Charged Semiconductor Defects: Structure, Thermodynamics and Diffusion (Engineering Materials and Processes)

Engineering Materials and Processes

Series Editor Professor Brian Derby, Professor of Materials Science Manchester Materials Science Centre, Grosvenor Street, Manchester, M1 7HS, UK

Other titles published in this series Fusion Bonding of Polymer Composites C. Ageorges and L. Ye

Fuel Cell Technology N. Sammes

Composite Materials D.D.L. Chung

Casting: An Analytical Approach A. Reikher and M.R. Barkhudarov

Titanium G. Lütjering and J.C. Williams

Computational Quantum Mechanics for Materials Engineers L. Vitos

Corrosion of Metals H. Kaesche Corrosion and Protection E. Bardal Intelligent Macromolecules for Smart Devices L. Dai Microstructure of Steels and Cast Irons M. Durand-Charre

Modelling of Powder Die Compaction P.R. Brewin, O. Coube, P. Doremus and J.H. Tweed Silver Metallization D. Adams, T.L. Alford and J.W. Mayer Microbiologically Influenced Corrosion R. Javaherdashti

Phase Diagrams and Heterogeneous Equilibria B. Predel, M. Hoch and M. Pool

Modeling of Metal Forming and Machining Processes P.M. Dixit and U.S. Dixit

Computational Mechanics of Composite Materials M. Kamiński

Electromechanical Properties in Composites Based on Ferroelectrics V.Yu. Topolov and C.R. Bowen

Gallium Nitride Processing for Electronics, Sensors and Spintronics S.J. Pearton, C.R. Abernathy and F. Ren

Modelling Stochastic Fibrous Materials with Mathematica® W.W. Sampson

Materials for Information Technology E. Zschech, C. Whelan and T. Mikolajick

Edmund G. Seebauer • Meredith C. Kratzer

Charged Semiconductor Defects Structure, Thermodynamics and Diffusion

123

Edmund G. Seebauer, BS, PhD Meredith C. Kratzer, BS, MS University of Illinois at Urbana-Champaign Department of Chemical and Biomolecular Engineering 600 S. Mathews Avenue Urbana, Illinois 61801-3792 USA

ISBN 978-1-84882-058-6

e-ISBN 978-1-84882-059-3

DOI 10.1007/978-1-84882-059-3 Engineering Materials and Processes ISSN 1619-0181 A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008936490 © 2009 Springer-Verlag London Limited Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: eStudio Calamar S.L., Girona, Spain Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface

Defect charging can affect numerous aspects of defect properties, including physical structure, rate of diffusion, chemical reactivity, and interactions with the electrons that give the semiconductor its overall characteristics. This book represents the first comprehensive account of the behavior of electrically charged defects in semiconductors. A comprehensive understanding of such behavior enables “defect engineering,” whereby material performance can be improved by controlling bulk and surface defect behavior. Applications are important and diverse, including fabrication of microelectronic devices, energy production from solar power, catalysis for producing chemical products, photocatalysis for environmental remediation, and solid-state sensors. The scope of this book is quite large, which helps to identify classes of behavior that are not as readily evident from an examination of defect charging in a narrower material- or application-specific context. The text summarizes current knowledge based on experiments and computations regarding defect structure, thermodynamics, and diffusion for both bulk and surfaces in an integrated way. Indeed, defect charging effects continue to be a fertile area of scientific research, with new phenomena coming to light during the past decade. Such effects include ion-induced defect formation, photostimulated surface and bulk diffusion, and electrostatically-mediated surface interactions with bulk defects. The present work outlines key aspects of these new findings. The most sophisticated forms of practical defect engineering have developed within the context of microelectronic device fabrication, particularly in silicon. Yet such engineering will almost certainly spread more broadly into other domains such as semiconductor-based sensors and solar energy devices. The present work does not attempt to review these advances in detail, but does point to more extensive reviews where they exist.

v

vi

Preface

In general, though, we hope that the scope and integration found in this book will stimulate new scientific findings and offer a new basis for new forms of defect engineering. Urbana, Illinois, USA, July 2008

Edmund G. Seebauer Meredith C. Kratzer

Acknowledgments

The authors would like to thank the following individuals: • Richard Braatz for his advice and insight pertaining to maximum likelihood approximation. • Susan Sinnott for sharing her unpublished findings regarding TiO2 defect ionization levels. • Alumni and alumnae of the Seebauer research group including Charlotte Kwok, Rama Vaidyanathan, Andrew Dalton, Kapil Dev, Mike Jung, and Ho Yeung Chan, for their intellectual contributions over the years to our knowledge of defect charging. • Patrick McSorley for his literature research and administrative assistance.

vii

Contents

1

Introduction............................................................................................. References.................................................................................................

1 3

2

Fundamentals of Defect Ionization and Transport.............................. 2.1 Introduction ................................................................................... 2.2 Thermodynamics of Defect Charging ........................................... 2.2.1 Free Energies, Ionization Levels, and Charged Defect Concentrations................................. 2.2.2 Ionization Entropy............................................................ 2.2.3 Energetics of Defect Clustering ....................................... 2.2.4 Effects of Gas Pressure on Defect Concentration ............ 2.3 Thermal Diffusion ......................................................................... 2.4 Drift in Electric Fields ................................................................... 2.5 Defect Kinetics .............................................................................. 2.5.1 Reactions.......................................................................... 2.5.2 Charging........................................................................... 2.6 Direct Surface-Bulk Coupling ....................................................... 2.7 Non-Thermally Stimulated Defect Charging and Formation ........ 2.7.1 Photostimulation .............................................................. 2.7.2 Ion-Defect Interactions .................................................... References.................................................................................................

5 5 5

3

7 13 15 17 19 24 25 25 29 31 32 32 33 34

Experimental and Computational Characterization ........................... 39 3.1 Experimental Characterization ...................................................... 39 3.1.1 Direct Detection of Bulk Defects ..................................... 39 3.1.2 Indirect Detection of Bulk Defects .................................. 43 3.1.3 Diffusion in the Bulk........................................................ 44 3.1.4 Direct Detection of Surface Defects................................. 45 3.1.5 Diffusion on the Surface .................................................. 46

ix

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Contents

3.2

Computational Prediction .............................................................. 47 3.2.1 Density Functional Theory............................................... 47 3.2.2 Other Atomistic Methods................................................. 50 3.2.3 Maximum Likelihood Estimation .................................... 51 3.2.4 Surfaces and Interfaces .................................................... 56 References................................................................................................. 56 4

Trends in Charged Defect Behavior...................................................... 4.1 Defect Formation........................................................................... 4.1.1 Effects of Crystal Structure and Atomic Properties ......... 4.1.2 Effects of Stoichiometry .................................................. 4.2 Defect Geometry ........................................................................... 4.3 Defect Charging............................................................................. 4.3.1 Bulk vs. Surface ............................................................... 4.3.2 Point Defects vs. Defect Aggregates................................ 4.4 Defect Diffusion ............................................................................ References.................................................................................................

63 63 63 66 68 69 70 71 71 72

5

Intrinsic Defects: Structure.................................................................... 5.1 Bulk Defects .................................................................................. 5.1.1 Silicon .............................................................................. 5.1.2 Germanium ...................................................................... 5.1.3 Gallium Arsenide ............................................................. 5.1.4 Other III–V Semiconductors ............................................ 5.1.5 Titanium Dioxide ............................................................. 5.1.6 Other Oxide Semiconductors ........................................... 5.2 Surface Defects.............................................................................. 5.2.1 Silicon .............................................................................. 5.2.2 Germanium ...................................................................... 5.2.3 Gallium Arsenide ............................................................. 5.2.4 Other III–V Semiconductors ............................................ 5.2.5 Titanium Dioxide ............................................................. 5.2.6 Other Oxide Semiconductors ........................................... References.................................................................................................

73 73 76 84 86 92 95 100 105 106 111 112 116 120 122 123

6

Intrinsic Defects: Ionization Thermodynamics .................................... 6.1 Bulk Defects .................................................................................. 6.1.1 Silicon .............................................................................. 6.1.2 Germanium ...................................................................... 6.1.3 Gallium Arsenide ............................................................. 6.1.4 Other III–V Semiconductors ............................................ 6.1.5 Titanium Dioxide ............................................................. 6.1.6 Other Oxide Semiconductors ...........................................

131 131 131 144 148 156 160 166

Contents

xi

6.2

Surface Defects.............................................................................. 6.2.1 Silicon .............................................................................. 6.2.2 Germanium ...................................................................... 6.2.3 Gallium Arsenide ............................................................. 6.2.4 Other III–V Semiconductors ............................................ 6.2.5 Titanium Dioxide ............................................................. 6.2.6 Other Oxide Semiconductors ........................................... References.................................................................................................

173 173 176 178 181 183 185 187

7

Intrinsic Defects: Diffusion .................................................................... 7.1 Bulk Defects .................................................................................. 7.1.1 Point Defects.................................................................... 7.1.2 Associates and Clusters.................................................... 7.2 Surface Defects.............................................................................. 7.2.1 Point Defects.................................................................... 7.2.2 Associates and Clusters.................................................... 7.3 Photostimulated Diffusion............................................................. 7.3.1 Photostimulated Diffusion in the Bulk............................. 7.3.2 Photostimulated Diffusion on the Surface ....................... References.................................................................................................

195 195 196 212 215 215 222 222 223 225 226

8

Extrinsic Defects ..................................................................................... 8.1 Bulk Defects .................................................................................. 8.1.1 Silicon .............................................................................. 8.1.2 Germanium ...................................................................... 8.1.3 Gallium Arsenide ............................................................. 8.1.4 Other III–V Semiconductors ............................................ 8.1.5 Titanium Dioxide ............................................................. 8.1.6 Other Oxide Semiconductors ........................................... 8.2 Surface Defects.............................................................................. 8.2.1 Silicon .............................................................................. 8.2.2 Gallium Arsenide ............................................................. 8.2.3 Titanium Dioxide ............................................................. References.................................................................................................

233 233 234 249 255 260 265 271 277 278 280 281 281

Index ................................................................................................................. 291

List of Abbreviations

AIMPRO APF BOV CBM CDB DFT DLTS EELS ENDOR EPR FC FE FIM GGA HR IETS KLMC KPFM LDA LDLTS LEED LMCC LSDA ML ODEPR OED PACS PAS PBE POV

Ab initio modeling program Atomic packing fraction Bridging oxygen vacancy Conduction band minimum Coincidence Doppler broadening Density functional theory Deep level transient spectroscopy Electron energy loss spectroscopy Electron-nuclear double resonance Electron paramagnetic resonance Faulted corner Faulted edge Field ion microscopy Generalized gradient approximation Hartree–Fock Inelastic electron tunneling spectroscopy Kinetic lattice Monte Carlo Kelvin probe force microscopy Local density approximation Laplace deep level transient spectroscopy Low energy electron diffraction Local moment countercharge Local spin density approximation Maximum likelihood Optically detected electron paramagnetic resonance Oxidation enhanced diffusion Perturbed angular correlation spectroscopy Positron annihilation lifetime spectroscopy Perdew–Burke–Ernzerhof In-plane oxygen vacancy xiii

xiv

PR QMC RAS RDS RHEED SDRS SHM SRDLTS SRH STM TB TED TEM UFC UFE VBM VEPAS XAFS

List of Abbreviations

Photoreflectance spectroscopy Quantum Monte Carlo Reflectance anisotropy spectroscopy Reflectance difference spectroscopy Reflection high-energy electron diffraction Surface differential reflectance spectroscopy Second harmonic microscopy Synchrotron radiation deep level transient spectroscopy Shockley–Read–Hall Scanning tunneling microscopy Tight-binding Transient enhanced diffusion Transmission electron microscopy Unfaulted corner Unfaulted edge Valence band maximum Variable-energy positron annihilation spectroscopy X-ray absorption fine structure

Chapter 1

Introduction

The technologically useful properties of a semiconductor often depend upon the types and concentrations of the defects it contains. For example, defects such as vacancies and interstitial atoms mediate dopant diffusion in microelectronic devices (Hu 1994; Bracht 2000; Dasgupta and Dasgupta 2004; Jung et al. 2005; Fahey et al. 1989). Such devices would be nearly impossible to fabricate without the diffusion of these atoms. In other applications, defects also affect the performance of photo-active devices (Guha et al. 1993; Chow and Koch 1999; Lutz 1999) and sensors (Fergus 2003), the effectiveness of oxide catalysts (Zhang et al. 2004; Baiqi et al. 2006), and the efficiency of devices for converting sunlight to electrical power (Green 1996; Kurtz et al. 1999). To improve material performance, various forms of “defect engineering” have been developed to control defect behavior within the solid (Jones and Ishida 1998), particularly for applications in microelectronics. Examples include surface oxidation (Cohen et al. 1998), various protocols for ion implantation and annealing (Townsend et al. 1994; Pearton et al. 1993; Wang et al. 1996; Roth et al. 1997; Williams 1998) and the incorporation of impurity atoms (Pizzini et al. 1997). Crystalline surfaces support native defects in the same way that the bulk solid does (Wilks 2002), with many close analogies between the two cases. Understanding surface defects is becoming increasingly important in practical applications – for example, as electronic devices shrink closer to the atomic scale (with the attendant increase in surface-to-volume ratios), and as molecular-level control of catalytic reactions becomes increasingly feasible. Of particular importance is defect-mediated surface diffusion, which plays an important role in crystal growth, heterogeneous catalysis, sintering, corrosion, and microelectronics fabrication. Considerably less is known about the behavior of surface defects than bulk defects. (Even less is known about defects at solid–solid interfaces, but some analogies with the bulk and free surface still hold.) Recent research has also indicated that surfaces or interfaces can directly influence point defect behavior in the bulk (Dev et al. 2003; Seebauer et al. 2006), and that bulk properties can couple directly into the behavior of surface defects (Ditchfield et al. 1998, 2000). E.G. Seebauer, M.C. Kratzer, Charged Semiconductor Defects, © Springer 2009

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1 Introduction

It has long been known that bulk defects in semiconductors can be electrically charged. Charging of surface defects has been identified and studied rather more recently. In either case, this charging can affect defect structure (Centoni et al. 2005; Chan et al. 2003), thermal diffusion rates (Allen et al. 1996; Lee et al. 1998; Tersoff 1990), trapping rates of electrons and holes (Mascher et al. 1989; Puska et al. 1990), and luminescence quenching rates (Tasker and Stoneham 1977). More interestingly, defect charging also introduces new phenomena such as nonthermally photostimulated diffusion (Ditchfield et al. 1998, 2000; Seebauer 2004). The use of a chemically active surface that can selectively remove self-interstitials over dopant interstitials simultaneously improves profile spreading and sheet resistance in ultrashallow junctions. Such phenomena offer completely new mechanisms for defect engineering, as well as new means to study the charging phenomenon itself. Semiconductors contain not only native atomic defects, but also defects that arise from the incorporation of foreign atoms into the crystal lattice. Although numerous review articles and books have been published on the general subject of semiconductor defect structure and behavior for both the bulk (Jarzebski 1973; Nishizawa and Oyama 1994; Stoneham 1979; Hu 1994; Fahey et al. 1989; Sinno et al. 2000; Pichler 2004; Cohen 1996; Smyth 2000; Kosuge 1994) and the surface (Henrich 1994; Ebert 2001), a comprehensive treatment of semiconductor defect charging is lacking. Correspondences and contrasts in charging behavior on surfaces and in the bulk have not been clearly delineated. The same lacuna exists for the various semiconductor types (Group IV, Group III–V, and oxide semiconductors). The present work fills those gaps based on currently available literature, and in so doing, identifies broad trends in behavior, some of which do not appear to have been identified before. Crystal properties such as atomic packing fraction (which depends on unit cell type, size, and ionicity of the constituent atoms) and mismatches in the radii of basis atoms of compound and oxide semiconductors inhibit the formation of certain types of defects. When comparing the magnitude and direction of bulk defect-induced relaxations, trends related to electron-lattice interaction and ionicity are observed. For a given material, surface defects do not typically take on the same configurations or range of stable charge states as their counterparts in the bulk. Similarly, only modest correspondence exists between the stable charge states of isolated point defects and the corresponding defect associates. At a given Fermi energy, the charge state of a defect associate does not necessarily equal the sum of the charges of the constituent defects. Although the formation energies, symmetry-lowering relaxations, and diffusion mechanisms of bulk and surface defect structures often depend strongly on charge state, typically those effects cannot be predicted a priori. Available literature delimits the focus of this book primarily to elemental, III–V compound, and oxide semiconductors. There exists very little literature regarding defect ionization in other important classes of semiconductors such as II–VI (e.g., CdSe) and ternaries (e.g., Hg1–xCdxTe). The notation for describing point defects varies widely through the literature. For example, most literature for oxide semiconductors uses “Kröger–Vink” notation to

References

3

represent charged crystal defects (Kröger and Vink 1958). The literature for silicon and III–V defects employs a substantially different notation. To foster a uniform treatment, this book will employ a single notation for all types of charged defects in all types of materials.

References Allen CE, Ditchfield R, Seebauer EG (1996) J Vac Sci Technol, A 14: 22–29 Baiqi W, Liqiang J, Yichun Q et al. (2006) Appl Surf Sci 252: 2817–2825 Bracht H (2000) MRS Bull 25: 22–27 Centoni SA, Sadigh B, Gilmer GH et al. (2005) Phys Rev B: Condens Matter 72: 195206 Chan HYH, Dev K, Seebauer EG (2003) Phys Rev B: Condens Matter 67: 035311 Chow WW, Koch SW (1999) Semiconductor-Laser Fundamentals: Physics of the Gain Materials, Berlin, Springer Cohen RM (1996) Diffusion and native defects in GaAs. In: 1996 Conference on Optoelectronic and Microelectronic Materials and Devices. Proceedings (Cat. No.96TH8197) 107–13 (IEEE, Canberra, Australia, 1996) Cohen RM, Li G, Jagadish C et al. (1998) Appl Phys Lett 73: 803–805 Dasgupta N, Dasgupta A (2004) Semiconductor Devices: Modeling and Technology, New Delhi, Prentice-Hall Dev K, Jung MYL, Gunawan R et al. (2003) Phys Rev B: Condens Matter 68: 195311 Ditchfield R, Llera-Rodriguez D, Seebauer EG (1998) Phys Rev Lett 81: 1259–1262 Ditchfield R, Llera-Rodriguez D, Seebauer EG (2000) Phys Rev B: Condens Matter 61: 13710– 13720 Ebert P (2001) Curr Opin Solid State Mater Sci 5: 211–50 Fahey PM, Griffin PB, Plummer JD (1989) Rev Modern Phys 61: 289–384 Fergus JW (2003) J Mater Sci 38: 4259–4270 Green MA (1996) High efficiency silicon solar cells. In: 1996 Conference on Optoelectronic and Microelectronic Materials and Devices. Proceedings (Cat. No.96TH8197) 1–7 (IEEE, Canberra, Australia, 1996) Guha S, Depuydt JM, Haase MA et al. (1993) Appl Phys Lett 63: 3107–3109 Henrich VE (1994) The Surface Science of Metal Oxides, Cambridge, Cambridge University Press Hu SM (1994) Mater Sci Eng, R 13: 105–92 Jarzebski ZM (1973) Oxide Semiconductors, New York, Pergamon Press Jones EC, Ishida E (1998) Mater Sci Eng, R 24: 1–80 Jung MYL, Kwok CTM, Braatz RD et al. (2005) J Appl Phys 97: 063520 Kosuge K (1994) Chemistry of Non-Stoichiometric Compounds, New York, Oxford Science Publications Kröger FA, Vink HJ (1958) J Phys Chem Solids 5: 208–223 Kurtz SR, Allerman AA, Jones ED et al. (1999) Appl Phys Lett 74: 729–731 Lee WC, Lee SG, Chang KJ (1998) J Phys: Condens Matter 10: 995–1002 Lutz G (1999) Semiconductor Radiation Detectors, Berlin, Springer Mascher P, Dannefaer S, Kerr D (1989) Phys Rev B: Condens Matter 40: 11764–11771 Nishizawa J, Oyama Y (1994) Mater Sci Eng, R 12: 273–426 Pearton SJ, Ren F, Chu SNG et al. (1993) Nucl Instrum Methods Phys Res, Sect B 79: 648–650 Pichler P (2004) Intrinsic Point Defects, Impurities, and their Diffusion in Silicon, New York, Springer-Verlag/Wein Pizzini S, Acciarri M, Binetti S et al. (1997) Mater Sci Eng, B 45: 126–133 Puska MJ, Corbel C, Nieminen RM (1990) Phys Rev B: Condens Matter 41: 9980–9993 Roth EG, Holland OW, Venezia VC et al. (1997) J Electron Mater 26: 1349–1354

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Seebauer EG (2004) New mechanisms governing diffusion in silicon for transistor manufacture. In: International Conference on Solid-State and Integrated Circuits Technology Proceedings, ICSICT 2:1032–1037 (IEEE, Beijing, China, 2004) Seebauer EG, Dev K, Jung MYL et al. (2006) Phys Rev Lett 97: 055503 Sinno T, Dornberger E, von Ammon W et al. (2000) Mater Sci Eng, R 28: 149–198 Smyth DM (2000) The Defect Chemistry of Metal Oxides, New York, Oxford University Press Stoneham AM (1979) Adv Phys 28: 457–92 Tasker PW, Stoneham AM (1977) J Phys C: Solid State Phys 10: 5131–40 Tersoff J (1990) Phys Rev Lett 65: 887–890 Townsend PD, Chandler PJ, Zhang L (1994) Optical Effects of Ion Implantation, Cambridge, Cambridge University Press Wang ZL, Zhao QT, Wang KM et al. (1996) Nucl Instrum Methods Phys Res, Sect B 115: 421–429 Wilks SP (2002) J Phys D: Appl Phys 35: R77–R90 Williams JS (1998) Mater Sci Eng, A 253: 8–15 Zhang Y, Kolmakov A, Chretien S et al. (2004) Nano Lett 4: 403–407

Chapter 2

Fundamentals of Defect Ionization and Transport

2.1 Introduction Native atomic defects include vacancies, interstitials, and antisite defects. Antisite defects, which consist of atoms residing in improper lattice sites, are relevant only for binary compounds such as III–V or oxide semiconductors. One such example is a gallium atom occupying an arsenic atom lattice site, denoted as GaAs, rather that its proper gallium atom lattice site. Defect clusters or complexes are formed when two or more of the atomic defects mentioned above join together. Examples of clusters include divacancies, trivacancies, di-interstitials, vacancy-interstitial pairs, etc. Clusters on the surface may be referred to as vacancy or adatom islands. The basic defect thermodynamics are the same for the bulk and surface. For an explicit discussion of the correspondence in defect structure and behavior between the two, the reader should refer to Table 5.2 in Chap. 5. In addition to native or intrinsic defects, extrinsic defects may also exist in the crystal lattice. These defects formed either intentionally (via doping or ion implantation, for instance) or accidentally by the introduction of foreign atoms into the semiconductor. In boron-doped silicon, for example, the two most likely extrinsic defects are boron in a silicon lattice site, donated as BSi, and boron in an interstitial location, Bi.

2.2 Thermodynamics of Defect Charging The thermodynamics of defect charging have been discussed in numerous journal articles and books (Van Vechten 1980; Van Vechten and Thurmond 1976b, a; Fahey et al. 1989; Pichler 2004; Jarzebski 1973). Note that the thermodynamic parameters, including band gaps, ionization energies, and energies of defect formation and/or migration, are not the eigenvalues of a Schrodinger equation describing the crystal (Van Vechten 1980). The thermodynamic parameters are defined statistically in terms of reactions occurring among ensembles of all possible configurations of the E.G. Seebauer, M.C. Kratzer, Charged Semiconductor Defects, © Springer 2009

5

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2 Fundamentals of Defect Ionization and Transport

system. Confusion over this distinction sometimes exists particularly with reference to ionization levels. When thermally generated or artificial point defects are introduced into a perfect semiconductor crystal, they increase the Gibbs free energy G of the system. The equilibrium concentration [X] of a neutral point defect X0 can be expressed as f ⎡X 0 ⎤ ⎣ ⎦ = θ 0 exp ⎡ −GX 0 ⎢ X [S ] ⎢⎣ kT

⎤ ⎡S f 0 ⎥ = θ X 0 exp ⎢ X ⎥⎦ ⎢⎣ k

⎤ ⎡ −H f 0 X ⎥ exp ⎢ ⎥⎦ ⎢⎣ kT

⎤ ⎥ ⎥⎦

(2.1)

where [S] is the concentration of available lattice sites in the crystal, θ X 0 is the number of degrees of internal freedom of the defect on a lattice site, and G Xf 0, H Xf 0 , and S Xf 0 are respectively the standard Gibbs free energy, enthalpy, and entropy of neutral defect formation (Fahey et al. 1989; Bourgoin and Lannoo 1981; Swalin 1962). The parameters k and T respectively represent Boltzmann’s constant and temperature. A defect may have several degrees of freedom due to spin degeneracy or equivalent geometric configurations at the same site (Pichler 2004). Typically only the spin degeneracy is of direct interest for defect charging. For simplicity, therefore, the discussion henceforth will focus upon the spin-degeneracy g rather than other degrees of internal freedom of the defect. In the case that two identical defects bind together to form a defect pair or complex, the concentration of the combined defect X2 is given by

[ X2 ] = θX

[ X ][ X ] exp ⎡ E Xb ⎢ [S ] ⎢⎣ kT

2

2

⎤ ⎥ ⎥⎦

(2.2)

where E Xb 2 denotes the binding energy of the X2 defect, and the degeneracy factor θ X 2 equals the number of equivalent ways of forming the X2 defect at a particular site (Fahey et al. 1989). The thermodynamics of defect clustering will be discussed in greater detail in Sect. 2.1.3. For oxide semiconductors, which typically exhibit small deviations from stoichiometry on the order of a few parts per thousand, it is possible to rewrite Eq. 2.1 to explicitly reflect the dependence of [X] upon the ambient oxygen pressure PO2. However, it becomes necessary to incorporate several additional variables including [MM], the concentration of metal in the metal sublattice, and [OO], the concentration of oxygen in the oxygen sublattice (Jarzebski 1973). It then follows that the concentration of vacancies in the metal sublattice is given by the mass-action expression

[VM ] [M M ]

⎡ ΔS ⎤ ⎡ Δ H1 ⎤ = PM −α exp ⎢ 1 ⎥ exp ⎢ − ⎥ ⎣ k ⎦ ⎣ kT ⎦

(2.3)

and that of vacancies in the oxygen sublattice by

[VO ] = P −α exp ⎡ ΔS2 − ΔS1 ⎤ exp ⎡ − ΔH 2 − ΔH1 ⎤

[OO ]

O2

⎢ ⎣

k

⎥ ⎦

⎢ ⎣

kT

⎥ ⎦

(2.4)

2.2 Thermodynamics of Defect Charging

7

where the constant α derives from the ratio of oxygen to metal in the MO or MO2 crystal (Jarzebski 1973). For instance, α theoretically equals ½ for a perfectly stoichiometric oxide semiconductor of type MO. Generally α takes the form α = 1/n, where n is an integer. n must always be an integer in order to preserve bulk charge neutrality. Values for n ranging anywhere from 2–8 appear in the literature. As the stoichiometry of most oxide semiconductors is highly temperature dependent, the empirical values of n are typically determined from temperature-dependent electrical conductivity measurements. ΔS1 and ΔS2 may contain contributions from the vibrational entropy of the crystal resulting from the addition of VM, VO, and extra oxygen atoms, as well as the standard Gibbs free entropy of the oxygen molecule in the gas phase ΔSOD 2 . The ΔH parameters contain the enthalpies associated with the same defect processes (Jarzebski 1973). For neutral defects, the equilibrium concentration of point defects does not depend upon the value of the chemical potential (or more colloquially, “Fermi energy” EF, even for T > 0 K) in the bulk. This is not the case for charged defects.

2.2.1 Free Energies, Ionization Levels, and Charged Defect Concentrations Neutral defects almost always have unsaturated bonding capabilities (e.g., dangling bonds). These capabilities facilitate the transfer of electronic charge between the host matrix and the defect, and often occur to the point that the defect becomes fully ionized. The degree and direction of electron transfer (toward or away from the defect, respectively, for acceptors and donors) naturally depend upon the electron richness of the host, as quantified by the host’s Fermi energy (i.e., chemical potential) in the vicinity of the defect. In semiconductors, the host’s electron richness can be adjusted readily by doping, imposed electric fields, photostimulation, and other factors. Thus, the ionization state of the defect can often be controlled. If the defect possesses significant capacity to store excess charge within its structure, the range of ionization states can be quite large. For example, a monovacancy in silicon nominally incorporates four unsaturated dangling bonds, and permits charge states ranging from (–2) to (+2) (Fahey et al. 1989; Schultz 2006). Some defects have eigenstates close to the edges of the valence band or conduction band; these states can be described by a hydrogenic model with a ground state and a series of bound excited states described by hydrogen atom wavefunctions, with full ionization occurring into the energy continuum of the valence or conduction band. This simple picture must, of course, be modified to account for the interactions of the electrons and holes with the lattice, which alters their effective mass. Also, the crystal reduces the binding potential, which incorporates a dielectric constant (Queisser and Haller 1998). For defects having eigenstates deeper within the band gap of the semiconductor, a more detailed quantum mechanical treatment is needed.

8

2 Fundamentals of Defect Ionization and Transport

For many purposes, the concentration of defects in a given charge state must be known. This concentration requires use of Fermi statistics, whose application to semiconductors is reviewed briefly here. Electrons in solids obey Fermi–Dirac statistics, for which the distribution of electrons over a range of allowed energy levels at thermal equilibrium is f (E) =

1 1+ e

( E − EF ) / kT

(2.5)

where k is again Boltzmann’s constant, f (E) the probability that an available energy state at E will be occupied by an electron at absolute temperature T. The physical interpretation of the chemical potential EF is that the probability of electron occupation is exactly 0.5 in an energy state lying at EF. In the limit of zero temperature, the chemical potential equals the Fermi energy. Although the Fermi energy is a concept that has formal meaning only in this limit, colloquial terminology commonly uses “chemical potential” and “Fermi energy” interchangeably at all temperatures, and the present treatment will follow that practice. In an ideal intrinsic (undoped) semiconductor, the Fermi energy EF takes the value EF =

EC + EV kT NV + ln NC 2 2

(2.6)

where EC is the energy at the bottom of the conduction band, EV is the energy at the top of the valence band, and NV (NC) is the effective density of states in the valence (conduction) band. For intrinsic material, the Fermi level lies approximately in the middle of the band gap. The product of the two charge-carrier concentrations is independent of the Fermi level and obeys ni2 = pi2 = n ⋅ p

(2.7)

where ni ( pi) is the intrinsic concentration of electrons (holes). Clearly, in undoped material, the concentrations of electrons and holes are equal. For reference, the intrinsic concentration for Si at room temperature is approximately 1.5 × 1010 cm–3. In doped material, the electron and hole concentrations are no longer identical. Boltzmann statistics can be used under most conditions to approximate Fermi statistics and obtain a probability that a state is occupied by an electron. The electron and hole concentrations can also be approximated by: ⎛ E − EC ⎞ n = N C exp ⎜ F ⎟ ⎝ kT ⎠

(2.8)

⎛ E − EF ⎞ p = NV exp ⎜ V ⎟ ⎝ kT ⎠

(2.9)

and

2.2 Thermodynamics of Defect Charging

9

with parameters identical to those in Eq. 2.6. When n and p are varied by doping, the Fermi level either rises toward the conduction band (made more n-type) or falls toward the valence band (made more p-type). This variation in Fermi energy must be taken into account when calculating the concentration of charged defects in the bulk. Fermi–Dirac statistics apply to the calculation of charged defect concentrations as follows. Take, for instance, the ionization of an acceptor defect X to X−1, which can be represented by the reaction:

X 0 ↔ X −1 + h+1 .

