction distance (composition) is known, or inversely, the extinction distance (composition) when thickness is known.

4 DEFECT PROCESSES u.

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235

Chemical Mapping

A lattice image is sensitive to the “projected potential,” which is determined by the sample thickness and composition at each atom-column. A chemical lattice image is obtained under conditions that are maximally sensitive to compositional changes. When chemical reflections are present, operating windows can be found such that the “signal” is insensitive (to first order) to thickness changes over the field of view. In the absence of chemical reflections, the relationship between the image properties and the composition changes rapidly with thickness, making the operating windows very narrow. The TEM samples are wedge-shaped, with atomically rough surfaces, causing the sample thickness to change from point to point. This makes it difficult to operate within the narrow windows available when chemical reflections are not present. Chemical reflections, such as the 002 for zinc blende system, occur because of chemical differences between the occupants of the different lattice sites. When the electron beam enters a crystal along a low index orientation, a number of diffracted beams get excited, each of the beams corresponding to a reflection in the diffraction pattern. They are coupled together and with the undiffracted beam, and exchange intensity while propagating through the sample. To first order, this multiple scattering process may be viewed as the scattering of electrons from the undiffracted beam to diffracted beams, and vice versa (the so-called “pendellosung” effect). “Structural” reflections are strongly coupled to the undiffracted beam, and thus exchange intensity rapidly with increasing sample thickness. This intensity transfer is less pronounced for the more “weakly allowed” chemical reflections. Because of the pendellosung effect, at certain sample thicknesses the signal is insensitive to changes in thickness, but varies (to first order) linearly in response to changes in composition (Fig. 5). Moreover, the severe aberrations of electromagnetic lenses impart the character of a bandpass filter to the objective lens, whose characteristics can be controlled by the lens defocus. Thus, judicious choice of defocus allows the lens to select, and thus further enhance the sensitivity of the image to chemical reflections. A chemical lattice image encodes the compositional changes in the sample into changes in the local patterns that form the image mosaic. These changes can be quantified in terms of noise, and hence confidence level, to form chemical maps. A chemical map is obtained by assuming that an image cell of intermediate composition is a linear superposition of the end-point compositions. Thus, as an example, an image unit cell of Al,,,Ga,,,As is assumed to be a 40:60 admixture of the image unit cells of AlAs and GaAs. Technically, the image unit cell vectors of the endpoint compositions are either obtained from simulation or, preferably, from the experimental image

236

P. SCHWANL)ER et al.

0

50

100

150 200 Thickness (A)

250

300

350

FIG. 5. Variation of intcnsity with sample thickness and composition for (002) (solid curves) and (022) (dotted curves) beams in AI,Ga, -.xAs. Dynamical calculation for 400 keV electrons parallel to [loo].

itself by averaging over many unit cells in areas of known composition. The composition of an intermediate “target” image unit cell is determined by finding the relative admixture of the end-point compositions present in the target unit cell image. In order to determine this, we use the angular distance of the in-plum component of the intermediate image vector ( e g , of Al, ,Ga, 6 A ~ as ) projected on the plane defined by the endpoint vectors (Fig.6). By exploiting the angular position of the vector rather than its length as metrics, efficient noise reduction is achieved. Noise that affects the intensity distribution within the image unit cell effectively leaves the angular position nearly unchanged (Ourmazd et nl., 1989a). The validity of this approach has to be established afresh for each material system, that is, the operation windows have to be determined by simulation. This is dcmonstrated in Fig. 7, where the angular position of the image vectors from the templates (i.e., the “chemical signal”) is plotted vs sample thickness and A1 concentration in AlGaAs viewed along the [ I l O ] direction. For a broad range of thickness (150-25OA) the chemical signal is a function of A1 content only and changes almost linearly with composition. As an experimental example, Fig. 8 (bottom part) shows the composition change across a GaAs/AlGaAs quantum well as measured by Chemical Mapping (Ourmazd et ul., 1990). Composition is represented as height in

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237

FIG. 6. Determination of composition by vector pattern recognition. Simulated templates (image unit cell patterns) are shown for pure GaAs and AlAs and an intermediate composition of A1,,,Ga0,,As. The angular distance of the projection of the intermediate image vector on the plane defined by the end-point composition vectors is used as a metric Tor the A1 concentration measurement

AlGaAs

Value o f defocus : -250

Angstrom

FIG. 7. Chemical signal, that is, the angular position of the projection of the image unit cell vector from the end-point templates, as a function of composition and sample thickness. Note that over a broad range of practical sample thicknesses (- 150-250 A), the angular position is a sensitive function of A1 concentration only (dcfocus = -250A on a JEOL 4000EX electron microscope).

238

FIG. X. Top: Chemical lattice image o f a CaAs quantum well between its two AI,,,Ga,,,As barriers. Bottom: Three-dimensional plot showing the result of vector pattern analysis of the lattice image obtained by Chemical Mapping shown above. Height represents the local composition and gray scale changes represent statistically significant changes in composition.

the chemical map. The corresponding high-resolution chemical lattice image is shown in the top part. As demonstrated by Baumann et (11. (1992), the composition of individual atom-columns can often be measured with near-atomic sensitivity. To summarize, Chemical Mapping requires two essential conditions: (a) the presence of chemical reflections. This allows one to identify realistic operating windows, within which the image is essentially independent of thickness and simply related to composition. (b) The validity of the assumption that the image of a material of intermediate cornposition is a linear superposition of the images of the end-point compositions. In spite of these two seemingly restrictive conditions, chemical mapping has been applied to a variety of materials to yield valuable quantitative information (Section IV.l).

4 DEFECT PROCESSES IN SEMICONDUCTORS

b.

239

QUANTITEM

In the absence of chemical reflections, it is not obvious how to distinguish changes in composition from changes in thickness; each affects the projected potential (Fig. 9). In that case it is not clear if chemical composition can be measured at all. For the illustration of the basic principle behind QUANTITEM, let us first consider a sample of uniform composition but changing thickness. This is usually the case in practice because TEM samples are always wedgeshaped and atomically rough. The basic idea is to determine the function F”(P) from Section 111.1 that describes the path of the image I with changing projected potential P or sample thickness. This function is obtained as follows. Each unit cell represents a random snapshot of the effect of the projected potential P on the image intensity I . These snapshots cover all values assumed by the potential over the field of view-precisely the range needed to relate I to P for the image at hand. Moreover, as we will see, the path is nearly periodic with sample thickness. The rate is given by the extinction distance corresponding to the periodicity of the “pendellosung”

FIG.9. Simulated lattice image of a Geo,2sSio,,5quantum well in a wedge-shaped sample. The thickness gradient runs at 35“ to the vertically oriented quantum well. Note that similar image patterns are obtained both in the quantum well and in pure Si (see circles), but at different thicknesses.

240

P. SCHWANDER E't d.

described in Section III.2.a. Given the random snapshots represented by the image unit cells and their path rate with changing projected potential, the function F"(P) is entirely determined from a single lattice image, without knowledge of the imaging conditions. The path described by the tip of the unit cell vector can be determined by plotting the vectors representing the image unit cell over the field of view. The F" simply represents the curve that is described by the vector tips. This is graphically shown in Fig. 10 from data obtained from an experimental image of a wedge-shaped Si sample. The problem of calibrating this path in terms of (known) changes in projected potential can be tackled in different ways. Here we will concentrate on lattice images that are taken from high symmetry directions of semiconductor materials in the absence of chemical reflections. In this case a convenient parameterization of the path yields a path variable, which changes linearly with the projected potential. As an example, consider Si for the (110) zone axis. Figure 11 shows points each representing the tip of an image unit cell vector, as obtained from a simulated image of Si wedge-shaped sample. For thickness changes of up to 3/4 of the period (extinction distance), the path is well-approximated by an ellipse. This strongly suggests using the ellipse phase angle me as the path variable. Here, ae is defined by the relation X = cos and Y = sin me, where ( X , Y ) is a point on the ellipse plane. Figure 12a shows the dependence of the ellipse phase angle on the sample projected potential

FIG. 10. Lattice image unit cells and their vector representation R, for three diKerent sample thicknesses. The cloud of points represents the tips of the vectors (not shown), drawn from an experimental image of a wedge-shaped Si sample. The path described by the image vectors represents the function F", which relates the image intensity to the sample projectcd potential.

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SIMULATION At = 3.8 A

- ELLIPSE FIT

ty

Si 41O>

X = A cos @e Y = B sin @e FIG. 11. QUANTITEM analysis of a simulated image of a Si wedge. Each point is the tip of a vector representing an image unit cell. As the sample thickness increases in 3.8A increments, the unit cell vectors describe a path that is almost exactly an ellipse for thickness changes of up to 3/4 of an extinction distance (periodicity).

0

0

0

0

0

-8

Qe 16 0

x=.lO

A x=.15

4

0 x=.20 X x=.25

0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.0

1.2

1.4

2 1.6

FIG. 12. Variation of the ellipse phase angle Qe with sample thickness t, normalized to the extinction distance 5 % ) intensity has been determined, and the corresponding simulated images analyzed by QUANTlTEM for thickness range of 3/4 of the extinction distance [a round-robin comparison of available image simulation packages has been initiated by Van Dyck and Op de Beeck (1994)l. The maximum error incurred by using the ellipse approximation is listed in the final column of the table. As can be seen, the error is less than 2.1 Yn for the Ge,Si, - x (type IV) semiconductors. For the Ga, -,Al,As (III/Vs) more than two Bloch waves are excited so that the ellipse approximation may not be justified. However, for the thickness range considered, the analysis reveals that the maximum error introduced is less than 3 % for an A1 content up to 0.4. Therefore it is established that the ellipse approximation is sufficient for most type IV and III/V semiconductors. However, we point out that each new material system should be evaluated afresh according to the recipe given in the preceding. Mupping Chemicul Composition. When changes of chemical composition are of interest, we may either use Chemical Mapping or QUANTITEM. Chemical Mapping requires the presence of distinct chemical reflections. They occur when compositional variations involve changes in the atomic occupancy of an ordered subset of the lattice sites. They are only present in

TABLE I APPLICABILITY OF QUANTITEM

FOR

SELECTED SEMICONDUCTOR MATERIALS P

Material

Orientation

Thickness Range (A)

Number of Bloch Waves I > 0.05

Intensity in Strongest 2 Bloch Waves

Extinction Distance

0.998 0.917 0.997 0.993 0.996 0.985 0.820 0.888 0.820 0.883 0.822 0.870

440 315 342 212 256 152 270 160 310 180 370 210

(4

Maximum Error

R ?I

m 0

1

~

90-397 42-196 90-394 44-197 87-240 12-90 60-220 46- 154 60-220 44- 160 60-220 50- 180

2 3 2

2 2 2 3 3 3 3 3 3

2.1 Yo 1% 1.8% 1.4% 1% 2% 2.5% 1.5% 3Yo 6'/o 7% 10%

2

246

P.ScnwANveR et at.

the lattice image when the atom-columns differ in composition when projected along the beam direction. This is the case in III/V semiconductors for projections along (loo), (110) but not along (111). The projection directions (100) are preferred because they contain the strongest chemical reflections. Then Chemical Mapping windows must be found that can be accessed in the experiment. This is achieved by image simulation based on multislice calculations of dynamical electron scattering and image formation (see, e.g., Spence, 1988 and Stadelmann, 1987). Finally, it must be established whether an adequate linear relationship can be obtained between the concentration range of interest and the angular distance of the in-plane component (see Section 111.2.~). In the absence ofchemical reflections QUANTITEM must be used. Here we refer to the section on the validity of the ellipse approximation above. As noted in Section II1.2.h, the chemical signal is encoded in the extinction distance. In order to obtain the best possible signal-to-noise ratio, the variation of extinction distance due to compositional changes should be as large as possible. This is achieved by choosing the optimal beam direction. For Ge,Si, --x the beam direction of choice is [I lo], as can be verified from Table I. It should be noted that although we restricted ourselves to two rather simple implementations of real-space methods, the chemical transition in both type IV as well as III/V semiconductor systems can be evaluated in most cases. One exception is AlAs for the beam direction [ l l O ] where a more general approach is required, as pointed out by Maurice et ul. (1996). In order to obtain good quantitative results from high-resolution lattice images certain technical prerequisites and skills are absolutely essential. The main important points are summarized in the following: Sample Preparation. High-quality sample preparation is necessary to keep the noise in the lattice images to an absolute minimum. Noise present in the image lowers the confidence level for any signal of interest. The noise stems mainly from amorphous overlayers created during the preparation procedure. Semiconductor samples prepared by ion-milling usually produce lattice images of poor quality (Ourmazd, 1993). Chemical thinning techniques can perform significantly better and are therefore preferable. An excellent technique that relies solely on chemical thinning has been developed for III/Vs (Kim and Ourmazd, 1994). However, for some systems, chemical etching cannot be applied due to differential thinning effects. Then ion-milling is often the only possibility. In that case we suggest a subsequent chemical cleaning step after ion milling. Needless to say, high-quality sample preparation is not only a prerequisite for QUANTITEM and Chemical Mapping, but for any type of quantitative HRTEM.