(2.10)

Equation 2.10 is equally valid for point defects such as vacancies and selfinterstitials as it is for divacancies and substitutional extrinsic defects. The law of mass action implies that ⎡ X −1 ⎤ = ⎣ ⎦

(

1+ g X

−1

)

[X ] ⎡ E −1 − EF exp ⎢ X kT ⎣

⎤ ⎥ ⎦

,

(2.11)

where [X] is the concentration of the defect in all charge states, g is an overall degeneracy factor, and E X −1 is the ionization level for the singly ionized acceptor. This expression can be simplified when │ E X −1 – EF│>> kT. Also, in the case that defect X has only two charge states, g is simply the ratio of the degeneracy of X−1 to that of X0, as shown in Eq. 2.12: ⎡ X −1 ⎤ θ −1 ⎣ ⎦ = X exp ⎡ EF − E X −1 ⎤ , ⎢ ⎥ 0 kT ⎡X ⎤ θX0 ⎣ ⎦ ⎣ ⎦

(2.12)

where θ X −1 and θ X 0 respectively denote degeneracy factors for X−1 and X0. In the same way, the single ionization of a donor defect can be represented by the reaction

X 0 ↔ X +1 + e−1,

(2.13)

+1

where the concentration of X can be determined from Eq. 2.14: ⎡ X +1 ⎤ = ⎣ ⎦

(

1+ g X

+1

)

[X ] ⎡ E − E X +1 ⎤ exp ⎢ F ⎥ kT ⎣ ⎦

(2.14)

or Eq. 2.15, when │EF – E X +1│>> kT: ⎡ X +1 ⎤ θ +1 ⎣ ⎦ = X exp ⎡ E X +1 − EF ⎢ 0 kT ⎡X ⎤ θX0 ⎣ ⎣ ⎦

⎤ ⎥. ⎦

(2.15)

The ionization levels in Eqs. 2.11 and 2.14 do not represent the eigenvalues of a Schroedinger equation, but rather thermodynamic quantities based on occupation

10

2 Fundamentals of Defect Ionization and Transport

statistics. In particular, the ionization level equals the value of the Fermi energy at which the concentrations of the two charge states are identical (to within a degeneracy factor). For example, if θ X −1 = θ X 0 in Eq. 2.12, then [X–1] = [X0] when EF = E X −1. The degeneracy factors in Eqs. 2.11 and 2.14 are usually concerned with differences in net electron spin among the charge states. For both acceptor and donor defects, the value of the overall degeneracy factor g can be deduced by applying the principle of equal occupation of states when EF is equal to the ionization level under consideration. As an example, neutral vacancy defects have no spin degeneracy, as they have no bound carriers. However, if one additional singly charged state exists (either X+1 or X−1), that singly charged state is twofold spin degenerate with electron spins that can be either up or down. Thus, for the specific case of a positive vacancy, we must have [V+1] = 2[V0] or alternatively [V+1] = 2/3 [V0], so g(V+1) = ½. The same argument gives g(V−1) = ½. Analogs of Eqs. 2.12 and 2.15 for charge states of two or higher can be constructed by induction from the single charge states. As an example, the concentration of the multiply charged acceptor X−2 with ionization level E X −2 is: ⎡ X −2 ⎤ θ −2 ⎣ ⎦ = X exp ⎡ − E X −2 + E X −1 − 2 EF ⎢ 0 kT ⎡ X ⎤ θX0 ⎣ ⎣ ⎦

⎤ ⎥, ⎦

(2.16)

⎡ X +2 ⎤ θ +2 ⎣ ⎦ = X exp ⎡ − 2 EF − E X +2 − E X +1 ⎤ . ⎢ ⎥ kT ⎡ X 0 ⎤ θX0 ⎣ ⎦ ⎣ ⎦

(2.17)

while that of the doubly ionized donor X+2 is:

Clearly the charged defect concentrations vary with T. Figure 2.1 shows the concentration of charged vacancies in silicon at 300 and 1,400 K as determined by Van Vechten and Thurmond (1976b). The fact that the concentration of a charged defect depends upon its charge state and the position of the Fermi energy implies related dependencies in the defect’s formation energy. After all, there is work involved in moving an electron from the Fermi energy into the energy state associated with the defect. At first glance, the formation of X−1 can be written as G Xf −1 = G Xf 0 + E X −1

(2.18)

where G Xf 0 = H Xf 0 − TS Xf 0 (Fahey et al. 1989). However, this expression neglects the fact that for each ionized defect, an appropriate number of charge carriers are generated. Thus, it is more accurate to generalize the formation energy of the charged defect to GXf q + e− q = G Xf q − qEF ,

(2.19)

2.2 Thermodynamics of Defect Charging

11

Fig. 2.1 Variation with EF of the concentration of various vacancy charge states in silicon relative to the neutral. The majority species change with temperature. For example, the neutral state exists at 300 K for EF between Ev + 0.14 eV (ionization level for (+2/0)) and Ev + 0.35 eV (ionization level for (0/−1)). However, at 1,400 K only the neutral vacancy is never the majority charge state. Note that a smaller range of EF is shown for 1,400 K than for 300 K because of band gap narrowing with increasing temperature. Reprinted figure with permission from Van Vechten, JA (1986) Phys Rev B: Condens Matter 33: 2678. Copyright (1986) by the American Physical Society.

where q is the charge state of defect X and EF is the Fermi energy. When considering a surface defect as opposed to a bulk defect, all the same basic principles apply except that the value of the Fermi energy at the surface (which often differs from that in the bulk) determines the concentrations of various ionization states. For practical purposes, it is often more useful to examine the work (or Gibbs free energy) associated with ionizing the defect. For the case of X−1 the change in free energy associated with ionization, Eq. 2.19, can be rearranged and combined with Eq. 2.18 to yield ΔG Xf −1 = G Xf −1 − G Xf 0 = E X −1 − EF .

(2.20)

The origin of Eq. 2.11 should now be explicitly clear. The corresponding free energy of ionization for the doubly ionized acceptor and singly ionized donor are given by ΔG Xf −2 = G Xf −2 − G Xf 0 = E X −2 + E X −1 − 2 EF

(2.21)

ΔG Xf +1 = G Xf +1 − G Xf 0 = EF − E X +1 .

(2.22)

and

12

2 Fundamentals of Defect Ionization and Transport

These free energies of defect ionization can be decomposed into corresponding enthalpies and entropies of ionization, ΔH Xf q and ΔS Xf q : ΔGXf q = ΔH Xf q − T ΔS Xf q .

(2.23)

Note that ΔG Xf q , ΔH Xf q , and ΔS Xf q all depend on temperature. The enthalpy of ionization is strongly affected by the degree of localization of the remaining bound carrier of the ionized state. A greater value of ΔH Xf q corresponds to an ionization level deeper within the band gap and a remaining carrier that is more loosely bound to the defect center. The value of ΔH Xf q at non-zero temperatures can be obtained from an empirical expression due to Varshni (1967) for the band gap energy Eg (equivalent to the free energy of electron-hole pair formation): E g (T ) = E g ( 0 ) −

αT 2 (T + β )

(2.24)

where α and β are empirical constants. Since Eg is the increase in free energy, ΔGcv, when an electron-hole pair is created, its temperature derivative is the negative standard entropy of that reaction (Van Vechten 1980), ∂Eg ≡ −ΔScv . ∂T

(2.25)

Then the definition ΔG = ΔH – TΔS implies ΔH cv (T ) = Eg (T ) − T

∂Eg (T ) ∂T

(2.26)

where ΔHcv is the enthalpy of electron-hole pair formation. Substitution of the derivative of Eq. 2.24 into the expression above yields the following empirical expression for enthalpy of electron-hole pair formation at non-zero temperatures: ΔH cv (T ) = E g ( 0 ) +

αβ T 2

(T + β )2

.

(2.27)

As an example, for Si the relevant constants are α = 0.000473 eV/K, Eg(0) = 1.17 eV and β = 636 K (Thurmond 1975). The enthalpy of ionization obtained from Eq. 2.25, when combined with ΔH Xf q at T = 0 K as deduced from experiment or DFT calculations (as ΔGcv at 0 K equals ΔHcv) (Dev and Seebauer 2003), is then used to describe the variation in enthalpy as a function of charge state according to ΔH Xf q (T ) = ΔH Xf q ( 0 ) + ΔH cv (T ) .

(2.28)

The enthalpy of ionization at 0 K, ΔH Xf q ( 0 ), is charge state-dependent, thus the enthalpies of multiply charged defects phenomenologically track with each other as a function of temperature, yet have different maxima and minima, as shown in Fig. 2.2.

2.2 Thermodynamics of Defect Charging

13

Fig. 2.2 Variation of the enthalpies of silicon vacancy ionization levels (and of the band gap) as a function of temperature. Reprinted figure with permission from Van Vechten, JA (1986) Phys Rev B: Condens Matter 33: 2677. Copyright (1986) by the American Physical Society.

2.2.2 Ionization Entropy Formation entropies for defects can contain several contributions, including configurational degeneracy, lattice mode softening due to bond cleavage, and ionization (Van Vechten and Thurmond 1976b, a). Our principal concern here is the ionization contribution, which helps govern charge-mediated effects. There exists significant theoretical and experimental evidence to suggest that the ionization entropy ΔS Xf q can be very large for certain kinds of native defects such as vacancies. The main contribution to ΔS Xf q originates from electron-phonon coupling near the vacancy, leading to lattice-mode softening (Van Vechten and Thurmond 1976a; Dev and Seebauer 2003). The magnitude can be calculated by considering either the effect of thermal vibrations upon the electronic defect levels or the effect of the thermally excited electronic states upon the lattice vibration mode frequencies (Van Vechten 1980), although the latter method has proven more useful for simple estimates (Van Vechten and Thurmond 1976a). In this perspective, the band gap energy Eg of a bulk semiconductor crystal corresponds to the standard chemical potential for creating a delocalized hole at the valence band maximum and a delocalized electron at the conduction band minimum.

14

2 Fundamentals of Defect Ionization and Transport

Such creation might occur thermally or by photoexcitation. The magnitude of Eg can be obtained from the empirical Varshni relation given in Eq. 2.24. Standard thermodynamic relations require that the entropy change Eg for formation of the electron-hole pair obeys (Thurmond 1975): ΔScv (T ) = −

∂ΔEcv α T (T + 2 β ) . = ∂T (T + β )2

(2.29)

Ionization of a defect represents another mechanism for creating two new carriers of opposite charge. One of the carriers roams the crystal in a delocalized way, while the other remains bound in the vicinity of the defect. The delocalized carrier contributes to ΔScv the way any delocalized carrier would. The effect of the bound carrier depends upon its degree of localization, however. If that carrier is loosely bound to the defect and therefore largely delocalized, the entropy for the ionization event clearly matches ΔScv. If the carrier is tightly bound to the defect, however, and hovers close to it, the contribution to ΔScv is more difficult to estimate a priori. To make such an estimate, Van Vechten and Thurmond examined experimental data for the entropies of optical transitions in Si, Ge, GaAs and GaP between various points in the Brillouin zone. These data were derived from the temperature dependence of the various gaps as determined by optical reflectance. For Si the reported entropies suffered considerable uncertainties, but values remained within a factor or two of ΔScv. Since that compilation, more data have become available for Si that confirm the early results, including data for the E2 and E0′ direct gaps up to 1,000 K (Jellison and Modine 1983) and for the E2, E0′, E1 and E1′ critical points up to 600 K (Lautenschlager et al. 1987). The optical results indicate that, at least for the four semiconductors examined, mode-softening effects from e––h+ pair formation are insensitive to the final state charge distribution, so that, like the case of charges loosely bound to the defect, ΔS Xf q (T ) ≈ ΔScv (T )

(2.30)

for single ionization events regardless of whether ionization results in a positive or negative vacancies (Van Vechten and Thurmond 1976a). Note that this argument should apply quite directly to the surface as well as the bulk, since the reflectance data on which the argument rests are sensitive primarily to surface optical susceptibilities. (Linear optical susceptibilities typically lie close to those of the bulk in any case.) Unlike the argument used for loosely bound carriers, however, Eq. 2.30 depends on data only for specific semiconductors – data that verify the conclusion only approximately. These arguments suggest that ΔScv(T) can be used to estimate ΔS Xf q (T ) regardless of the degree of localization of the bound charge. However, the reliability of the estimate does depend upon the degree of localization, which fortunately can be obtained with ease from DFT calculations. A consequence of the correspondence between ΔScv(T) and ΔS Xf q (T ) is that, as T increases and Eg decreases, free energies referenced to the valence band maximum

2.2 Thermodynamics of Defect Charging

15

Fig. 2.3 (a) Formation energies of various dimer vacancy charge states on Si(100)–(2×1) as a function of Fermi energy at 0 K. The formation energy is referenced to the neutral dimer vacancy and the Fermi energy is referenced to the valence band maximum. The charge state with the lowest formation energy at a given Fermi energy has the highest concentration. (b) Variation of the dimer vacancy ionization levels with temperature.

for vacancy ionization levels remain at a constant energy below the conduction band for negatively charged vacancies and remain a constant energy above the valence band for positively charged (Van Vechten and Thurmond 1976a). This consequence makes the ionization levels quite easy to calculate from DFT results. An example for the divacancy on the Si(100) surface is shown in Fig. 2.3.

2.2.3 Energetics of Defect Clustering It is important to remember that the enthalpy of formation need not refer simply to the enthalpy of formation of a point defect such as a vacancy or interstitial. An expression must also exist to describe the enthalpy of defect cluster formation. The term “cluster” encompasses a wide variety of defects including the divacancy, diinterstitial, vacancy-dopant pair, etc. Numerous methods and approximations for calculating the formation enthalpy of a defect pair exist in the literature; this section will summarize the primary approaches and discuss their validity. Consider a pair formed from two identical charged defects Xq and Xq according to the fairly simple reaction Xq + Xq → (XX)2q.

(2.31)

Associated with this reaction is an enthalpy of pair formation or “binding energy.” For simplicity, the following discussion will distinguish between two components of the binding energy of (XX)2q. The Coulombic interaction and the shortrange “chemical” interaction between defects Xq and Xq sum to yield the binding energy of the pair:

(

)

(

)

ΔH b XX 2 q = ΔH b ,Coulombic XX 2 q + Φ X q X q .

(2.32)

16

2 Fundamentals of Defect Ionization and Transport

This treatment applies to both bulk and surface clusters. For example, Kudriavtsev et al. have calculated surface binding energies with a similar model that takes into account both covalent and ionic contributions (2005). A first-order approximation of the binding energy ΔHb of the pair is obtained from the Coulomb interaction energy of the two defects as estimated in the fully screened, point charge approximation. The fully screened, point charge approximation is a reasonable estimate of the binding energy as long as the eigenstates of the two defects reside within the band gap and are, therefore, localized electronic states (Dobson and Wager 1989). In reality, only a fraction α of the Coulombic energy contributes to ΔHb according to

(

)

ΔH b,Coulombic XX 2 q =

q 2 e 2α , 4πε 0ε r r

(2.33)

where q is the integral value of the charge on each defect (i.e., (+1), (−1), etc.), e is the unit electronic charge, r is the equilibrium nearest neighbor separation distance, ε0 is the permittivity of free space, and εr is the relative dielectric constant of the material in question. A negative value of binding energy indicates that defect clustering is energetically favorable. The fraction α corresponds to the amount of association energy it takes to push the defect ionization levels out of the band gap. Notice that the Coulomb energy between the two charges must be modified to account for the consequent polarization of the surrounding ions in the lattice. For some defect complexes, this can be accounted for with the static dielectric constant of the semiconductor, εr, which is a measure of the polarizability of the lattice. In other instances, especially when defects are situated on adjacent lattice sites, the continuum quantity εr does not sufficiently account for the local effects of lattice polarization. In such cases, the Coulombic binding energy is typically lower than the experimentally determined binding energy. There is one additional portion of the overall pair binding energy to be considered, the short-range “chemical” interaction between Xq and Xq, Φ X q X q . This component is especially important for semiconductors having primarily covalent bonding character, as the concept of a Coulombic potential necessitates that a point defect be treated as a fixed core (Fahey et al. 1989). When charges arise from bound carriers with wave functions that extend to neighboring sites, this approximation is clearly not applicable. The non-Coulombic interactions between the defects can be summed into the term Φ X q X q , for which several different estimates exist. For instance, Ball et al. cite the applicability of the Buckingham potential model to defect modeling in CeO2 and other oxide semiconductors (2005). According to this model ⎛ r q q Φ ( rX q X q ) = AX q X q exp ⎜ − X X ⎝ ρX qX q

⎞ CX q X q , ⎟− 6 ⎠ rX q X q

(2.34)

where rX q X q is the nearest neighbor distance between Xq and Xq, and AX q X q , ρ X q X q , and C X q X q are adjustable parameters. The parameters were selected to reproduce

2.2 Thermodynamics of Defect Charging

17

the unit cell volumes of various relates oxides. Also, the pair interaction in a covalent material as a function of radial separation can be expressed as Φ ( r ) = Φ 0 exp ⎡⎣ − β ( r / rX q X q − 1) ⎤⎦ ,

(2.35)

where rX q X q is the nearest neighbor distance between Xq and Xq, and β and Φo are adjustable parameters (Cai 1999). β and Φ0 can be determined by fitting experimental data consisting of elastic constants, lattice constants, and cohesive energy.

2.2.4 Effects of Gas Pressure on Defect Concentration In many compound semiconductors, one of the constituent elements typically exists in gaseous form under laboratory or processing conditions. For example, the oxygen in metal oxides exists as O2 gas. Upon heating in an environment having a low partial pressure of oxygen, some of the lattice oxygen escapes from the crystal structure and diffuses through the material into the gas phase, leaving behind oxygen vacancies and (depending upon reactions among defects) other kinds of defects as well. A reverse process can also take place if the ambient partial pressure of oxygen is high enough; oxygen can diffuse into the material and annihilate oxygen vacancies. Analogous phenomena occur in other compound semiconductors such as GaAs; at sufficiently high temperatures, both Ga and As have significant vapor pressures and can exchange with the corresponding vacancies within the GaAs crystal structure. Since As is the more volatile species, GaAs tends to lose As more readily when heated in vacuum. Point defect concentrations in such cases depend upon ambient conditions (Kroger and Vink 1958; Sasaki and Maier 1999b, a), and can be calculated from equations derived via mass-action principles applied to all the relevant defects and charge carriers (Jarzebski 1973; Sasaki and Maier 1999b). This approach has been applied quite extensively in the case of metal oxides. The equilibrium between a crystal MO and the gas phase is described according to g MO ≡ M M0 + OO0 ↔ MO ( )

(2.36)

1 g g M M0 + OO0 ↔ M ( ) + O2( ) . 2

(2.37)

At a fixed temperature, the concentration of defects in the bulk can be varied by altering the partial pressure of the ambient. When a neutral oxygen atom is added to the MO crystal lattice a new pair of lattice sites is created; the cation site remains vacant, creating a metal vacancy: 1 (g) O2 ↔ OO0 + VM0 . 2

(2.38)

18

2 Fundamentals of Defect Ionization and Transport

If the metal vacancy were to subsequently ionize to VM−1, the concentration of be described as a function of oxygen partial pressure according to

VM−1 could

K1 PO1/2 2 ⎡VM−1 ⎤ = ⎣ ⎦ p ⎡O ⎤ , ⎣ O⎦

(2.39)

where K1 is the equilibrium constant for Eq. 2.39 and p is the concentration of free hole carriers. Equations of this form can be written for other charge states as well. In the case of the vacancy in the (−2) state, the term p2 would appear in the denominator, whereas for the neutral state p would not appear at all. Oxides can also exchange metal atoms with the gas phase, although most experimental configurations do not allow independent control of metal gas phase pressure. However, metal vapor pressures are typically low, so that experiments that allow independent control of metal partial pressures still give good approximations to equilibrium conditions. When PM is high, metal atoms fill metal vacancies in the bulk and create vacancies in the oxygen sublattice: g M ( ) ↔ M M0 + VO0 .

(2.40)

Once oxygen vacancies ionize into the (+1) charge state, their concentration is given by K 2 PM ⎡VO+1 ⎤ = ⎣ ⎦ n ⎡M 0 ⎤ . ⎣ M⎦

(2.41)

Additionally, it should be noted that the electroneutrality condition must always be obeyed. This condition accounts for the fact that the overall crystal has no electrical charge, even though charged defects exist in the bulk: n + ⎡⎣VM−1 ⎤⎦ = p + ⎡⎣VO+1 ⎤⎦ .

(2.42)

One final equilibrium expression, n ∗ p = Ki

(2.43)

arises from the equilibration of electrons and holes in the crystal. The ionized defects in MO are now described by a series of algebraic equations containing seven variables: n, p, ⎡⎣VM−1 ⎤⎦, ⎡⎣VO+1 ⎤⎦, PM, PO2, and T, where T is the absolute temperature of the system. Normally T and PO2 are taken as independent variables; PM is then a dependent variable. This treatment can be generalized to materials with a larger variety of charged defects and electrical states. In this treatment, the charge state dependence arises from the equilibrium constants; no separate contributions to the free energy, entropy, or enthalpy of ionization are broken out. This approach differs decidedly from that of Van Vechten, who explicitly references concentrations of charged species to the corresponding concentrations of neutral species.

2.3 Thermal Diffusion

19

Fig. 2.4 Brouwer diagram of a pure oxide in complete equilibrium at 800ºC from (Sasaki and Maier 1999), where the concentrations of charged and neutral oxygen vacancies and interstitials are shown as a function of oxygen partial pressure. Reprinted with permission from Sasaki K, Maier J (1999) J Appl Phys 86: 5427. Copyright (1999), American Institute of Physics.

Based on equations such as 2.36 through 2.43, a plot of species concentrations vs. oxygen partial pressures can be subdivided into regions in which various defects predominate. To make this subdivision, it is necessary to know (either exactly or approximately) the values of all of the equilibrium constants at a given temperature. This complicated system of equations is often visualized by applying the graphical method proposed by Brouwer (1954). It is thus common to find the concentration of charged defects in TiO2, ZnO, UO2, and CoO plotted as a function of oxygen partial pressure as illustrated in (Fig. 2.4), where either the valence band maximum or conduction band minimum is used as the reference for the Fermi energy. In this instance, the temperature is held constant at 800ºC. The change in the diagram as a function of temperature will be related to the enthalpies of the various defect formation reactions.

2.3 Thermal Diffusion Bulk diffusion in semiconductors is typically mediated by point defects such as vacancies and interstitials, which may exchange with the lattice and with defect clusters (which sometimes play a role as reservoirs of defects) (Pichler 2004; Chang et al. 1996; Fu-Hsing 1999; Seeger and Chik 1968; Hu 1973; Casey et al. 1973). The situation is similar for surface diffusion, although the relevant defects are typically surface vacancies and adatoms, which can exchange with surface lattice sites and islands.

20

2 Fundamentals of Defect Ionization and Transport

For both the bulk and the surface, an atomistic description of the diffusion rate exists for native point defects, based on the work of Einstein and Smoluchowski. That description quantifies the defect motion (before exchange with the lattice or other reservoirs) in terms of a diffusion coefficient D. In a single dimension (say, x), component Dx of the diffusion coefficient in that direction is defined in terms of the mean square x-displacement Δx 2 of the diffusing species and the time interval t during which diffusion takes place, according to Δx 2 . 2t

Dx =

(2.44)

In the case of diffusion in N dimensions (2 for surface, 3 for bulk) with mean square displacement Δr 2 , this equation generalizes to D=

Δr 2 . 2 Nt

(2.45)

Treatments of diffusion in this context often examine random hopping motion between well-defined, energetically favorable sites (Pichler 2004). If Г represents the hopping frequency between sites and L is the hop length between them, then the diffusion coefficient can be recast as: D=

ΓL2 . 2N

(2.46)

Since thermal diffusion of defects on or within semiconductors generally involves some form of bond stretching or breakage, the hopping frequency typically incorporates a temperature dependence in Arrhenius form. Zener (1951), Vineyard (1957), Rice (1958), and Flynn (1968) have all presented theories for the jump frequency of a diffusing defect in the bulk, where Г can be expressed as (for temperatures above the Debye temperature): ⎛ ΔS Γ = Γ 0 ⋅ exp ⎜ m ⎝ k

⎞ ⎛ ΔH m ⎞ ⎟ ⋅ exp ⎜ − ⎟ ⎠ ⎝ kT ⎠

(2.47)

where Г0 stands for a weighted mean frequency, often called an “attempt frequency,” and ΔSm and ΔHm respectively represent the entropy and enthalpy of migration, T is the absolute temperature, and k is Boltzmann’s constant. Related descriptions exist for surface diffusion (Gomer 1996). D is sometimes written in simpler Arrhenius form D (T ) = D0 exp ( − Ea / kT )

(2.48)

where Ea is the activation energy for diffusion, D0 is the pre-exponential factor. Clearly Ea = ΔHm in this treatment. Comparing Eqs. 2.47 and 2.48 indicates that D0 =

Γ 0 exp(ΔS m / k ) L2 . 2n

(2.49)

2.3 Thermal Diffusion

21

For both bulk and surface intrinsic diffusion, Γ0 usually lies near a vibrational frequency (about the Debye frequency) of 1012 s−1, while L is typically an atomic bonding distance near 0.3 nm. Thus, in the absence of significant entropy effects during diffusion, the pre-exponential factor lies near 10–3 cm2/s both in the bulk and on the surface. Significant deviations from this value are often observed, however, particularly for surface diffusion (Seebauer and Jung 2001). An individual type of defect can sometimes diffuse by more than one pathway. Self-diffusion of the silicon interstitial is a primary example, where numerous possible pathways have been identified (Lee et al. 1998b; Kato 1993; Munro and Wales 1999; Sahli and Fichtner 2005). For the case of oxide semiconductors, where significant deviations from stoichiometry often occur, the dominant diffusion mechanism may depend on the partial pressure of the ambient (Bak et al. 2003; Diebold 2003; Hoshino et al. 1985; Jun-Liang et al. 2006; Kohan et al. 2000; Millot and Picard 1988; Oba et al. 2001; Tomlins et al. 1998; Erhart and Albe 2006). Defects of different types can sometimes bind together (while retaining their individual identities) to diffuse as a pair. For example, vacancies can be attracted to substitutional impurities by various long and short range electrostatic and strain forces, leading to binding energies on the order of 1 eV or more (Nelson et al. 1998). The exchange of places between the impurity and the vacancy induce diffusional motion of both species (Belova and Murch 1999). As another example, interstitials can bind to intrinsic or extrinsic defects and migrate via a “pair diffusion” mechanism, frequently also called an “interstitialcy” mechanism (although the former will be used throughout the remainder of this text). The impurity atom exchanges with the lattice on every hop, instead of squeezing between lattice sites for multiple hops before exchanging. The impurity atom does not necessarily carry the same host atom with it; over time, the impurity atom can exchange with any number of different host atoms. The atomistic perspective outlined above constitutes the basis of most theoretical methods for estimating diffusion coefficients, as well as interpretation of experimental methods that directly image atomic motion. On length scales longer than a few atomic diameters, however, the total rate of mass transport within the bulk or on the surface depends not only upon the mobility of defects, which the equations given above describe, but also upon the number of those defects available to move. This total rate is the focus of primary concern in many practical applications, such as the diffusion of dopants within microelectronic devices and heterogeneous catalysis, sintering, and corrosion on surfaces. In principle, several types of defects can contribute to the overall motion. Such behavior has been reported for bulk silicon, where certain measurements have been interpreted in terms of separate diffusional pathways involving vacancies and interstitial atoms (Ural et al. 1999, 2000; Bracht and Haller 2000). However, more typically a single defect type dominates the transport. For example, bulk diffusion can be mediated by the directional migration of vacancies or interstitials, the displacement of lattice atoms into interstitial sites, or the interchange of diffusing atoms between substitutional and interstitial sites in the crystal lattice (Sharma 1990). The latter mechanism is often referred to as “kick-out” diffusion.

22

2 Fundamentals of Defect Ionization and Transport

Fig. 2.5 Sketch of an Arrhenius plot for mesoscale surface mass transport on metals, showing two temperature regimes.

Similar principles apply to surface diffusion. Migration on semiconductor substrates has not been studied to the extent of that in the bulk. In analogy to bulk diffusion, however, different migration mechanisms dominate on the surface as a function of temperature, Fermi energy, and stoichiometry. For metals, a large body of aggregated experimental data for nickel, tungsten, silicon, germanium, aluminum oxide, and other materials indicates that two distinct temperature regimes of Arrhenius behavior exist for surface self-diffusion (Doi et al. 1995; Bonzel 1973; Seebauer and Jung 2001; Seebauer and Allen 1995; Mills et al. 1969; Plummer and Rhodin 1968; Binh and Melinon 1985; Tsoga and Nikolopoulos 1994; Fukutani 1993), as sketched schematically in Fig. 2.5. The trends appear to arise from adatom-dominated transport at low temperatures and vacancydominated transport at high temperatures. It is reasonable to suppose that similar mechanisms operate on semiconductor surfaces, although insufficient experimental data currently exist to verify that idea. A continuum approach often proves more useful for quantifying diffusion in many kinds of experimental measurements taken at length scales longer than a few atoms (Seeger and Chik 1968). When a spatially inhomogeneous distribution of defects exists in the bulk or on the surface, the species migrate to reestablish equilibrium. In the continuum description, a diffusion coefficient D can be defined assuming that the chemical potential of the diffusing species X scales linearly with its concentration [X]. (Note that other factors sometimes influence the chemical potential gradient, such as strong curvature in surface scratch decay experiments.) In the absence of electric fields, the flux J of the diffusing species obeys Fick’s 1st law: J = − D∇ [ X ].

(2.50)

The dependence of [X] on time and space is described by Fick’s 2nd law ∂[X ] ∂t

= ∇ • ( D∇ [ X ] ).