4

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247

Taking H R T E M Images. Several excellent commercial medium voltage (200-400 kV) high-resolution instruments are currently available. They have a point-to-point resolution below 2 A and sufficient electrical and mechanical stability so that high-contrast lattice images can routinely be obtained. Nevertheless, for quantitative analysis some special considerations have to be taken into account. We restrict ourselves to the two most important ones, namely, errors due to sample mistilt and degradation by radiation damage due to electron dose. Special care has to be taken when aligning the specimen zone axis parallel to the beam direction. In QUANTITEM a deviation from the exact zone axis will lower the symmetry so that more Bloch waves get excited. This may introduce a systematic deviation from the ellipse. For Sir 1001, for example, a misalignment as small as 1 mrad can lead to systematic errors of about 25% for the measurement of the ellipse phase angle, that is, projected potential. The alignment to the zone axis is carried out in the diffraction mode by monitoring the intensity of the diffracted beams on the phosphor screen of the microscope. The crystal is tilted until equal intensities are obtained for symmetry equivalent beams. This procedure is thus limited by the accuracy with which the eye can detect relative intensities on the phosphor screen. In practice, this procedure yields a misalignment of typically 0.8 mrad. In that case QUANTITEM allows for an optional correction that can limit the systematic deviation of the ellipse angle to less than 10% (for details see Kisielowski et al., 1995). Lattice images with larger deviations from the zone axes should not be analyzed by the methods outlined in this chapter. A procedure to identify lattice images with inadequate mistilts has been suggested by Schwander and Schneider (1996). Radiation damage can degrade the information content of the lattice image, or cause physical changes in the sample. Increasing the electron dose increases the noise present in the lattice images. Chemical Mapping and QUANTITEM deduce the information content of the image, including the noise. Thus the effect of increased noise due to irradiation damage is automatically included in the confidence levels obtained by the algorithm. The possible physical changes induced due to radiation damage depend on the particular system under consideration. To assess this effect we suggest analyzing a series of images with increasing electron dose obtained from the same sample area. The signal of interest is then observed as a function of the electron dose. As an example we refer to a study of the interfacial width in GaAs/AlAs as described by Ourmazd (1993). Real-space Methods Software. For quantitative analysis the images must first be digitized. A careful correction of any imaging distortions of the

248

P. SCHWANDER et al.

digitizing system used has to be performed before applying vector pattern recognition. The image should then be resampled to remove moirt effects stemming from the incommensurate ratio of the image unit cell size to the pixel size. Software programs for Chemical Mapping and QUANTITEM have becn written in F77, to run on Silicon Graphics Workstations under IRIX. All the examples given in this chapter have been analyzed by either of these programs. They are available to academic and research institutions at no cost.

IV. Applications

After the applicability has been validated for a specific semiconductor system, either the real-space technique Chemical Mapping or QUANTITEM may be used to measure chemical composition and interfacial roughness with near-atomic spatial resolution and sensitivity. From carefully designed and v:ell thought-out experiments, high-level information, such as on the reaction of point defects, can be obtained. For a demonstration of the capabilities, a few examples are presented in this section.

1. DIFFUSION I N AlGaAs Using the Chemical Mapping technique described in Section 111.24 it is straightforward to make sensitive measurements of interdiffusion on individual interfaces. Semiconductor multilayers are increasingly used in materials of everyday life, such as C D lasers. One has to keep in mind that such layers are far from thermal equilibrium and thus represent highly metastable systems. For example, on crossing a modern GaAslAlGaAs interface, the A1 concentration changes by several orders of magnitude within a few lattice spacings. Intermixing can occur on this scale by direct exchange of atoms, or via interaction with an intrinsic defect, such as a vacancy, an interstitial, or an antisite defect (Ga on an As site, for example). Normally, the large energy barriers for atomic motion and the small equilibrium concentrations of intrinsic point defects prohibit significant intermixing at room temperature. However, thin films, whether destined for fundamental experimentation or for commercial use, are often subjected to processing steps, such as annealing, which can inject significant concentration of point defects. The passage of such defects through interfaces can cause substantial intermixing. It is thus important to investigate the stability of thin films against intermixing during typical processing steps. Here we discuss briefly the effect

4

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249

SEMICONDUCTORS

05

" I _ -

as grown annealed

03

02

7

0.1

0

0

10

20

30

40

50

€4

70

distance in Angstrom

FIG.14. (a) Chemical lattice image of an AI,,Ga,,,As/GaAs interface before annealing. (bj Composition profiles before and after annealing at 650 'C for 2 h. Each point gives the composition of a 100 A segment of an atomic plane. The error bars are smaller than the dots. The solid and dotted lines are solutions of the linear dilfusion equation used to fit the interdiffusion coefficient D.

-

of annealing on thin-film multilayers, highlighting those phenomena that are revealed by our microscopic approach. Such experiments consist of mapping composition profiles across single interfaces with atomic resolution, before and after appropriate annealing (Fig. 14). The interdiffusion coefficient D is then deduced by fitting the experimental data to the (linear) diffusion equation. In this way it is possible to measure interdiffusion coefficients as small as cm2/s, at individual interfaces in sampling volumes as small as lo-'' cm3. As an example, consider the intermixing of an AlGaAs multilayer. Figure 15 shows an Arrhenius plot of 1nD vs l/kT for three GaAs/Al, ,Ga, ,As interfaces, each at different depths from the surface of the multilayer stack. Remarkably, a change of 3000 A in the interface depth changes the interdiffusion coefficient by up to a factor of 20 (Rouvikre rt al., 1992). The depth of such a layer therefore strongly influences its stabiIity. This behavior has also been observed by Chemical Mapping in the HgCdTe/CdTe system (Kim et nl., 1989) and by other techniques (Guido et al., 1989) for AlGaAs.

250

P. SCHWANVER et al.

FIG. 15. Arrhenius plot of the interdiffusion coefficient D at GaAs/AIGaAs interfaces at three different depths from the sample surface (300A, 1500 A. 2800 A).

More generally, the depth dependence of interdiffusion is related to the injection of point defects from the sample surface during the anneal (Fig. 16). Such diffusion studies on the atomic-scale therefore allow the use of thin multilayered semiconductor films as microscopic laboratories for systematic investigations of the microscopics of point defect diffusion. As will be shown, the depth dependence of the interdiffusion coefficient may be used to measure and separate the formation and migration energies of a given native defect type. Such investigations are especially valuable for process control in semiconductor device fabrication in order to predict the diffusion phenomena that occur where intrinsic point defects play a crucial role. Accurate measurements of the formation (H,) and migration enthalpies ( H , ) for intrinsic point defects are thus of scientific and technological importance. To illustrate the theoretical and experimental state of affairs, we use the example of point defects abundant in GaAs during typical anneals in As-rich ambients. Theory indicates that the dominant defect species injected into n-type GaAs during anneals in As-rich ambients is the triply negatively charged Vtia. Quantitatively, the formation enthalpies H , f for point defects in GaAs have been calculated. However, standard diffusion experiments measure the total “activation enthalpy” Q = H , H , for defect diffusion, which includes the migration enthalpy H , . (Tan et al., 1991). Thus, quantitative comparison between experiment and theory has not been possible.

+

4 DEFECTPROCESSES IN SEMICONDUCTORS

251

As or Ga Ambient x

e

-

*

* .

*

*

L

FIG. 16. Schematic diagram, showing the injection of point defects into a rnultilayer system from the surface during an anneal. The concentration of point defects, and hence the interdihsion coefficient decay with distance from the surface.

Our measurement procedure augments an old approach with our new technique: it consists of injecting a selected point defect type by annealing in an appropriate ambient, and observing the broadening in series of marker layers, caused by the arrival and passage of defects from the surface. Specifically, the Al/Ga intermixing in a GaAs/Al,,,Ga,,,As multilayer annealed in an As ambient can be monitored by Chemical Mapping. Due to the high sensitivity and resolution of the technique, one can measure atomic-scale changes in the marker layers long before steady state is reached. This enables one to extract the formation and migration energies separately. To appreciate the principle of this approach, consider the diffusion of V,, introduced at the surface, through a sample containing a series of ideal marker layers. The passage of the defects causes intermixing between the marker and host atoms with coefficients

D

= C(Z, t, T )d

(1)

As defined in Alexander (1986), c(z, t, T ) is the V,, concentration at depth

252

P. SCIIWANDER er ol.

z, time t and temperature 7: In thermal equilibrium the defect concentration The interdiffusion can be explicitly written as (Crank, is c, = coe-‘Hf-7Sf’IkT. 1989)

where do, is a pre-exponential factor, ci the grown-in defect concentration; the other symbols have their usual meanings. The first term describes the intermixing caused by the vacancies arriving from the surface, the second, the intermixing due to the vacancies already present in the bulk before any injection from the surface. The experimental observable is the total intermixing at each marker layer, given by ALz = L2 - L2 o - {tanncal 0 D(z, t, T ) & where Lo is the initial width of the interface between the matrix and the marker layer. Due to lack of sensitivity and spatial resolution, experiments are usually carried out in the “steady-state” regime, where the erfc = 1 and c, >> ci. Under such conditions, Eq. (2) reduces to

Measurements in this regime therefore only yield the total activation energy Q. Chemical Mapping, however, allows one to measure D vs depth and temperature before steady state has been reached, when the full expression in Eq. (2) is operative. This yields each of the activation, formation, and migration enthalpies Q , H,, and H , separately. Following Eq. (2), the total intermixing at each layer results from two sources: the defects “frozen in” during growth and the defects arriving from the surface. A judicious choice of a cap layer thcrefore allows one to distinguish between the two. Experimentally, this requires the measurement of the intermixing at interfaces with different depths, annealed at differing temperatures in two sets of samples: one with an appropriate cap layer, preventing the penetration of vacancies from the sample surface, and one without a cap. A simultaneous fit of Eq. (2) to the measured intermixingnormalized to the time of the anneal- then yields experimental values for Q, H,, H,, and S,, respcctively. Table I 1 summarizes the results as reported system. Because a more by Rouviere ef al. (1992) for the GaAs/AI,,,Ga,,,As rigorous discussion of these results is beyond the scope of this chapter, we refer the reader to the more detailed review by Ourmazd (1993). To summarize thc results presented here: (a) Thin films of the highest quality are atomically rough. Chemical Mapping can be used to map such interface roughness in cross section at near-atomic resolution and sensitivity.

4 DEFECT PROCESSES IN

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SEMICONDUCTORS

TABLE I1 THERMODYNAMIC PROPERTIES OF V,, IN GaAs (REFERRED TO INTRINSIC MATERIAL AT PAS4 = lo5 Pa) ~~

Parameter Activation enthalpy Formation enthalpy Migration enthalpy Formation entropy Migration pre-exponential factor “Grown-in” V,, concentration Equilibrium V,, concentration

Symbol (Unit)

Q (eV) H , (ev) H , (eV) S,(k,) do, (cm21s) c, (cm 3, c,(crn-’) -

Value 5.1 +/-0.2 4.0 +/-0.5 I .I+/-0.5 33+/-5 3.9 x 10-4 1.2 x 10Ih 7.2 x 10l6

(b) Thin films relax during typical processing steps such as annealing or implantation, in ways often not encountered in bulk solids. The influence of thermal treatment on such thin layers can be investigated by Chemical Mapping on a microscopic scale for specific interfaces of interest. (c) Chemical Mapping is able to use thin films as microscopic laboratories to investigate the fundamental properties of intrinsic point defects. Nevertheless, careful design of such experiments is necessary and the interpretation of the results has to be performed under the close guidance of theory and in combination with results from other techniques and on different length scales. 2. DIFFUSION IN GeSi

The application of GeSi alloys as the base channel in ultrafast heterojunction bipolar transistors (HBT) is expected to lead to promising advantages in future ultrafast semiconductor devices. Careful design of the Ge concentration profile and a high doping level in the base can lead to considerable advantages in terms of higher emitter injection efficiency, decreased base resistance, and reduced base transit time (see, e.g., Wolf, 1990). For a successful application of such devices with GeSi layer thicknesses of several nanometers, stability during typical thermal processing is of utmost importance. QUANTITEM investigations of interdiffusion in Si/SiGe layers at the atomic scale can therefore determine valuable input parameters for process modeling of such devices.

254

P. SCHWANDER et ul.

The interdiffusion in GeSi alloys has been investigated by Gribelyuk et al. (1997). The Si,,,5Geo.,5 quantum well stacks with step-like deposited profiles and widths of 50A have been analyzed before and after typical annealing treatments in argon, oxygen, and vacuum ambients. Figure 17 presents concentration profiles of such a quantum well before and after annealing at 800 “C for 5 h in vacuum. The composition profile in the as-grown material reveals a compositional change that occurs within a few monolayers. The measured interdiffusion in the annealed material cannot be explained by a standard linear diffusion model, that is, with a concentration independent diffusion coefficient. Instead, the nonlinear diffusion model following D(c) = do exp( - Q/kT) exp(AQc/kT) was found to agree well with the experimental data. This model assumes both that the system is in equilibrium and that the activation energy of a defect mediating the interdiffusion changes linearly with Ge concentration c. It has to be pointcd out, however, that strain relaxation as typically measured by X-ray observations in such systems is not accounted for by this simple model. Moreover, the interdiffusion has been found to be depth dependent, indicating that the system is still in a nonequilibrium state. As a technical result though, these investigations confirmed that for typical thermal budgets involved in processing SiGe HBTs, the influence of interdiffusion on the final device performance is negligible.

0.3 C

.-c0

2 0.2

lr

C

8C

s 0.1 s 0.0 0

50

100 Distance [A]

150

200

FIG. 17. Composition profiles of a Si, 75Ge02 5 quantum well before and after annealing ( 5 h at 800‘C in vacuum). The profiles have been averaged over a distance of 400A along the SiiSiGe interface.

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DEFECT PROCESSES

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IN SEMICONDUCTORS

It is also revealed that in spite of the high spatial resolution and sensitivity of quantitative real-space methods in HRTEM, the interpretation of diffusion experiments is in general not unique. In fact, the interpretation must be based on results from more than one technique and in any case be guided closely by theory.

3.

Si/SiO, INTERFACIALROUGHNESS

One major feature of QUANTITEM is its unique ability to measure (crystalline) sample thickness in samples of uniform composition at the level of one unit cell and at a precision approaching a few monolayers (see Table 111). In what follows, we describe how QUANTITEM can be used to measure the atomic roughness of buried interfaces from samples in planview, revealing topographic maps of interfacial roughness in Si/SiO,. Due to its key role in integrated circuit technology, the Si/SiO, system is one of the most studied interfaces (see, e.g., Helms and Deal, 1988). Microscopic roughness at this interface (e.g., MOSFET gate oxides with thicknesses below 100 A) affects the carrier mobility and device reliability. Valuable information on the atomic configuration of this interface has been obtained by lattice imaging in cross section (see, e.g., Goodnick et ul., 1985). However, cross-sectional investigation provides only a projected view of the interface. This is a limitation encountered whenever information from buried interfaces is required. QUANTITEM on the other side, measures the thickness of the crystalline part of the sample, with any amorphous overlayer adding noise. If the crystalline part is bounded by two identical interfaces (such as in a Si0,-Si-SiO, sandwich), or if one of its interfaces has known or negligible roughness, QUANTITEM may be used to measure the atomic configur-

TABLE 111 QUANTITEM SPATIALRESOLUTION AND SENSITIVITY

Material

Si

Ge0.25%75

Orientation

(100) (110) (111) (110)

Spatial Resolution

2.7 3.8 2.2 3.8

x x x x

2.7A2 S.4A2 3.8 A 2 5.4'4'

Sensitivity Typical

Best

15.0A 5.3 A 11.0'4 5.4 at% Ge

3.2 8, 2.0 A 3.2 A 2.3 at% Ge

ation of the interface of interest (Fig. 18a). Figure 18b shows a plan-view lattice image and Figs. 18c and d contain QUANTITEM maps of the particular SiO2-Si(100)-Si0, sandwich formed by chemical etching of the Si sample from both sides and subsequent formation of a native oxide. In the three-dimensional representations, height represents the local roughness of the two interfaces (top and bottom), viewed in superposition. Such roughness maps can be used to calculate the autocorrelation function that quantifies the spatial extent of interfacial undulations. In summary, QUANTITEM can thus provide plan-view maps of interfacial roughness for buried interfaces. These images resemble the topographic maps produced by the scanning tunneling microscope from surfaces. A combination of the two measurement techniques therefore can lead to an enhanced understanding of interface properties.

Fi 2V) bias. For lower bias (Jager et al., 1996), tip induced band-bending is responsible for the fact that only the valence-band minimum can be seen for n-type GaAs and that the registry does not change with polarity. For low-bias voltages and p-type GaAs, valence-band states are imaged and both sublattices can even appear simultaneously (Fig. 4).

5 SCANNING TUNNELINGMICROSCOPY OF DEFECTS IN SEMICONDUCTORS 271

FIG.3. The STM images of GaAs recorded simultaneously at sample voltages of (a) + 1. 9 V and (b) - 1. 9 V. (c) Top view of the (1 10) surface atoms. As atoms are represented by open circles and Ga atoms by closed circles. The rectangle indicates a unit cell, whose position is the same in all three figures. Note in the STM images that there is one maximum per unit cell and the apparent shift of the maxima between (a) and (b). Reprinted from Feenstra et al. (1987b) with permission from the authors.

FIG. 4. The STM image of p-type GaAs. The upper half was recorded at + 25 mV and the lower half at -25 mV. Note the two maxima per unit cell and no apparent difference between the two polarities. Reprinted from Jager et al. (1996) with permission from the authors.

272

NIKOSD. JAGERAND EICKER. WEBER

IV. Shallow Defects The unique capability of XSTM, a “three-dimensional” imaging of bulk defects on the atomic scale in real space, was first demonstrated for shallow dopants.

1.

DOPANTS IN GaAs

Johnson et ul. (1993a) were the first to image beryllium and zinc dopants in GaAs. Figure 5 is a filled-state, constant-current 28nm x 28nm image

M i l

I : Arc., undcr i I

1

5

4

7 01 m c e ( A*

h

)

FIG, 5. The STM image of a (110)-cleaved, 5 x 10’9cm-3, Zn-doped Ga..j surface. The image displays 20 x 28 nm of the As sublattice taken with sample voltage -2.1 V and current 0.1 nA. Six type A hillocks, six type B hillocks, and one type C hillock can be seen corresponding to Zn-dopants at various depths from the surface. (b) Tip traces along the [l-lo] and [OOl] directions of a selection of the hillocks of type A, B, and C identified in (a). (c) Scatter plot of area under the [I-101 tip height traces (integrated intensity) for all hillocks. Reprinted from Johnson et al. (1993a) with permission from the authors.

5

SCANNING TUNNELING

MICROSCOPY OF DEFECTS IN SEMICONDUCTORS

273

of p-type GaAs, displaying the As sublattice and ionized Zn dopants in the top several surface layers. The dopants appear as individual circular hillocks of about 2 nm in diameter. The authors advanced the following argument as to why the dopants appear bright. With negative bias applied to the sample the bands are bent downward, that is, the p-type dopants near the surface must be fully ionized and distort the local tip-induced band bending. Holes tunneling to the sample are attracted to the negatively charged dopants and thus see an enhanced hole-state density. This results in enhanced current, forcing the tip to rise to maintain a constant current. Thus the dopants appear bright in a filled-state image. The different depth of the dopants manifests itself in different sizes and symmetries of the hillocks as can be seen in the profiles along [OOl] and [l-lo] for three types of hillocks. Type A is in the topmost layer. It displays the brightest hillock and has two bright As atoms along [I-lo]. An active Be acceptor is substitutional on a G a site. Therefore, the center of the Coulomb potential responsible for the enhanced current is between the nearest neighbor As atoms along the topmost [l-lo] “zig-zag” chain, Thus an even number of surface As atoms must appear bright. The number of atoms affected is determined by the range of the distortion. Type B is Be on a G a site in the second layer. Its nearest neighbor in the topmost layer is only one As atom. This is why hillocks of type B are centered on one As atom along [l-lo]. Type C and even weaker features are dopants lying deeper. When one counts the number of hillocks in Fig. 5, one can find type A and B six times. This corresponds closely to the average concentration of five dopants per layer expected for the known bulk concentration of 5 x 10l9 ~ m - ~ . Empty state images of Zn are shown in Fig. 6 (Zheng et al., 1994~).The Zn dopants there exhibit a -4nm wide triangle protrusion with a (110) mirror symmetry plane, which can be used as a signature for Zn, because no other defect including beryllium has been reported showing such a feature.

2.

Si IN GaAs

In Fig. 7(a), SiGadonors in subsurface GaAs layers appear as delocalized 2.5 nm diameter protrusions superimposed on the background lattice (Zheng et al., 1994a). As can be seen from the profile and histogram in Figs. 7(b) and (c), at given bias the height and the diameter of the protrusions depend on the depth of the dopant. As can be seen in Fig. 8, the apparent height of any dopant is strongly bias dependent for both polarities and ranges from a tenth of an Angstrom to a few Angstroms. This supports the assumption that the contrast is of electronic and not of structural origin.

274

NIKOSD.

JACER AND

EICKER. WEBER

Empty State Image, Sample bias= +20 V

FIG. 6 . A 200A square STM image of cleaved GaAs (1 10) at 0.5nA and +2.0V sample bias. Triangle-shaped features (A, B, and C) are observed, corresponding to subsurface Zn,, acceptors at different depths. Reprinted from Zheng rt ctl. ( 1 9 9 4 ~ )with permission from the authors.

The authors have argued that, for positive sample bias, the Coulomb potential of the ionized donor locally decreases the band bending and effectively increases the number of states available for tunneling, that is, the tip rises in a constant current image. The height of the protrusions decreases with increasing bias, more conduction band states can contribute to the tunneling current at high bias, and the relative effect of the perturbation of the SiGadonor is reduced. For negative bias, electrons accumulate at the surface. Although the Coulomb potential is now screened by conduction band electrons, it still perturbs the local band structure. A detailed quantitative theory for the contour formation mechanism has not yet been proposed. Si on a Ga site in the top (1 10) surface layer shows very localized features (Fig.9) (Zheng et al., 1994a), because the surface electronic structure is modified by the dangling bond, which introduces a localized midgap level. This level traps the donor electron and thus forms a half-filled localized dangling bond, which is imaged as a local maximum in a filled state image. The half-filled state appears dark in the empty state image, because it traps an electron and thus attenuates the current compared to the completely

5

SCANNING TUNNELING MICROSCOPY OF DEFECTS IN SEMICONDUCTORS 275

0

40

80

Distance

120

180

(A)

;l I 4

20

1 .o

Height

I

(,

FIG. 7. (a) 250A x 250A STM image of the cleaved GaAs ( I 10) surface, acquired with 0.5 nA tunneling current at a sample voltage of - 1.5 V. Two types of features marked D (donor) and B (black) are seen. The D features appear as bright protrusions of different apparent heights. They correspond to Sic&,in various SUbSUrfaCe layers that enhance the local tunneling probability. Feature B is a vacancy. (b) Surface profile across the centers of features D and D’ in (a). (c) Histogram of different heights of D features acquired from many images of D features in a large area of a (1 10) cleaved surface demonstrating that Si donors up to five layers below the surface can be imaged. Reprinted from Zheng et ul. (1994a) with permission from the authors.

I

I

P

FIG. 8. The STM images of the protrusion region D in Fig. 7(a) taken with tunneling current of 0.5nA at various sample voltages. The sample voltages are (a) - 3 V; (b) -2.0 V; (c) - 1.5 V; (e) +3.0 V: (f) +2.0 V, and (g) 1.5 V. The topographic profiles along the [l-101 direction (indicated by the arrows) across the features are shown in (d) and (h). All images are 60 A x 60A in size. The varying height proves the electronic rather than the structural nature of the protrusions. Reprinted from Zheng er ul. (1994a) with permission from the authors.

+

5


FIG. 9. Images of a substitutional donor Si,, on the top layer of the GaAs ( I 10) cleavage surface. (a) Filled-states image acquired with 0.5 nA tunneling current and at -2.5 V sample bias. (b) Empty-states image acquired with 0.5 nA tunneling current and at +2.5 V sample bias. The location of the Si dangling bond state is marked by the letter A. The defect structure is localized to a couple of nearest neighbors. Reprinted from Zheng et al. (1994a) with permission from the authors.

empty neighboring G a dangling bond states (Wang et al., 1993). The simulated filled-states image is shown in Fig. lO(a) and the simulated empty states image is shown in Fig. 10(b).