(2.51)

Note that “species” must be defined carefully. For example, for purposes of Fick’s laws, an interstitial dopant atom constitutes a different species than a substitutional one. Failure to make this distinction sometimes leads to erroneous discussion of “non-Fickian diffusion” when kick-in/kick-out reactions interconvert the interstitial and substitutional species. Diffusion profiles measured at short times (before the mobile species has exchanged a significant number of times

2.3 Thermal Diffusion

23

with the lattice) yield non-Fickian shapes such as exponentials (Vaidyanathan et al. 2006b). In surface diffusion, these considerations apply as follows. The literature often defines a mesoscale diffusivity DM (Seebauer and Jung 2001; Bonzel 1973) (called the “mass transfer diffusivity” in older literature) that incorporates both the hopping diffusivity DI (called “intrinsic diffusivity” in older literature) of a defect and the concentration of mobile defects [Xmobile], normalized by the concentration of substrate atoms [substrate] (not to be confused with [S], the concentration of available lattice sites in the crystal: DM = DI

[ X mobile ]

[ substrate]

.

(2.52)

Experimental techniques that are at least indirectly sensitive to the creation and migration of mobile defects generally measure DM rather than DI. Example techniques (Seebauer and Jung 2001) include scratch decay and low-energy electron microscopy, wherein mobile atoms are typically formed from step, kink, or terrace sites. DI is commonly measured by methods that can track individual atoms, such as scanning tunneling microscopy (STM) and field ion microscopy (FIM). DM typically has a stronger temperature dependence than DI due to the added temperature dependence of Ceq, which varies due to the creation and exchange of adatoms or vacancies with the bulk (Kyuno et al. 1999) or other surface features such as steps (Ehrlich and Hudda 1966; Breeman and Boerma 1992), islands (Beke and Kaganovskii 1995), and extrinsic defects (Heidberg et al. 1992; Hansen et al. 1996). The kink sites at steps or island edges can mediate both adatom and vacancy diffusion mechanisms that operate in parallel. Without being destroyed themselves, kinks can independently create both adatoms and terrace vacancies (Blakely 1973). Although adatoms and vacancies can form and annihilate as pairs on terraces, equilibrium between these species does not require their coverages to be equal. Moreover, one species can dominate mass transport through superior numbers even if its mobility falls below that of the other species. The definition of DM given in Eq. 2.52 contains two temperature-dependent factors for the mobile species: the hopping diffusivity DI = D0 I exp( − E I / kT ) ,

(2.53)

and the equilibrium concentration Ceq = Csub exp( −ΔG f / kT ) .

(2.54)

Substitution into Eq. 2.52 for DI and Ceq and decomposition of ΔGf into its constituent enthalpy ΔHf and entropy ΔSf yields: ⎛ ΔS f ⎞ ⎛ − ( ΔH f + E I ) ⎞ DM = D0 I exp ⎜ ⎟ exp ⎜ ⎟. kT ⎝ k ⎠ ⎝ ⎠

(2.55)

24

2 Fundamentals of Defect Ionization and Transport

For mesoscale diffusion, an effective pre-exponential factor can be defined together with a corresponding activation energy D0 M = D0 I exp( ΔS f / k )

(2.56)

E M = ΔH f + E I .

(2.57)

The form of Eq. 2.57 shows that a diffusion mechanism with high EM can dominate a mechanism with low EM if the former has a much higher value of D0M and if the temperature is sufficiently high. Such diffusion mechanisms working in parallel can produce the temperature dependence shown in Fig. 2.5. Regardless of the specific bulk or surface diffusion mechanism, the charge state of the primary diffusing defect can affect its rate of hopping. For example, changing the charge state of a bulk interstitial atom affects not only its effective size (and therefore its ability to squeeze between lattice atoms) but also its ability to chemically bond to the surrounding atoms. In silicon, for example, the energy barrier for the migration of V+2 differs from that of V−2 (Bernstein et al. 2000; Kumeda et al. 2001; Watkins 1967, 1975, 1986; Watkins et al. 1979). Such effects can, in principle, show up in the pre-exponential factor as well as the activation energy. Thus, there are two ways for charge state to affect the rate of motion of a defect over length scales greater than atomic: changes in concentration and changes in hopping rate. When multiple charge states for a defect exist simultaneously, their effects are typically additive. For example, an effective diffusivity of selfinterstitials can be expressed as Dieff

⎛ ⎡ X i0 ⎤ ⎞ ⎛ ⎡ X i+1 ⎤ ⎞ ⎛ ⎡ X i−1 ⎤ ⎞ ⎣ ⎦ ⎣ ⎦ ⎦ ⎟ + ... ⎜ ⎟ ⎜ ⎟ = Di0 + Di +1 + Di −1 ⎜ ⎣ ⎜ [ Xi ] ⎟ ⎜ [ Xi ] ⎟ ⎜ [ Xi ] ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(2.58)

where [Xi] is the total concentration of interstitials (in all charge states). The relative importance of each of these terms depends upon the position of the Fermi energy.

2.4 Drift in Electric Fields Semiconductor pn junctions and heterojunctions are the foundation of most major microelectronic devices, and these structures contain appreciable built-in electric fields. Such fields act on mobile charged defects (Sheinkman et al. 1998) during processing and subsequent device use. These fields, as well as their interactions with electrically active defects introduced during the fabrication process, can dramatically degrade device performance (El-Hdiy et al. 1993). The reduction in size scale of these devices has caused the magnitudes of these electric fields to progressively rise.

2.5 Defect Kinetics

25

When an electric field ε of 104 to 106 V/cm is applied along with thermal diffusion from a constant source, field-aided diffusion takes place (Sharma 1990) according to: J = −D

∂[ X ] ∂x

+ qμ [ X ]ε ( x )

(2.59)

where [X] is the concentration of defect X and q is the charge of defect X. The mobility μ can be approximated roughly as qD/kT. The transport rate of charged defects can be either retarded or enhanced depending on the direction of the field. For a complete solution of the equations of motion for the defects, this transport equation must be solved together with Poisson’s equation for the electrostatic potential Ψ: ∇2 Ψ =

e⎛ q ⎞ ⎜ n − p + ∑ qi ⎡⎣ X i ⎤⎦ ⎟ ε⎝ i ⎠

(2.60)

where n and p are the number of electrons and holes in the conduction and valence bands, respectively, qi is the charge associated with the defect Xi with concentration [Xi], and e is the electronic charge. For example, qi would take on a nominal charge of (+1) for singly ionized acceptors and (−1) for singly ionized donors. Field-assisted diffusion also occurs on semiconductor surfaces (Yagi et al. 1993; Kawai and Watanabe 1997). Such behavior has been observed most notably in the imaging of charged defects with scanning tunneling microscopy. Tipinduced electric fields affect the electronic structure of a semiconductor surface containing native defects (Ness et al. 1997b, a; Saranin et al. 1997).

2.5 Defect Kinetics The rate expressions that govern the generation, destruction and clustering of point defects are important for predicting and interpreting transient behavior that commonly occurs during semiconductor processing as well as certain experimental techniques designed to detect defects. The following sections outline some basic principles of defect reaction kinetics, as well as the kinetics of defect charging.

2.5.1 Reactions Although rate expressions for defect reactions can be developed in the abstract, it is perhaps more instructive to set such a presentation in the context of a specific case to bring out the nuances of kinetic integration that typically characterize a reaction network of defects within a typical semiconductor. The focus here will be on defects within the bulk, but analogous descriptions apply to surfaces.

26

2 Fundamentals of Defect Ionization and Transport

The specific example we will use is boron acting as a p-type dopant within silicon (Jung et al. 2004b). Microelectronic devices comprise an obvious application. When boron is the primary impurity, it resides primarily in substitutional sites in a (–1) charge state. However, the substitutional boron can interact with interstitial silicon through a kick-out mechanism and become interstitial itself. In typical p-type material, interstitial boron exists as Bi+1, interstitial silicon as Sii+2. Quantum calculations have also pointed to the existence of a well-defined complex of substitutional boron and the silicon interstitial:(BSiSii)+1. Boron can diffuse (and participate in reactions such as clustering) either as the free interstitial or through pair diffusion via (BSiSii)+1. Similar expositions detail the kinetics of defect formation and annihilation in GaN (Tuomisto 2005), ZnO (Kotlyarevsky et al. 2005), and Si (Pichler 2004). Figure 2.6 details the interaction among the various boron and silicon species. Breakup of the (BSiSii)+1 complex to yield interstitial species occurs by two pathways that are each kinetically first-order in the concentration of (BSiSii)+1. Dissociation to yield free Sii+2 is denoted by the rate rdis, while dissociation via kick-out to yield free Bi+1 is denoted by rko. The reverse reaction of kick-in is also fundamentally first-order, depending only on the concentration [Bi+] because each Bi is completely surrounded by lattice Si atoms with which it can react. The association reaction between Sii+2 and BSi–1 is second-order, however, because BSi–1 is by far the minority species in terms of lattice site occupation. Although an activation barrier may exist in principle when these species get close enough to react, the opposite charges on the reactants and the negative free energy of formation for the complex give reasons to believe that the complex forms with no barrier. A rate expression describing standard diffusion limitation by reactants (Laidler 1980) therefore is warranted.

Fig. 2.6 Composite reaction network for reactions of boron defects in silicon, incorporating the kick-out and pair diffusion mechanisms.

2.5 Defect Kinetics

27

The reaction stoichiometries are:

( BSi Sii )+1 → Bi+1 + SiSi

(kick-out)

(2.61)

Bi+1 + SiSi → ( BSi Sii )

(kick-in)

(2.62)

( BSi Sii )+1 → BSi−1 + Sii+2

(dissociation)

(2.63)

BSi−1 + Sii+2 → ( BSi Sii )

(association).

(2.64)

+1 rko = kko exp ( − Eko / kT ) ⎡( BSi Sii ) ⎤ ⎣ ⎦

(2.65)

rki = kki exp ( − Eki / kT ) ⎡⎣ Bi+1 ⎤⎦

(2.66)

+1 rdis = kdis exp ( − Edis / kT ) ⎡( BSi Sii ) ⎤ . ⎣ ⎦

(2.67)

rassoc = kassoc ⎡⎣ BSi−1 ⎤⎦ ⎡⎣ Sii+2 ⎤⎦

(2.68)

kassoc = 4π aDassoc

(2.69)

+1

+1

The corresponding rate expressions are:

with where Dassoc = DBSi−1 + DSii+2, with DBSi−1 << DSii+2 . Here, a represents a reaction distance or “capture radius.” From these expressions, along with maximum likelihood estimates for the corresponding activation energies and pre-exponential factors, it is possible to determine the conditions under which the interstitial diffusion mechanism dominates pair diffusion (Jung et al. 2004b). The kinetic treatment of point defects and defect pairs in boron-implanted Si can be extended to describe the further associations of interstitials into clusters of indefinitely large sizes during postimplant annealing (Jung et al. 2004a). The association reaction between Sii or Bi and a cluster is second-order. A rate expression describing standard diffusion limitation is appropriate: rassoc = k assoc [cluster ][ Bi or Sii ]

(2.70)

rassoc = 4π aDBi or rassoc = 4π aDSii

(2.71)

with where clusters have been assumed to be much less mobile than free interstitials. Dissociation kinetics is more problematic, since the rate expressions used in the literature sometimes have inadequacies. The rate expression commonly employed in literature for dissociation kinetics assumes the following form (Stolk et al. 1997): rdis =

Dint exp(− Eb / kT )[cluster ] a2

(2.72)

28

2 Fundamentals of Defect Ionization and Transport

where Eb denotes the binding energy and Dint the interstitial diffusivity. Eq. 2.72 is problematic in several ways, but its greatest flaw originates from its assumption that dissociation involves two sequential steps: interstitial release from the cluster followed a diffusional hop away from the cluster. However, except for (BSiSii)+1, the barrier to interstitial release (for which the binding energy sets only a lower bound) is significantly larger than that for diffusional hopping. Since the preexponential factor is likely to be the same for both release and hopping (i.e., close to a Debye frequency), interstitial release must be rate limiting in this reaction sequence. It can be easily shown that in a reaction sequence including a ratelimiting step, steps after the rate limiting one exert no influence whatever on the overall rate. Thus, rate expressions that include parameters from steps following the rate limiting one must in principle be incorrect. In general, simple models for elementary dissociation reactions employ first-order kinetics and cast rate constants as the mathematical product of an attempt frequency and an exponential Boltzmann factor containing a transition-state activation barrier (which may not be the same as the binding energy). Regarding the pre-exponential factor, the Debye frequency for the host material represents a good first approximation. Since in this particular case, cluster dissociation energies generally increase with cluster size, a general model for interactions among defects and clusters can telescope the entire cluster dissociation cascade (from large to small clusters) into a computationally manageable set of events There are many cluster dissociation pathways due to the varying stoichiometries of larger clusters. Consequently, a nearly continuous distribution of dissociation energies is well suited to approximating the activation energies of the pathways. For practical applications, such a model has proven capable of providing insight into how the initial conditions and subsequent heating steps in rapid thermal annealing affect transient enhanced diffusion (Gunawan et al. 2003). These can be tailored to minimize the size and number of defect clusters that diffuse within the bulk. Use of such a model can be accomplished within the context of a differential equation solver that produces numerical results for the coupled equations just outlined. However, some engineering applications employ unsteady-state heating protocols such as linear ramps. Numerical simulations can certain handle such situations, but sometimes an analytical or nearly-analytical approach is more useful for quick estimates of defect behavior. For example, annealing to remove ion implantation damage in pn junction formation is limited in part by transient enhanced diffusion (TED) of dopants, often leading to unacceptable spreading of the original dopant profile (Jung et al. 2003). Annealing usually employs a linear ramping program that raises the solid temperature over 1,000°C within about 1 s. Although the system is transiently heated, some straightforward approximations permit important insights to be gained into defect behavior and the resulting dopant diffusion. Under common heating conditions, it can be shown that the diffusion of boron in TED is mediated by the motion of free Bi (Jung et al. 2004b). The motion of free Bi is hampered by boron exchange with lattice sites. When this exchange mechanism dominates, boron moves while it is a free interstitial, but is rapidly immobilized when (BSiSii) releases Sii. The trapped boron can move again

2.5 Defect Kinetics

29

only after lengthy period of waiting for association with another free Si interstitial. The time constant describing this waiting period is (kassoc[Sii])−1, where kassoc denotes the rate constant for the association reaction between BSi and Sii. By defining tmax as a characteristic time over which the wafer remains near the peak temperature, the number of liberation events can be estimated from the ratio of tmax to the characteristic time (kassoc[Sii])−1 for the association reaction. Then the degree of profile spreading can be estimated from Gunawan et al. as x2 =

6 Ddiff , Bi ⎛1− b ⎞ kassoc [ Sii ] tmax ⎜ ⎟ kki ⎝ b ⎠

(2.73)

where kki and Ddiff , Bi represent respectively the rate constant for kick-in and the diffusion coefficient for Bi hopping. The branching ratio b describing the pathways for the dissociation reaction of BSiSii to form Bi and Sii is given by b=

rBSi Sii → BSi + Sii . rBSi Sii → BSi + Sii + rBSi Sii → Bi + Si

(2.74)

Several easily satisfied assumptions are required for these equations to hold. First, dissociation must occur in the equivalent of a single step. Second, interstitial reassociation with the most actively dissociating clusters must be neglected. Lastly, the distribution of cluster dissociation energies must be wider than 1.5kT. Under these circumstances, a closed-form analytical expression can be obtained connecting each temperature in the linear ramp with the dissociation energy E* of the most active dissociating species. Simulations can then be used to understand why heating rate affects cluster dissociation energy and thus cluster concentration.

2.5.2 Charging Defects exchange charge with the conduction and valence bands via either thermal or radiative processes. In the thermal processes, the defect captures or emits a charge carrier directly, with a rate that is generally quite fast for defect eigenstates close to the bands, but sometimes much slower (many seconds) for levels lying deep within the band gap. A detailed discussion of such kinetics can be found in Landsberg’s book (Landsberg 1991). Radiative processes involve the absorption or emission of a photon. When the semiconductor is illuminated, the rate of electron-hole pair generation gE obeys two continuity (mass balance) equations: d Δn Δ n 1 ⎡ ( n0 + n1 + Δn )( Δn − Δp ) Δnn1 ⎤ = = + ⎢ ⎥ dt N n0 + n1 ⎦ τ n τ n0 ⎣

(2.75)

Δpp1 ⎤ d Δp Δp 1 ⎡ ( p0 + p1 + Δp )( Δp − Δn ) = = + ⎢ ⎥, dt N p0 + p1 ⎦ τ p τ p0 ⎣

(2.76)

gE −

gE −

30

2 Fundamentals of Defect Ionization and Transport

where n0 and p0 are the electron and hole concentrations in thermal equilibrium, τn and τp are the electron and hole lifetimes, and N is the recombination center density (Schmidt et al. 1998). The time constants τn0 and τp0 for capture of electrons and holes by defects (inducing a change in ionization state), depend upon the capture cross sections σn and σp of the defect, and also upon the defect concentration and the thermal velocities of electrons and holes υth equal to 3kT / m∗ . The times constants are given by:

τ n0 =

1 Nσ nν th

τ p0 =

1 Nσ pν th

(2.77) .

(2.78)

The statistical factors n1 and p1 are the equilibrium concentrations of electrons and holes, respectively, when the Fermi level coincides with the eigenstate of the defect ET: ⎛ E − Ec ⎞ n1 = N C exp ⎜ T ⎟ ⎝ kT ⎠

(2.79)

⎛ Ec − Eg − ET ⎞ p1 = NV exp ⎜ ⎟, kT ⎝ ⎠

(2.80)

where Ec is the conduction band edge energy and Eg the band gap. Under steady-state conditions, the time-dependent terms in these equations vanish, and various general analytic solutions exist (Shockley and Read 1952). Shockley and Read presented one such solution, with the underlying assumption of spin nondegeneracy. Additional simplifications lead to the Shockley–Read–Hall (SRH) model commonly used today (Macdonald and Cuevas 2003), and involve eliminating N, rearranging, and using the identities n0/(n0 + n1) = p0 / ( p0 + p1) and p1n1 = p0n0 to obtain: gE =

Δn

τn

=

Δp

τp

=

p0 Δn + n0 Δp + ΔnΔp . τ p 0 ( n0 + n1 + Δn ) + τ n 0 ( p0 + p1 + Δp )

(2.81)

Further manipulation of this equation, along with the assumption of negligible excess carrier densities, leads to the following expression for the charge carrier lifetime:

τ SRH = τ n = τ p =

τ p 0 ( n0 + n1 + Δn ) + τ n 0 ( p0 + p1 + Δp ) n0 + p0 + Δn

.

(2.82)

The concept of a “demarcation level” is also sometimes used to describe exchange of charge with the valence and conduction bands. Whereas the ionization level relates to thermodynamic equilibria, the demarcation level focuses on kinetic rates of charge capture. The hole demarcation level is defined as the ionization

2.6 Direct Surface-Bulk Coupling

31

level at which the rate of hole emission equals the rate of electron capture. That is, a hole at a level equal to the hole demarcation level has an equal probability of being thermally excited to the valence band as it does recombining with a electron in the conduction band. Depending on the position of a defect ionization level relative to the demarcation level, one process or the other will be more likely to occur. The location of the hole demarcation level is extracted from the following definition ⎡n ⎛σ ν EF − EDp = kT ln ⎢ ⎜ n n ⎣⎢ p ⎝ σ pν p

⎞⎤ ⎟⎥ ⎠ ⎦⎥

(2.83)

where σ are the capture cross sections and υ thermal velocities of electrons (n) and holes ( p), n and p the concentrations of electrons and holes, and EDp the hole demarcation level (Bube 1992). If a measurable phenomena occurs that changes a particular set of defect levels from recombination centers to hole traps, a measurement of electron density allows for the determination of the demarcation level. A corresponding expression can be derived from the electron demarcation level.

2.6 Direct Surface-Bulk Coupling Researchers have expended substantial effort comparing the physics of bulk solids with those of free surfaces and solid interfaces. Curiously, much less attention has focused upon the direct coupling between these phenomena. Only a handful of papers have considered topics such as bulk quenching of surface exciton emission (David et al. 1989), bulk-influenced surface state behavior at steps on metals (Baumberger et al. 2000), and bulk doping effects on surface band bending in semiconductors (Vitomirov et al. 1989; Kuball et al. 1994; King et al. 2007). While Vitomirov et al. considered the effects of the semiconductor bulk on surface electronic properties; other work has shown that surface and interface electronic properties affect bulk semiconductor behavior. Two mechanisms can lead to direct surface-bulk coupling: near-surface band bending and defect exchange with dangling bonds. Band bending near a free surface or solid-solid interface occurs when dangling bonds in those regions exchange charge with the semiconductor bulk. This charge exchange sets up a space charge region (and associated electric field) within the semiconductor. When point defects within the semiconductor are charged, this electric field provides an electrostatic coupling mechanism between the surface or interface and the bulk defects (Dev et al. 2003). For a semiconductor such as silicon, for which surfaces and interfaces typically support Fermi level pinning near the middle of the band gap, the direction of the electric field typically repels charged defects within the bulk. At high semiconductor doping levels, the electric field within the space charge region is so strong that field-induced drift dominates diffusion of the defects. Thus, the ability of the surface or interface to absorb defects from deep within the bulk is greatly diminished. However, the variation in

32

2 Fundamentals of Defect Ionization and Transport

Fermi level near the surface or interface can also change the average charge state of charged defects in that vicinity, leading to complex effects such as the pileup of implanted dopants (Jung et al. 2005). The degree of band bending can, in principle, be controlled through adsorption at a free surface (Rosenwaks et al. 2004) and ion bombardment at a solid-solid interface (Dev et al. 2003; Rosenwaks et al. 2004). Defect exchange with dangling bonds at a surface also provides a method to withdraw defects from the underlying semiconductor bulk (Kirichenko et al. 2004). An atomically clean surface can annihilate interstitial atoms by simple addition of the interstitials to dangling bonds. But if the surface is decorated with a strongly bound adsorbed species, defect annihilation requires the insertion of interstitials into existing bonds. This process should have a higher activation barrier and a correspondingly reduced probability of occurrence (Vaidyanathan et al. 2006a). Such effects have been demonstrated recently in silicon implanted with silicon isotopes (Seebauer et al. 2006) and arsenic (Vaidyanathan et al. 2006a). The latter case points to the possibility of creating very shallow p-n junctions with exceptionally high levels of electrically active dopant. This exchange mechanism can also inject defects into the semiconductor whose defect concentration is below that indicated by thermodynamics. Seebauer et al. recently showed that point defect concentrations as deep as 0.5 μm within a semiconductor can be controlled over several orders of magnitude through manipulation of surface chemical state through gas adsorption (Seebauer et al. 2006).

2.7 Non-Thermally Stimulated Defect Charging and Formation The creation of defects in particular charge states can be stimulated non-thermally either by photostimulation or (in a near-surface region) by ion bombardment. For the most part, such phenomena represent largely uncharted territory. However, some important underlying physics have been uncovered, especially in recent years.

2.7.1 Photostimulation Photostimulation changes the steady-state concentrations of charge carriers, which in turn can alter the average charge state of defects that are present. Such effects can propagate through into phenomena such as defect diffusion as described in Chap. 7. A non-equilibrium steady-state model can be formulated (Kwok 2007) to describe the electronic occupation of defect levels under photostimulation. The model is based on Simmons and Taylor (1971) who extend Shockley–Read statistics (Shockley and Read 1952) from the case of one distinct defect level to an arbitrary distribution of levels. In brief, the electron occupancy of any defect level is determined by the interplay of four electron transitions between the defect level, the conduction band, and

2.7 Non-Thermally Stimulated Defect Charging and Formation

33

Fig. 2.7 Electronic transitions under photostimulation between the conduction band (a and b), valence band (c and d), and a defect level at energy Et.

the valence band, as shown in Fig. 2.7. The probability of occupation f of the ith defect level under photostimulation can be calculated by solving the mass balance equations for electrons and holes, dn = G − ∑ν nσ ni nN ti ( Eti )[1 − f ( Eti )] + ∑ eni N ti ( Eti ) f ( Eti ) dt i i

(2.84)

en = ν nσ n N c exp[( Et − Ec ) / kT ]

(2.85)

e p = ν pσ p N v exp[( Ev − Et ) / kT ]

(2.86)

where

with Nt and Et denoting respectively the defect density and the defect level. Ec and Ev are respectively the conduction band minimum and the valence band maximum, with Nc and Nv denoting the corresponding density of states. The quantity ν is the thermal velocity, σ is the capture cross section, where the subscripts “n” and “p” denotes respectively electrons and holes. G(x) is the photo-generation rate of electron-hole pairs at a particular depth x (Blood and Orton 1992). Under steady-state conditions, dn/dt = dp/dt = 0. Moreover, the occupancy of any defect level Et is constant, therefore,

vnσ n nN t ( Et )[1 − f ( Et )] − en Nt ( Et ) f ( Et ) − ... v pσ p pN t ( Et ) f ( Et ) + e p N t ( Et )[1 − f ( Et )] = 0 .

(2.87)

The unknowns include n, p, and f of each defect level. Satisfaction of the charge neutrality condition then gives a unique solution.

2.7.2 Ion-Defect Interactions Ion bombardment, even at energies as low as about 10 eV, can also stimulate defect formation in semiconductors. Such phenomena play a key role in determining material properties in applications such as plasma enhanced deposition, reactive

34

2 Fundamentals of Defect Ionization and Transport

ion etching, ion beam assisted deposition, and ion implantation. Adjustment of the solid temperature during ion exposure affects the results; such effects have been well studied in the context of plasma etching (Wong-Leung et al. 2001; Tsujimoto et al. 1991; Gregus et al. 1993), ion implantation (Nitta et al. 2002; Turkot et al. 1995; Shoji et al. 1992), and beam-assisted deposition (Nastasi et al. 1996). Ions act mainly by knocking atoms out of their lattice sites on the surface or in the bulk, thereby creating surface or bulk vacancies as well as adatoms or interstitials. In principle, defect charging can affect the dynamics of such processes. For example, consider a case wherein a vacancy or interstitial produced by ion impact can assume different charge states depending upon the position of the Fermi energy. If the defect can ionize rapidly (even during the impact), it is reasonable to suppose that parameters such as the threshold energy for defect formation might depend upon the charge state the defect finally assumes, and thereby indirectly upon the Fermi energy. Consequently, variations in Fermi level would show up in the defect formation probability. Additionally, Wang and Seebauer indicate that some of the excess defects formed via ion-stimulation can either annihilate or form bound complexes at a rate influenced by electrostatic forces (2005). In this instance, charged defect concentration would be affected not by formation dynamics, but rather by the kinetics of annihilation with preexisting defects. Such effects could be quite pronounced at low incident ion energies, or at higher energies near the final stopping point of the ion when most of the incident energy has been lost to the solid. These effects are incompletely explored and likely to be complicated. For example, measurements of beam-assisted deposition (Dodson 1991; Kuronen et al. 1999; Rabalais et al. 1996; Lee et al. 1998a; Marton et al. 1998) and surface diffusion (Ditchfield and Seebauer 1999, 2001) have suggested that solid temperature may directly affect the dynamics of defect formation when ion energies fall below about 100 eV. This effect is surprising, given the fact that thermal energies of the target atoms in the solid are only a few tens of meV. Wang and Seebauer have outlined a mechanism explaining such effects (2005) based on measurements of surface diffusion. However, molecular dynamics simulations suggest that the effects operate for a wide variety of crystalline solids and defect types, both on the surface and in the bulk (Wang and Seebauer 2002). Such effects could ultimately be exploited in a variety of applications, such as modulating the dynamics of defect formation near the pn junction of semiconductor devices during the ion implantation of dopants. Judicious tuning of temperature and ion energy may also enable the selection of specific defect formation or sputtering processes during ion beam-assisted deposition and reactive etching.