3. DOPANT PROFILING

For semiconductor devices of the near future, dopant distribution will need to be controlled down to the atomic scale. Techniques such as secondary ion mass spectrometry (SIMS) and electrical techniques such as capacitance-voltage (C-V) measurements provide average information over a certain lateral area and are limited in their depth resolution to about 2 to 5 nm. Johnson et al. (1993b) compared XSTM to SIMS and C-V measurements on Be doped layers with varying doping concentrations. Figure 11 shows such a comparison. Figure 1l(a) shows their intended growth plan, that is, the intended doping concentration, the points where AlGaAs marker layers were grown, the points of growth interruption to change the doping or to introduce a &doped layer, and the calculated band-structure. Figure 1l(b) shows a topographic STM image 100 nm x 50 nm in size. Atomic corrugation is observed but, due to the size of the figure, it is hard to see. The dark spotted lines correspond to the AlGaAs marker layer; the varying contrast from bright to dark and to bright again reflects the different doping of the layers. Individual dark spots are vacancies (see Section 111, p. 00) and individual bright spots are due to individual dopants. In Fig. 11(c), individ-

278

NIKOSD. JAGER

AND

EICKER. WEBER

(b)

FIG. 10. Theoretical STM images of Si at a Ga site (SiGa)in the top GaAs (110) surface layer. The bias for the images are (a) - 1. 2 V, and (b) + 1.9 V. Reprinted from Wang et d. (1993) with permission from the authors.

ual profiles at the positions indicated in (b) are given, while the solid line in Fig. ll(d) gives an averaged profile of (b), which is compared to SIMS results for Be (dashed line). In Fig. 1 l(e), the number of dopants appearing in a 10nm wide stripe of the XSTM image is counted (solid line) and compared to the SIMS analysis (dashed line). The carrier density obtained from the C-V data along with SIMS data is displayed in Fig. ll(f). From

5

SCANNING TUNNELING

MICROSCOPY OF DEFECTS IN SEMICONDUCTORS 279

FIG. 11. The STM image of GaAs layers with variable dopant concentration and various depth profiles. (a) Calculated bulk-band structure for the intended heterostructure. In registry here, (b) shows the topographic image, 100 x 50nm in size, of the layers taken with a sample voltage -2.1 V and current 0.1 nA. The relative tip height is given by gray scale, from 0 to 0.2 nm. (c) Several topographic line scans across (b) along the [loo] direction (AIGaAs layers are vertically aligned); atomic corrugation is shown in the inset. (d) Line scan (solid) corresponding to the averaged line scan over the top half of (bj smoothed slightly to take out atomic corrugation and compared to the Be concentration measured by SIMS (dotted, right axis). (e) Dopant histogram determined by counting the white hillocks (dopants) in a series of rectangles across (b) (solid, left axis) and compared to the measured Be concentration measured by SIMS (dotted, right axis). (f) Be concentration (solid), A1 counts (dotted) and Be concentration (solid) measured by SIMS and dopant concentration determined by Polaron C-V (dashed). Reprinted from Johnson (1993b) with permission from the authors.

280


these figures, the following can be concluded about the capability of XSTM for dopant profiling. The tip height is directly sensitive to the varying doping concentration. The tip height difference between a 1018cm-3 and a 1019cm-3 doped layer is about 0.1 nm. Individual active dopants can be mapped, thereby allowing the direct observation of dopant activity and distribution on the atomic scale. As a further example of the degree of profiling achievable with XSTM, measurements of Be &doped layers are shown in Fig. 12, from the work by

FIG. 12. The STM images of Be &doped layers. (a) The horizontal scale (300nm) is strongly compressed. Tunneling conditions: sample bias - 1.0 V, current 20 PA. The gray-scale range is 0.08 nm with a [OOl] corrugation of approximately 0.03 nm. Electrically active Be dopants appear as white hillocks approximately 2.5 nm in diameter and up to 0.05 nm high. (b) Enlarged view (54 x 31 nm') of the section of the two lowest &doped layers outlined in (a). Atomic corrugation in both directions is clearly observed, allowing dopants to be counted and their positions measured to the nearcst bilayer iii the plane shown. Reprinted from Johnson et ul. (1995) with permission from thc authors.

5 SCANNING TUNNELING MICROSCOPY OF DEFECTS IN SEMICONDUCTORS 281

Johnson et al. (1995). Figure 12(a) is a large-scale (300nm x 120nm) filled-state image of GaAs with four 6-doped layers incorporated. Note that the x-axis is strongly compressed. Figure 12(b) shows an uncompressed view of two layers. In Figs. 13(a)-(d) the distribution of dopants in the &layers along the [00 I ] growth direction is statistically evaluated. The intended width of these layers is, of course, one atomic row and the intended areal density is 3 x 10'2cm-2 for layer 1, 1 x 1013cm-2 for layer 2, 3 x l O I 3 cm-2 for layer 3, and 1 x 10'4cm-2 for layer 4. However, the XSTM image and analysis show that a &layer width of 1 nm is possible only for low areal densities (10'3cm-2). At higher concentrations, the Be dopants spread symmetrically within up to approximately 5 nm. The dis-

[WljGrowth direction

-

--+

12 10

8;;

8 8 2 6

6

-k5

4

w 5

24

2 6

0

0

-e

-

10

s

8

0

6

6

5 4 u J 2

4

2

0

0 15

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6

10

5

4

5

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2 :

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8

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> m 4

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-30

z

w

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0

0

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20

- 10

E

-

10

g

2

0 30 0 2 4 6 8 10 12 Peak height (x 0 0065nrn) IOOl] Position (bilayets) x 0 28 nm -20

-10

0

10

20

FIG. 13. Position-related histograms based on the STM images in Fig. 12(a). (a)-(d) Histograms of the [OOl] positions of the dopants counted for the &layers, with growth direction indicated. A Gaussian fit (dotted) is used to determine the average position (solid) and width of the distribution. The intended position of the &layer is a dotted vertical line. (e)-(h) Histograms of dopant height occurrence for the layers in Fig. 12. Horizontal scale denotes apparent peak height in nanometers. Note that the relative height is reproducible within a given experiment, but might vary slightly on an absolute scale because of tip-to-tip variations. Reprinted from Johnson el a/. (1995) with permission from the authors.

282

NIKOSD. JAGER AND EKKE R. WEBER

tance between two dopant atoms is not randomly distributed, but a deficit of closely spaced dopants is seen. Taking this into account, the spreading can be ascribed to the coulombic repulsion between ionized dopant atoms. In Fig. 13, the intended position of the &layer is also indicated. The shift of the center of the dopant distribution can be explained by the presence of drift of the ionized Be-dopants that is driven by a surface E-field. Such an E-field can be built up by the pinned [OOl] growth surface.

V. Deep Level Defects 1.

ARSENICANTISITEIN GaAs

The STM images of the arsenic antisite (As,;,) presented by Feenstra et ul. (1993a) settled a decade-old debate as to whether the EL2 deep level, commonly found in semi-insulating bulk GaAs, is an isolated arsenic atom sitting on a gallium site or a defect complex involving an arsenic antisite and a nearby As interstitial (Kaminska and Weber, 1993; Baraff, 1992; and references therein). An isolated As antisite would have a tetrahedral symmetry, while a lower symmetry requires at least one additional defect. Figure 14 shows filled-state STM images of the EL2 defect at various depths below the cleavage plane. As expected for an antisite-related defect, the central core of the feature is on a gallium site, that is, the core is in between the As rows appearing in the filled-state image. The two satellites about 15 A away from the core give EL2 a fingerprint. Note that the only symmetry operation for all features present is a single (1-10) mirror plane passing through the central core. If the defect had a lower than tetrahedral symmetry in the bulk, this would -assuming random orientation - violate this rule. As no such violation is observed, the EL2 defect must have tetrahedral symmetry in the bulk. Details of the deep-level wavefunctions not previously known are shown in Fig. 15(a) [Zheng et a / . (1994e)l and compared to a theoretical model [Fig. 15(b)] by Capaz et ul. (1995). Spectroscopic data from the defect core for various background doping levels are given in Fig. 16 (Feenstra et uf., 1993a). A spectrum far away from defects would only show tunneling out of valence band states for large negative voltage, and into conduction band states for large positive voltage, similar to the spectra shown but without states in the bandgap. However, the given spectra demonstrate that the EL2 defects reveal, in an n-type matrix, their well-known donor state 0.5 eV above the valence band. In a p f matrix the Fermi level is lower and falls into a band of deep-level donor states. In a degenerately doped p-type matrix the spectrum depends on the

5

SCANNING TUNNELING MICROSCOPY OF DEFECTS I N S~MICONDUCTORS 283

t

iio

-

1 nm

FIG. 14. The STM images of arsenic antisites in CaAs at various depths below the (1 10) surface. Reprinted from Feenstra et ul. (19934 with permission from the authors.

local environment of the defects. Coulombic interactions between neighboring AsGs defects strongly determine the states that the defect will exhibit (Feenstra et ul., 1994). The spectra also explain the contrast observed in the images. It arises from defect states in the bandgap. Spectroscopy on the satellites is identical to data taken on the core. This proves the satellites are due to extended wavefunctions of the arsenic-antisite core and do not originate from another point defect, which should locally show other electronic levels. In conclusion, the real-space direct atomic resolution STM data on EL2 are inconsistent with an additional As interstitial along [ l l 11.

2. VACANCIES Surface vacancies are the defects most often observed on the UHV cleaved (1 10) surface of III-V semiconductors. Their surface concentration is typi'~ (Cox et ul., 1990b) and they are created either cally 10'' ~ r n - ~ - l Ocm-2 by the cleaving process or over time by Langmuir evaporation (Ebert et al., 1995).

284

N I K ~D. S JAGEKAND EICKER. WEBER

Fic;. 15. (a) l h e arsenic antisite in bulk n-GaAs showing dctails of the wavefunction. Figure lS(a) was reprinted from Zheng et al. (IY94e)with permission from the authors and Fig. 15(b), iheorctical model of the As,, was reprinted from Capaz et a/. (1995), also with permission from the authors.

Surface vacancies cannot be identified by counting features and comparing the number to a known bulk concentration, because on the surface they are also generated by the cleaving process, over time, and by the tip scanning. A correlation between observed depressions and vacancies can be made by excluding all other possibilities, in particular other intrinsic defects

5

SCANNING TUNNELING MICROSCOPY OF DEFECTS IN SEMICONDUCTORS 285 I

I

I

I

6

4

0

2 0

-2

-1 0 1 2 SAMPLE VOLTAGE (V) = E-EF (eV)

FIG. 16. Tunneling spectra acquired from layers of LT-GaAs containing varying amounts of compensating shallow dopants. The valence-band maximum (E,) and the conduction-band minimum (E,) are indicated by dashed lines in each spectrum. An intense band of states arising from arsenic-antisite defects appears within the bandgap. The states of a bulk arsenic-antisite defect are shown in the upper part of the figure, relative to the band edges of spectrum (a). Reprinted from Feenstra et ul. (1993a) with permission from the authors.

and adsorbates. As the distribution of depressions is not homogeneous right after cleavage, and their concentration near cleavage steps is much higher than in smooth areas, one can exclude adsorbates as their origin, because adsorbates are expected to be homogeneously distributed. Charge-state arguments and tip-induced migration make antisites unlikely to be the cause of the depressions. A clear identification of subsurface vacancies is still missing. In GaAs the donor type-As vacancy (VAJ is found mostly on p-type, while the gallium vacancy (V,,) is mostly found on n-type material (Lengel et a/., 1993, 1994). The As vacancy appears in a filled state, that is, the arsenic sublattice revealing, image as a highly localized depression (-0.7 A), see Fig. 17 (upper left). In the empty-state, that is, the gallium-revealing, image, the V,, causes its nearest-neighbor Ga dangling bonds to appear raised about 0.7A above the Ga sublattice, but apparently does not induce a substantial lateral shift from their usual position (Fig. 17, upper right). Lengel rt ul. (1994) argued on the basis of tight-binding molecular dynamics simulations that the apparent rise of the G a dangling bonds is due to an

286

NIKOSD. JAGER AND E I C KR. ~ WEBER As suhlaitics

(iii

sublat~icc

FIG. 17. Simultaneously acquired filled- and empty-state images of the arsenic (top) and gallium (bottom) vacancies on p- and n-type GaAs(ll0). A setpoint current of 0.1 nA and bias voltages of - 2.0 V, + 2.5 V and 2.5 V, 2.0 V were employed for atom-selective imaging o n p - and n-type material, respectively. Note the apparent symmetry in the topographic features ofthe defects on both types of material when the roles of the cation and anion are interchanged. [Similar to material from Lengel er ul. (1993); reprinted from Lengel et NI. (1997) with permission from the authors.] ~

+

outward relaxation of the Ga atoms in the nearest neighborhood driven by occupied defect levels. No in-plane shift is observed, because the unsatisfied dangling bonds do not rebond. Total energy minimization using ah initio pseudopotentials (Zhang and Zunger, 1996, 1997; Kim and Chelikowsky, 1996, 1997; Yi et al., 1995),


however, suggest vertical inward movement of the Ga atoms next to the arsenic vacancy and rebonded geometries. These calculations explain the apparent rise of the Ga dangling bonds in the STM images as a pure electronic effect, rather than structural. It has been counterargued that, for an electronic effect, this feature should be bias-dependent to a degree higher than observed. In addition, no in-plane displacements expected for rebonded G a dangling bonds are visible in the experimental data (Harper et a/., 1997a, 1997b). The positive charge on the V,, induces a local band bending causing a weak (-0.15 I$), but more extended ( - 5 lattice periods) apparent “depression” of neighboring As atoms in the filled-states images, and an apparent “rise” of the Ga atoms in empty-states images. The positive-charge state of the As vacancy is also evident in the spectroscopic data taken on the vacancy (Fig. 18b) compared to the pristine p-GaAs surface (Fig. 18a). The defect band (D) from the shallow acceptors is diminished because of reduced band bending in the presence of a deep positively charged state (Lengel et al., 1994). The exact charge state is still an open matter (Lengel et cil., 1994; Zhang and Zunger, 1996, Harper et a/., 1997a, Kim and Chelikowsky, 1997), although the compensation of the local band-bending of a V,, by an singly ionized p-dopant is interpreted to strongly support a charge of 1 for the V,, (Chao et a/., 1996). The negatively charged Ga vacancies appear symmetric, with the role of the anion and cation reversed. They appear in the G a sublattice as a localized depression (Fig. 17, lower right), cause two bright As dangling

+

FIG. 18. (a) Spectroscopy of the pristine p-GaAs (1 10) surface (left) compared with (b) an isolatcd As defect (right). The spectra are acquired at two tip-sample separations (0,O) differing by 2.1 A (left) and 2.8A (right). Reprinted from Lengel et al. (1994) with permission from the authors.

288

NIKOSD. JACER AND

EICKE

R.

WEBEK

bonds in the filled state image (Fig. 17, lower left), and induce an extended apparent depression of Ga atoms and an elevation of As atoms.

VI. Complex Defects COMPLEXES 1. Zn-VACANCY DEFECT

IN

InP

The XSTM can be used to investigate not only individual defects, but also interactions, such as the attraction between defects, and the resulting creation of defect complexes. After cleaving and annealing for 5min at 480 K, four main types of features can be found on the (110) surface of p-type Zn-doped InP (Fig. 19, Ebert et a/., 1996b): isolated negatively chargcd Zn dopants; isolated positively charged phosphorous vacancies (c); Zn-vacancy dipoles of various distances between the Zn atom and the vacancy (d)-(f); and Zn-vacancy complexes (a) (b), where the Zn atoms sit in closest proximity to a vacancy. Because individual Zn-dopants are negatively charged and individual vacancies are positively charged, they appear in filled-state images as extended bright protrusions and extended dark depressions, respectively. The extent ranges over several lattice periods. Figure20(a) shows a profile through a Zn dopant and Fig. 20(b) through a vacancy. The Zn-vacancy complex [Fig. 19(a) and Fig. 20(d)] is a localized hole that has the size of one missing phosphorus dangling bond, and one or two apparently raised dangling bonds. The complex shows no surrounding overall height changes, that is, it causes no band bending, proving that the

+

FIG, 19. Vacancy-Zn defect complexes and dipoles. (a) and (b) show two complexes, (c) shows an isolated vacancy for comparison, and (d) -(f) show dipoles with differcnt separations. Reprinted from Ebert er ul. (1996b) with permission from the authors.

5


lattice spacings in [ 17 01 direction FIG. 20. Height profiles through (curve a) a negatively charged Zn-dopant atom; (curve h) a positively charged P vacancy; (curve c) the sum of a and b showing the structural relaxation of the vacancy; (curve d) a charge-compensated vacancy (defect complex); (curve e) the sum of a and b offset by seven lattice spacings; and (curve f) the profile of a dipole consisting of a vacancy and a Zn atom separated hy seven lattice spacings. All linecuts have been measured with the same tip at -2.1 V and 0.8 nA. Thc profilcs were measured along the [-1101 direction (along the apparent atomic rows). Reprinted from Ebert et ul. (1996b) with permission from the authors.

complex is uncharged. One can also look at the complex as a (partially) charge-compensated vacancy showing only structural contributions to the contrast. This can be seen by adding the hillock caused by the negative charge of a Zn dopant [Fig.20(a)] to the profile of a positively charged vacancy [Fig.20(b)], which leaves only the structure of a vacancy [Fig. 2O(c)]. The apparent similarity to the profile of the complex Fig. 20(d) is striking. When one adds the profile of a Zn dopant and a vacancy displaced by seven lattice spacings [Fig. 20(e)], agreement with the profile of a Zn-vacancy dipole (profile f ) is found. If one subtracts the structural properties of a vacancy from a vacancy dipole, that is, subtracting profile d from f, only the electronic effect remains. The integrals of the charge accumulation and of the depletion zones in the dipole, which were obtained

290

NIKWD. JAGER AND EICKER. WEBER

by subtraction, are smaller than the integrated charge of individual defects, demonstrating the partial compensation. The compensation increases with decreasing distance until it reaches near completion for the complex. Chao et a / . (1996) used a similar method to determine the charge on an arsenic vacancy in GaAs. One can also demonstrate an attraction between the positively charged vacancies and the negatively charged Zn-dopants, because the spacing between vacancies and Zn atoms is not random, but an increased number of closely separated pairs, that is, dipoles can be found.

2. ARSENIC CLUSTERS IN LT-GaAs

The STM images of arsenic precipitates found in low-temperature-grown n-GaAs after annealing are shown in Fig. 21 by Feenstra et a/. (1993b). The LT-GaAs is grown at tcmperatures around 200'C, causing an excess As . annealing at -600 C gives the concentration of 10'' ~ r n - ~Subsequent material a high resistivity ( > 10' R-cm) and causes the As to cluster. The Fermi level is then found to be pinned near midgap. In the topography

FK;. 21. STM image of GaAs, showing (a) topography and (b) conductivity, acquired at a sampic voltage of -2.5 V. As indicated in (b) the structure contains a low-temperature n-type GaAs layer, surrounded by p-type layers grown at higher temperature. The sample was annealed at 600-C. Two steps occur in the LT-layer, as marked by arrows in (a). The bright protrusions in the LT-layer are arsenic precipitates. Reprinted from Feenstra et a/. (1993b) with permission from the authors.

5 SCANNING TUNNELING MICROSCOPY OF DEFECTS IN SEMICONDUCTORS 291 10

,

,

1 ~

GaAs LTA-n

I

I

I

,

,

I

,

j Ec

I I l 1 2 SAMPLE VOLTAGE (V) = E-EF (eV)

I

-2

I

i EV

I

-1

,

I

l

I

0

FIG.22. The STM spectra acquired from the low-temperature n-type GaAs layers, annealed at 600’C. Spectra are shown for measurements made on the arsenic precipitates (solid line) and on regions of bare GaAs in between the precipitates (dashed line). The valence- and conduction-band edges are marked by E, and E,, respectively. Reprinted from Feenstra (1993b) with permission from the authors.

-

image (a), As precipitates appear as protrusions of 158, height with a diameter of 50 A.In the conductivity image (b) they are shown by areas of high conductivity. Figure 22 shows a spectrum acquired directly from the precipitate (solid line) and, for comparison, at a position in between precipitates. The Fermi level is in both cases near midgap, although it is closer to the valence band edge on the precipitate. For the spectrum measured on the precipitate, one can also observe a considerable density of states in the bandgap. These results and depletion areas found around As precipitates support the idea that the high resistivity of n-type LT-GaAs may be due to carrier depletion around metallic arsenic precipitates (Warren et a/., 1990).

-

3.

COMPENSATION

MECHANISMS IN Si-DOPED GaAs

In highly doped n-GaAs the free electron concentration tends to saturate with increasing silicon dopant concentration. The origin of this effect was identified on the atomic scale by Domke et al. (1996). There had been

292

NIKOS D. JAcm AND EICKER. WEBEK

controversy over the cause of the decrease in the doping efficiency of Si at high concentrations. The amphoteric nature of Si [Si can be incorporated on a Ga site (SiGa)where it acts as a donor and on an As site (SiAs)where it acts as an acceptor] only partially explains the electrical deactivation of up to 99% of incorporated Si. Additional models have been proposed, such as the formation of Si pairs, Si clusters, and complexes involving Si and an intrinsic defect. The existence of a nonhydrogenic Si level resonant with the conduction band has also been suggested, among other mechanisms. Figure 23 shows all the different types of defects found on highly Si-doped GaAs( 110) samples with Si concentrations ranging from 2.7 x 10'' cm-32.5 x 1 0 1 9 ~ ~ nshowing 55-95%, compensation. Panels (al) to (el) show filled-state images, and panels (a2) to (e2) show empty-state images. Frames (al) and (a2) exhibit a gallium vacancy (see V.2), (bl) and (b2) a Si,, donor (see IV.2), (cl) and (c2) an Si,, acceptor [as can be demonstrated by its symmetry and charge state], ( d l ) and (d2) a Si,,-VG, dopant vacancy complex (see forementioned), and (el) and (e2) show narrow trenches along [l-lo]. These trenches arise from planar Si clusters on (111) planes intersecting with the [I 101 cleavage plane. Close to the trenches are dislocations (frames f). Counting the occurrence of all these defects in many STM images, their individual bulk concentration was determined. Figure 24 shows these concentrations as a function of the total incorporated Si concentration obtained by SIMS. The crosses represent the sum of all defects seen in STM, which matches the concentration seen by SIMS. Note that the number of vacancies and vacancy-defect complexes increases with time. In the figure the concentrations extrapolated to the moment of the

FIG. 23. The STM images of occupied (upper frames) and empty (lower frames) density of statcs uf the major defects on Si-dopcd GaAs ( 1 10) surfaces (with the exception of frame f2). ( a l ) and (a2) show a G a vacancy, (bl) and (b2) a Si,, donor, ( c l ) and (c2) a Si,, acceptor, (dl) and (d2) a SiGa-Vc;acomplex, (el) and (e2) the intersection line of a planar Si cluster, and (fl) a dislocation close to a Si cluster (f2 is a zoom of the occupied states of the stacking fault in fl). Reprinted fi-om Domke et a/. (1996) with permission from the authors.


I

I

I

1018

1019

1020

incorporated S i concentration (cm-3) FIG. 24. The Si concentration present in the form of Si,, donors, Si,, acceptors, Sin clusters, and (SiGZ3-VcJcomplexes as a function of the Si doping concentration incorporated into the crystals during growth. The Si doping concentration incorporated during growth has been measured by secondary ion mass spectrometry (SIMS). The sum of the Si concentrations determined for the different defects measured in the STM images (x) agrees well with that measured by SIMS (solid line). The horizontal error bars originate from the SlMS measurements. They should be applied to 311 of the respective data points. All vertical error bars show the reproducibility of the STM measurements. The data are based on more than 3000 observations or Si atoms. Reprinted from Domke et ul. (1996) with permission from the authors.

cleave are shown. It was extrapolated that the number of vacancies at the time of the cleave is negligible; it is therefore not shown, but there is a significant number of Si-V pairs at the time of the cleave. In conclusion, the figure reveals that with increasing Si concentration the concentration of Si,, donors is almost constant, while it is consecutively electrically deactivated by Si,, acceptors, Si clusters, and SiGa-VG,complexes.

VII.

Conclusion

The experimental and theoretical results presented in this chapter show clearly the extraordinary possibilities offered by direct observation of point-like defects in semiconductors with XSTM. So far, this relatively

294

NIKOS D. JACERAND EICKER. WEBER

young technique has mostly confirmed conclusions reached using other spectroscopic techniques. Although a solid theory of the contrast formation mechanism is still to be developed, XSTM already provides new insights, However, the success in deciding controversial questions directly on the atomic scale in cases such as arsenic antisite and the determination of the Si-dopant compensation mechanisms in GaAs give confidence that XSTM will play an increasing role in identifying defects in semiconductors.