References Bak T, Nowotny J, Rekas M et al. (2003) J Phys Chem Solids 64: 1043–56 Ball JA, Grimes RW, Price DW (2005) Modell Simul Mater Sci Eng 13: 1353–1363 Baumberger F, Greber T, Osterwalder J (2000) Phys Rev B: Condens Matter 62: 15431–4

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Kotlyarevsky MB, Rogozin IV, Marakhovsky AV (2005) in Zinc Oxide – A Material for Microand Optoelectronic Applications Nickel N & Terukov E (Eds.) Netherlands, Springer Kroger FA, Vink HJ (1958) J Phys Chem Solids 5: 208–223 Kuball M, Kelly MK, Cardona M et al. (1994) Phys Rev B: Condens Matter 49: 16569 Kudriavtsev Y, Villegas A, Godines A et al. (2005) Appl Surf Sci 239: 273–278 Kumeda Y, Wales DJ, Munro LJ (2001) Chem Phys Lett 341: 185–194 Kuronen A, Tarus J, Nordlund K (1999) Nucl Instrum Methods Phys Res, Sect B 153: 209–12 Kwok CTM (2007) Advanced methods for defect engineering in silicon Ph.D. diss. University of Illinois at Urbana-Champaign Kyuno K, Cahill DG, Averback RS et al. (1999) Phys Rev Lett 83: 4788–91 Laidler KJ (1980) Chemical Kinetics, New York, Wiley Landsberg PT (1991) Recombination in Semiconductors, Cambridge, Cambridge University Press Lautenschlager P, Garriga M, Vina L et al. (1987) Phys Rev B: Condens Matter 36: 4821–30 Lee SM, Fell CJ, Marton D et al. (1998a) J Appl Phys 83: 5217 Lee WC, Lee SG, Chang KJ (1998b) J Phys: Condens Matter 10: 995–1002 Macdonald D, Cuevas A (2003) Phys Rev B: Condens Matter 67: 075203 Marton D, Boyd KJ, Rabalais JW (1998) J Vac Sci Technol, A 16: 1321 Millot F, Picard C (1988) Solid State Ionics 28–30: 1344–1348 Mills B, Douglas P, Leak GM (1969) Trans Met Soc AIME 245: 1291–1296 Munro LJ, Wales DJ (1999) Phys Rev B: Condens Matter 59: 3969–3980 Nastasi M, Mayer JW, Hirvonen JK (1996) Ion-Solid Interactions: Fundamentals and Applications, New York, Cambridge University Press Nelson JS, Schultz PA, Wright AF (1998) Appl Phys Lett 73: 247–9 Ness H, Fisher AJ, Briggs GAD (1997a) Surf Sci 380: L479–L484 Ness H, Fisher AJ, Briggs GAD (1997b) Surf Sci 380: L479–L484 Nitta N, Taniwaki M, Suzuki T et al. (2002) Materials Transactions 43: 674–80 Oba F, Nishitani SR, Isotani S et al. (2001) J Appl Phys 90: 824–8 Pichler P (2004) Intrinsic Point Defects, Impurities, and their Diffusion in Silicon, New York, Springer-Verlag/Wein Plummer EW, Rhodin TN (1968) J Chem Phys 49: 3479–3496 Queisser HJ, Haller EE (1998) Science 281: 945–50 Rabalais JW, Al-Bayati AH, Boyd KJ et al. (1996) Phys Rev B: Condens Matter 53: 10781–92 Rice SA (1958) Phys Rev 112: 804–811 Rosenwaks Y, Shikler R, Glatzel T et al. (2004) Phys Rev B: Condens Matter 70: 085320 Sahli B, Fichtner W (2005) Phys Rev B: Condens Matter 72: 245210 Saranin AA, Numata T, Kubo O et al. (1997) Phys Rev B: Condens Matter 56: 7449–7454 Sasaki K, Maier J (1999a) J Appl Phys 86: 5434–5443 Sasaki K, Maier J (1999b) J Appl Phys 86: 5422–5433 Schmidt J, Berge C, Aberle AG (1998) Appl Phys Lett 73: 2167 Schultz PA (2006) Phys Rev Lett 96: 246401 Seebauer EG, Allen CE (1995) Prog Surf Sci 49: 265–330 Seebauer EG, Jung MYL (2001) in Landolt-Bornstein Numerical Data and Functional Relationships: Adsorbed Layers on Surfaces Bonzel Hp (Eds.) New York, Springer Verlag Seebauer EG, Dev K, Jung MYL et al. (2006) Phys Rev Lett 97: 055503 Seeger A, Chik KP (1968) Phys Status Solidi B 29: 455–542 Sharma BL (1990) Diffus Defect Data, Pt A 70/71: 1–102 Sheinkman MK, Kashirina NI, Kislyuk VV (1998) Electric field-caused redistribution of mobile charged donors in semiconductors. 1:89–92 (IEEE, Sinaia, Romania, 1998) Shockley W, Read JWT (1952) Phys Rev 87: 823–842 Shoji K, Fukami A, Nagano T et al. (1992) Appl Phys Lett 60: 451–3 Simmons JG, Taylor GW (1971) Physical Review B (Solid State) 4: 502–11 Stolk PA, Gossmann HJ, Eaglesham DJ et al. (1997) J Appl Phys 81: 6031 Swalin RA (1962) J Phys Chem Solids 23: 154–156 Thurmond CD (1975) J Electrochem Soc 122: 1133–1141

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Tomlins GW, Routbort JL, Mason TO (1998) J Am Ceram Soc 81: 869–876 Tsoga A, Nikolopoulos P (1994) J Am Ceram Soc 77: 954–960 Tsujimoto K, Okudaira S, Tachi S (1991) Jpn J Appl Phys 30: 3319–3326 Tuomisto F (2005) Vacancy defects in semiconductor materials for opto and spin electronics Ph.D. diss. Helsinki University of Technology Turkot BA, Forbes DV, Robertson IM et al. (1995) J Appl Phys 78: 97–103 Ural A, Griffin PB, Plummer JD (1999) Phys Rev Lett 83: 3454–3457 Ural A, Griffin PB, Plummer JD (2000) Phys Rev Lett 85: 4836–4836 Vaidyanathan R, Seebauer EG, Graoui H et al. (2006a) Appl Phys Lett 89: 152114 Vaidyanathan R, Jung MYL, Braatz RD et al. (2006b) AlChE J 52: 366–370 Van Vechten JA (1980) in Handbook on Semiconductors Keller Sp (Eds.) New York, NorthHolland Van Vechten JA, Thurmond CD (1976a) Phys Rev B: Condens Matter 14: 3539–50 Van Vechten JA, Thurmond CD (1976b) Phys Rev B: Condens Matter 14: 3551–7 Varshni YP (1967) Physica 34: 149–154 Vineyard GH (1957) J Phys Chem Solids 3: 121–127 Vitomirov IM, Waddill GD, Aldao CM et al. (1989) Phys Rev B: Condens Matter 40: 3483 Wang Z, Seebauer EG (2002) Phys Rev B: Condens Matter 66: 205409–1 Wang Z, Seebauer EG (2005) Phys Rev Lett 95: 015501 Watkins GD (1967) Phys Rev 155: 802–815 Watkins GD (1975) EPR studies of the lattice vacancy and low-temperature damage processes in silicon. In: Lattice Defects in Semiconductors, 1974 1–22 (Inst. Phys, Freiburg, West Germany, 1975) Watkins GD (1986) Mater Sci Forum 10–12: 953–60 Watkins GD, Troxell JR, Chatterjee AP (1979) Vacancies and interstitials in silicon. In: International Conference on Defects and Radiation Effects in Semiconductors 16–30 (Inst. Phys, Nice, France, 1979) Wong-Leung J, Jagadish C, Conway MJ et al. (2001) J Appl Phys 89: 2556–2559 Yagi K, Yamanaka A, Yamaguchi H (1993) Surf Sci 283: 300–308 Zener C (1951) J Appl Phys 22: 372–375

Chapter 3

Experimental and Computational Characterization

3.1 Experimental Characterization Defect properties, including thermodynamic ionization levels, can be derived directly from experimental techniques such as electron paramagnetic resonance, positron annihilation spectroscopy, and deep level transient spectroscopy experiments. Some of these techniques can be modified to deal with novel materials such as CdSe nanocrystals (Kuçur et al. 2005), quantum dots (Wang et al. 2002), and organic semiconductors (Chen et al. 2004). Additional methods exist for probing ideal and defected semiconductor surfaces. As point defects and defect clusters often exist at low concentrations, scientists frequently utilize diffusion measurements to indirectly characterize their charge-dependent properties. Each of these techniques provides distinct information, and not all of them are well suited to every semiconductor.

3.1.1 Direct Detection of Bulk Defects Electron paramagnetic resonance (EPR) spectra provide detailed structural information about a semiconductor defects, including their symmetry, atomic, and lattice configurations (Weber 1983; Watkins 2000; Rowan 1990; Stich et al. 1995; Kennedy and Wilsey 1985). EPR is equally useful for defect clusters as it is for isolated point defects. For instance, investigations have shed light upon multi-vacancy, interstitial-antisite, and Frenkel pairs in SiC (Ilyin et al. 2007; von Bardeleben et al. 2000). The technique, based upon the absorption of electromagnetic waves in the microwave frequency domain, involves the creation of magnetic dipoles in crystals subjected to magnetic fields. The point symmetry of a defect can be deduced from the angular dependence of its spectrum, while its atomic and

E.G. Seebauer, M.C. Kratzer, Charged Semiconductor Defects, © Springer 2009

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3 Experimental and Computational Characterization

lattice structure are inferred from hyperfine interactions. EPR data can be described by the spin Hamiltonian Hˆ S = μ B B ( g parallel cos θ S Z + g perpendicular sin θ S X ) + Aparallel S Z I Z

(3.1)

1 ⎡ ⎤ + Aperpendicular ( S X I X + SY IY ) + D ⎢ S Z2 − S ( S + 1) ⎥ 3 ⎣ ⎦

where μB is the Bohr magnetron, B is the magnetic field, S and I are the electronic and nuclear spin operators, respectively, D is the axial fine-structure parameter, and g and A are the g-tensor and the hyperfine structure tensor, respectively. The label Z (or parallel) applies for the c-axis and X,Y (or perpendicular) apply for all axes perpendicular to it. θ is the angle between the c-axis and the applied external magnetic field B. The variation in the EPR spectrum intensity with the measurement temperature provides information about the magnetic state of the sample. Feher introduced the technique of electron-nuclear double resonance (ENDOR), where the resonance of nearby lattice atomics can be detected as a change in the EPR signal of a defect, approximately 50 years ago (1956). He also revealed its applicability to mapping out the wavefunction of the bound electron of the shallow donor in silicon (Feher 1959). Its utility in probing the wave function of shallow donors in ZnO nanoparticles has recently been demonstrated (S. B. Orlinskii 2005). Long spin-lattice relaxation times and low defect concentrations place stringent conditions on the experimental techniques required to observe defects in silicon with EPR. To be able to analyze the hyperfine spectra of 29Si, one must be able to observe defect concentrations of less than 3 × 1013 cm−3. Additionally, the defects in silicon are observable only at low sample temperatures (4–40 K) and magnetic field modulation frequencies of less than 1 kHz (Brower 1977). Up until the mid1980s, the use of EPR for the characterization of group III–V semiconductors was also not as widespread; the high nuclear spins of the constituent atoms in a III–V lattice cause experimental difficulties (Kennedy and Wilsey 1985). This is no longer the case, and EPR is well-suited to investigating solids such as Mn-doped GaAs, where g factors are helpful in distinguishing between neutral and ionized acceptors as well as MnGa-containing complexes (Weiers 2006). Several reviews have been written about the application of positron annihilation lifetime spectroscopy (PAS) to the identification of defects in solids (Siegel 1980; Seeger and Banhart 1990; Puska and Nieminen 1994; Dupasquier and Mills 1995; Eldrup and Singh 1997; Kauppinen 1997; Krause-Rehberg and Leipner 1999). According to a Monte Carlo study of defect sensitivity limits using positron lifetime spectroscopy, defect detection levels of approximately 1013/cm3 are possible when high statistics spectra and high-intensity beams are used (Chin et al. 2001). Positrons injected into a semiconductor become preferentially trapped in features such as vacancies, vacancy clusters, and negative charge centers, where they subsequently

3.1 Experimental Characterization

41

annihilate from the trapped state in the defect. The positron annihilation rate can be written as

λ = π re2 c ∫ dr Ψ + ( r ) n ( r ) γ ⎡⎣ n ( r ) ⎤⎦ 2

(3.2)

where re is the classical radius of the electron, c is the speed of light, Ψ+(r) is the positron wave function, n(r) is the electron density, and γ[n] is the enhancement factor of the electron density of the positron (Kauppinen 1997). The positron trapping rate at a defect is defined as

κd =

μd [ X ] N

(3.3)

where µd is the defect-specific positron trapping coefficient, [X] is the concentration of defect X, and N is the atomic density of the lattice (Kauppinen 1997). The lifetime of positrons trapped in defects is long in comparison to those that annihilate at defect-free regions (Dutta et al. 2005). The lifetime of positrons trapped at a defect varies with the defect charge state (Saarinen 1991; Makinen et al. 1992). In GaAs, for example, the lifetime of a positron trapped in VAs−1 is 37 ps less than one trapped in VAs0 (Saarinen et al. 1991). Similarly, the (−1) charge state of PSiVSi in P-doped Si has a 18 ps smaller positron lifetime than that of the same pair in the neutral charge state (Makinen et al. 1992). Additionally, annihilation characteristics can be used to shed light upon the nature, structure, and abundance of defects in the bulk (Alatalo et al. 1993; Eldrup and Singh 1997; Ghosh et al. 2000; Dutta et al. 2005). PAS, as well as another variant of the technique, variable-energy positron annihilation spectroscopy (VEPAS), have been used to identify multi-vacancy clusters and negative VSi-related complexes in irradiated Si (Coleman et al. 2007) and SiC (Aavikko et al. 2007). Coincidence Doppler broadening (CDB), a complementary technique to PAS, is useful for characterizing the momentum distribution of the annihilation electrons in a material (Szpala et al. 1996; De et al. 2000; Ghosh et al. 2000; Chakrabarti et al. 2003; Dutta et al. 2005). In compound semiconductors, vacancies in different sub-lattices can be differentiated and vacancy-impurity pairs can be identified (Wang et al. 2007). Deep level transient spectroscopy (DLTS), introduced by Lang in 1974 (Lang 1974), is widely used to study electrically active deep level defects in semiconductors. Under ideal circumstances, it can be used to detect defect concentrations as low as 10–6 times the carrier concentration of the semiconductor (Auret and Nel 1991; Stievenard and Vuillaume 1986; Kaminski 1993; Auret and Deenapanray 2004). It can also be used for the identification of radiation-induced defects (Khan and Yamaguchi 2007), extended defects (Gelczuk et al. 2005), and nearsurface defects, where the technique is instead influenced by the position and shape of the Fermi energy at the Schottky diode interface (Auret and Nel 1991).

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The experimentally obtained DLTS signature can be compared to libraries of known electronic signatures in order to positively identify a specific defect. DLTS is a space charge technique requiring the formation of p+n (n+p) or Schottky barrier junctions. As the voltage applied over the junction is pulsed, capacitance transients are measured (Svensson et al. 1995). These transients occur because carriers trapped at deep levels in the bandgap are thermally emitted when the junction bias returns to the steady state value. As the transients display an exponential time dependence, the activation energies for electronic transitions, capture cross sections, and concentrations of defects within the bandgap can be resolved. The key feature of DLTS is the specification of an emission rate window so that the measurement apparatus only responds to values of the capacitance transient with this range. As the temperature is varied, a peak in the instrument response is observed at the temperature associated with the trap emission rate. The transient capacitance of the defect filled during the time pulse tp is ΔC(t) = S0 exp[–en(T)t]

(3.4)

where en(T ) is the emission rate at a temperature T that reflects the majority carrier capture cross section and the energy level of the defect (Stievenard and Vuillaume 1986). The shift in the DLTS peak position as a function of the rate window determines the dependence of the emission time constant on temperature. As the transients display an exponential time dependence, the activation energies for electronic transitions, capture cross sections, and concentrations of defects within the bandgap can then be resolved. The parameter S0 can be broken down into S0 = C0 ([X]/2Nd)*f(W)

(3.5)

where [X] is the concentration of defect X, ND the free carrier concentration, and C0 the capacitance for the polarization V0. The function f(W) describes the region in the bulk of the semiconductor where the trap is filled: f(W) = (W0 – λ0)2 – (W1 – λ1)2/W02

(3.6)

where W0, W1 are the space-charge regions for the reverse bias V0 and V1, respectively, and λ0 and λ1 are the depths for which the defect energy level crosses the Fermi level for the polarizations V0 and V1. Experimental studies can be complemented with computational predictions to identify DLTS levels where the capture is inhibited due to band bending (Fleming et al. 2007). The sensitivity of DLTS, when used in the capacitive mode, decreases linearly with the distance towards the junction, reaching zero at the junction. Small uncertainties in the determination of the position of a DLTS peak can cause significant errors in the evaluation of the parameters of electronic states deep within the band gap; it is difficult to distinguish defects that are very close in energy. Several modifications have been developed to improve the accuracy of the analysis and its sensitivity to noise, including fitting and correlation methods (Su and Farmer 1990; Dmowski 1992; Doolittle and Rohatgi 1994; Istratov 1997; Hanine et al. 2004). Also, “positron-DLTS,” high resolution Laplace DLTS (LDLTS), and synchrotron radiation DLTS (SRDLTS) can be used in place of the conventional

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43

technique to overcome sensitivity issues and glean information about closely spaced electronic levels in the band gap and the origin of deep-level defects (Beling et al. 1997; Evans-Freeman and Abdulgader 2004; Fujioka et al. 2004; Dobaczewski et al. 2004). “Positron-DLTS,” which monitors the transients in the electric field of the depletion region, rather than the inversely related depletion width, as deep levels undergo ionization, can be used to complement conventional DLTS (Beling et al. 1997; Beling 2002; Deng et al. 2002). It may allow for the determination of extra microstructure information about deep-level defects, such as whether a deep-level defect is vacancy-related. High resolution Laplace DLTS (LDLTS) overcomes some sensitivity issues of DLTS and provides information on closely spaced electronic levels in the bandgap by recording and analyzing nonexponential capacitance transients (Evans-Freeman and Abdulgader 2004; EvansFreeman et al. 2002; Dobaczewski et al. 2004; Makarenko and Evans-Freeman 2007). Synchrotron radiation DLTS (SR-DLTS) replaces the electrical pulses of conventional DLTS with pulses of synchrotron radiation, and can be used to glean information about the origin of deep-level defects (Fujioka et al. 2004). It was partially inspired by combination of x-ray absorption fine structure (XAFS) with synchrotron radiation, which has been used for local structural analysis of semiconductor defects (Ishii et al. 1999; Ishii et al. 2000). Lastly, Goto and co-workers have developed a direct method for observing neutral and charged vacancies in crystalline silicon using low-temperature ultrasonic measurements (2007). Ultrasonic pulse echo method was utilized for sound velocity measurements of silicon. The magnetic dependence of the low-temperature softening in B-doped Si yields information about the charge states of the defects in the bulk.

3.1.2 Indirect Detection of Bulk Defects The charge states of bulk defects can also inferred indirectly from physical and electrical measurements. For example, measurements of TiO2 electrical conductivity versus oxygen partial pressure have been used to develop defect models that account for charged vacancies and interstitials (Bak et al. 2002; Nowotny et al. 2007). The ionic conductivity is proportional to the oxygen partial pressure raised to an exponent that depends upon oxygen vacancy charge state (Kofstad 1972). Thermopower measurements, which provide information about the conductivity components arising due to both electrons and holes, can be utilized in a similar fashion (Rekas 1986). The defect models under examination detail a series of charged defect formation reactions and corresponding equilibrium constants. When charge neutrality conditions are assumed, intrinsic equilibrium constants and defect concentrations can be determined. More complicated expressions can be developed to account for impurities as well as the associations between impurities and native point defects.

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3 Experimental and Computational Characterization

3.1.3 Diffusion in the Bulk As the direct detection of semiconductor defects is not always feasible, diffusion measurements can serve as a valuable tool for inferring information about the type and concentration of bulk point defects. Techniques for diffusion measurement are typically easier to use than those for direct defect detection. For example, self-diffusion within a semiconductor can be measured directly with the use of an isotopic tracer together with secondary ion mass spectroscopy (Bracht et al. 1998; Ural et al. 1999; Seebauer et al. 2006). The diffusion coefficient is obtained by fitting the distribution of tracer atoms resulting from a heat treatment to theoretical distribution profiles (Seeger and Chik 1968). These experiments can be performed on both intrinsic and doped material. When a tracer method is used, an allowance is made for a correlation factor, f, where the self-diffusion DSD coefficient is related to the tracer diffusion coefficient DT according to DT = f DSD.

(3.7)

Correlation factors are related to both defect type and crystal lattice structure. For example, f is equal to 0.5 and 0.78 for the vacancy mechanism in the diamond and face-centered cubic lattice, respectively (Compaan and Haven 1956, 1958; Jarzebski 1973). The total tracer diffusion coefficient can be broken down into contributions of individual types of point defects such as vacancies and self-interstitials (Pichler 2004).

DT = f I ⋅ D I + f V ⋅ DV .

(3.8)

The respective contributions of interstitial and vacancy diffusion can be identified by an assortment of experimental methods, usually in which the concentration of one type of defect is varied. Early investigations were based on oxidation-enhanced diffusion, in which surface oxidation injects excess interstitials into the bulk. The oversaturation of self-interstitials can be estimated from the growth of stacking faults (Pichler 2004). Gosele and Tan developed a method that correlates the retarded migration of extrinsic dopants to the fraction of self-interstitials contributing to diffusion (1983). Also, it is possible to compare the diffusivities of multiple dopants under non-equilibrium conditions in the bulk of the semiconductor. The best example of this technique, including how to extract numerical diffusivity values from experimental measurements, has been described by Fahey et al. (1985). Self-diffusion in silicon has been monitored using both the stable isotopes 29Si and 30Si and the unstable isotope 31Si (Pichler 2004). Compound semiconductors can be probed with isotopes of either one of their constituent elements. For example, the diffusion of Ga and As tracers in GaAs has been studied, as well as that of dopants such as aluminum, sulfur, and antimony, typically as constituents of a superlattice (Schultz et al. 1998; Egger et al. 1997). Similarly, measurements of the diffusion kinetics of both metal and oxide can be obtained for the metal oxide semiconductors such as TiO2, ZnO, UO2, and CoO. An alternate method for the determination of point-defect migration energies involves high-energy electron,

3.1 Experimental Characterization

45

photon, or ion irradiation. The irradiation creates point defects at temperatures low enough to prevent their migration in the bulk. By monitoring the annealing behavior of these excess defects at higher temperatures, one can deduce their migration energies (Fahey et al. 1989). Point defects created in the bulk by the ion implantation of dopant atoms can also be monitored to obtain information about the energetics of diffusion. During annealing, transient enhanced diffusion occurs in the bulk. If this phenomenon is caused by the point defects created by implantation, the migration energy of the defect responsible for diffusion equals the activation energy of diffusion at the beginning of the transient time period (Fahey et al. 1989).

3.1.4 Direct Detection of Surface Defects The surface geometries and reconstructions of defect-free semiconductor surfaces have been investigated with low-energy-electron-diffraction (LEED) (Van Hove et al. 1986), reflection high-energy electron diffraction (RHEED) (Joyce et al. 1985; Clarke and Vvedensky 1988; Farrell and Palmstrom 1990), electron energy loss spectroscopy (EELS) (Dubois et al. 1987; Jones 1992), inelastic electron tunneling spectroscopy (IETS) (Salace et al. 2002; Petit et al. 2003; Huang et al. 2006; Lye et al. 1997; Balk et al. 1991; He and Ma 2003), scanning tunneling microscopy (STM), and transmission electron microscopy (TEM). Defects can be explored by both averaging and targeted imaging methods. Averaging methods examine the whole surface, while targeted imaging methods examine individual surface defects, and include techniques such as STM. Surface diffusion measurements have been used in a fashion akin to bulk diffusion measurements to shed light upon surface charging effects. Optical spectroscopies based on the reflectance of ultraviolet and visible light, such as photoemission spectroscopy, surface differential reflectance spectroscopy (SDRS), and reflectance anisotropy spectroscopy (RAS, also called reflectance difference spectroscopy (RDS) by some authors), and photoreflectance spectroscopy (PR) provide information on the structure and electronic nature of large-area surfaces, including changes induced by temperature or adsorption (Borensztein 2005; Palumbo et al. 2002). Conventional photoemission spectroscopy, based on the energy analysis of photoelectrons emitted by a surface bombarded with ultraviolet or x-ray photons, provides information about the localized electronic states that are responsible for Fermi energy pinning at certain semiconductor device interfaces and chemisorption bonds (Margaritondo and Franciosi 1984). Photoemission spectroscopy can also be coupled with a synchrotron radiation source to explore electronic band structure, surface sensitivity, surface core-level shifts, and the nature of electronic surface states (Margaritondo and Franciosi 1984). SDRS can be used only with samples that display a crystalline or morphological anisotropy on a dimension range equivalent to the size of the optical beam. As such, its use is limited to surfaces on which adsorption induces a change in surface optical anisotropy (Borensztein and Witkowski 2004).

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3 Experimental and Computational Characterization

Photoreflectance is one of a class of modulation spectroscopies in which a semiconductor is periodically perturbed and the resulting change in dielectric constant is detected by reflectance (Carlson et al. 1993). For both SDRS and RAS, photogenerated minority carriers migrate to the interface and recombine with charge stored there. The resulting change in built-in field affects the surface reflectance in narrow regions of wavelength corresponding to optical transitions of the substrate material. The presence of a non-zero PR spectrum demonstrates unequivocally the existence of surface band bending, and experiments as a function of temperature and pump intensity can yield useful information about the electronic band structure of the surface. For example, this laboratory has used photoreflectance to probe the band bending, or defect-induced changes in the position of the surface Fermi energy, at Si(100)-SiO2, Si(111)-SiO2, and Si(100)-Si3N4 interfaces (Dev et al. 2003; Dev and Seebauer 2005, 2004). The atomic resolution afforded by scanning tunneling microscopy enables the characterization of defects, including their atomic configuration and real-time migration, on semiconductor surfaces (Stievenard 2000). Researchers have also recently demonstrated the applicability of Kelvin probe force microscopy (KPFM) to obtaining high-resolution defect images and local surface charge densities (Lagel et al. 1999; Rosenwaks et al. 2004). As point defects induce localized perturbations that give rise to shallow or deep ionization levels in the band gap, voltage-dependent depressions surrounding STM and KPFM-imaged surface defects serve as charge state signatures (Ebert 2002; de la Broise et al. 2000). Attention has to be paid to the interaction between the substrate and the tip-sample system, however, as the charge state of the defect can be influenced by the presence of the tip (de la Broise et al. 2000). Also, electrons can tunnel directly between the defect gap states and the tip states, causing a permanent current between the electrodes and affecting the spectroscopy of the charged defect (de la Broise et al. 2000). Despite these limitations, STM may still be useful for probing the hole capture rate of single deep levels with atomic-scale resolution (Berthe et al. 2008). Berthe and co-workers have illustrated the applicability of this method for exploring the capture and relaxation of charge carriers by point-defect states in nanostructures. Kubby and Boland have published an extensive review article on the principles, experimental methods, and characterization of semiconductor surfaces with STM (1996). A more pertinent discussion on the theory and issues behind imaging charged defects can be found in an article by de la Broise (2000). STM images of neutral and charged surface defects on silicon, germanium, gallium arsenide, and other III–V semiconductors will appear throughout this review (Brown et al. 2002; Lee et al. 2000; Lengel et al. 1994; Semmler et al. 2000; Ebert et al. 2001).

3.1.5 Diffusion on the Surface Numerous methods exist to measure surface diffusion (Seebauer and Jung 2001), although only a few have been employed to examine surface charging on semiconductors.

3.2 Computational Prediction

47

Indeed, most methods for measuring diffusion on Si, Ge, and GaAs surface carry serious drawbacks (Allen et al. 1996). Standard electron or ion probes often lack sufficient spatial resolution or induce structural damage. While several authors have explored the diffusion of surface vacancies with STM, the migration is sometimes artificially triggered by the STM tip (Ebert et al. 1992). In addition, high-resolution, non-damaging probes such as STM cannot follow coverage dependence in the diffusivity. Second harmonic microscopy (SHM) has been used to measure concentration-dependent diffusivities on semiconductor surfaces (Schultz et al. 1992; Schultz et al. 1993). SHM uses spatially resolved surface second harmonic generation (SHG) to monitor, without damage, adsorbate coverages at a resolution approaching the optical diffraction limit (~1 µm). Results have been interpreted in terms of surface point defect charging (Allen et al. 1996).

3.2 Computational Prediction It is not always possible to use the experimental characterization techniques discussed in Sect. 3.1. Charged defects can be unstable or experimentally invisible, induced or altered by cleaning and processing procedures, or difficult to probe in novel structures and devices. For these systems, theoretical calculations can give valuable insight into their electronic and structural properties. The sophistication of these methods has steadily increased over the last 50 years (Drabold and Estreicher 2007). For instance, charged defects on the surface of C nanotubes (McGuire and Pulfrey 2006) and in the bulk of BN nanotubes (Piquini et al. 2005) have been explored using multi-scale modeling and density functional theory calculations, respectively. Cluster models and DFT can also be used to calculate electronic structure and hyperfine coupling constants, rendering the interpretation of EPR spectra significantly easier (Esteves et al. 2008). However, the variety of density functional theory and atomistic methods available for calculating defect ionization levels also complicates the interpretation of results; different methods often yield inconsistent findings (Mahanti et al. 2007). For this reason, systems-based methods such as maximum-likelihood estimation are especially valuable for distilling the results of disparate methods into a single “best” value for a defect formation energy, ionization level, or activation energy of diffusion.

3.2.1 Density Functional Theory Ab initio density functional theory (DFT) is one of the most frequently used atomic scale tools for investigating the properties of defects in semiconductors. Indeed, a large fraction of the available literature for charged defect behavior originates from DFT calculations. The accuracy of DFT continues to improve, though the method has not yet shed some significant shortcomings. For example,

48

3 Experimental and Computational Characterization

solid crystals are usually modeled via an approximation in which a limited domain of atoms is defined and replicated with periodic boundary conditions. Finite availability of computational power often constrains the size of the domain and how the corresponding domain in momentum space is sampled, thereby introducing artifacts whose behavior can be unpredictable. Also, DFT is a ground-state theory that is not ideally suited for calculations of excited states. At 0 K where DFT calculations are performed, all states within the conduction band and many defect states within the band gap constitute antibonding, excited states. Indeed, it is well known that DFT significantly underestimates the value of a typical semiconductor band gap, sometimes by a factor of two. These issues lead to problems for neutral point defects and additional problems for charged defects. Few authors address these issues directly, and few compensate for them fully. One helpful exception is recent work due to Schultz, who has presented a novel modification to DFT that enforces electrostatic boundary conditions accounting for net charge effects. The method eliminates the need for ad hoc procedures to calibrate the electron chemical potential for each calculation, and handles defect dispersion with a new occupation scheme for populating states (Schultz 2006). This approach has successfully obtained ionization levels for a variety of silicon defects that span the full experimental band gap, suggesting that the band gap problem has been significantly mitigated. The work draws explicit attention to the fact that standard DFT must be coupled with modifications in order to replicate experimentally observed charged defect levels. Typically DFT is employed to calculate the defect formation energy G as follows:

( )

GXf q + e− q = ETot X q − ETot ( bulk ) − ∑ ni μi + q [ EF + Ev + ΔV ]

( )

(3.9)

i

where ETot X q and ETot ( bulk ) respectively denote the total energy of the domain in the presence and absence of the charged defect X with charge q. The defect is created by the addition or removal of ni atoms having chemical potential μi. The quantity EF represents the Fermi energy referenced to Ev, the valence band edge, and the correction term ΔV aligns the reference potential in the computational domain to that in the bulk. The domain usually employs a “supercell” geometry with periodic boundary conditions and a plane wave basis set (Payne et al. 1992). This approach typically describes the band structure of the crystal more accurately than cluster calculations, which often suffer from spurious quantum confinement effects that alter the band structure (Van de Walle and Neugebauer 2004). Prior to the use of supercells, the Mott–Littleton method was employed, which divided the lattice into an inner region where atomic relaxations were explicitly calculated, and an outer region where the continuum approximation was invoked (Harding 1990). DFT does not solve the Schroedinger equation but instead works in the realm of electron density while approximating quantum exchange-correlation effects by various approaches such as the local density approximation (LDA), local spin density approximation (LSDA), and generalized gradient approximation (GGA) (Jones and Gunnarsson 1989; Hafner et al. 2006). Such approximations lead to

3.2 Computational Prediction

49

a significant (~50%) underestimation of the semiconductor band gap (Mori-Sánchez et al. 2008). Deep electronic levels associated with point defects incur similar errors (Eaglesham 1995; Windl et al. 1998). Correction schemes to improve these errors have been proposed by numerous authors, many of which have been reviewed and assessed in a recent article by Castleton et al. (2006). These methods include hybrid functionals, the exact-exchange approach, and modified pseudopotentials (Segev et al. 2007; Deak et al. 2007). The Perdew–Zunger self-interaction correction to account for the LDA and LSDA yields improved values of total energy, binding energy, electron density, and orbital eigenvalues (Perdew and Zunger 1981; Chan et al. 2003). The Makov–Payne correction scheme, published in 1995, works well for atomic or molecular systems calculated in otherwise empty supercells, although it often leads to overly high corrections for defects in semiconductors (Van de Walle and Neugebauer 2004; Schwarz et al. 2000; Makov and Payne 1995; Kantorovich and Tupitsyn 1999; Schultz 1999, 2000; Castro et al. 2003; Nozaki and Itoh 2000). Even though these correction schemes pervade the computational literature, it has been suggested that they still introduce large errors in the calculated total energies for charged defects in semiconductors (Lento et al. 2002; Van de Walle and Neugebauer 2004). Another problem is that DFT calculations typically incorporate only 100–1,000 atoms within the supercell, resulting in the imposition of unnatural boundary conditions (Castleton et al. 2006). In supercells containing a defect, the introduction of periodic boundary conditions in effect populates the “infinite” solid with a substantial concentration of regularly spaced defects. With small supercells, the interval between defects becomes much shorter than in a real solid. The defects interact with each other in an artificial way that can be eliminated only in the limit of an infinitely large supercell. The interactions sometimes have a large effect on the calculated defect formation energies (Shim et al. 2005). For charged defects in particular, the supercell approximation leads to a size-dependent defect–defect Coulomb interaction energy (Lento et al. 2002). Consequent errors in ionization level scale linearly with the charge, q. Computations performed with small supercells typically overestimate ionization level positions with respect to the valence band maximum (Puska et al. 1998). Various methods for deconvoluting spurious supercell size and shape effects from the charged defect formation energy have been proposed (Wright and Modine 2006). A uniform, neutralizing “jellium” background charge (Lento et al. 2002) is commonly employed. More sophisticated approaches sometimes employ a multipole expansion to reduce spurious defect–defect interactions (Price and Platzman 1991), but errors arising from multipole interactions still reduce the accuracy of the formation energy (Castleton and Mirbt 2004; Makov and Payne 1995). Other methods include the local moment countercharge (LMCC) method by Schultz, based on the linearity of the Poisson equation (1999; 2000), and an alternate supercell modification method that identifies a common electron reservoir for all defects and mitigates with defect energy banding (Schultz 2006). Modified gallium pseudopotentials for use in DFT calculations of the zincblende cubic phases of GaAs, GaP, and GaN (von Lilienfeld and Schultz 2008).