ACKNOWLEDGMENT We would like to acknowledge the authors of our references for permission to reprint their figures and for providing originals. And we would like to thank Eve Edelson for revising the manuscript.

REFERENCES Albrektsen, O., Arent, D. J., Meier, H. P., and Salemink, H. W. M. (1990). Appl. Phys. Lett., 57, 3 I . Alves, J. L. A,, Hebenstreit, J., and Schemer, M. (1991). Phys. Rev. B, 44, 6188. Baraff, G . A. (1992). Deep Centers in Semiconductors, S . T. Pantelides (ed.). Gordon and Breach Science Publishers, New York, p. 547. Binnig, G., Rohrer, H., Gerber, Ch., and Weibel, E. (1982). Phys. Rev. Lett., 49, 57. Capaz, R. B., Cho, K., and Joannopoulos, J. D. (1995). Phys. Reii. Lett., 75, 1811. Chao, K.-J., Smith, A. R., and Shih, C. K. (1996). Phys. Rev. B, 53, 6935. Cox. G., Szynka, D., Poppe, U., Urban, K., Kisielowski-Kemrnerich, C., Kriiger, J., and Alexander, H. (1990a). Phys. Rev. Lett., 64, 2402. Cox, G., Szynka, D., Poppe U., and Urban, K. (1990b). Vucuum. 41, 591. Domke, C., Ebert, Ph., Heinrich, M., and Urban, K. (1996). Phys. Rei:. B, 54. 10288. Ebert, Ph., Cox, G., Poppe, U., and Urban, K. (1992). UltrLimicroscopy, 42-44, 871. Ebert, Ph., and Urban, K. (1993a). Ultramicroscopy, 49, 344. Ebert, Ph., Lagally, M. G., and Urban, K. (1993b). Phys. Rev. Lett., 70, 1437. Ebert, Ph., Urban, K., and Lagally, M. G. (1994). Phys. Rru. Lerr., 72,840. Ebert, Ph., Heinrich, M., Simon, M., Urban, K., and Lagally, M. G. (1995). Phys. Rev. B, 51, 9696. Ebert, Ph., Chen, X., Heinrich, M., Simon, M., Urban, K., and Lagally, M. G. (1996a). Phys. Rev. Lett., 76, 2089. Ebert, Ph., Heinrich, M., Simon, M., Domke, C., Urban, K., Shih, C. K., Webb, M. B., and Lagally, M. G . (1996b). Phys. Rev. B, 53,4580. Ebert. Ph., Engels, B., Richard, P., Schroeder, K., Bliigel S., Dornke, C., Heinrich, M., and Urban, K. (1996~).Phys. Rev. Lett., 77, 2997. Feenstra, R. M. and Fein, A. P. (1985). Phys. Rev. B, 32, 1394. Feenstra, R. M., Stroscio, J. A,, and Fein, A. P. (1987a). Surj: Sci., 181, 295. Feenstra. R. M., Stroscio, J. A,, Tersoff, J., and Fein, A. P. (1987b). Pliys. Rev. Lett., 58, 1192. Feenstra, R. M., Yu, E. T., Woodall, J . M., Kirchner, P. D., Lin, C. L., and Pettit, G. D. (1992). A p p l . Phys. Lett., 61, 795.

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SCANNING TUNNELING MICROSCOPYOF DEFECTS IN SEMICONDUCTORS 295

Feenstra, R. M., Woodall, J. M., and Pettit, G. D. (1993a). Phys. Rev. Lett., 71, 1176. Feenstra, R. M., Vaterlaus, A., Woodall, J. M., and Pettit, G. D. (1993b). Appl. Phys. Lett., 63, 2528. Feenstra, R. M., Woodall, J. M., and Pettit, G. D. (1994). Mat. Sci. For., 143-147, 1311. Frohn, J., Wolf, J. F., Besocke, K. H., and Teske, N. (1989). Rev, Sci. Instrum., 60, 1200. Gebauer, J., Krausc-Rehberg, R., Domke, C., Ehert, Ph., and Urban, K. (1997). Phys. Xeu. Left., 78. 3334. Gwo, S., Smith, A. R., Shih, C. K., Sadra, K., and Streetman, B. G. (1992). Appl. Phys. Lett., 61, 1104. Gwo S., Smith, A. R., and Shih, C. K. (1993). J . Vac. Sci. Techno/. A , 11, 1644. Harper, J., Lengel, G., Allen, R. E., and Weimcr, M. (1997a). Phys. Reo. Lett., 79, 3312. Harper, J., Lengel, G., Allen, R. E., and Weimer, M. (1997b). Phys. Rev. Lett., 79, 3314. Jager, N. D., Liu, X., Zheng, J. F., Newman, N., Ogletree, D. F., Weber, E. R., and Salmeron, M. (1996). Pr0ceeding.c.23rd Intwzutioncrl Conference on the Physics of Semiconductors, Berlin 1996. M. Schemer and R. Zimmcrman (eds.), World Scientific, Singapore, 2, p. 847-850. Jiiger, N. D., Toporowski, M., Salmeron, M., and Weber, E. R. (1997). Unpublished. Johnson, M. B. and Halbout, J. M. (1992). J . Vuc. Sci. Techno/. B, 10, 508. Johnson, M. B., Albrektsen, O., Feenstra. R. M., and Salemink, H. W. M. (1993a). Appl. Phys. Lett., 63, 2923. Johnson, M. B., Meier, H. P., and Salemink, H. W. M. (1993b). Appl. Phys. Lett., 63, 3636. Johnson, M. B., Koenrdad, P. M., van der Vleuten, W. C . , Salemink, H. W. M., and Wolter, J. H. (1995). Phys. Reo. Lett., 75, 1606. Kaminska, M. and Weber, E. R. (1993). Semiconductors and Sernirnetuls, 38, R. K . Willardson, A. C. Beer, and E. R. Weber (eds.), Academic Press, Boston, p. 59. Kim, H. and Chelikowsky, J. R. (1996). Phys. Reo. Lett., 77, 1063. Kim, H. and Chelikowsky, J. R. (1997). PIIJJS. Rev. Lett., 79, 3315. Lengel, G., Wilkins, R., Brown, G., and Weimer, M. (1993). J . Vuc. Sci. Technol. B, 11,1472. Lengel, G.. Wilkins, R., Brown, G., and Weimcr, M., Gryko, J., and Allen, R. E. (1994). Piiys. Rw. Lett., 72, 836. Lengel, G., Harper, J., and Weimer, M. (1996). Phys. Rev. Lett., 76, 4725. Lengel, G., Harper, J., and Weimer, M. (1997). Private communication. Pfister, M., Johnson, M. B., Alverado, S. F., Salemink, H. W. M., Marti, U., Martin, D., Morier-Genoud, F., and Reinhard, F. K. (1994). Appl. Phys. Lett., 65, 1168. Salemink, H. W. M.,Johnson, M. B., and Albrektsen, 0. (1994). J . Vuc. Sci. Technol. B, 12, 362. Samuelson, L., Lindahl, J., Montelius, L., and Pistol, M.-E. (1992). Physica Scripta, T42, 149. Smith, A. R., Gwo. S., and Shih, C. K . (1994). Rev. Sci. Instr., 65, 3216. Stroscio, .I. A,, Feenstra, R. M., and Fein, A. P. (1986). Phys. Rev. Lett., 57, 2579. Tersoff, I., and Hamann, D. R. (1983). Phys. Rev. Lett., 50, 1998. Tersof, J. and Hamann, D. R. (1985). Phys. Rev. B, 31, 805. van der Wielen, M. C. M. M., van Roij, A. J. A., and van Kempen, H. (1996). Phys. Rei:. Lett., 76, 1075. Vaterlaus, A,, Feenstra, R. M., Kirchner, P. D., Woodall, J. M., and Pettit, G. D. (1993). J . Vac. Sci. Technol. B, 11, 1502. Wang, J., Arias, T. A,, Joannopoulos, J. D., Turner, G . W., and Alerhand, 0. L. (1993). Phys. Rev. B,47, 10326. Warren, A. C., Woodall, J. M., Freeouf, J. L.. Grischkowsky, D., McInturff, D. T., Melloch, M. R., and Otsuka, N. (1990). Appl. Phys. Lett., 57, 1331. Whitman, L. J., Stroscio, J. A., Dragoset, R. A,, and Celotta, R. J. (1990). Phys. Rev. B, 42, 7288. Yi, J.-Y., Ha, J. S., Park, S.-J., and Lee, E.-H. (1995). Phys. Rev. B, 51, 11198.

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Yu, E. T., Johnson, M. B.,and Halbout, J. M. (1992). A p p l . Phys. Lett., 61, 201. Zhang, S. B. and Zunger, A. (1996). Phys. Reil. Lett., 77, 119. Zhang, S. B. and Zunger, A. (1997). Phys. Rev. Lrfr., 79, 3313. Zheng, J. F., Liu, X., Newrnan. N., Weber, E. R., Ogletree. D. F., and Salrneron, M. (1994a). Phys. Reu. Lett., 72, 1490. Zheng, J. F., Walker, J. D., Salmeron, M. B., and Weher, E. R. (1994b). Phys. Rev. Lrrt., 72, 2414. Zheng, J. F., Salmeron, M. B., and Weber, E. R. (1994~).Appl. P h p Leir., 64, 1836. Zheng, J. F., Ogletree, D. F., Walker, J., Salmeron, M., and Weber, E. R. (1994d). J. Vuc. Sci. Techno/. B, 12, 2100. Zhcng, J. F.. Salrneron, M., and Weher, E. R . (1994e). Unpublishcd.

SEMICONDUCTORS AND SEMIMETALS. VOL . 51B

CHAPTER 6

Perturbed Angular Correlation Studies of Defects Thomas Wichert TECHNITCHI. PIiYXK

.

SAAKHKUCKIN FkDERAl

K ~ P U B L I C0 1. (;I.K\.lhNY

I . INTKOUUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Anulyiical Techniques und Sensitivities . . . . . . . . . . . . . . . . 2 . Rudiouctive Ootopes . . . . . . . . . . . . . . . . . . . . . . . . . 11. EXPERlMbNTAL METHOD . . . . . . . . . . . . . . . . . . . . . . . . I . Hyperfine Inreructions . . . . . . . . . . . . . . . . . . . . . . . . 2 . The Perturbed y-y Angulur Cbrrrl(ition Technique . . . . . . . . . . . . 3 . Radiouctivr Probe A t o m . . . . . . . . . . . . . . . . . . . . . . . 4. Experimentul Aspects . . . . . . . . . . . . . . . . . . . . . . . 111. SUBSTITUTIONAL DOPANTS . . . . . . . . . . . . . . . . . . . . . . . . 1 . Donur-Accepror PLriring . . . . . . . . . . . . . . . . . . . . . . . 2 . Lutticr Sites of Isolated Probe Atoms . . . . . . . . . . . . . . . . . 3 . Clustering of Dopunis . . . . . . . . . . . . . . . . . . . . . . . . 4 . Effect o/ Clicvnicul TrunsmutLition . . . . . . . . . . . . . . . . . . IV . LIGHTELEMENTS A N D TRANSITION METALS. . . . . . . . . . . . . . . . 1. Hydrogen in Elemental Semiconcluctors . . . . . . . . . . . . . . . . 2 . Hydrogen in III- V Semiconduciors . . . . . . . . . . . . . . . . . . 3. Trunsirion Metals . . . . . . . . . . . . . . . . . . . . . . . . . . V . INTRINSICDEFECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. frrucliution und Quenching . . . . . . . . . . . . . . . . . . . . . . 2 . Single Recoil Process . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Oj~stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . VI . SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

.

291 298 299 300 302 . 305 316 320 322 324 . 331 340 . 342 . 351 . 352 . 365 312 382 382 388 393 400 401 401

.

Introduction

Progress in semiconductor technology is driven by progress in the knowledge and control of defects; these include intrinsic defects. such as vacancies and self.interstitials. and extrinsic defects. such as dopants and 297 Copylight 1 1YY9 by A'ddemi' Press All right? of rcproduction in any form rcaerved ISBN 0-12 752165.8 ISSN 0080.8784 53000

298

THOMAS WICHERT

impurity atoms in thcse materials. This chapter will illustrate to what extent radioactive probe atoms in combination with the perturbed y y angular correlation technique (PAC) are able to contribute to the identGcation of different types of defects in different semiconductors, elemental as well as Ill-Vs and 11-Vls.