50

3 Experimental and Computational Characterization

An additional issue for investigation of diffusion is that most quantum calculations are performed tacitly assuming a temperature of 0 K, but diffusion mechanisms can change at higher temperatures. For example, diffusion within Si at typical processing temperatures appears to be governed by collective atomic motions that do not operate at lower temperatures (Eaglesham 1995). Moreover, DFT calculations typically ignore entropic effects, some of which can change pre-exponential factors by many orders of magnitude.

3.2.2 Other Atomistic Methods The shortcomings of the DFT approach have been addressed by an assortment of other computational techniques, although most are only valid for the investigation of neutral defects. Classical molecular dynamics and total-energy pseudopotential calculations of neutral defects are also often limited to systems of less than 106 atoms and nanoscale-order time scales (Kohn and Sham 1965; Payne et al. 1990; de la Rubia 1996; Heine 1975; Vogel et al. 1997; Van de Walle and Neugebauer 2004; Stadele et al. 1999; Car et al. 1985). Green’s function calculations of the electronic structure of neutral defects in solids were common during the 1980s and early 1990s (Car et al. 1984; Baraff et al. 1983; Wachutka et al. 1992; Scheffler 1982). More recently, Green’s function implementation in linear muffin-tin orbital calculations has been demonstrated by Gorczyca et al., although these authors only considered unrelaxed substitutional defects in GaN, AlN, and BN (1997; 1999; 2002). By combining the supercell approach with the full-potential version of the linear muffin-tin orbital method, thermodynamic defect ionization levels can be estimated as the center of gravity of the impurity band determined by means of the density-of-states functions (Gorczyca et al. 2002). Hartree–Fock (HF) based models seek to approximately solve the electronic Schrödinger equation for quantum many-body systems. For example, Nolte has considered the discrete charge states associated with mesoscopic point-like semiconductor defects using the unrestricted Hartree–Fock approximation (1998). Ab initio HF calculations are highly resource demanding, fail to consider correlation effects, and can be applied only to systems with small numbers of atoms (Van de Walle and Neugebauer 2004). More recently, the ab initio HF approach has been largely replaced by less rigorous tight-binding (TB) schemes for the investigation of charged defects in semiconductors (Horsfield and Bratkovsky 2000; Colombo 2002; Van de Walle and Neugebauer 2004; Elstner et al. 1998; Jenkins and Dow 1989; Jenkins et al. 1992). Empirical TB methods suffer from variability in the selection of parameters (Jenkins et al. 1992; Jenkins and Dow 1989; Strehlow et al. 1986), while first-principles TB methods depend critically on the specifically designed local orbitals that are used to calculate the Hamiltonian matrix elements (Elstner et al. 1998; Van de Walle and Neugebauer 2004). The tight-binding method has proven useful for the study of shallow impurities in semiconductor nanostructures and nanowires, however (Diarra et al. 2007).

3.2 Computational Prediction

51

Monte Carlo calculations, which were first used to study diffusion processes, have also been utilized to explore the properties of defects and impurities in semiconductors (Van de Walle and Neugebauer 2004; Law et al. 2000; Bunea and Dunham 1998). The Quantum Monte Carlo method can be used to study diffusion since most thermal vibrations do not lead to a net movement of atoms with the crystal lattice (Bunea and Dunham 1998). The specifics of the Quantum Monte Carlo (QMC) method including use of periodic boundary conditions and pseudopotentials and sources of error can be found in a recent article by Needs (2007). The kinetic lattice Monte Carlo (KLMC) approach is used to consider only processes that lead to atomic jumps. The method can also be altered to ignore host atoms and consider only intrinsic and extrinsic point defects and their agglomerates (Bunea and Dunham 1998). Such simulations can be used to calculate charged defect concentration as a function of Fermi level and illustrate Fermi leveldependent defect-mediated diffusion phenomena (Qin and Dunham 2003). MartinBragado et al. have also recently developed an approach for the modeling of dopant diffusion in Si for implementation in an atomistic KMC process simulator that accounts for multiply charged defects and a local Fermi energy-dependent electric bias (2005a; 2005b; 2006). For charged defects, their model considers the dependencies of charge reactions, electric bias, pairing and break-up reactions on the local Fermi level. Also, Alfe and Gillan have reported quantum Monte Carlo calculations of the formation energies of (+2) Mg vacancies and (−2) oxygen vacancies in MgO (2005).

3.2.3 Maximum Likelihood Estimation Despite the variety of experimental and computational methods for studying defects and their charging behavior, the results have often proven conflicting and unreliable. For example, experimental results have been used to justify diffusion coefficients for the Si self-interstitial that vary by more than ten orders of magnitude at typical processing temperatures (Eaglesham 1995). In light of these problems, rationally defensible procedures for estimating ionization levels and other thermodynamic and kinetic parameters must be accepted in place of certain truth about their values. One method that deals directly with this problem is the statistical technique of maximum likelihood (ML) parameter estimation (Beck and Arnold 1977). This approach gives the most likely value for each parameter based on the available literature, and estimates the corresponding uncertainty. The maximum likelihood approach has proven quite useful in predicting ionization levels for interstitial atoms in Si (Jung et al. 2005), as well as their diffusion coefficients (Jung et al. 2004) and rate constants for their exchange with the lattice (Jung et al. 2004) and the surface (Kwok et al. 2005; Seebauer et al. 2006). This approach can be extended by a combination of parameter sensitivity analysis (Jung et al. 2003) and the method of maximum a posteriori estimation (Gunawan et al. 2003) to yield

52

3 Experimental and Computational Characterization

further refinements of these quantities for use in predictive models for phenomena such as transient enhanced diffusion in ion implantation technology (Wang and Seebauer 2005). In ML estimation, the most likely value y for a given parameter is obtained by minimizing the objective function (Gunawan et al. 2003)

Φ ( y ) = ∑ wi ( yi − y ) , 2

(3.10)

i

where yi denotes the estimate for the parameter drawn from a particular report i in the literature, and wi is a weighting factor that accounts for the accuracy of yi . Setting the derivative of Φ( y ) with respect to y equal to zero yields an analytic formula for y :

∑ wi yi y= i . ∑ wi

(3.11)

i

The weighting factors wi can be computed based on the common assumption (Beck and Arnold 1977) that the uncertainty in yi obeys a normal (Gaussian) distribution. Each weighting factor wi is set to equal the inverse of the variance σ i 2 , where σ i is the standard deviation of yi : wi =

1

σ i2

.

(3.12)

The uncertainty in the most likely value y is quantified through an averaged standard deviation given by

σ=

1

∑ σ1 i

i

.

(3.13)

2

Parameter estimates can then be reported in the form y ± σ . In some experimental papers, particularly for temperature-dependent diffusion, the values of the activation energy and pre-exponential factor must be extracted directly from linear fits of ln(D) versus reciprocal temperature (1/T) using an ordinary least square estimator. Assuming that errors are additive and satisfy standard Gauss–Markov assumptions (i.e., that the errors are uncorrelated, have zero mean, and have constant variance), σ i for the paper then obeys (Beck and Arnold 1977)

σi =

min ∑ (Yk − b1 X k − b0 ) b1 ,b0

( m − 2) ∑ ( X k − X )

2

2

.

(3.14)

Here, X and Y respectively stand for 1/T and ln(D) for each data point; X denotes the mean of X; m denotes the total number of data points, and b1 and b0 represent the fitting parameters, i.e., the activation energy and the prefactor.

3.2 Computational Prediction

53

3.2.3.1 Handling Difficulties with Confidence Intervals

Application of ML estimation requires specification of a quantitative uncertainty

σ i for each parameter value yi employed in the computation. When a report pro-

vides the parameter value and corresponding confidence interval, the numerical values can usually be employed as-is for purposes of ML estimation. However, in some cases the report provides no confidence interval. In other cases, the confidence interval is so narrow that the corresponding weighting coefficient computed from Eq. 3.12 dominates those from all other reports. The ML parameter value is thereby unnaturally skewed. This phenomenon occurs especially often when the distribution of reported parameters includes values that lie near zero. If the relative errors for the various reports are similar, than the absolute values of σ i for the yi values near zero will be small. For example, consider two ionization level reports, 0.05 eV and 0.5 eV, each given with a 20% confidence interval. The absolute values of the confidence intervals will be 0.01 eV and 0.1 eV, so that the corresponding weighting factors from Eq. 3.12 will differ by an order of magnitude. The “most likely” value y for the parameter given by Eq. 3.11 will therefore be skewed close to 0.05 eV. One way to estimate confidence intervals for these various situations is to employ a suitably chosen average (to supply σ i where a value is lacking, or to replace an unnaturally small reported value of σ i ). In the present work, an average value σ has been calculated (from two or more reports) for the major experimental methods employed to measure ionization levels. One exception is IR spectroscopy, for which only one exceedingly small (0.003 eV) error estimate has been found. Consequently, σ for defect ionization levels obtained using IR spectroscopy has been set to the average of all of the other experimental techniques, 0.05 eV. Table 3.1 shows the values of σ employed in this work for each experimental technique, along with the references from which they derive. Table 3.1 Average maximum likelihood ionization level standard deviation for an assortment of experimental techniques Technique

σ

Defect formation and annealing Diffusion measurements DLTS EPR Hall effect

0.06

IR spectroscopy PACS PAS Photoconductivity

(eV) References

0.1 0.02 0.05 0.01 0.003 (0.05) 0.03 0.03 0.05

(Emtsev et al. 1989; Lugakov and Lukashevich 1989; Srivastava and Singh 1996) (Awadalla et al. 2004; Bracht et al. 1995) (Pensl et al. 2003; Deng et al. 2002; Bliss et al. 1989) (Son et al. 2002; Trueblood 1967; Pereira et al. 2003) (Krause-Rehberg et al. 2000; Krause-Rehberg and Leipner 1999; Krause-Rehberg et al. 1998) (Dornen et al. 1989) (Haesslein et al. 1997; Haesslein et al. 1998) (Corbel et al. 1988; Deng et al. 2002; Shan et al. 1997) (Benkhedir et al. 2004; Benkhedir et al. 2006)

54

3 Experimental and Computational Characterization

Computational papers cannot be treated in such a straightforward fashion. DFT-based papers rely on various approximations that contain biases, utilize varying supercell sizes and Brillouin zone sampling, and sometimes provide only defect formation energies rather than the ionization levels of interest here. In cases where a single paper produces several estimates for a parameter using a family of closely related methods (such as the local density approximation and the generalized gradient approximation), yi has been taken here as the average of the individual estimates yij . The standard deviation can then be calculated using the formula

σi =

∑ ( yij − yi )2 j

n −1

,

(3.15)

where n is the total number of estimates in the paper. This procedure tacitly assumes that all estimates reported by a particular laboratory have equal probability of being accurate. In computational studies that give multiple ionization level estimates using different supercell sizes and Brillouin zone sampling, there is no commonly accepted procedure for calculating σ i when it is not given. It is best to examine the relationship between the independent variables and ionization levels. The work of Puska et al. gives an example of the issues involved (1998). In that work, the authors used four different supercells containing 32, 64, 128, and 216 atomic sites. Larger supercells should minimize artifacts arising from spurious defect-defect interactions. Additionally, the authors used four different k-point sets: Γ, Γ + L, 23, and 33. Different k-point sampling should also influence the defect–defect interactions within the supercell. Table 3.2 illustrates how the ionization levels of the silicon vacancy determined by Puska et al. varied with supercell size and k-point sampling. While the position of the (+2/0) level converges toward a single value with increasing supercell size, the position of the (0/−2) level does not. In cases where the ionization level converges with increasing computational sophistication, yi has been taken in this work as the value from the most sophisticated Table 3.2 Supercell size dependence of the calculated ionization levels of the Si vacancy Size

(+2/+1)

(+2/0)

(+1/0)

(0/−1)

(−1/−2)

(0/−2)

64 & Γ 64 & Γ + L 64 & 23 64 & 33 128 & Γ 216 & Γ Experimental

(0.50) (0.40) − − (0.32) (0.19) (0.13)

0.43 0.39 − − 0.28 0.15 0.09

(0.35) (0.38) − 0.05 (0.24) (0.11) (0.05)

(0.70) 0.62 0.26 0.41 0.39 (0.57) −

(0.59) 0.66 0.33 0.53 0.52 (0.40) −

0.65 (0.64) (0.30) (0.47) (0.64) 0.49 −

Adapted from (Puska et al. 1998) where experimental results from (Watkins 1986) are provided for purposes of comparison. Numbers in the parentheses correspond to the transitions between a thermodynamically stable and an unstable charge state. Notice how the computational defect ionization levels change as a function of supercell size and k-point set. All of the values are in eV.

3.2 Computational Prediction

55

calculation, while σ i has been set to equal the average of standard deviations from other laboratories using a similar computational method. This average is approximately 0.1 eV for both GGA- and LDA-based DFT reports. If the position of the defect ionization level does not converge with increasing supercell size or k-point sampling, yi has been taken as the average of the individual estimates, yij . The standard deviation can then be calculated using Eq. 3.15. The tendency for DFT-based methods to underestimate semiconductor lattice constants and band gaps has been discussed in Sect. 3.2. In some cases, this shortcoming directly influences the defect ionization levels reported in the literature (and, consequently, the values and errors obtained with maximum likelihood estimation). The primary cases involve reports that attempt to “improve” the accuracy of their estimates by adjusting or scaling the computational lattice constant or band gap to match the experimental value. For example, one study reports Ge divacancy ionization levels generated with the theoretical lattice constant (5.6078 Å, obtained by relaxation of a defect-free cluster) as well as the experimental lattice constant (5.6579 Å) (Coutinho et al. 2006). The levels obtained from the two approaches differ by 0.01 to 0.07 eV (depending on the particular level). In such cases, the ML estimates made here assume that the two approaches have an equal probability of occurring. Some computational reports reference defect ionization levels to the conduction band minimum, underestimate the band gap, and then attempt pinpoint “true” ionization levels (Foster et al. 2001; Schick et al. 2002). For example, the calculated band gap in the DFT-LDA calculations of charged defects in GaAs performed by Schick et al. is approximately 0.6 eV less than the experimental band gap. The authors attempt to shift the conduction band states to correct the band gap; defect states with “valence band character” are left fixed to Ev, while those with “conduction band character” are shifted by the necessary correction factor. This sort of band gap correction is also encountered for oxide semiconductors (Lee et al. 2001). As DFT is primarily a ground state theory, uncorrected ionization level (referenced to the valence band maximum) have been employed for ML estimation in the present work (where available). When ionization levels were referenced to the conduction band minimum, they have been adjusted here by the experimental band gap value at 300 K. As ionization levels are computed from charged defect formation energies, the standard deviation of an ionization level (not always included in a computational report) can also be calculated from these formation energies. An ionization level is defined as the Fermi energy at which the formation energies of a defect in two different charge states are equivalent. The ionization level error is thus related to the errors of two defect formation energies according to 2 2 σ i = σ FE , q + σ FE , q 1

2

(3.16)

where σ FE ,q1 is the standard deviation of the formation energy for the defect with charge q1 and σ FE , q2 is the standard deviation of the formation energy for the defect with charge q2.

56

3 Experimental and Computational Characterization

Note that for both experimental and computational estimates of a parameter, the presence of systematic bias errors undercuts the primary assumptions of ML estimation. If the bias errors can be estimated quantitatively for all reports within a given set of results, suitable corrections can be applied before ML estimation begins. If the magnitude of the bias error is unknown, however, alternate techniques of greater sophistication are required that generally give results of uncertain reliability.

3.2.4 Surfaces and Interfaces Computational methods can apply in equal measure to surface defect phenomenology, though their application for this purpose is comparatively recent. DFT has been the method of choice. For example, the charging of surface vacancies on Si(100) and Si(111) has been explored with density-functional theory by Dev and co-workers (Chan et al. 2003; Lautenschlager et al. 1987; Sasaki and Maier 1999; Lengel et al. 1996). This laboratory adapted the approach of Van Vechten and Thurmond for estimating ionization entropies to estimate surface vacancy ionization levels at processing temperatures (Dev and Seebauer 2003b; Dev and Seebauer 2003a, 2004). With regard to other materials, considerable attention has focused upon III–V surfaces such as GaAs(110), InP(110), InAs(110), and InSb(110) (Kim and Chelikowsky 1996; Cox et al. 1990; Qian et al. 2002; Ebert et al. 2000; Zhang and Zunger 1996; Schwarz et al. 2000; Domke et al. 1998; Lengel et al. 1996). Models of scanning tunneling microscopy images for III–V surfaces have been proposed (Kim and Chelikowsky 1996; Zhang and Zunger 1996) that seek to replicate the experimentally observed surface defect geometries and electronic structure. The defect chemistry of oxide semiconductor surfaces is still poorly understood, although recent DFT investigations have yielded some preliminary results (Menetrey et al. 2004; Rasmussen et al. 2004; Fox et al. 2006; Wang et al. 2005). Spin-polarized calculations performed using the generalized gradient approximation and the embedded-cluster numerical discrete variational method have also been used to explore oxide semiconductor surface defects (Perdew et al. 1996; Perdew et al. 1992; Chen et al. 2001).

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Chapter 4

Trends in Charged Defect Behavior

Chapters 5–8 serve as a compendium of published knowledge regarding the structure, thermodynamics and diffusion of charged semiconductor defects. As a preamble to that presentation, it seems appropriate to separately summarize broad generalizations in defect behavior that operate across several classes of semiconductor materials and across surfaces vs. bulk. The present chapter serves as that summary. Some trends described here do not appear to have been previously identified, especially in such a comprehensive manner; those trends are indicated below.

4.1 Defect Formation Certain crystal properties such as close-packed cells and mismatches in the radii of compound and oxide semiconductor basis atoms inhibit the formation of interstitials and antisites. Trends in the defect chemistry of group IV, III–V, and oxide semiconductors cannot be explained merely by crystal lattice structure and basis, however. Stoichiometry can also influence the formation energies of defects in the bulk.

4.1.1 Effects of Crystal Structure and Atomic Properties Interstitial and antisite defects are most likely to form in “loosely packed” semiconductors, or those with a considerable degree of structural openness. Intuitively, crystal structures containing more open space are more likely to accommodate excess atoms (interstitials) or displaced atoms (antisites). Consequently, the formation energies of these defects in structurally open semiconductors are lower than in closely-packed semiconductors. Openness can be quantified by the atomic packing E.G. Seebauer, M.C. Kratzer, Charged Semiconductor Defects, © Springer 2009

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Table 4.1 Summary of easily or normally grown crystal structures for group IV, III–V, and oxide semiconductors along with their unit cell dimensions and atomic packing fraction Material

Crystal Structure

a (Å)

c (Å)

Volume (Å3)

APF

Si Ge GaAs GaP GaSb BN BN AlN GaN InN TiO2 TiO2 ZnO UO2 CoO

Diamond Diamond Zincblende Zincblende Zincblende Zincblende Hexagonal Hexagonal Hexagonal Hexagonal Rutile Anatase Hexagonal Fluorite Rocksalt

5.431 5.6579 5.6533 5.4508 6.09593 3.6155 2.504 3.112 3.1896 3.548 4.59–4.64 3.764 3.2495 5.471 4.26

– – – – – – 6.6612 4.982 5.1855 5.76 2.95–2.97 9.46646 5.2069 – –

160.2 181.1 180.7 161.9 226.5 47.3 36.2 41.8 45.7 62.8 63.0 134.1 47.6 163.8 77.3

0.34 0.34 0.35 0.34 0.35 0.31 0.20 0.46 0.42 0.50 0.78 0.74 0.55 0.71 0.70

The lattice constants a and c (T = 300 K) in Å and unit cell volume in Å3 are also included. Lattice constants are taken from (Adachi 2005).

fraction (APF), which represents the ratio of the volume of the atoms considered as touching spheres to the volume of the crystal unit cell (Callister 2002). Packing fraction is a function of lattice structure and size, as well as the ionicity, coordination, and radii of the constituent basis atoms. Table 4.1 lists the packing fractions of numerous group IV, III–V, and oxide semiconductors along with crystal structures and unit cell dimensions. For the most part, group IV and III–V semiconductors possess a large degree of openness. The crystal structures of the oxide semiconductors are considerably less open, except for that of ZnO, which has been mentioned in the literature as an advantageous material property (Nainaparampil and Zabinski 2001). It is interesting to compare the two types of BN presented in the table; the APF of zincblende BN is 0.31 while that of hexagonal BN is 0.20. In hexagonal BN under N-rich conditions, Ni in the (+3) or (+2) charge state has a low formation energy in comparison to the other defects in the bulk (Orellana and Chacham 2001). In the zincblende structure under the same conditions, positively charged nitrogen vacancies and negatively charged boron vacancies dominate for p-type and n-type material, respectively. Clearly, the greater degree of structural openness in hexagonal BN allows for excess nitrogen atoms to be accommodated more easily. Similar trends are observed in the defect chemistry of zincblende versus hexagonal AlN (Van de Walle and Neugebauer 2004). In comparing rutile versus anatase titanium dioxide, consider the temperature at which titanium interstitials overtake oxygen vacancies as the dominant defect in the bulk. At 10–1 Pa, the switch occurs

4.1 Defect Formation

65

at 1,100ºC in rutile in comparison to 580ºC in anatase. The more favorable formation of interstitials in anatase is attributed to the fact that the crystal lattice structure has a slightly lower APF than rutile (Weibel et al. 2006). At least to a rough approximation, the formation of defects in semiconductors not discussed in this book may be predicted by examining unit cell type and coordination. The number of atoms within the unit cell and the coordination of those atoms affect the APF. Depending on the coordination number of an atom, even in instances when atoms are in contact along the unit cell axis, large volumes are left empty along them in the interior of the cells. Thus, trends in unit cell types may have direct implications on structural openness. Many III–V semiconductors have zincblende structures containing 4 cations and 4 anions per unit cell, although those containing nitrogen and boron are often more stable in hexagonal structures. The hexagonal wurtzite structure is also preferred by some monoxides such as ZnO and BeO. The hexagonal crystal lattice has 2 cations and 2 anions per unit cell. Monoxides such as CoO typically crystallize in the face-centered cubic rocksalt structure, which contains 4 cations and 4 anions in each unit cell; there are some exceptions to this rule. The fluorite structure preferred in oxide semiconductors tetravalent atoms with large radii such as Zr, Hf, Ce, Pr, Tb, Po, Th, Pa, U, Np, Pu, and Am (Jarzebski 1973). The fluorite structure contains 4 cations and 8 anions per unit cell. Dioxides having formula MO2 usually possess either a rutile or fluorite crystal structure. The rutile structure is found with oxides of tetravalent metals such as Ti, Nb, Ta, Cr, Mo, W, Mn, Ru, Os, Ir, Ge, Sn, Pb, and Te (Jarzebski 1973). Lastly, the anatase structure adopted by TiO2 contains 4 cations and 8 anions. In the diamond, zincblende, and wurtzite lattices the atoms are all fourfold coordinated. In the rocksalt lattice, both the cations and anions are sixfold coordinated. The fluorite and anatase structures contain cations that are eightfold coordinated and anions that are fourfold-coordinated. In the rutile structure, each cation is sixfold coordinated whereas each anion is surrounded by three cations. Even in semiconductors with low atomic packing fractions, the formation of certain defects is unfavorable when the crystal basis comprises atoms with large differences in radii. The radii of the elements in common group IV, III–V, and oxide semiconductors are shown in Table 4.2. Notice the large covalent radius of Ga and small covalent radius of N in conjunction with the packing factor of wurtzite GaN. The moderate degree of openness in the unit cell as well as the mismatch in radii between the gallium and nitrogen atoms makes the formation of self-interstitial and antisite defects unfavorable; only vacancies have formation energies that are low enough to ensure appreciable concentrations in the bulk. When a large gallium atom seeks to form an interstitial or antisite defect such as Gai or GaN (for the case of GaN), the nearby atoms move away from the defect and cause energetically unfavorable strains in the stiff crystal lattice (Van de Walle and Neugebauer 2004). Similarly, when a small nitrogen atom is introduced into the crystal, strain-inducing displacements occur to accommodate its presence. These strains lead to high defect formation energies.

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Table 4.2 Covalent and ionic radii of the elements in common group IV, III–V, and oxide semiconductors Element

Covalent radius (Å)

Ionic radius (Å)

Si Ge B Al Ga In N P As Sb O Ti Zn U Co

1.17 1.22 0.81 1.25 1.25 1.5 0.7 1.1 1.21 1.41 – – – – –

– – – – – – – – – – 1.4 0.745 0.74 1.14† 0.79

Radii are from (Solyom 2007) and (Basak et al. 2003) (where indicated with †). In covalent crystals, the covalent radius is the relevant dimension. The covalent radius is merely the measured radius when to atoms are pulled closely together to the point at which they are just touching. Conversely, in ionic crystals, the ionic radius is pertinent. Ionic radii depend on two factors: the degree of ionization and the number of neighboring atoms (or coordination). The length of an ionic bond reflects a balance between the electrostatic Coulomb attraction of the unlike charges and the mutual Born repulsion of the positively charged nuclei.

4.1.2 Effects of Stoichiometry The defect types that exist in III–V and oxide semiconductors can be predicted with surprising accuracy based on the ambient partial pressures of the volatile constituents of the solid. This is not a new finding, although it sometimes remains unmentioned when discussed the charged defects in compound and oxide semiconductors. For that reason, it is worthwhile to underscore how partial pressure affects bulk defect chemistry. For compound and oxide semiconductors, the gas-phase composition of the system strongly affects the nature of charged bulk and surface defects. The partial pressures of the volatile constituents within the solid exert the most effects. The important defects in III–V materials vary from group III element-rich to group V element-rich conditions (Hurle 1999). For example, Ga-rich GaAs contains a negligible concentration of arsenic interstitials and arsenic antisites; the crystal incorporates mainly gallium interstitials and antisites. But under arsenic-rich conditions, the arsenic antisite increases in concentration, as does the gallium vacancy.

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67

Table 4.3 Semiconducting properties of binary metal oxides of non-stoichiometric composition as detailed by (Bak et al. 1997)

Deficit of metal Excess of oxygen Excess of metal Deficit of oxygen

p-type semiconductors

n-type semiconductors

CoO, NiO, FeO, MnO, Cu2O UO2 – –

– – TiO2, ZnO, CdO TiO2, ZrO2, Nb2O5, Ta2O5, WO3, CeO2, PuO2, SnO2, Bi2O3, PbO2

Most metal oxides are non-stoichiometric under typical laboratory and processing conditions, and the extent and nature of their non-stoichiometry alters the electrical properties of the material. As detailed in Table 4.3, oxides can be classified into two main categories, p-type and n-type, and four subcategories, metal deficient, metal excess, oxygen deficient, and oxygen excess. Kosuge (1994) and Jarzebski (1973) have discussed such phenomena at length in Chemistry of NonStoichiometric Compounds and Oxide Semiconductors. Both books include valuable sections on the derivation of Kröger–Vink diagrams, which employ principles of chemical equilibria to depict the concentration of ionized defects in the bulk as a function of oxygen partial pressure, as well as oxide semiconductor case studies. A metal-deficient p-type semiconductor such as CoO, for example, is typically represented as Co1–xO, where for any non-zero value of x, the crystal has an assortment of atomic defects. For the case of x > 0, the crystal has metal vacancies, VM, whereas for the case of x < 0, the crystal has extra metal atoms that are situated at interstitial positions, Mi (Kosuge 1994). Charged metal vacancies, VZn and VU, are the predominant defects in O-rich ZnO and UO2. Under metal rich or stoichiometric ambient conditions, these semiconductors are more likely to contain ionized oxygen vacancies and metal interstitials. Titanium dioxide falls into both the metal excess and oxygen deficient categories, however; its defect chemistry is characterized by a predominance of Tii+3 at low PO2 and VO+2 at high PO2, respectively. Behavior in reducing atmospheres has been quantified in considerable detail. TiO2 loses oxygen and becomes an oxygendeficient compound according to the following equilibrium: TiO2 ↔ TiO2–x + 0.5(x)O2

(4.1)

where x is the deviation from stoichiometry in the oxygen sublattice (Nowotny et al. 1997). The relationship between the concentration of defects in the bulk and the deviation from stoichiometry, x, in TiO2–x, is then given in its most thorough form by:

x=

(

)

2 ⎡⎣Tii+3 ⎤⎦ + ⎡⎣Tii+4 ⎤⎦ − ⎡⎣VTi−4 ⎤⎦ + ⎡⎣VO+2 ⎤⎦ . 1 + ⎡⎣Tii+3 ⎤⎦ + ⎡⎣Tii+4 ⎤⎦ − ⎡⎣VTi−4 ⎤⎦

(4.2)

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4.2 Defect Geometry Creation of a defect disturbs the translational symmetry of a solid or surface and ruptures bonds. Neighboring atoms readjust their positions by relaxations of various kinds, and sometimes form new bonds with other neighbors. The relaxations and rebonding produce new orbitals localized in the vicinity of the defect that often have the capacity to capture excess charge to varying degrees from the valence or conduction bands. The capture represents defect ionization, and depends upon charge availability within the solid as quantified by the Fermi level. Sometimes a wide array of rebonding and relaxation configurations is possible, with the most stable variety depending upon the number of charges captured. Indeed, the charge on the defect can strongly affect the magnitude, direction and symmetry of the ionized structure. The variety permits only a few generalizations connected mainly to how ionic the semiconductor is. Not surprisingly, the ion cores of ionic semiconductors interact electrostatically with localized charge in ways that purely covalent semiconductors do not (Robert et al. 1991; Hoglund 2006). A semiconductor can be classified covalent or ionic by comparing the electronegativities of its constituent atoms (Bouhafs et al. 1999; Martins et al. 2007). When the difference between the electronegativities of the elements in a compound is relatively large, the compound is classified as ionic. As the group IV semiconductors silicon and germanium possess only one type of element, they are purely covalent. Differences in electronegativity greater than 1.8 are great enough to classify a compound as “ionic” while differences less than 1.2 are found in “covalent” compounds. The more electronegative element of the pair will attract electrons away from the less electronegative element of the pair. On the basis of the Pauling electronegativities found in Table 4.4, the group III–V semiconductors are covalent while the oxide semiconductors are ionic. In covalent semiconductors such as Si and Ge, interstitial atoms typically cause neighboring atoms to move away from the extra atoms to relieve strain, while vacancy formation generally induces surrounding atoms to relax into the empty lattice site. The magnitude of the relaxation often depends upon charge state, as exhibited by VSi, for example (Lento and Nieminen 2003). Rebonding to reduce the number of dangling bonds is common. For example, interstitial atoms in both Si and Ge can form <110> split-interstitial or dumbbell configurations. Rebonding also occurs around VSi, though not for VGe. The bond strengths and solid cohesive energy are lower in Ge (Holland et al. 1984), so free dangling bonds cost less energy than they do in Si. The large strain energies associated with full rebonding become less necessary to incur. The large relaxations around VSi compared to VGe can be viewed in terms of weaker electron-lattice coupling in Ge versus Si. When a vacancy forms in an ionic material such as TiO2 or ZnO, there is stronger electron-lattice coupling because of electrostatic interactions with the nearby atom cores. These interactions repel the surrounding nearest neighbors away from the empty lattice site. For example, the defects VZn−2 and VO+2 in ZnO force the respective nearest-neighbor oxygen and zinc atoms away from the defect site.

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Table 4.4 Electronegativities of the elements in common group IV, III–V, and oxide semiconductors Element

Electronegativity

Si Ge B Al Ga In N P As Sb O Ti Zn U Co

1.90 2.01 2.04 1.61 1.81 1.78 3.04 2.19 2.18 2.05 3.44 1.54 1.65 1.38 1.88

Surfaces also rearrange by relaxations of various kinds following defect formation. Localized rebonding leads to the formation of additional defects or long-range energy-lowering surface reconstructions. For instance, charged dimer vacancy on Si(100) results from the weak rebonding of the exposed second-layer atoms (Wang et al. 1993). On a prototypical III–V semiconductor such as GaAs, reconstructions minimize the number of anion and cation dangling bonds and surface dimers under cation- and anion-rich conditions, respectively. Alternatively, the existence of a high concentration of defects can trigger energetically favorable surface reconstructions (Srivastava 1997). The removal of a silicon atom from the Si(111)–(7×7) surface produces small local modifications in surface geometry (Lim et al. 1996). Vacancies exist in multiple charge states on Si(111) (Dev and Seebauer 2003), so the modifications in surface geometry may well be charge state dependent; this dependence has yet to be investigated. Lastly, defects on semiconductors surfaces in excess of 1012 defects/cm2 can induce pinning of the surface Fermi level (Dev et al. 2003). Adsorption can sometimes remove the pinning, and thereby affect the concentration of charged surface defects (Seebauer 1989).

4.3 Defect Charging With regards to defect charging, it is difficult to predict the behavior of surface defects or defect associates by examining the charging of point defects in the bulk. Charging of surface defects has been identified and studied fairly recently; the ionization levels of surface defects are rarely mentioned in the literature. Defect

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aggregates have been examined in more detail. Experimental and computational work indicates that their charge states and ionization levels differ from those of point defects.

4.3.1 Bulk vs. Surface Although both bulk and surface defects ionize, for a given semiconductor the two types do not necessarily take on the same range of charge states. There are typically fewer stable charge states on the surface than in the bulk; sometimes donor or acceptor states present in the bulk will not be manifested on the surface. To illustrate this point, Table 4.5 lists the stable bulk and surface defect charge states a selection of common semiconductors.

Table 4.5 Summary of the most likely stable charge states of surface and bulk vacancy, interstitial, and antisite defects for several elemental, III–V, and oxide semiconductors Material

Regime VIII/IV/metal

VV/oxygen

IIII/IV/metal IV/oxygen

VIII

IIIV

Silicon

Bulk

–

+2, 0, −1, −2 –

–

–

–

–

–

–

–

–

–

Surface Germanium Bulk

GaAs

Surface Bulk

GaP

Surface Bulk Surface

GaN

TiO2 ZnO

Bulk

+2, 0, −1, −2 +2, 0, −1 (l), −2 (u) +2, +1, 0, −1, −2 <0 0, −1, −2, −3 +1, 0, −1 +1, 0, −1, −2, −3 –

0, −1, −2, −3 Surface – Bulk −4 Surface – Bulk 0, −1, −2 Surface −

– –

+2, +1, 0, −1 – – +1, −1, −3 +3, +2, +1, 0 +1, 0, −1 – +1, −1, −3, +3, +2, −5 +1 +1, 0, – −1 (InP) +3, +1, −1, +3, +1 −2, −3 – – 2 +3 or +4 +2, +1, 0 – +2, 0 +2, +1, 0 2 −

– +1, 0, −1

– +2, +1, 0 – – +3, +1, 0, −1 +2, 0

– 0, −1, −2

–

–

– +2, 0, −1, −2 –

+3, +2, +1, 0, −1 – n/a – +1, 0, −1, −2 −

+2, +1, 0, −1 – – – – −

+3, +2, +1, 0, −1 – – – +2, +1, 0 −

For the silicon surface, the parenthetical (l) and (u) refer to (lower) and (upper) – the two different types of vacancies on the Si(100) – respectively. Also, charge states of defects on InP are given in lieu of those in GaP. In some instances dashes reflect the fact that no such defects form. For example, there are no charged interstitials on the Si and Ge surfaces (partly due to the fact that the concept of a surface “interstitial” is somewhat amorphous). In another entry, the “<” symbol indicates that the literature agrees only upon the fact that negative VGe charge states exist.

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4.3.2 Point Defects vs. Defect Aggregates Only modest correspondence exists between the stable charge states of isolated point defects and the corresponding defect associates. For example, the monovacancy and divacancy in silicon adopt almost identical charge states. The monovacancy is stable in the (+2), (0), (−1), and (−2) charge states, while the divacancy exists as (+1), (0), (−1), and (−2). Yet VGe is stable in five charge states, while VGeVGe likely exists only as (0), (−1), and (−2). Certainly the concept of “charge additivity” cannot explain the charge states of a semiconductor defect associate. That is, at a given Fermi energy, the charge state of a defect associate does not necessarily equal the sum of the charges of the constituent defects. For example, in highly n-type Si, the majority point vacancy state VSi−2 does not dimerize to form (VSiVSi)−4; (−2) is the preferred charge state of the divacancy. If “charge additivity” were to apply in GaAs, the mixed divacancy VGaVAs would exist in the (+1) charge state for Fermi energies from the VBM to 0.10 eV, as (0) between Ev + 0.10 and 0.38 eV, as (−1) between Ev + 0.38 and 0.55 eV, and as (–2) above Ev + 0.55 eV. Yet in actuality, VGaVAs exists in the (−2) charge state for all values of the Fermi energy above Ev + 0.19 eV.

4.4 Defect Diffusion The diffusion rates and pathways of bulk and surface defects can depend strongly on charge state. The prefactors associated with the mesoscale diffusion of Sii+1 and Sii0 in silicon are 3.33 ± 2.81 × 102 cm2/s and 9.29 ± 8.17 cm2/s, respectively (Bracht et al. 1998; Silvestri et al. 2002). As the diffusional pre-exponential factor comprises the product of the intrinsic prefactor together with an entropic formation term, large ionization entropies translate to large diffusional prefactors. The migration energy barriers for the site-to-site hopping of gallium vacancies in GaAs exhibit a clear charge state dependence (El-Mellouhi and Mousseau 2007); certain migration pathways are more favorable for defects of a given charge state. Similar dependencies are observed in GaN (Ganchenkova and Nieminen 2006) and TiO2 (Iddir et al. 2007). Measurements of mesoscale surface diffusion on both Si(100) and Si(111) have shown a dependence on native defect charge state. Pre-exponential factors for mesoscale surface diffusion of groups III and V adsorbates on Si(111) increase for both positive and negative defects (Allen et al. 1996). Site-tosite migration barriers on surfaces also depend on defect charge state; VGa−1 diffuses more readily on GaAs(110) than VGa0, for example (Yi et al. 1995). In stoichiometric and hypostoichiometric oxide semiconductors, oxygen always diffuses via a vacancy mechanism because few oxygen interstitials exist in the bulk. Under oxygen-rich conditions, however, an interstitial-mediated mechanism of anion diffusion comes into effect. For instance, in O-rich ZnO (Erhart and Albe 2006) and UO2+x (Matthews 1974), anion migration is mediated by interstitials.

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For all ranges of stoichiometry, metal cation diffusion in ZnO, UO2, and CoO is governed by vacancies, while that of Ti in TiO2 is mediated by interstitials. Metal interstitials dominate diffusion when the migration rate of metal vacancies in the bulk is very low and limited by their formation at the gas/solid interface.

References Allen CE, Ditchfield R, Seebauer EG (1996) J Vac Sci Technol, A 14: 22–29 Basak CB, Sengupta AK, Kamath HS (2003) J Alloys Compd 360: 210–216 Bouhafs B, Aourag H, Ferhat M et al. (1999) J Phys: Condens Matter 11: 5781–5796 Bracht H, Haller EE, Clark-Phelps R (1998) Phys Rev Lett 81: 393–396 Callister W (2002) Materials Science and Engineering, San Francisco, John Wiley and Sons Dev K, Seebauer EG (2003) Surf Sci 538: 495–499 Dev K, Jung MYL, Gunawan R et al. (2003) Phys Rev B: Condens Matter 68: 195311 El-Mellouhi F, Mousseau N (2007) Physica B 401–402: 658–61 Erhart P, Albe K (2006) Phys Rev B: Condens Matter 73: 115206 Ganchenkova MG, Nieminen RM (2006) Phys Rev Lett 96: 196402 Hoglund RF (2006) in 3D Laser Microfabrication Misawa H & Juodkazis S (Eds.) Germany, Wiley-VCH Holland B, Greenside HS, Schlüter M (1984) Phys Status Solidi B 126: 511–515 Hurle DTJ (1999) J Appl Phys 85: 6957–7022 Iddir H, Ogut S, Zapol P et al. (2007) Phys Rev B: Condens Matter 75: 073203 Jarzebski ZM (1973) Oxide Semiconductors, New York, Pergamon Press Kosuge K (1994) Chemistry of Non-Stoichiometric Compounds, New York, Oxford Science Publications Lento J, Nieminen RM (2003) J Phys: Condens Matter 15: 4387–4395 Lim H, Cho K, Capaz RB et al. (1996) Phys Rev B: Condens Matter 53: 15421–15424 Martins R, Barquinha P, Pereira L et al. (2007) Appl Phys A 89: 37–42 Matthews JR (1974) Mechanical Properties and Diffusion Data for Carbide and Oxide Fuels. Ceramics Data Manual Contribution. United States Nainaparampil JJ, Zabinski JS (2001) J Mater Res 16: 3423–9 Nowotny J, Radecka M, Rekas M (1997) J Phys Chem Solids 58: 927–937 Orellana W, Chacham H (2001) Phys Rev B: Condens Matter 63: 125205 Robert JL, Mosser V, Contreras S (1991) Physics of AlGaAs compounds for sensing applications. In: TRANSDUCERS’91. 1991 International Conference on Solid-State Sensors and Actuators. Digest of Technical Papers (Cat. No.91CH2817-5) 294–9 (IEEE, San Francisco, CA, USA, 1991) Seebauer EG (1989) J Vac Sci Technol A 7: 3279–3286 Silvestri HH, Sharp ID, Bracht HA et al. (2002) Dopant and self-diffusion in extrinsic n-type silicon isotopically controlled heterostructures. In: Materials Research Society Symposium – Proceedings 719:427–432 (Materials Research Society, San Francisco, CA, United States, 2002) Solyom J (2007) in Fundamentals of the Physics of Solids, Springer Berlin Heidelberg Srivastava GP (1997) Rep Prog Phys 60: 561–613 Van de Walle CG, Neugebauer J (2004) J Appl Phys 95: 3851–79 Wang J, Arias TA, Joannopoulos JD (1993) Phys Rev B: Condens Matter 47: 10497–508 Weibel A, Bouchet R, Knauth P (2006) Solid State Ionics 177: 229–236 Yi JY, Ha JS, Park SJ et al. (1995) Phys Rev B: Condens Matter 51: 11198–11200

Chapter 5

Intrinsic Defects: Structure

5.1 Bulk Defects When an atom is removed from or introduced into a semiconductor crystal lattice, the defect that forms may take on an assortment of possible geometries (Fig. 5.1). Isolated vacancies can be described by two geometric configurations. A simple lattice vacancy, VL, results when an atom is removed from the lattice, leaving a void in its place (Fig. 5.1a). A “split” vacancy, VB, results when an atom resides at the bond center between two empty sites (Fig. 5.1d). In most semiconductors VL is the dominant configuration, whereas VB typically describes the transition state in vacancy migration (Centoni et al. 2005). For simplicity, the present exposition will drop the “L” subscript for the lattice vacancy but retain the “B” subscript. Analogs to the lattice and “split” vacancy exist for interstitial defects, although self-interstitials can occupy a considerably larger number of geometries than vacancies. Well known examples in silicon include tetrahedral (Fig. 5.1e), split-<110> (Fig. 5.1c), hexagonal (Fig. 5.1f), extended, split-<100>, and bond-centered configurations (Needs 1999; Sahli and Fichtner 2005; Centoni et al. 2005). In addition, several lower-symmetry configurations have been postulated based mainly upon evidence from quantum calculations. In particular, a highly asymmetric “caged” silicon self-interstitial has been identified by Clark and Ackland (1997). Marques et al. have found evidence for an extended self-interstitial in silicon, consisting of four displaced atoms and three empty lattice sites along the (110) plane, and a newly identified asymmetric configuration, formed from three displaced atoms and two lattice sites (2005). Foreign atoms can also adopt interstitial configurations in the crystal lattice; diffusion is often mediated by these interstitials. The charge states of these species can vary with n- or p-type doping, translating into variations in overall migration rate. The last major type of point defect is the “substitutional” defect, which forms when a foreign atom replaces a lattice atom. In compound III–V and oxide semiconductors such as GaAs or TiO2, the “foreign” atom may already be present in E.G. Seebauer, M.C. Kratzer, Charged Semiconductor Defects, © Springer 2009

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Fig. 5.1 Geometries of defects in relation to a conventional unit cell of Si where (a) is the lattice vacancy (b) is the undefected unit cell (c) is the <110> split-interstitial (d) is the split vacancy (e) is the tetrahedral self-interstitial and (f) is the hexagonal self-interstitial. Reprinted figure with permission from Centoni SA, Sadigh B, Gilmer GH et al. (2005) Phys Rev B: Condens Matter 72: 195206-2. Copyright (2005) by the American Physical Society.

the material, but lie on the “wrong” site in the crystal structure. Such structures constitute “antisite” defects. As an example, two types of antisites are possible in GaAs–AsGa, where the As atom resides on a Ga lattice site, and GaAs, where the Ga atom resides on an As lattice site. These defects frequently introduce strain (either tension or compression) into the crystal lattice, as the original atom and substitutional antisite atom typically have different atomic radii. The symmetry of a point defect after relaxation, as well as the magnitude and direction of the relaxation, typically depends upon charge state. For many defects, both neutral and ionized, the Jahn–Teller distortions responsible for structural relaxation lead to a breaking of local symmetry. The distortions occur when a degenerate set of orbitals is occupied unevenly with electrons, leading to unevenly balanced forces (Coutinho et al. 2005; Fazzio et al. 2000; Janotti et al. 1999). Crystal class and unit cell type govern the types of symmetries available to a given defect, as summarized in Table 5.1. Defects with tetrahedral symmetry (Td) can relax and adopt new symmetries such as D2d, where the tetragonal lattice distortion has two equal and shorter interatomic distances and four approximately equal and longer distances, and C2v, where the orthorhombic lattice distortion has one shorter interatomic distance and five approximately equal and longer distances arise (Fazzio et al. 2000). In some instances, symmetry-lowering relaxations can be so energetically favorable that negative-U behavior results (Centoni et al. 2005; Baraff et al. 1979; Watkins and Troxell 1980; Car et al. 1984). For a negative-U system, the capture of an electron or hole is immediately followed by a capture of a second carrier of the same type; thus, the intermediate charge state is energetically unstable at all possible chemical potentials (Lento and Nieminen 2003).

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Table 5.1 Point-group symmetries as a function of semiconductor crystal class Crystal Class

Point-group Symmetry

Cubic Tetragonal Orthorhombic Monoclinic Triclinic Trigonal Hexagonal Non-crystallographic

T, Th, O, Td, Oh C4, S4, C4h, D4, C4v, D2d, D4h D2, C2v, D2h C2, C5, C2h C1, Ci C3, S6, D3, C3v, D3d C6, C2h, C6h, D6, C6v, D3h, D6h C∞, C∞h, C∞v, D∞h, C5, S8, D5, C5v, C5h, D4d, D5d, D5h, D6d, I, Ih

Isolated point defects of all kinds can join together to form defect complexes, associates, and clusters. For the purposes of this discussion, the classification of Smyth will be utilized to distinguish between these different species (2000). A complex is different from an associate in that it consists of both an extrinsic and intrinsic defect. These will be discussed in detail in Chap. 8. On the other hand, a defect associate is formed from two intrinsic defects. The term “cluster” is used to refer to any grouping of two or more defects. The stability and concentration of these defects depend upon a variety of factors including charge state, deviation from stoichiometry, and attractive and repulsive atomic interactions within the semiconductor. The divacancy is the most well-studied defect associate; in almost all of the materials discussed here, the geometry of some type of divacancy has been examined. A wider assortment of associates and clusters is possible in compound semiconductors than in simple elemental semiconductors due to the existence of two different elements in the basis of the compound crystal lattice. In GaAs, for instance, reports exist concerning the Ga divacancy (VGaVGa), As divacancy (VAsVAs), and mixed divacancy (VGaVAs). Especially in CoO and UO2, defect clusters often contain four or more point defects. Extended defects can also be charged. The category of extended defects includes line defects such as screw and edge dislocations as well as planar defects including grain boundaries, platelets, and stacking faults. Colombo has identified an assortment of stable extended defects in silicon and utilized kinetic lattice Monte Carlo (KLMC) simulations to simulate defect evolution in the bulk on the basis of TBMD-derived interactions (2002). Unfortunately, charging behavior of extended defects has been studied only rarely. As one example, Simpkins et al. distinguished between neutral and negatively charged dislocations within GaN using scanning Kelvin probe and conductive AFM (2002). There was no mention, however, of the dependence of dislocation charge state on Fermi level. Point defect-dislocation coupling may lead to quantitative variation in the relative stability of different point defect charge states (De Araujo et al. 2007). It appears that the dependence of this coupling on dislocation charging has not been explored. DFT can also be employed to study the bond angles, lengths, coordination, and charging of threading and screw dislocations (Haugk et al. 2000). Another type of extended defect, the mesoscopic point-like defect, retains many of the features of

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a point defect, including multiple charge states within the band gap (Nolte 1998). A mesoscopic point-like defect possesses a characteristic energy scale that extends continuously from the point-defect limit to the classical limit. They can be distinguished from other extended and plane defects by their bulk volumes, which are enclosed by surfaces of minimal curvature. Unlike true point defects, however, mesoscopic defects possess extremely high charge state multiplicity (on the order of ±10 rather than ±4), which requires their energy levels to be calculated within a many-electron theoretical framework (Nolte et al. 2000). The mesoscopic pointlike defect has not been investigated in extensive detail; the remaining extended defects retain little of the characteristics of their component point defects.

5.1.1 Silicon Crystalline silicon has a diamond structure with a lattice constant of 5.43072 Å. The diamond structure can be thought of as the union of two superimposed facecentered cubic crystal lattices where one is displaced by a quarter of the length of the unit cell in the [100], [010], and [001] directions. Each silicon atom has four valence electrons that it shares to form tetrahedral covalent bonds with its neighbors. 5.1.1.1 Point Defects When a single atom is removed from the diamond crystal lattice, it leaves behind four dangling bonds located at the vertices of a tetrahedron. Typically, the four neighboring atoms to the vacancy site rebond (to form two pairs) in order to eliminate the dangling bonds (Antonelli et al. 1998). This leads to the formation of the lattice vacancy. The lattice vacancy is the preferred configuration for VSi+1, VSi0, and VSi−1 (Centoni et al. 2005), which exist for all but strongly n-type doping. As early as 1974, however, Bourgoin and Corbett suggested that under strong n-type conditions, VB−2 would form and would adopt a split vacancy ground-state configuration rather than the lattice vacancy structure (1974). In the split vacancy, one of the neighboring Si atoms relaxes towards the center of the vacancy to form a symmetric configuration in which the displaced atom bonds to six neighboring atoms as illustrated in Fig. 5.2. In the (–2) charge state, the overall vacancy contains a total of 12 electrons available for bonding: one from each of the 6 neighbors, four from the central atom, and two extra drawn from the conduction band to create the net (–2) charge. These 12 electrons pair up into six covalent bonds with the central atom, whose atomic orbitals rehybridize in an unusual way to permit this high degree of coordination. The energetic benefit from this unusual coordination originates from that fact that all electrons are paired in the split configuration (with no dangling bonds). As the extra electrons due to ionization can delocalize over the six bonds, there is less electron repulsion for VB−2 than for VL−2. Centoni et al.

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Fig. 5.2 Doubly negative “split” silicon vacancy, where the dark spheres give the relaxed positions of the atom cores with respect to ideal bulk bonds denoted by sticks. The intersections of the sticks represent the positions of lattice atoms in the absence of the vacancy. The isosurface of the electron density for the highest deep level occupied by four electrons is depicted by the light gray regions. Reprinted with permission from Puska MJ “Point defects in silicon, first-principles calculations,” (2000) Comput Mater Sci 17: 368. Copyright (2000) with permission from Elsevier.

have recently confirmed the fact that the vacancy in strongly n-type material exists in the split, rather than lattice, configuration via first-principles calculations; they posit that many researchers are unaware of this phenomenon because it occurs only at low temperatures, whereas most technologically useful processes involve elevated temperatures. As the split-vacancy is the saddle point for vacancy migration, the interchange between VL and VB provides a natural pathway for athermal diffusion. Consequently, in n-type material, the vacancy can migrate through the silicon lattice without having to overcome a thermally-activated barrier. At standard processing temperatures, thermal diffusion dominates and this phenomenon is obscured. Thus, many computational studies consider only the lattice configuration of the defect that has been observed experimentally and occurs at temperatures above the 0 K characteristic of DFT. The lattice monovacancy in bulk Si can support charge states of (–2), (–1), (0) and (+2) (Van Vechten 1988). Experiments and computations have shown that the strong Jahn–Teller distortions associated with rebonding lead to a breaking of Td symmetry for all charge states except (+2) (Tarnow 1993; Ögüt et al. 1997; Estreicher 2000). Using DFT, Lento and Nieminen have calculated transitions to tetragonal D2d symmetry for all but VSi−1, which adopts a D2 configuration (2003). According to Watkins’ linear combination of atomic orbitals model, when electrons are added to VSi+2, the symmetry is reduced to D2d for VSi+1 and VSi0 and then to C2v for VSi−1 and VSi−2. More recent LDA calculations suggest, instead, that VSi−1 and VSi−2 assume D3d symmetry (Wright 2006). In all of these approaches, the symmetry lowering lattice relaxation from Td to D2d for the (+2) to (0) charge state transition leads to negative-U behavior, destabilizing the (+1) charge state of the defect; this phenomenon has been predicted since the 1970s (Car et al. 1984; Baraff et al. 1979; Watkins and Troxell 1980). With a 256-atom supercell, Lento and Nieminen observed large inward relaxations of 32.3–47.6% for charge states from (+1) to (–2). 256- and 33-atomic supercells give a −9.7 and 12.9% contraction for VSi+2, respectively; the inward relaxation of the doubly positively charged

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vacancy is only obtained with supercells larger than ~200 atoms (Ögüt et al. 1997; Puska et al. 1998). Interestingly, Centoni et al. found that VB does not experience symmetry-breaking distortions as the lattice vacancy does. For interstitials in silicon, the question of which structures are most stable is still under debate. The relatively symmetric tetrahedral, hexagonal, and split-<110> configurations have received the most attention (Lee et al. 1998; Leung et al. 1999; Maroudas and Brown 1993a, b; Ungar et al. 1994; Tang et al. 1997; Chadi 1992). An alternative, low-symmetry caged interstitial structure has been proposed by Clark and Ackland (1997), although subsequent investigators have questioned its stability (Leung et al. 1999; Needs 1999). The tetrahedral interstitial atom is fourfold coordinated with four fivefold-coordinated nearest-neighbor atoms. The atoms that form the <110>-split interstitial are also fourfold coordinated, although only two of the neighboring atoms are fivefold coordinated. The pair of Si atoms at a substitutional site directed in the <110> direction has been shifted from the ideal lattice site by 0.76 Å in the <001> direction (Lee et al. 1998). The displacement is energetically favorable because it minimizes the relaxations associated with the neutral defect. Most computational investigations suggest that tetrahedral Sii+2 dominates in p-type silicon, with a switch to Sii0, Sii−1, or Sii−2 in a <110>split configuration for n-type material (Jung et al. 2005; Chadi 1992; Lee et al. 1998). While some investigators have found the highly symmetric hexagonal geometry to be energetically favorable for Sii+2, regardless of defect charge state, the hexagonal configuration is usually assumed to be the saddle point for interstitial diffusion (Bar-Yam and Joannopoulos 1984; Lee et al. 1998; Needs 1999; Centoni et al. 2005; Goedecker et al. 2002). Among the high-symmetry configurations, the <100>-split and bond-centered configurations are generally acknowledged to be unstable (Goedecker et al. 2002; Needs 1999; Leung et al. 1999). Little quantitative discussion exists in the literature on the relaxations associated with the different charge states of these defects (Balboni et al. 2002; Needs 1999). According to Lee et al., small, charge state-independent relaxations (~1 Å) are associated with the tetrahedral and hexagonal self-interstitial sites (1998). Centoni et al. calculated relaxation volume tensors only for neutral defects, which are pertinent to I<110> but not IT (2005). For the split-interstitial, the two fivefoldcoordinated neighbors expand outwards in the <110> direction by approximately 5%, while the two fourfold-coordinated neighbors are almost the same as the bulk bond length. 5.1.1.2 Associates and Clusters The native point defects present in bulk Si readily combine to form a wide variety of vacancy, interstitial, and vacancy-interstitial associates. The charging of such species has attracted quite a bit of study. Vacancies aggregate mainly into divacancies, though trivacancies also exist. Interstitials aggregate into entities ranging continuously from di-interstitials through multiatom clusters to extended [311]

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79

defects of arbitrarily large sizes. Mixed vacancy-interstitial combinations seem limited to the pairing of lone interstitials with monovacancies. The geometry and structure of the silicon divacancy (V2) has been studied extensively in the literature. V2 can be generated through the irradiation-induced recombination of VSi, but exists in significant concentrations even in non-irradiated material, where it is thought to affect both self- and impurity diffusion in silicon (Stein et al. 1969; Khivrich et al. 1996; Van Vechten 1986). The divacancy was first identified by Corbett and Watkins (1965; 1961) and later Bemski et al. (1963) using EPR. The silicon divacancy consists of two nearest neighbor vacancies, each surrounded by three lattice atoms, as shown in (Fig. 5.3). Indeed, it is now well established that the divacancy is stable in the charge states of (+1), (0), (−1) and (−2) as the Fermi level is progressively raised from the valence band to the conduction band (Sugino and Oshiyama 1990; Pesola et al. 1998; Szpala and Simpson 2001). Watkins and Corbett were the first to suggest that the divacancy behaves as an amphoteric center (1965); namely, that it can act either as a donor or as an acceptor depending on the electron availability from the surrounding medium. Pesola et al. have shown conclusively using spin density studies that the V2+2, unlike VSi+2, is not thermodynamically stable for any value of the Fermi level in the band gap (1998). Consequently, the silicon divacancy does not exhibit the characteristic negative-U behavior of VSi, which is stable as VSi+2 but not as VSi+1. No one has looked explicitly at V2−4, which may be worthwhile, as Mueller et al. have recently suggested an additional (−2/−4) level for the silicon monovacancy (2003).

Fig. 5.3 The silicon divacancy where dashed circles labeled A and B represent the two adjacent vacancy lattice sites axes for spin-resonance data are also indicated in the figure. The electrons associated with the defect can be regarded as forming the bonds shown in the figure. Reprinted figure with permission from Corbett JW, Watkins GD (1961) Phys Rev Lett 7: 314. Copyright (1961) by the American Physical Society.

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The unrelaxed divacancy is undistorted and has D3d symmetry (Ögüt and Chelikowsky 1999). The removal of two silicon atoms from the crystal lattice leaves six under-coordinated Si atoms and 18 associated bonds that can compensate for the resulting strain. The structure produces two doubly degenerate levels in the bandgap (Ögüt and Chelikowsky 2001; Saito and Oshiyama 1994). This nominal geometry automatically suggests that symmetry-lowering (and energy lowering) Jahn–Teller relaxations occur. Experiments have shown that Jahn–Teller distortions indeed occur for the divacancy (+1), (0), and (–1) states, but not for (–2). At low temperatures near 20 K, Watkins and Corbett have reported lower symmetry C2h structures for V2−1 and V2+1 by studying EPR spectra (1965). Recall that C2h relaxations can be of two varieties: pairing and resonantbond. These two varieties, including the six-nearest neighbors, are schematically illustrated in (Fig. 5.4). In the large-pairing (LP) structure, atoms 1 and 2 are pulled towards each other such that d12 < d13 = d23. In the resonant-bond structure (RB) d12 > d13 = d23, so there is a more even distribution of bond strain. The actual low-temperature symmetry turns out to be LP (Watkins and Corbett 1965). However, at higher temperatures near 110 K, studies by ENDOR (electron-nuclear double resonance) spectroscopy showed that the C2h symmetry of V2−1 and V2+1 changes to a motionally averaged D3d symmetry due to rapid oscillations between various Jahn–Teller configurations (Sieverts et al. 1978; Wit et al. 1976). Nagai et al. used two-dimensional angular correlation of positron annihilation spectroscopy (2D-CAR) to show that the neutral divacancy, V20, exhibits the same pairing type Jahn–Teller relaxations as V2−1 (2003). The doubly negative divacancy, V2−2, may also have D3d symmetry with slight inward relaxations (Dobaczewski et al. 2002). Dobaczewski et al. combined high-resolution DLTS in the presence of uniaxial stress (at temperatures around 140 K) to show that V2−2 does not exhibit the Jahn–Teller relaxations that characterize the other three charge states of the divacancy.