1. ANALYTICAL TECHNIQUES AND SL:NSITIVITIES

One of the key problems in the control of electrical properties of semiconductors concerns the influence of the doping procedure and thermal treatment of a sample on the degree of electrical activation of the dopants. Furthermore, the mechanisms that cause the compensation or passivation of the dopants should be understood. A typical lower bound for the conductivity of an intentionally doped semiconductor with CT = 1 Scmand a mobility of ,u = lo3 cm2/Vs would correspond to a dopant concentration of about 10i6cm-3. As a similar concentration of defects would bc sufficient to compensate or passivate these dopant atoms, analytical techniques should be sensitive to defect concentrations down to loi6cm-3 and below. A major portion of the analytical methods commonly used are electrical and optical techniques. These techniques, including electrical transport measurements (Hall effect and conductivity), capacitance-voltage (C-V) measurements, deep-level transient spectroscopy (DLTS), and photoluminescence spectroscopy (PL), are distinguished by their high sensitivities to defect concentrations well below 10'hcm-3 as is indicated in Fig. 1. However, several of these techniques do not yield easily accessible microscopic information about the identity of the extrinsic or intrinsic defects that are involved. Alternatively, those experimental techniques that are well suited for the chemical identification of defects often lack the required ~. these sensitivity to defect concentrations of the order of 1016~ r n - Among are techniques, such as Raman scattering, extended X-ray absorption fine-structure measurements (EXAFS), Rutherford backscattering spectroscopy (RBS) including ion channeling (Chann.), and nuclear magnetic resonance (NMR), which do not allow routine investigations of defects below concentrations of about 10" cm-3. Here, the magnetic resonance techniques, electron paramagnetic resonance (EPR) and electron nuclear double resonance (ENDOR), along with their different relatives, represent an important exception. The magnetic resonance techniques are able to identify the chemical nature of a defect at concentrations around or below 1 0 " ~ m - if~ the defect is involved in a paramagnetic center. Although this list cannot be regarded as exhaustive, in many cases one is left with the choice of using either a technique that possesses the needed

6

PERTURBED

ANGULARCORRELATION STUDIES OF DEFECTS Electrical and optical properties: Chemical identification: - using stable atoms - radioactive probes

I

299

----

-

1-

10’2

ll@ olo~

FIG. 1, Sensitivities of different experimental techniques to defects in semiconductors. The abbreviations used are explained in the text (SIMS = secondary ion mass spectroscopy). Reprinted from Wichert (1996) with permission from the author.

sensitivity but is blind to the chemical nature of the involved dopants or defects, or one that is sensitive to the chemical nature of dopants or defects but lacks the required sensitivity for low concentrations. Of course, with regard to this classification, there are always exceptions to the rules and it should be noted that Fig. 1 can represent only an overview of the sensitivities of various techniques, which should be a guide for routine investigations. Thus, most of the detection limits given in Fig. 1 apply to 1 cm2 samples with a thickness of 1 pm. If applied to a 1 mm thick, homogeneously doped sample -for example, infrared spectroscopy (IR) would still be sensitive to concentrations down to 10’ cm- ’. For more detailed information about the different techniques, the reader is referred to Stradling and Klipstein (1990), Feldman and Mayer (1986), Spaeth et ul. (1992), and Langouche (1992) and to other chapters in this volume.

2. RADIOACTIVEISOTOPES

The use of radioactive isotopes opens the way for new analytical techniques, sometimes improves the limited sensitivities of previously used methods, and can also add chemical specificity to a chemically insensitive

300

THOMAS WICHERT

method. Besides Mossbauer spectroscopy (MS), a new technique is the perturbed y;’ angular corrclation technique (PAC), which measures the local hyperfine interaction at the site of a radioactive probe atom (Fig. 1). From PAC measuremcnts, the identity of defects can be determined similarly to the way results are obtained with EPR and ENDOR techniques, however, without the requirement of a paramagnetic center and the restriction to low temperatures. In combination with radioactive isotopes, the channeling technique, called “Blocking,” and NMR, called “b-NMR,” shows significantly higher sensitivities. The first systematic studics of defects in semiconductors using radioactive probe atoms started in the 1980s using MS (Weyer et ul., 1981), b-NMR (Grupp et a/., 1982), Blocking (Lindner et ul., 1986), and PAC (Witthuhn, 1985; Wichert ei ul., 1989) and have been discussed in recent reviews by Williamson et a/. (1992); Ackerman et a/. (1992); Wichert et ul. (1992), and Hofsiss and Lindner (1991). The use of radioactive isotopes in combination with chemically insensitive methods, such as DLTS and PL measurements, is discussed by Wichert (1996), Magerle (1997), and in chapters elsewhere in this volume. The present contribution will focus on the PAC technique and will present selected experiments that illustrate the pros and cons of PAC and that have contributed to shaping the field of identifying point defects with the help of radioactive isotopes. Because analytical techniques are not applicable in all cases or under all conditions and their dekct-related information is never complete, most defect problems are solved by the application of complementary methods, with each providing a piece of the puzzle. In this sense, the use of radioactive isotopes provides a new kind of information, even for defects that have been well studied by more conventional techniques.

11.

Experimental Method

The basic principles of the perturbed yy ungulur c‘orrelution technique (PAC) are illustrated in Figs. 2 and 3. The radioactive parent probe atom creates, usually through its /j-decay, an excited duuyhter probe atom. The excited nucleus subsequently releases its excitation energy through the emission of a y-cascade. The property of the 7-rays, in which they always carry their angular momentum vector either along or opposite to their propagation direction, is what makes PAC possible. By detecting both prays (yl and ; 1 2 ) in a coiricidence experirnent, information is obtained on the respective orientations of the same nuclear spin I at two times that are defined by the emission of y 1 (yielding the initial orientation) and y z (yielding the orientation after the time At). From the change of the spin

6 PERTURBED ANGULARCORRELATIONSTUDIES OF DEFECTS

301

FIG.2. In a PAC experiment, the probability of detecting ?,and y z in coincidence Is modulated by the precession frequency o of the nuclear spin I. The actual value of w IS determined by the strength of the electric or magnetic field at the site of the nucleus of the (daughter) probe atom.

42

electric field gradient FIG. 3. A more detailed nuclear decay scheme of the 8-decay shown in the inset of Fig. 2. For the case of the PAC probe ' l l I n / ' l l C d (nuclear spin I = 5/2), along with the corresponding magnetic quantum numbers A4 = 5/2, i3/2, and 1/2 the level splitting is shown that is induced by the electric field gradient ( Vzz)at the intermediate nuclear state with quadrupole moment Q and lifetime 7 (see also Fig. 10). Note that this energy splitting, characterized by (11 = AE/h, is on the order of IO-'eV, whereas the p-transition energies are on the order of 10' eV.

*

*

302

THOMAS WICHERT

orientation AI, measured for an ensemble of probe atoms during the time At elapsed between the emission of both y-rays, the spin precession frequency LC) is deduced. The way in which the change in A1 is reflected by the y-y coincidence probability detected as a function of the elapsed time At will be discussed here. It should be noted that there are probe atoms, where the excited state that emits y1 forms a long-lived nuclear state as will be shown for the probe lllmCdin Fig. 10; in this case, parent and daughter probes are chemically identical. Keeping in mind that the presence of a defect givcs rise to an electric field gradient, Fig. 3 shows that the defect interaction with the quadrupole moment Q of the intermediate state gives rise to a characteristic energy splitting of this nuclear state. This splitting, which is proportional to the largest component of the electric field gradient V,,, relates to a precession of the spin I with the frequencies wi in a classical picture. For experimental reasons, Ar has to be observable for a time duration that is comparable with the time for about one revolution of the nuclear spins. Therefore, the intermediate nuclear state, populated by the emission of yl, has to possess an adequate lifetime T besides the spin I and quadrupole moment Q. In quantum mechanics, with respect to the quantization axis z,the frequencies LO, lead to phase shifts (wi.At) of the nuclear spins I, which depend on the magnitude of the clectric field gradient (see Fig. 3). This section will introduce the relevant part of the hyperfine interaction that governs these phenomena and will present the theoretical description of the experimental PAC spectra needed for the identification of defects.

1. HYPERFINE INTERACTIONS In PAC experiments, similar to Mossbauer spectroscopy or NMR, the hyperfine interaction arising from the interaction of an electromagnetic field with a nuclear moment is observed at the site of a nucleus. In general, the Hamiltonian H , describing the energy shift of nuclear states by this interaction, is composed of a magnetic and an electric part

The first term describes the magnetic part of the interaction and gives rise to the Larmor frequency wL, which is proportional to the magnetic field B. The second term in Eq. (1) arises from the interaction between the nuclear electric charge distribution and the charge distribution of the surrounding electrons and positive ions. In a multipole expansion of He,it turns out that the monopole term causes the same energy shift for all sublevels of the

6 PERTURBED ANGULARCORRELATION STUDIES OF DEFECTS

303

nuclear spin I (designated by the magnetic quantum number M ) , the dipole term vanishes, but the quadrupole term HQ causes an energy shift that depends on the quantum number M (see Fig. 3). This last term comes from the interaction of the nuclear electric quadrupole moment Q with the 3 x 3 tensor of the electric field gradient with the components which are the second derivative of the electrostatic potential V(r)

vj,

d2 v(r) v..= ___ xi,xj = x, y,z ' J axiaxj >

at the site of the nucleus (r = 0). For PAC this quadrupole interaction has been the relevant one for the study of defects in semiconductors so far. It is described by

H -

eQ

c K j [t(IiIj + IjIi)

a - 61(21 - 1) i , j

- 6,,121

(3)

where x i , x j represent the Cartesian coordinates x, y, z and hij is the Kronecker delta. The tensor K j is traceless and symmetric and is therefore completely described by five components (see Fig. 4).

Electric Field Gradient Tensor

FIG. 4. Correspondence between the components K j of the traceless electric field gradient tensor and the quadrupole coupling constant v, [Eq. (15)], the asymmetry parameter q [Eq. ( 5 ) ] ,and the Euler angles ai.Note the F 3dependence of the electric field gradient, where I is the distance of the charge q to the nucleus.

304

THOMAS WICHEKT

The 3 x 3 tensor representing the electric field gradient is simplified by transformation into a principal axis system so that y j = 0 for i # j . In this system, Eq. (3) becomes

Equation (4) shows that two parameters are sufficient to characterize the field gradient tensor in its principal axis system, that is, the largest component of the electric field gradient t z and the difference Vxx- V,, which, usually, is expressed by the asymmetry parameter

with I t z ] 3 IVY,/ 2 lVxxl so that 0 < y~ 6 1 holds. The remaining three parameters necessary to describe the tensor within an arbitrarily chosen system of coordinates, for cxample, the coordinates of the host crystal, correspond to the three Euler angles ui that described the transformation to the principal axes system [see Eqs. (3) and (4) and Fig. 41. As will be explained in what follows, besides V,, and q, two of these angles and the third angle within +rc are determined by PAC so that the strength, symmetry and orientation- strictly speaking the alignment -of the field gradient tensor can be determined; the third angle remains uncertain within -tx because in a 11-1’ correlation experiment the sign of the quadrupole interaction is not accessible. The eigenvalues of the Hamiltonian in Eq. (4) correspond to the cnergy shifts E M of a nuclear state with spin I and z-component M . Their cvaluation is most straightforward if V’, = Yy, that is, for the case of an i i x i d l y symmetric field gradient ( q = 0)

E M = [3M2 - I(1

+ I)]

eQ V,, 41(21 - 1)

(6)

and the energy difference between two different sublevels M and M’ is

with the abbreviation R)

eQ K z

a

- 41(21 - l)h

Equation (7) shows that the energy difference A E ( M , M ‘ ) corresponds to

305

6 PERTURBED ANGULARCORRELATION STUDIES OF DEFECTS I =512

4 2 1 ,

,

0.0

0.2

.