Fig. 5.4 Views of the silicon divacancy rotated so the <111> direction is perpendicular to the plane of the page including (a) the D3d structure (b) the LP structure (C2h) (c) the RB structure (C2h). Solid circles represent the undercoordinated Si atoms surrounding the silicon divacancy whereas dashed circles represent missing Si atoms. Reprinted figure with permission from Wixom RR, Wright AF (2006) Phys Rev B: Condens Matter 74: 205208-3. Copyright (2006) by the American Physical Society.

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The structural behavior observed experimentally for divacancies has been largely confirmed by computational work. Recently, disagreement has arisen as to the lowest energy configurations and relaxations of the divacancy due to discrepancies between computational methods. Saito and Oshiyama employed 64-atom total energy calculations with the LDA to conclude that V2+1 undergoes a LP-type relaxation while V2−1 undergoes a RB relaxation (1994). Similar 64-atom supercell calculations were performed by Seong and Lewis, who found the neutral divacancy to exhibit a resonant-bond type relaxation (1996). Pesola et al. employed local spin density approximations using a 216-atom supercell to propose S2 symmetry for the various charge states of V2; the differences, however, are small, and may not be distinguishable from C2h symmetry in experiments (1998). These same authors identified pairing structures as the lowest energy C2h configuration for V2+1 and V20, versus a resonant bond structure for V2−1. Cluster calculations by Coomer et al. found pairing relaxations to be favorable for the divacancy in the (+1), (0), and (−1) charge states (1999b). Ögüt and Chelikowsky studied the structural relaxations of the divacancy using a cluster method instead of the more commonly used supercell approach (2001). The authors reasoned that the defectinduced atomic relaxations may not decay rapidly, and the supercell containing even up to 216 atoms may be inadequate to capture the relaxations. Their results predict pairing type Jahn–Teller relaxations for V2+1, V20 and V2−1, which is consistent with the experiments. Recent DFT calculations performed by Wixom and Wright (2006), however, may explain many of these contradictory findings. These authors obtained supercell total energies with both the local density approximation (LDA) and Perdew, Burke, Ernxerhof (PBE) formation of the generalized-gradient approximation (GGA) for electronic exchange and correlation. Both methods showed D3d point-symmetry to be stable for V2−2; with the LDA, RB configurations were favored for the (0) and (−1) charge states while the LP configuration was preferred for the (+1) charge state. In contrast, the computations performed with the PBE found the large-pairing C2h configuration to be the most stable for the (+1), (0), and (−1) charge states. Evidence is sparse for vacancy clusters larger than the divacancy, and charge states have not been examined closely. Most experimental studies revolve around EPR spectra of neutron-irradiated silicon. Lee and Corbett assigned the Si-A4 center to the trivacancy (V3) with a tentative charge assignment of (–1) (1973; 1974). Si-P3 and Si-A4 centers have been attributed to the tetravacancy (V4), again with a possible charge of (–1) (Lee and Corbett 1973, 1974; Jung and Newell 1963). The Si-P1 center has been assigned to the pentavacancy (V5) (Lee and Corbett 1973, 1974; Nisenoff and Fan 1962). The pentavacancy has also been observed in plastically deformed silicon (Brohl et al. 1987). Lee and Corbett assigned a negative charge state to V5. Computational studies by molecular dynamics and Hartree– Fock methods confirm the stability of various larger vacancy clusters, particularly V6 and V10, but do not examine charging effects (Chadi and Chang 1988a; Hastings et al. 1997; Estreicher et al. 2001). Interstitial clusters also form in bulk silicon, especially under non-equilibrium conditions produced by electronic device processing techniques such as ion

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Fig. 5.5 Ball and stick models of various structural configurations of the di-interstitial. The stability of these different geometries has been investigated for the (+1), (0), and (−1) charge states of SiiSii. Reprinted from Eberlein TAG, Pinho N, Jones R et al., “Self-interstitial clusters in silicon,” (2001) Physica B 308–310: 456. Copyright (2001) with permission from Elsevier.

implantation (Richie et al. 2004). Most studies relating to interstitial clusters focus primarily on their high mobility and contribution to OED and TED (Cowern et al. 1999; Martin-Bragado et al. 2003; Law et al. 1998; Hane et al. 2000); charging behavior has only been examined only for small clusters. Computational studies suggest that the silicon di-interstitial is stable in three charge states, (+1), (0), and (−1). The P6 center observed by EPR measurements in low-energy ion-implanted, proton- or neutron-irradiated silicon arises from the silicon di-interstitial defect. The point-symmetry of the defect is temperature dependent; at low temperature (200 K), the symmetry is either C2 or C1h; a thermally activated symmetry transition to D2d occurs at room temperature (300 K) (Kim et al. 1999). Support for the tetragonal C2 structure (form L) comes from early EPR experiments where the positively charged di-interstitial (presumably (SiiSii)+1, as no other candidates are mentioned) was projected to lie long the <100> crystal axis (Lee et al. 1976). More recently, Kim et al. (1999) and Eberlein et al. (2001) have performed ab initio total energy and density functional calculations that reveal lower formation energies for the low temperature defect with C1h symmetry for several charge states ((+1), (0), and (−1)). The structure for (SiiSii)0 proposed by Kim and co-authors comprises a center atom and dumbbells where the dumbbell structure is aligned parallel to the [110] direction (form K). Eberlein et al. also looked at one additional C1h structure (form C) for the neutral di-interstitial where the [011] dimer is added to a site near a bond center, maintaining a (011) mirror plane. The various forms are shown schematically in Fig. 5.5. The tri-interstitial, I3, is thought to exist in a (+1) and (0) state. Coomer et al. assigned the W photoluminescence center with the zero-phonon line at 1.0182 eV to I3 (1999a). Carvalho et al. (2005) and Richie et al. (2004) have looked at five possible structures for I3 with C3v, Td, C2 and C3 symmetries. The structures described by Carvalho and co-authors are shown in Fig. 5.6. The so-called I3-IV

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Fig. 5.6 Ball and stick models of I3 and I4 in silicon where Si atoms at the defect core are represented as gray spheres. Reprinted figure with permission from Carvalho A, Jones R, Coutinho J et al. (2005) Phys Rev B: Condens Matter 72: 155208-2. Copyright (2005) by the American Physical Society.

mixed vacancy-interstitial structure with C2 symmetry is the most stable for I30, while the I3-I structure with C3v symmetry is the most stable for I3+1. Similar to the tri-interstitial, the tetra-interstitial (I4) is also thought to exist in the (+1) and (0) charge states (Carvalho et al. 2005; Coomer et al. 1999a, 2001). The defect attracts interest as a potential nucleation site for larger, extended defects. Until recently, it has been difficult to obtain structural information about the defect from EPR spectra; for instance, reports from 1997 and 1999 claim that the tetra-interstitial is not experimentally identifiable (Arai et al. 1997; Kohyama and Takeda 1999). Kohyama and Takeda attributed this lack of experimental evidence for I4 to the fact that the defect has no electronic states within the band gap when explored with density-functional calculations using the ab initio pseudopotential method. Several articles, including those with new insight into the hyperfine structure of the defect, now unambiguously assign the Si-B3 center observable via electron paramagnetic resonance to the tetra-interstitial defect (Pierreux and Stesmans 2003; McHedlidze and Suezawa 2003; Coomer et al. 2001). The Si-B3 spectra is unique as it has tetragonal or D2d point-symmetry, a result of the molecular structure of the defect, which involves the replacement of four neighboring atoms in the (001) plane by four [001]-oriented split-interstitial pairs (see Fig. 5.6f). The most studied interstitial-vacancy associate in Si is that formed from two lone species (the so-called “Frenkel pair”), which can be (+1) or (0). Emtsev et al. recently suggested that Frenkel pairs are positively charged in p-type Si and neutral in n-type Si (2007). Diffuse X-ray experiments led Ehrhart and co-workers to stress the importance of “separated” Frenkel pairs in irradiated silicon; they proposed that these defects are either intrinsically neutral, or donor-acceptor pairs

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with an overall neutral charge (Ehrhart and Zillgen 1997; Partyka et al. 2001). Additionally, the separation distance between the interstitial and vacancy is dependent upon the type and amount of irradiation; low doses of electron and He ion irradiation result in separation distances of approximately 0.8 and 1.2 nm, respectively. Energetically favorable Frenkel pair formation volumes obtained via ab initio calculations, however, contradict the positive formation volumes obtained by Ehrhart et al. for “separated” VSiSii pairs and propose closer defect pairing than that suggested by experiments (Centoni et al. 2005). The binding energy and bond structure of the neutral Frenkel pair has been investigated by tight-binding molecular dynamics and ab initio Hartree–Fock calculations (Cargnoni et al. 1998), as well as ab initio calculations using the GGA (Hobler and Kresse 2005). All of the computational investigations agree that the “bond-defect” configuration identified by Cargnoni et al. is the equilibrium structure of the VSiSii defect. The structure is obtained by placing a vacancy at the third nearest-neighbor position of a selfinterstitial dumbbell along the <110> direction (with a separation of about 6 Å); larger bond distances lead to smaller, less favorable binding energies.

5.1.2 Germanium Similar to silicon, germanium exists in the diamond crystal lattice structure. The lattice constant of Ge is slightly larger than that of Si, 5.6579 Å versus 5.4310 Å. As Ge belongs to the same group of the periodic table as Si, it also has four valence electrons with which to form covalent bonds within the solid. 5.1.2.1 Point Defects The vacancy defect in germanium behaves differently from that in silicon; no donor negative-U behavior is observed, and VGe−1 and VGe−2 relax to different point symmetries. To date, it appears that only the lattice vacancy has been considered; there has been no investigation of VB−2 in Ge. For VGe0 and VGe+1, the distortions in the bond lengths that induce a reduction to D2d point symmetry are three times smaller in Ge than in Si. This is due to the weak electron-lattice coupling in germanium, which is about six times smaller than in silicon, and causes the charge states (+2), (+1), and (0) to not form a negative-U system (Fazzio et al. 2000; Ögüt and Chelikowsky 2001). The vacancy in Ge does not rebond to reduce the number of dangling bonds from four to two (as in silicon). Both Janotti et al. (1999) and Fazzio et al. proposed C2v symmetry for the negatively charged vacancy states using density functional theory with the LDA, a 128-atom supercell, an energy cutoff of 12 Ry, and Brillouin zone sampling using one k-point. However, in a spin density functional study using the AIMPRO code, Coutinho et al. recently published results suggesting that VGe−1 relaxes to the D2 or Td structure, while VGe−2 undergoes a tetragonal Jahn–Teller distortion that leads to D2d symmetry (2005).

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Fig. 5.7 Schematic of the structural distortions of a Ge vacancy for several charge states of interest. The symmetry of each structure is shown within parenthesis. Reprinted figure with permission from Coutinho J, Jones R, Torres VJB et al. (2005) J Phys: Condens Matter 17: L523. Copyright (2005) by the Institute of Physics Publishing.

For VGe−1 the proposed D2 deformation follows the occupancy of a bonding state between atom pairs whereas for VGe−2, the distorted D2d structure allows the four available electrons to be accommodated in the t2 state. The method utilized by Coutinho et al. may provide more realistic results, as vacancy dangling bonds induce long range strain fields that cause the structures predicted by DFT to depend heavily on supercell size and shape, as well as the Brillouin zone sampling scheme. A schematic of their calculated structural distortions in Ge for several charge states of interest is shown in Fig. 5.7. In contrast to the vacancy, the germanium interstitial resembles silicon in many respects. Gei can exist in either a <110> split-interstitial or dumbbell, distorted hexagonal or caged, or tetrahedral configuration. For example, similar to Sii0, the dumbbell configuration of Gei0 has the lowest formation energy (Janotti et al. 1999; da Silva et al. 2001; Schober 1989; Moreira et al. 2004). The stable geometry of the self-interstitial, the <110> dumbbell, has also been called a “kite-defect” due to its symmetrical configuration. Da Silva et al. arrived at this designation based on the delocalization of the charge density among the four-atom ring formed by the two Ge atoms in the dumbbell and their two nearest neighbors along the zigzag (2001; 2000). For the neutral charge state of the germanium dumbbell self-interstitial, these authors find that all of the atoms along the <110> zigzag direction relax outward. The relaxations for the first-, second-, and third-neighbor atoms to the dumbbell are about 2.9, 5.4, 3.7% of the nearest-neighbor distance, respectively. The stability of various geometries for the positive charge states of Gei has also been explored. The defect prefers to situate at a tetrahedral site when doubly positive (+2) and the caged C3v configuration between a hexagonal and tetrahedral site when singly positive (+1) (Carvalho et al. 2007; Carvalho et al. 2008; Birner et al. 2000). At temperatures higher than about 200 K, the self-interstitial also has enough energy to overcome the barrier between the dumbbell and caged sites. This is one explanation for the suggested negative-U behavior that leads to the destabilization of the thermodynamically unstable hexagonal Gei+1 at higher temperatures. While the germanium self-interstitial likely doesn’t exist in any negative charge states, structural calculations propose that the split-<110> configuration is favored for Gei−1 (da Silva et al. 2000; Moreira et al. 2004).

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5.1.2.2 Associates and Clusters Germanium vacancy associates have received far more treatment in the literature than self-interstitial associates. The divacancy in germanium is stable in four charge states, (+1), (0), (−1) and (−2) (Coutinho et al. 2006; Janke et al. 2007; Ögüt and Chelikowsky 2001), similar to the divacancy in silicon. As such, the Ge divacancy actually has one more stable charge state than the Ge monovacancy; VGe has no donor levels within the band gap. For example, ion implantation produces relatively stable divacancies and vacancy agglomerates in crystalline Ge (Markevich 2006; Nagesh and Farmer 1988). Irradiation generates immobile defects comprising divacancies bound to interstitial germanium atoms, although comparatively less is known about their charging (Emtsev 2006; MacKay et al. 1959; Trueblood 1967; Callcott and Mackay 1967; Hiraki et al. 1969; Dostkhodzhaev et al. 1977). Pairs formed of monovacancies and interstitials (i.e., Frenkel pairs) are not observed in Ge, as the VGaGei defect annihilates at temperature above 65 K. Similar to the divacancy in silicon, the ideal, undistorted state of the divacancy in germanium has D3d symmetry (Ögüt and Chelikowsky 2001), but charge-dependent relaxations cause symmetry changes. The charge state-dependent Jahn–Teller distortions and relaxation energies of VGeVGe have been explored using ab initio and density functional theory calculations. Ögüt and Chelikowsky found resonantbond distortions to be slightly lower in energy than those with pairing distortion. They obtained Jahn–Teller energies for RB distortions of 0.09, 0.19, and 0.06 eV for the (+1), (0), and (−1) charge states of the divacancy, respectively. Using cluster calculations that accounted for spin-polarization effects, Coutinho et al. suggested different symmetry-lowering relaxations for (VGeVGe)+1 and (VGeVGe)−1 (2006). For these divacancy charge states, either pairing or resonant-bond relaxations occurred depending on whether the authors used the theoretical or experimental lattice constant for Ge. For example, when the experimental lattice constant was used, the (+1) and (−1) charge states of the divacancy exhibit weak resonant-bonding distortion leading to C2h point-symmetry, although the Jahn–Teller energies associated with these relaxations were almost negligible (< 0.01 eV). On the other hand, supercell and cluster DFT calculations suggest resonant-bond distortions for (VGeVGe)0 and symmetric relaxations for all other charge states (Janke et al. 2007). The magnitudes of distortions observed by Janke et al. were comparable to those reported by Coutinho et al. (2006).

5.1.3 Gallium Arsenide Gallium arsenide has a zincblende crystal lattice with a lattice constant of 5.65330 Å. The zincblende structure consists of two equivalent, interpenetrating face-centered cubic lattices, one containing Ga atoms and the other As atoms; these sublattices are separated by 2.44 Å along the body diagonal of the unit cube (Blakemore 1982).

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5.1.3.1 Point Defects The picture of charged defects in gallium arsenide, a group III–V binary semiconductor having the zincblende structure, differs from that of silicon and germanium because there exist two different types of vacancies, interstitials, and antisite defects. The vacancies and interstitials are symbolized by VGa, VAs, Gai, Asi, and antisite defects by GaAs and AsGa. An ideal Ga vacancy in GaAs is surrounded by four As atoms that remain in their bulk crystalline positions; defect creation results in four broken bonds pointing out from the four As atoms towards the Ga vacancy (Bachelet et al. 1981). The As dangling bonds do not form pairs in any charge state, and the defect maintains the same configuration for all charge states. All of the atoms move significantly inwards while maintaining Td symmetry, with charge state-independent volumetric relaxations on the order of 35% for the stable defects VGa0, VGa−1, VGa−2, and VGa−3 (Laasonen et al. 1992; Hyangsuk and Lewis 1995; El-Mellouhi and Mousseau 2005). By contrast, the geometric and electronic structure of the arsenic vacancy is far different. Except for the positively charged vacancy, there are substantial departures from the initial local symmetry; for VAs+1, tetrahedral symmetry is preserved with an associated inward 17.1% volume change. For the (+1) charge state of the arsenic vacancy, in contrast to the other stable charge states, there is no breathing mode displacement to break the local symmetry and no bond stretching and pairing to cause large volume deformations (El-Mellouhi and Mousseau 2005). For the neutral and (–2) vacancy, large Jahn–Teller distortions destabilize the defect, and negative-U behavior is observed (Northrup and Zhang 1994). El-Mellouhi and Mousseau calculated the point symmetries and percent volume changes for the stable (−1) and (−3) charge states as ~D2d and D2d-resonant and −51.2 and −62.9, respectively (2005). Figure 5.8 summarizes the volumetric relaxations of the charged gallium vacancy in comparison to those of the charged arsenic vacancy. Gallium and arsenic interstitials can exist in several structural forms. Tetrahedral, hexagonal, twofold-coordinated bridge-bond, and split (both <110> and <100>) geometries have been explored for Gai+3, Gai+2, Gai+1, and Gai0. For illustrative purposes, the eight initial configurations studied by Malouin et al. (2007) are depicted in Fig. 5.9. Recent DFT-LDA calculations using the local-orbital basis set program SIESTA at 0 K indicate that a tetrahedral configuration is most favorable for all Gai charge states (Malouin et al. 2007). The geometry entails the shifting of the interstitial atom in a tetrahedral position with respect to four Ga atoms. An alternate tetrahedral configuration wherein the Ga interstitial is surrounded by four lattice As atoms has the next highest formation energy for charge states (+2), (+1), and (0). Earlier studies yielded alternate results may be inaccurate due to the strong finite size effects of 32-atom and 216-atom unit cells and use of TB versus DFT potentials (Chadi 1992; Zollo and Nieminen 2003). For the arsenic interstitial, Asi+1 consists of an As atom and a Ga atom sharing a Ga lattice site while Asi0 and Asi−1 likely take on a split-interstitial configuration (Schick et al. 2002; Chadi 1992; Zhang and Northrup 1991). For the split-interstitial, the Asi-As bond expands by 3.5% and contracts by 3% in comparison to the neutral state bond length

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Fig. 5.8 Histogram of the change in volume of the relaxed tetrahedron formed by atoms surrounding the Ga or As vacancy in GaAs. Results are calculated by DFT for two densities of k points in the Brillouin zone, and expressed as a percentage change compared to the ideal volume. Reprinted figure with permission from El-Mellouhi F, Mousseau N (2005) Phys Rev B: Condens Matter 71: 125207-6. Copyright (2005) by the American Physical Society.

Fig. 5.9 (a) The eight initial configurations for the gallium interstitial in GaAs. The first six configurations, going from left to right, are viewed near the <110> direction, while the remaining two are viewed along the <100> direction. (b) The metastable configurations obtained after the full relaxation of Gai. Reprinted figure with permission from Malouin M-A, El-Mellouhi F, Mousseau N (2007) Phys Rev B: Condens Matter 76: 045211-3. Copyright (2007) by the American Physical Society.

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Fig. 5.10 The atomic geometry of the isolated arsenic antisite. Solid circles are As atoms and open circles are Ga atoms. Reprinted figure with permission from Dabrowski J, Scheffler M (1989) Phys Rev B: Condens Matter 40: 10391. Copyright (1989) by the American Physical Society.

when the system is allowed to relax in the (−1) and (+1) charge states, respectively (Schick et al. 2002). The arsenic antisite defect, which is illustrated in Fig. 5.10, has received a great deal of attention in conjunction with the oft-observed EL2 defect in GaAs, with which it has been associated (Lagowski et al. 1982; Weber 1983; Kaminska et al. 1985; Skowronski et al. 1985; Dabrowski and Scheffler 1988; Chadi and Chang 1988b; Chadi 2003; Hausmann et al. 1996; Hurle 1999; Overhof and Spaeth 2005). The defect can exist in two different forms: stable and metastable. Dabrowski and Scheffler identified the basic mechanism of EL2 metastability with that of the AsGa ↔ VGaAsi structural transition (1989). Further discussion of the metastable EL2 defect appears elsewhere (Dabrowski and Scheffler 1988). The stable EL2 defect has Td symmetry in all charge states and, upon relaxation, the distance to the next nearest neighbors increases by 4.7% for AsGa0 and decreases by 3% and 1.4% for AsGa+1 and AsGa+2, respectively (Overhof and Spaeth 2005; Dabrowski and Scheffler 1988). The gallium antisite is stable in both the neutral and (–2) state; the cation antisite defect likely acts as a negative-U system. GaAs0 prefers to adopt a threefoldcoordinated broken-bond configuration, while GaAs−2 is most stable in a fourfoldcoordinated substitutional configuration (Zhang and Chadi 1990). Large outward bond relaxations of 34% that, in turn, lead to the destabilization of GaAs−1, are calculated for the neutral charge state. Td symmetry, with small inward relaxations, is predicted for the (−2) charge state of GaAs (Zhang and Chadi 1990; Zhang and Northrup 1991; Baraff and Schlüter 1985; Hakala et al. 2002). The two geometries for the defect are depicted in Fig. 5.11. 5.1.3.2 Associates and Clusters Almost all of the isolated point defects in GaAs (including VGa, VAs, Gai, Asi, GaAs, and AsGa) join together to form defect associates and clusters, some of which

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Fig. 5.11 Structural models for the Ga antisite in GaAs where a) is the broken-bond configuration for GaAs0 and b) is the four-fold coordinated substitutional site model for the negative charge state. Reprinted figure with permission from Zhang SB, Chadi DJ (1990) Phys Rev Lett 64: 1790. Copyright (1990) by the American Physical Society.

are better understood than others. The divacancy defect in GaAs tends to take on the form of a “mixed” divacancy, or VGaVAs, where a gallium vacancy binds with an arsenic vacancy. Little discussion is found in the literature on the charging of the other defect associates in GaAs. For instance, three different forms of the neutral di-interstitial have been investigated, the As di-interstitial, the Ga di-interstitial, and the “mixed” di-interstitial, GaiAsi (Zollo and Nieminen 2003; Zollo et al. 2004). At least one report has looked at larger interstitial clusters in GaAs (Zollo and Gala 2007). The existence of Frenkel pairs (either VGaGai or VGaAsi) in GaAs has been suggested, although little is known about their stability or charge state (Pons and Bourgoin 1985; Stievenard et al. 1990; Reddy et al. 1996). Monovacancies can also bind to antisite defects to form associates such as VGaGaAs and VAsAsGa (Laine et al. 1999; Baraff and Schlüter 1986; Makinen et al. 1989). Historically, species such as these have been discussed in the context of the EL2 defect. Although the ODMR-visible EL2 defect is now almost conclusively associated with AsGa (Overhof and Spaeth 2005), earlier reports correlated it with more complicated defect configurations such as AsGaVAs (Baraff and Schlüter 1985), AsGaAsi (von Bardeleben et al. 1986), AsGaVGaVGa, and AsGaVGaVAs (Wang et al. 1988; Wager and Van Vechten 1987). The mixed divacancy in GaAs, VGaVAs, is stable in three charge states, (0), (−1), and (−2). Positron annihilation measurements reveal that divacancies are formed in high-energy (500 MeV) heavy ion implanted GaAs (Chen et al. 2000). Further experimental evidence for the existence of the divacancy comes from positron lifetime measurements (Dannefaer and Kerr 1986; Dannefaer et al. 1989) and local vibrational mode (VLM) spectroscopy (Skowronski 1992). Dannefaer and coworkers found VGaVAs to be stable for temperatures below 620 K (1986; 1989). The defect is likely responsible for the EL6 defect seen in the DLTS spectrum of electron irradiated material (Reddy et al. 1996) and the 0.93 eV photoluminescence band observed in n-type GaAs (Vorobkalo et al. 1973; Reshchikov et al. 1995, 1996). Charged divacancies where the missing species are both gallium atoms have also been considered (Logan and Hurle 1971). Positron lifetime measurements

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Fig. 5.12 Ball and stick models of the VGaVAs divacancy showing (a) the symmetry-conserving C3v structure corresponding to the (−2) charge state and (b) the symmetry lowering σ1v mode corresponding to the (0) and (−1) charge states. Reprinted figure with permission from Pöykkö S, Puska MJ, Nieminen RM (1996) Phys Rev B: Condens Matter 53: 3817. Copyright (1996) by the American Physical Society.

suggest that similar, but larger, clusters of gallium and arsenic vacancies (e.g. VGaVAsVGa, VAsVGaVAs, or VGaVAsVGaVAs) may also occur, albeit in a transient manner, in bulk GaAs (Gebauer et al. 2000; Mascher et al. 1988). For instance, neutron or electron irradiation, as well as plastic deformation, are known to produce agglomerates containing 6, 10, and even 12 vacancies. According to calculations employing a self-consistent charge DFT-based tight-binding method, each contains an equal number of gallium and arsenic atoms; V12 is especially stable due to the presence of dimers that reduce the total number of dangling bonds (Staab et al. 1999; Leipner et al. 2000; Bjorkas et al. 2006; Haugk et al. 2000). In its ideal, unrelaxed state, VGaVAs possesses C3v symmetry (Pöykkö et al. 1996; Xu 1992b, a), but distortions occur depending upon charge state. Although Xu performed tight-binding calculations that focused on electronic structure rather than charge state-dependent geometry, he proposed negligible lattice distortions accompanying the (0), (−1), and (−2) charge states of the defect (1992b). Ab-initio calculations provide additional insight into the charge state-dependent structural relaxations (Pöykkö et al. 1996). Pöykkö and co-authors found C3v symmetrypreserving relaxations for (VGaVAs)−2 and energy lowering σ1v point-symmetry for (VGaVAs)−1 and (VGaVAs)0. Both of these modes are shown in Fig. 5.12. For all three charge states, the Ga and As atoms surrounding the divacancy relax inwards; this is similar to the behavior of Ga and As monovacancies in GaAs. On average, relaxations shorten the bond lengths by ~10%. The interatomic distances between the As atoms at the Ga vacancy end are about 2 to 3% shorter than the corresponding distances between Ga atoms at the As vacancy end. For the σ1v symmetry, the As atoms at the Ga vacancy end exhibit pairing type Jahn–Teller relaxations; the Ga atoms at the As vacancy end, on the other hand, exhibit resonant-bond type relaxations. One should note that these values all derive from one source; reports on the preferred relaxations of the divacancy in Si and Ge (pairing versus resonant-bond type) frequently conflict. The stable configurations of the three types of GaAs di-interstitial, AsiAsi, GaiGai, and GaiAsi, are illustrated in Fig. 5.13. In each of the three diagrams, the di-interstitials are arranged in a triangular structure, where the two interstitial

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Fig. 5.13 Equilibrium atomic configurations for a) As di-interstitial, b) Ga di-interstitial, and c) mixed di-interstitial. Reprinted figure with permission from Volpe M, Zollo G, Colombo L (2005) Phys Rev B: Condens Matter 71: 075207-4. Copyright (2005) by the American Physical Society.

atoms and one displaced lattice atom share one lattice site (Volpe et al. 2005). The arrangement can also be described as an associate of an atom in a tetrahedral site with a <110> dumbbell. The four atoms that surround the shared lattice site adopt a distorted tetrahedral configuration that increases the open volume accompanying the di-interstitial defect. Volpe et al. also computed binding energies for the neutral As di-interstitial (–2.49 eV), Ga di-interstitial (–1.67 eV), and mixed di-interstitial (–3.07 eV); the presence of the As self-interstitial, in particular, makes these defects energetically favorable. Zollo and Gala obtained a similar value for the binding energy of (AsiAsi)0, as well as a prediction of C1h point-symmetry (2007).

5.1.4 Other III–V Semiconductors While some structural similarities exist between charged defects in GaAs and those in other III–V semiconductors, many differences are also observed. These differences are not restricted to those compounds that contain a second row element such as boron or nitrogen. As a reminder, some III–V semiconductors possess a zincblende crystal structure similar to GaAs, while others such as BN and AlN that include second-row elements assume a hexagonal crystal structure. In this section, the geometries of the point defects and defect associates and clusters that arise in GaSb, GaN, and BN will be compared and contrasted to those of similar defects in GaAs.