,

~

0.4

, . , 06

rl

08

,

1 1.0

rl

FIG 5 . Left: Energy splitting E M of an I = 5/2 nuclear state in units of hoQ [Eq. (7)] as a function of the asymmetry parameter (V,, = const.). Right: Dependence of the ratio m2/(01 on the parameter ti along with the parameter ber,El2 Defect in GaAs David C. Look, Defects Relevant for Compensation in Semi-Insulating GaAs R. C. Newman, Local Vibrational Mode Spectroscopy of Defects in llI/V Compounds Andrzej M. Hennel, Transition Metals in 111,W Compounds Kevin J. Miilluy und Ken Khucha/uryun, DX and Relatcd Defects in Semiconductors V. Swwninathan and Andrew S. Jordan, Dislocations in fll/V Compounds Krzysztof’ W. Nuuka, Deep Level Defects in the Epitaxial IIljV Materials

Volume 39 Minority Carriers in 111-V Semiconductors: Physics and Applications Niluy K. Durn, Radiative Transitions in GaAs and Other I11 V Compounds Richard K. Ahrenkiel, Minority-Carrier Lifetime in Ill-V Semiconductors Tomofumi Furuta, High Field Minority Electron Transport in p-GaAs Murk S. Lundstrum, Minority-Carrier Transport in 111-V Semiconductors Richard A . Ahram, Effects of Heavy Doping and High Excitation on the Band Structure of GaAs David Yevirk and Witold Bardyszew.rki, An Introduction to Non-Equilibrium Many-Body Analyses of Optical Processes in 111-V Semiconductors

Volume 40 Epitaxial Microstructures E. F. Schubrrt, Delta-Doping of Semiconductors: Electronic, Optical, and Structural Properties of Materials and Devices A. Gossard, M. Sundaram, and P. Hupkins, Wide Graded Potential Wells P. Petroff, Direct Growth of Nanometer-Size Quantum Wire Superlattices E. Kapon, Lateral Patterning of Quantum Well Heterostructures by Growth of Nonplanar Substrates H. Ternkin, D. Gershoni, and M. Puni.sh, Optical Properties of Gal-,In,As/InP Quantum Wells

CONTENTS OF VOLUMES IN THISSERIES

43 1

Volume 41 High Speed Heterostructure Devices f? Capasso, F. Beltrum, S. Sen, A. Pahlevi, and A. Y. Cho, Quantum Electron Devices: Physics and Applications P. Solomon, D. J. Frunk. S. L. Wright, and F Canora, GaAs-Gate Semiconductor-InsulatorSemiconductor FET M. H. Hashemi und U. K. Mishru, Unipolar InP-Based Transistors R. Kiehl, Complementary Heterostructure FET Integrated Circuits T. Ishibashi, GaAs-Based and InP-Based Heterostructure Bipolar Transistors H. C. Liu and T. C. L. C. Sollner, High-Frequency-Tunneling Devices H. Ohnishi, T. More, M. Takatsu, K. Imamura, and N. Yokoyama, Resonant-Tunneling Hot-Electron Transistors and Circuits

Volume 42 Oxygen in Silicon f? Shimura, Introduction to Oxygen in Silicon W. Lin, The Incorporation of Oxygen into Silicon Crystals T. J. Schaffner and D. K. Schroder, Characterization Techniques for Oxygen in Silicon W M. Bullis, Oxygen Concentration Measurement S. M.Hu, Intrinsic Point Defects in Silicon B. Pujot, Some Atomic Configurations of Oxygen J. Michel and L. C. Kimerling, Electical Properties of Oxygen in Silicon R. C. Newman and R. Jones, Diffusion of Oxygen in Silicon T. Y. Tan and W. J. Taylor, Mechanisms of Oxygen Precipitation: Some Quantitative Aspects M. Schrems, Simulation of Oxygen Precipitation K Simino and I. Yonenaga, Oxygen Effect on Mechanical Properties W. Bergholz, Grown-in and Process-Induced Effects F. Shimura, Intrinsic/Internal Gettering H. Tsuyo, Oxygen Effect on Electronic Device Performance

Volume 43 Semiconductors for Room Temperature Nuclear Detector Applications R. B. James and T. E. Schlesinger, Introduction and Overview L. S. Darken and C. E. Cox, High-Purity Germanium Detectors A. Burger, D. Nason, L. Van den Berg, and M. Schieber, Growth of Mercuric Iodide X J. Bao, T. E. Schlesinger, and R B. James, Electrical Properties of Mercuric Iodide X J. Bao, R. B. James, and T. E. Schlesinger, Optical Properties of Red Mercuric Iodide M. Hage-Ali and P. Stffert, Growth Methods of CdTe Nuclear Detector Materials M. Huge-Ali and P Szffert, Characterization of CdTe Nuclear Detector Materials M. Huge-Ali and P. Siffert, CdTe Nuclear Detectors and Applications R. B. James, T. E. Schlesinger, J. Lund, and M. Schieber, Cd,_xZn,Te Spectrometers for Gamma and X-Ray Applications D. S. McGregor, J. E. Kammeruad, Gallium Arsenide Radiation Detectors and Spectrometers J. C. Lund, F Olschner, and A. Burger, Lead Iodide M. R. Squillante, and K. S. Shah, Other Materials: Status and Prospects V. M. Cerrish, Characterization and Quantification of Detector Performance J. S. Iwunczyk and B. E. Putt, Electronics for X-ray and Gamma Ray Spectrometers M. Schieber, R. B. James, and T. E. Schlesinger, Summary and Remaining Issues for Room Temperature Radiation Spectrometers

432


Volume 44 11-IV BluelGreen Light Emitters: Device Physics and Epitaxial Growth J. Hun and R. L. Gunshor, MBE Growth and Electrical Properties of Wide Bandgap ZnSe-based I1 VI Semiconductors Shizuo Fujittr und Shigeo Fujitu, Growth and Characterization of ZnSe-based 11-VI Semiconductors by MOVPE Etrsen Ho trnd Leslie A. Kolodziejski, Gaseous Sourcc UHV Epitaxy Technologies for Wide Bandgap 11-VI Semiconductors Chris G. Vfin de Wulle, Doping of Wide-Band-Gap I1 -VI Compounds -Theory Roberto Cingoluni, Optical Properties of Excitons in ZnSe-Based Quantum Well Heterostructures A . Ishihashi untt A. V. Nurmikko, 11-VI Diode Lasers: A Current View of Device Performance and Issues Suprutik Guha und John Prtruxllo, Defects and Degradation in Wide-Gap 11-VI-based Structures and Light Emitting Devices

Volume 45 Effect of Disorder and Defects in Ion-ImplantedSemiconductors: Electrical and Physiochemical Characterization Heiner Ryssrl, Ion Implantation into Semiconductors: Historical Perspectives You-Nian Wang und Teng-Cui Mu, Electronic Stopping Power for Energetic Ions in Solids Suchiko T. Nakagawu, Solid Effect on the Electronic Stopping of Crystalline Target and Application to Range Estimation G. Miiller, S. Kolbirxr and G. N. Greuvrs, Ion Beams in Amorphous Semiconductor Research Jumanu Bousscy-Said, Sheet and Spreading Resistance Analysis of Ion Implanted and Annealed Sciniconductors M. L. Polignano and G. Qircirolo, Studies of the Stripping Hall Effect in Ion-Implanted Silicon J. Stoemeno.r.Transmission Electron Microscopy Analyses Roberta Nipori and Marco Servidori, Rutherford Backscattering Studies of Ion Implanted Semiconduciors P. Zazmmseil, X-ray DilTraction Techniques

Volume 46 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Optical and Photothermal Characterization M . Fried 7: Lohner trnd J . Gyultri, Ellipsometric Analysis Anronios Seas and Constnntinos Chrisrojides, Transmission and Reflection Spectroscopy on Ion

Implanted Semiconductors Anclreas Othonos nnd Constiintinos Christufdes, Photoluminescence and Raman Scattering of Ion Implanted Semiconductors. Influence of Annealing Cunsluntinos Chrisfqfides, Photomodulated Thermoreflectance Investigation of Implanted Wafers. Annealing Kinetics of Defects U. Zammit, Photothermal Deflection Spectroscopy Characterization of Ion-Implanted and Annealed Silicon Films Andreas Mundeh. A r k f Budirnan and Miguel Vurgas, Photothermal Deep-Level Transient Spectroscopy of Impurities and Defects in Semiconductors R. K u h h and S Churbonneuu, Ion Implantation into Quantum-Well Structures Alexundre M . Myusnikov und Nikoluy N. Grrasimenko, Ion Implantation and Thermal Annealing of 111-V Compound Semiconducting Systems: Some Problems of 111-V Narrow G a p Semiconductors


Volume 47

433

Uncooled Infrared Imaging Arrays and Systems

R. G. Busrr und M. P. Tornpsett, Historical Overview P . M! Kruse, Principles of Uncooled Infrared Focal Plane Arrays R. A . CVocid, Monolithic Silicon Microbolometer Arrays C. M . Hanson, Hybrid Pyroelectric-Ferroelectric Bolometer Arrays D.L. Pollu und J . R. Choi,Monolithic Pyroelcctric Bolometer Arrays N . Teranishi, Thermoelectric Uncooled Infrared Focal Plane Arrays M . F. Tornpsetl, Pyroelectric Vidicon 7: W KennJ,,Tunneling Infrared Sensors J . R. Vig, R. L. Filler und Y Kim, Application of Quartz Microresonators to Uncooled Infrared Imaging Arrays P. M! Kruse. Application of Uncooled Monolithic Thermoelectric Linear Arrays to Imaging Radiometers

Volume 48 High Brightness Light Emitting Diodes G. 8.Siringfdlow, Materials Issues in High-Brightness Light-Emitting Diodes M . G. Cruford, Overview of Device issues in High-Brightness Light-Emitting Diodes F . M . Steranku, AlGaAs Red Light Emitting Diodes C. H . Chen, S. A . Stockman, M . J . Peanusky, and C. P. Kuo, O M V P E Growth of AlGaInP for High Efficiency Visible Light-Emitting Diodes F. 4.Kish und R . M . Fletcher, AlGaInP Light-Emitting Diodes M . W Hodapg, Applications for High Brightness Light-Emitting Diodes I . Akusaki und H . Amano, Organometallic Vapor Epitaxy of G a N for High Brightness Blue Light Emitting Diodes S. Nukamuru, Group 111-V Nitride Based Ultraviolet-Blue-Green-Yellow Light-Emitting Diodes and Laser Diodes

Volume 49 Light Emission in Silicon: from Physics to Devices Duvid J. Lockwood, Light Emission in Silicon Grrhtrrd Ahstreiter, Band Gaps and Light Emission in Si/SiGe Atomic Layer Structures Thomas G. Brown trnd Dennis G. HUN, Radiative Isoelectronic Impurities in Silicon and Silicon-Germanium Alloys and Superlattices J. Michel, L. V. C. Axsuli, M. T. Mor.se, und L. C. Kinzerling, Erbium in Silicon Yo,shihiko Kunernitsu, Silicon and Germanium Nanoparticles Philippe M. Fauchet, Porous Silicon: Photoluminescence and Electroluminescent Devices C. Delerue, G. Allan, und M. Lrrnnoa, Theory of Radiative and Nonradiative Processes in Silicon Nanocrystallites Louis Brus, Silicon Polymers and Nanocrystals

Volume 50 Gallium Nitride (GaN) J. I. Pmkove urzd T. D. Moustukus, Introduction S. P. DenBuurs and S. Kel/er. Metalorganic Chemical Vapor Deposition (MOCVD) of Group I11 Nitrides W. A . Bryden und T. J. Kistenmucher, Growth of Group I11 -A Nitrides by Reactive Sputtering N . Newmon, Thermochemistry o f 111-N Semiconductors S. J. Peurton und R. J. Shul, Etching of I11 Nitrides S. M. Beduir, Indium-based Nitride Compounds A . Trumpert, 0. Brundt, und K. H. Ploog, Crystal Structure of Group III Nitrides

434


H. Morknr, F: Harniluni, and A. Sulvudoi; Electronic and Optical Properties of 111-V Nitride based Quantum Wells and Superlattices K . Doverspike and J. I. Punkuve, Doping in the 111-Nitrides T. Suski und P. Prrlin, High Pressure Studies of Defects and Impurities in Gallium Nitride B. Monrmar, Optical Properties of GaN W. R. L. Lamhrrrhc, Band Structure of thc Group 111 Nitrides N . E. Chrisfmsen und P. Perlin, Phonons and Phase Transitions in G a N S. Nukumuru, Applications of LEDs and LDs 1. Akusaki uwd H . Amano, Lasers J. A . Cooper, Jr., Nonvolatile Random Access Memories in Wide Bandgap Semiconductors

Volume 51A Identification of Defects in Semiconductors George D. Watkins, EPR and ENDOR Studies of Defects in Semiconductors J.-M. Spaeth, Magneto-Optical and Electrical Detection of Paramagnetic Resonance in Semiconductors T. A. Kennedy und E. R. Glusrr, Magnetic Resonance of Epitaxial Layers Detected by Photoluminescence K fl. Chow, B. Nitri, and R. F: Ki.fl. pSR on Muonium in Semiconductors and Its Relation to Hydrogen Kimrno Saurinen, Pekku Hautujiirvi, unil Catherine Corhel, Positron Annihilation Spectroscopy of Defects in Semiconductors R. Jone.7 und P. R. Briddun, The A b f n i t i o Cluster Method and the Dynamics of Defects in Semiconductors

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