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5.1.4.1 Point Defects The gallium vacancy in GaSb is stable in the (−1), (−2), and (−3) charge states. For the (−1) and (−2) charge states of the defect, symmetry-breaking distortions cause a transition to ~Td and ~D2d symmetry, respectively (Hakala et al. 2002). In contrast, an inward relaxation of 12% preserves Td symmetry for VGa−3. As a reminder, the inward structural relaxation of the gallium vacancy in GaAs is almost independent of charge state, with a magnitude of 13.5–14.5% that preserves Td symmetry in all cases (Laasonen et al. 1992; Hyangsuk and Lewis 1995; El-Mellouhi and Mousseau 2005). Similarly to VAs, however, the point-symmetry and relaxations of the antimony vacancy states VSb−1, VSb0, or VSb+1, depend strongly upon the charge state of the defect. According to Hakala et al., the singly negatively charged vacancy exhibits a very strong inward relaxation of about 20% to D2d symmetry, similar to that found for the VAs−1 in GaAs. On the other hand, for VSb+1, the symmetry group is ~Td with a small inward relaxation of 5%. The tetrahedral interstitial configuration is hypothesized to be the most favorable for all stable interstitial charge states in the band gap, Gai+1, Sbi+1, and Sbi+3. The (–1) gallium antisite defect, GaSb−1, relaxes inwards by 9% to preserve Td symmetry; in GaAs, GaAs−2 is also stable in a fourfold-coordinated substitional configuration (Hakala et al. 2002; Zhang and Chadi 1990). Conversely, SbGa behaves quite differently than the comparable antisite in GaAs and InP. Only the neutral charge state, which displays Td symmetry and outward relaxations of approximately 11%, is stable; there are no stable configurations of SbGa+2, SbGa+1, SbGa−2, or SbGa−1 within the band gap (Hakala et al. 2002). For gallium nitride and boron nitride, which both contain a second-row element, the formation energies of antisite defects in these compounds are prohibitively large. Although those of cation self-interstitials also make their spontaneous formation unlikely, the structure of the more energetically favorable anion (or nitrogen) interstitial will be discussed in brief. GaN has the wurtzite structure rather than the zincblende. Conflicting results exist in the literature regarding the relaxations of the stable charge states of the gallium vacancy, VGa0, VGa−1, VGa−2, and VGa−3. Neugebauer and Van de Walle (1996) indicated that the VGa−3-N bond length increases by 10.6–11.8%, while Limpijumnong et al. (2004) found that the N atoms move outward from VGa−3 by only about 4% of the ideal bond length. In an even earlier work, the former authors suggested charge state-dependent relaxations of the defect, with distances between the vacancy center and the furthest N atom of 3.7, 6.3, 8.3, and 10% for the (0), (−1), (−2), and (−3) charge states, respectively (Neugebauer and van de Walle 1994, 1996). In contrast, the relaxations of the boron vacancy in wurtzite BN are independent of charge state, and closer to the magnitude of those for VGa−3, or about 10% (Orellana and Chacham 1999). But when the charge state of the nitrogen vacancy in GaN changes from (+1) to (−3), the direction of the relaxation changes from outward (VN+1) to inward (VN−1, VN−2, VN−3), with an associated switch from Td to D2d symmetry (Limpijumnong and Van de Walle 2004; Gulans et al. 2005; Ganchenkova and Nieminen 2006). The nitrogen vacancy in BN exhibits exclusively outward relaxations that decrease in

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Fig. 5.14 Ball and stick model of atomic configurations of (a) Ni+3 and (b) Ni−1 defects in GaN in comparison with (c) the undefected GaN crystal lattice. The nitrogen interstitial is depicted in a split-interstitial configuration for both charge states and atomic relaxations are denoted as percentages. Reprinted figure with permission from Limpijumnong S, Van de Walle CG (2004) Phys Rev B: Condens Matter 69: 035207-5. Copyright (2004) by the American Physical Society.

magnitude from 19.8 to 7.7% for the charge states (+3) to (0), with only slight distortions from the Td symmetry (Orellana and Chacham 1999). In GaN, the gallium interstitial favors a site near the octahedral interstitial site for both stable charge states, Gai+1 and Gai+3 (Limpijumnong and Van de Walle 2004). This configuration differs considerably from the split-interstitial and tetrahedral geometries that are observed for Gai in GaAs and GaSb. Also, a small increase in the distance to the nearest neighbor nitrogen atoms, on the order of 0.1 Ǻ, is observed when the charge state decreases from (+3) to (+1). The boron interstitial in BN assumes two different lowest-energy configurations depending on the charge state of the defect. For the (+1) and (+2) charge states, the interlayer B atom is in line with the N and B atoms of the adjacent planes, while for the neutral, (−1), and (−2) charge states, the B interstitial binds with a B atom of the first layer and a B-N pair of the second layer; the latter charge states are associated with large distortions (Orellana and Chacham 2001). For the nitrogen interstitial in GaN, on the other hand, a split-interstitial configuration arises in which Ni forms a bond with one of the nitrogen host atoms, sharing its lattice site (Fig. 5.14). The gallium atoms relax outwards by a distance equal to 25% of the bulk Ga-N bond length for the (+3) charge state, while an inward relaxation of 6.1% is observed for Ni−1 (Limpijumnong and Van de Walle 2004). 5.1.4.2 Associates and Clusters Studies of larger defects in III–V materials other than GaAs have focused on associates and clusters of two to five vacancies, as well as vacancy-interstitial pairs. The mixed GaSb divacancy in the (−1), (−2), and (−3) charge states may mediate self-diffusion in GaSb (Hakala et al. 2002). In both GaP and GaSb, divacancy and trivacancy clusters (VGaVP, VGaVSb, VGaGaSbVGa, for example) have been correlated

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to intrinsic and extrinsic diffusion mechanisms (Young and Pearson 1970; Shaw 2003; Hakala et al. 2002). Similar defect associates have been studied in InP (Beling et al. 1998; Bretagnon et al. 1997; Dannefaer et al. 1996; Willutzki et al. 1995; Deng et al. 2003; Xu 1990; Puska 1989; Zhao et al. 2006b; Horii et al. 1992). Researchers have studied vacancy (VNVN and VBVN) and vacancy-antisite (NGaVN and NAlVN) associates in GaN, BN, and AlN using positron annihilation spectroscopy (Rummukainen et al. 2004; Tuomisto et al. 2005), 1D-CAR (Arutyunov et al. 2002), and ab initio calculations (Assali et al. 2006). Frenkel pairs are known to arise in InP (Hausmann and Ehrhart 1995; Polity and Engelbrecht 1997; Karsten and Ehrhart 1995) and GaSb (Hakala et al. 2002), while no mention has been found in the literature of di-interstitial defects in III–V semiconductors other than GaAs. The charge states of all of the aforesaid defects have not been explored. In a thorough study of GaSb, Hakala et al. found symmetry-lowering relaxations for the three charge states of VGaGaSb ((−1), (−2), (−3)) as well as for (VSbSbGa)+1 and (VGaSbi)0 (2002). All of the defects containing a gallium vacancy relax to exact or approximate C3v point-symmetry. The symmetry of the (VSbSbGa)+1 defect is lowered to C1h. VGa and VSb in GaSb relax inwards in a fashion similar to VGa and VAs in GaAs. Xu studied the electronic structure of divacancies in GaAs, GaSb and GaP using tight-binding calculations and found that the divacancy showed similar behavior in the three materials (1990; 1992a; 1992b). Therefore, one may conclude that the divacancy VGaVSb in GaSb, as well as the divacancy VGaVP in GaP, would also show inward relaxations of the Ga and Sb/As atoms surrounding the divacancy.

5.1.5 Titanium Dioxide Titanium dioxide naturally occurs primarily in three crystalline phases: brookite (orthorhombic), anatase (tetragonal), and rutile (tetragonal). Upon annealing, metastable anatase and brookite undergo first-order phase transformations towards the stable rutile phase (Burns et al. 2004). The structure of anatase TiO2 is considerably more open than that of rutile. Reduction, or removal of oxygen ions, facilitates the collapse from anatase to rutile, while the inclusion of interstitial titanium ions inhibits the same transformation (Shannon 1964). The rutile TiO2 structure consists of chains of TiO6 octahedra with each pair sharing opposite edges of the unit cell. Each Ti atom is surrounded by six oxygen atoms, whereas each oxygen atom is surrounded by three metal atoms arranged as corners of an equilateral triangle. This tri-coordination (for nominally divalent O atoms) signifies the predominantly ionic character of bonding in rutile TiO2. The octahedra within the structure are irregular; they show a slight orthorhombic distortion and are in contact with ten neighboring octahedra (Camargo et al. 1996). Each oxygen atom in the rutile TiO2 structure is shared by three octahedra, leading the oxygen atoms to have a triangular planar coordination with respect to the titanium atoms (Bursill and Blanchin 1984). Rutile TiO2 has lattice constants

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of 4.59–4.64 and 2.95–2.97 Å for the “a” and “c” sides of the unit crystal, respectively (He and Sinnott 2005). The anatase TiO2 structure consists of a titanium atom surrounded by six oxygen atoms in a distorted octahedral configuration (Diebold 2003). As in rutile, the oxygen atoms are threefold coordinated. When the anatase-to-rutile phase transformation occurs, the relatively open anatase structure collapses in volume by about 8%; the cell volumes of anatase and rutile are approximately 68 Å3 and 62 Å3, respectively (Shannon 1964). The collapse is accompanied by distortion of the oxygen framework, breaking of two of the six Ti-O bonds, and shifting of the Ti+4 ions. The crystal parameters of bulk anatase TiO2 calculated by Na-Phattalung et al. (2006), a = 3.764 Å, c/a = 2.515, and u = 0.208 Å are in good agreement with prior experimental values obtained by Howard et al. (1991). 5.1.5.1 Point Defects The structures associated with the charged defects in TiO2 are not nearly as well described as for those in group IV and III–V semiconductors. The effect of variable charge states on defect configuration and relaxation is not often explored in conjunction with titanium dioxide; only recently have the structural relaxations around charged Tii, VO, and VTi defects been elucidated. Charged antisite defects (OTi−4 and TiO+6) are rarely addressed. One recent study has calculated prohibitively large formation energies for these defects in anatase TiO2 (Na-Phattalung et al. 2006), a function of the large cation-anion size mismatch and strong ionicity of the material. This explanation has not been invoked anywhere else, however, to explain their absence from the overall defect chemistry in the bulk. Instead, point defect formation is typically explained in terms of Schottky or Frenkel defect pair formation. Intrinsic disorder of the Schottky type leads to the presence of a lone VTi−4 defect and two VO+2 defects within the crystal. Frenkel type disorder implies the formation of a vacancy and an interstitial of the same atom; in TiO2, the Frenkel defect comprises Tii+4 and VTi−4. This is not to say that lone point defects do not exist in bulk TiO2, but rather that the metal oxide community takes a distinctly different approach to describing their simultaneous formation with complementary defects in the crystal lattice. When it comes to describing the structure and geometry of both anion and cation vacancy defects in TiO2, a clear void exists in the literature. Due to the positive charge on the oxygen vacancy, the surrounding Ti cations are displaced outwards and the O anions relax inwards towards the defect site. Using the modified semi-empirical intermediate neglect of differential overlap (INDO) and firstprinciples methods, relaxations from 0.1 to 0.3 Å have been calculated, or approximately 5–15% of the bulk Ti-O bond length of 1.95 Å (Stashans et al. 1996; Cho et al. 2006). Recent DFT calculations performed using the GGA suggest nearest (and next-nearest) neighbor Ti and O atom relaxations of 13.6 (13.8)% and −1.0 (2.4)% surrounding VO+2 and 2.2 (3.4)% and 5.6 (9.7)% surrounding VTi−4 (He et al. 2007).

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Fig. 5.15 Possible geometries for octahedral Ti interstitial sites in rutile TiO2. Titanium and oxygen atoms are shown in dark and light gray, respectively. Reprinted figure with permission from He J, Sinnott SB (2005) J Am Ceram Soc 88: 740. Copyright (2005) by Blackwell Publishing.

The titanium interstitial, whose charge state has been the subject of much controversy, is preferentially located at an unoccupied octahedral site adjacent to its original lattice site (He and Sinnott 2005; Yu and Halley 1995). Configurations in which the Ti atom moves in the <100> or <010> direction to form the defect are more favorable than those in which the Ti interstitial moves along the <001> direction, as shown in Fig. 5.15. The presence of the interstitial titanium atoms causes additional distortion of the octahedra within the unit cell. Due to the open structure of the rutile phase along the <001> direction, relatively small relaxations are found around the Ti interstitial compared to the case of the oxygen vacancy. A first-principles study performed by Cho et al. found that the closest Ti atoms relax away from the interstitial site by 13% of the bulk Ti-O bond length, while the nearest four O atoms pull in toward the cation interstitial by 10% (2006). Slight (< 1% relative change from bulk bond distance) differences in oxygen and titanium atom relaxations surrounding Ti interstitials in the (+4) versus (+3) charge state are calculated (He et al. 2007). The oxygen interstitial and its associated geometry have not received a great amount of attention in the literature due to its high formation energy at Ti-rich, or reduced, conditions. Although, in principle, the interstitial could contribute to the defect chemistry of TiO2 at standard (1 atm) and high (105 Pa) oxygen partial pressures, it exhibits a strong tendency to bind to lattice oxygen, which causes O2 dimers to substitute for O lattice sites. Consequently, the presence of Oi is indicated by the neutral substitutional diatomic molecule (O2)O that forms on the O site (Na-Phattalung et al. 2006). The defect is reminiscent of the split-interstitial that occurs in silicon, although first-principles calculations show that its diatomic bond length, wave functions, and stretch frequencies better resemble those of a free molecule (Limpijumnong et al. 2005). No significant relaxation of the neighboring atoms is expected, as the charge neutral O2 is electrostatically equivalent to any O atom on a lattice site and the rutile structure is open enough to accommodate the diatomic molecule. Although charge state effects were not addressed explicitly, the diatomic molecule would not be expected to ionize easily.

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Fig. 5.16 Atomic structures of (a) bulk anatase TiO2 (b) Tii+4 (c) VTi−4 (d) (O2)O and (e) VO+2. The large spheres are the Ti atoms and the small spheres are the O atoms. Relaxation directions of neighboring atoms are indicated by the arrows. In some cases, ideal positions of the atoms before relaxation are shown by dashed circles. Reprinted figure with permission from Na-Phattalung S, Smith MF, Kwiseon K et al. (2006) Phys Rev B: Condens Matter 73: 125205-4. Copyright (2006) by the American Physical Society.

To touch briefly upon the defect chemistry in anatase versus rutile TiO2, a recent paper by Na-Phattalung et al. addresses several features of the defect geometry in anatase that remain unexplored for rutile TiO2 (2006). The three nearest-neighbor Ti atoms of the oxygen vacancy in anatase experience outward relaxations away from VO+2 and towards its five remaining O neighbors. VTi−4, a defect that arises in small concentrations, causes the surrounding O atoms to relax outwards substantially to a point halfway between their two remaining Ti neighbors. When Tii+4 is situated at an octahedral site, one vertical O neighbor moves towards the defect, while the surrounding Ti atoms relax outward as a result of the electrostatic repulsion between positively charged Ti ions. In anatase TiO2, the substitutional O2 molecule induces a small outward relaxation of the neighboring Ti atoms. The defect has a bond length of 1.46 Å, or about 21% greater than the bond length of free O2, 1.21 Å. The geometries of all of the aforementioned defects in anatase TiO2, along with the relaxation directions of neighboring atoms, are depicted in Fig. 5.16. 5.1.5.2 Associates and Clusters Point defect associates and clusters have been examined mostly in rutile, rather than anatase, titanium dioxide. The few articles that discuss VTi, VO, Tii, and Oi clustering in anatase TiO2 mostly relate to the magnetic properties of the semiconductor (Bai and Chen 2008; Olson et al. 2006; Yoon et al. 2006; Yoon et al. 2007). There has been no discussion of charging effects in these associates. In reduced TiO2–x, where the total concentration of defects is greater than in stoichiometric TiO2, defects prefer to join together and form ordered structures or clusters (Fergus 2003; Goodenough 1972). At sufficiently large concentrations

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Fig. 5.17 The atomic arrangement around the most stable divacancy in TiO2. For comparison, the initial configuration before relaxation is also displayed in the upper portion of the figure. The dashed circles indicate the cation vacancy sites. Titanium atoms are indicated by light gray circles while oxygen atoms are shown as dark gray circles. Reprinted with permission from Ahn H-S, Han S, Hwang CS (2007) Appl Phys Lett 90: 252908-2. Copyright (2007), American Institute of Physics.

(greater than 6%), computations have projected that vacancy associates actually decrease the size of the TiO2 band gap; this behavior may prove advantageous for the optimization of photo-electrochemical devices (Hossain et al. 2007). As with GaAs and the other III–V semiconductors, much attention has been focused upon the divacancy defect in TiO2. For rutile TiO2, however, the cation and anion divacancies, VTiVTi and VOVO, rather than the mixed divacancy, have received the most consideration (Catlow and James 1982; Yu and Halley 1995). Experimental and theoretical evidence also exists for Frenkel defects (He and Sinnott 2005), oxygen vacancy-titanium interstitial associates (Miyaoka et al. 2002), divacancyinterstitial clusters (Richardson 1982), and di-interstitials (Hasiguti 1972). Figure 5.17 illustrates the unrelaxed and fully relaxed structure for the neutral cation divacancy in oxygen-rich rutile examined via first-principles calculations (Ahn et al. 2007). When the two titanium atoms are removed from the crystal lattice, the oxygen atoms are left with dangling bonds. As a result, these atoms form O-O bonds with neighboring oxygen atoms after moving in the direction indicated by the arrows in the figure. The resulting O-O bond length of ~1.4 Å is only slightly larger than O-O bond lengths in O2 and O3 molecules (~1.25 Å). Calculations from the same laboratory shed light upon the possible configurations of the oxygen divacancy in rutile TiO2 (Cho et al. 2006). Two different configurations, where the oxygen atoms are either removed from nearest-neighbor or apical positions around a Ti atom, were examined. The apical divacancy was found to have ~0.60 eV higher formation energy compared to the nearest-neighbor divacancy. No information on the structural relaxations accompanying divacancy formation

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was provided. Ab-initio calculations have also been used to investigate the effect of oxygen vacancy associates on the band structure of TiO2, although relaxation effects were also neglected due to excessive computational demands (Hossain et al. 2007).

5.1.6 Other Oxide Semiconductors The crystal structures shared by ZnO are wurtzite, zincblende, and rocksalt; wurtzite is the only thermodynamically stable phase at room temperature. Although the zincblende ZnO structure can be stabilized by growth on cubic substrates and the rocksalt structure occurs at high pressure, their defects will not be discussed within this work (Özgür et al. 2005). In the hexagonal wurtzite crystal lattice, Zn atoms are tetrahedrally coordinated to four O atoms, where the Zn d electrons hybridize with the O p electrons. The unit cell measures 3.24 and 5.20 Å along the a and c-axes, respectively, indicating a slightly non-ideal hexagonal structure, as a/c = 1.602 rather than 1.633 (Pearton et al. 2004). UO2 crystallizes in the fluorite structure wherein U4+ ions form a face-centered cubic sublattice that meshes with a simple cubic O2− sublattice. Uranium ions occupy every other cube formed by oxygen anions. The experimental lattice constant of UO2 is 5.47 Å (Kudin et al. 2002). Interestingly, the fluorite structure is stable up to extremely high temperatures; this is clearly an advantage of UO2 as a nuclear fuel (as are many of its other thermophysical properties (Fink 2000)). The melting point of UO2 is 2,860ºC in comparison to 1,687ºC and 1,210ºC for Si and Ge, respectively. CoO is one of a category of p-type transition metal oxide semiconductors that also includes NiO, FeO, MnO, and Cu2O. The divalent rocksalt-structured CoO has a lattice parameter of 4.26 Å. The larger oxygen anions are close-packed while the smaller cobalt cations occupy octahedral interstices (Fryt 1976). 5.1.6.1 Point Defects The structure and geometry of the defects in other oxide semiconductors such as ZnO, UO2, and CoO have been explored in a similar fashion to those in Group IV and III–V semiconductors (Petit et al. 1998; Crocombette et al. 2001; Erhart et al. 2005). In UO2 and CoO, a full understanding of defect configuration has yet to be obtained (Chen et al. 2007; Gupta et al. 2007; Iwasawa et al. 2006). ZnO, on the other hand, has been characterized with more success; especially recently, its point defects have been explored in more detail than those in TiO2. When an oxygen atom is removed from the wurtzite ZnO lattice, an almost perfect tetrahedron of Zn atoms forms around the vacancy site. The magnitude and direction of the relaxations are highly dependent upon the charge state of the defect, as shown in Fig. 5.18. Although both VO0 and VO+2 experience large

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Fig. 5.18 Ball and stick representation of the local atomic relaxations around the oxygen vacancy in ZnO in the (0), (+1), and (+2) charge states. Reprinted with permission from Janotti A, Van de Walle CG (2005) Appl Phys Lett 87: 122102-2. Copyright (2005), American Institute of Physics.

displacements, VO0 displays an inward relaxation of the Zn atoms of approximately 11–12%, while VO2+ experiences an outward relaxation of about 19–23% due to electrostatic repulsion (Erhart et al. 2005; Janotti and Van de Walle 2005). As a reminder, for VO+2 in TiO2, small inward and outward relaxations of the neighboring oxygen and Ti atoms on the order of −1% and +2.4%, respectively, were calculated. The relaxations of Ti and O atoms around VO0 in TiO2 have not been investigated. The large relaxation associated with the formation of VO+2 in ZnO substantially reduces the formation energy of the defect with respect to VO+1, making the oxygen vacancy observed under zinc-rich conditions a negative-U defect (Janotti and Van de Walle 2005). The (+1) vacancy causes the four Zn nearest neighbors to be displaced outwards by only 2% (Janotti and Van de Walle 2005). In contrast to the relaxations surrounding VO, Erhart et al. found that those induced by VZn are almost independent of charge state (2005). The (0), (−1), and (−2) charge states of the zinc vacancy defect exhibit a symmetric 14% outward relaxation of the nearest O neighbors, while relaxations of farther neighbors are negligible. The structures of isolated oxygen interstitials as well as substitutional diatomic oxygen in ZnO have been examined. Limpijumnong et al. have considered the latter of these two species, which is stable only in the neutral charge state; the defect will not exist in any appreciable concentration, as it has a formation energy far higher than that of Oi in both zinc- and oxygen-rich material (2005). In oxygen-rich material with Fermi levels in the lower part of the band gap, however, the neutral dumbbell-shaped oxygen interstitial is the dominant point defect. Before arriving at this conclusion, Erhart et al. considered the relative importance of three Oi geometries, octahedral, dumbbell, and rotated dumbbell, as a function of charge states ranging from (+2) to (−2) (2005). As shown in Fig. 5.19, the octahedral defect is stable in the (0), (−1), and (−2) charge states, the dumbbell in the (0) charge state, and the rotated dumbbell in the (0) and (−2) charge states, respectively. The dumbbell interstitial is characterized by two oxygen atoms that form a homo-nuclear bond and jointly occupy a regular oxygen lattice site; it is conceptually similar to the well-known dumbbell interstitial defect in silicon and the nitrogen interstitial in GaN (Erhart et al. 2005; Colombo 2002; Limpijumnong and

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Fig. 5.19 Overview of the possible oxygen interstitial configurations in ZnO. The medium and dark gray spheres are the zinc and lattice oxygen atoms, respectively. The interstitial oxygen atom(s) are colored in light gray. The a) octahedral and b) dumbbell configurations are shown for the neutral charge state. The rotated dumbbell geometry that occurs for c) the neutral charge state is also shown for d) the (+2) charge state. Reprinted figure with permission from Erhart P, Klein A, Albe K (2005) Phys Rev B: Condens Matter 72: 085213-5. Copyright (2005) by the American Physical Society.

Van de Walle 2004). The rotated dumbbell is related to the non-rotated dumbbell by a rotation of the oxygen bond about the <10-10> axis by an angle of 105º; each oxygen atom is coordinated to three, rather than four zinc atoms. As with TiO2, antisite defects are rarely considered in ZnO. Intuitively, the formation energy of ZnO will be high under O-rich conditions and that of OZn high under Zn-rich conditions (Oba et al. 2001). Even though the formation energies of ZnO+4 and ZnO+2 are only slightly higher than those of vacancies and interstitials in heavily Zn-rich p-type and n-type material, they hardly exist in practice (Zhao et al. 2006a). 5.1.6.2 Associates and Clusters Defect associates and clusters also arise in other oxide semiconductors such as ZnO, UO2, and CoO. In zinc oxide, defects tend to agglomerate into vacancy pairs; our knowledge of these defects comes from photoluminescence, radiation damage, and Hall effect studies. More complicated charged clusters are known to arise in uranium dioxide and cobaltous oxide. To the best of our knowledge, no information is available on the dependence of these cluster geometries and their relaxations on charge state, let alone Fermi energy. Zinc vacancies, oxygen vacancies, and zinc interstitials bind together to form experimentally observable neutral and charged defect associates and clusters in ZnO. The mixed divacancy, VZnVO, is observable in photoluminescence spectra (Liu et al. 1992; Zhong et al. 1993; Brauer et al. 2007), and positron annihilation results suggest that clusters of up to three or four Zn vacancies occur in flash-annealed ZnO (Borseth et al. 2006). It is unlikely that isolated oxygen vacancies aggregate in ZnO, as large lattice relaxations upon VO formation and the presence of directed bonds leads to dimer vacancy instability (Carrasco et al. 2004, 2005).

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The “green band” observable in luminescence spectra of ZnO has been attributed to a neutral oxygen-related associate formed from VO+1 and VZn−1 (Studenikin and Cocivera 2002) or ZnO (Reynolds et al. 1997). The model of Studenikin and Cocivera accounts for the slow PL decay component, as the defect VO+1VZn−1 is suggested to emit light only when the associate captures a hole. This assignment is significant, as it implies that a neutral defect pair can form from constituent defects that retain their original charge states. Look et al. originally attributed the 30 meV shallow donor level observed in irradiated ZnO to a zinc sublattice defect, either a Zni or Zni-related cluster (1999a). A paper published shortly thereafter put forth more support for the chain-like character of the proposed Zni-related cluster (Look et al. 1999b). The authors accounted for the high radiation threshold (> 1.6 MeV) of the material via a stable defect such as VZnZnOOZnZni, which contains the three-displacement chain VZnZnOOZn. For this model to be valid, the positively charged Zni must be more than a nearest-neighbor distance away from the negatively charged VZn, preventing immediate recombination and rendering the VZnZni Frenkel pair unstable. The simpler Frenkel pair, VZnZni, has been studied via optically detected electron paramagnetic resonance; this defect can exist in the neutral or singly positive charge state depending on whether VZn−1 joins with Zni+1 or Zni+2 (Vlasenko and Watkins 2005, 2006). In cobaltous oxide, the so-called 4:1 cluster, comprising four Co vacancies and one Co interstitial, remains the most studied defect agglomeration with charge states of (–3), (–4), or (–5) (Nowotny and Rekas 1989; Logothetis and Park 1982; Catlow and Stoneham 1981; Petot-Ervas et al. 1984). 2D-ACAR measurements provide experimental evidence for positron trapping at this cluster (Chiba and Akahane 1988). According to the embedded-molecular cluster model of Khowash and Ellis, the 4:1 cluster has a Co+3 cation sitting in the center of a cube with four nearest O and Co vacancies, as seen in Fig. 5.20; the cluster can exist in the (−3), (−4), or (−5) charge state (Khowash and Ellis 1987; Gesmundo et al. 1988; Grimes et al. 1986). From an investigation of stoichiometry-dependent electrical conductivity data, Nowotny and Rekas also support the existence of the 4:1 cluster in the (−5) charge state (1989). It appears that the Fermi energy dependence of these charge states has not been explored; the studies considered only binding energies, not Fermi level-dependent formation energies, so there is no indication that the (−3) charge state is stable for energies close to the valence band maximum, for example.

Co +2 Co+3 Fig. 5.20 The 4:1 cluster in CoO. Reprinted figure with permission from Nowotny J, Rekas M (1989) J Am Ceram Soc 72: 1216. Copyright (1989) by Blackwell Publishing.

VCo

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Fig. 5.21 Ball and stick model of the 2:2:2 defect cluster for hyperstoichiometric UO2. Figure from Willis BTM (1987) J Chem Soc, Faraday Trans II 83: 1076. Reproduced by permission of The Royal Society of Chemistry.

When uranium dioxides tends towards hyperstoichiometric UO2+x, the increasing concentration of oxygen interstitials is stabilized by defect clusters of oxygen interstitials and vacancies (Ruello et al. 2004). The 2:2:2 Willis cluster and the cuboctahedral cluster are often mentioned in the literature in order to explain the defect structure of hyperstoichiometric UO2+x (Iwano 1994; Hubbard and Griffiths 1987; Murray and Willis 1990). The 2:2:2 cluster consists of two vacant oxygen sites, two oxygen interstitials displaced away from the cubic-coordinated interstitial sites along <110>, and two oxygen atoms displaced along <111>, as shown in Fig. 5.21 (Willis 1978, 1987). The charge state of the 2:2:2 cluster in UO2+x depends upon ambient temperature and pressure. By comparing experimental conductivity results with the equation for pressure dependence, Kang et al. have been able to derive the effective charge of the (2:2:2) cluster; in the high PO2 regime of hyper-stoichiometric UO2+x, a (–1) charged cluster prevails (2000). Explanations for chemical diffusion coefficient measurements in uranium dioxide also invoke the (−1) charge state of the defect cluster (Ruello et al. 2004). The cuboctahedral cluster in UO2+x is an aggregate of one interstitial oxygen ion, twelve oxygen ions displaced to interstitial sites, and eight oxygen vacancies, as shown in Fig. 5.22. It has been investigated by single and multiple charge state models, with a focus on the (−4), (−5), and (−6) charge states (Iwano 1994). The

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Fig. 5.22 The cuboctahedral cluster with open circles as normal oxygen ions, open squares as oxygen vacancies, black circles as uranium ions, gray circles as oxygen ions at the center of the vacancy cube, and light grey circles as oxygen ions at a position displaced along the <110> direction from the mid-point of each edge of the oxygen vacancy cube. Reprinted from Iwano Y, “A defect structure study of nonstoichiometric uranium dioxide by statistical-mechanical models,” (1994) J Nucl Mater 209: 80. Copyright (1994) with permission from Elsevier.

formation energy and vibrational entropy of the defect vary with charge state. The formation energies obtained via the two models (single versus multiple charge states), however, reveal different trends as a function of defect charge state.

5.2 Surface Defects Characterizing the geometries of point defects on semiconductor surfaces involves considering additional complexities not relevant to bulk defects. For instance, a semiconductor can have more than one cleavage plane. Additionally, semiconductor surfaces often reconstruct to reduce the number of dangling bonds. These reconstructions can differ greatly among the various crystallographic orientations and may have complicated geometries themselves. Defect geometries differ accordingly among the crystal planes, and relatively few generalizations can be made. The terminology used to describe surface defects sometimes differs from that used to describe bulk defects. The surface science community tends to refer to “surface interstitials” as “adatoms”; other communities rely upon a mix of the two. A large agglomeration of vacancy defects in the bulk is a “vacancy cluster”; the analogous feature on the surface is often called a “vacancy island.” Table 5.2 summarizes the correspondences in defect structure and behavior for the bulk and surface.

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Table 5.2 Correspondence in defect structure and behavior for the bulk and surface Bulk

Surface

Interstitial atom Vacancy Interstitial cluster Vacancy cluster Kick-in/kick-out Vacancy-interstitial formation

Adatom or surface interstitial Vacancy Adatom island Vacancy island Exchange diffusion Vacancy-adatom formation

5.2.1 Silicon The Si(100) surface that is technologically relevant to integrated circuit fabrication exhibits a reconstruction with (2×1) periodicity, creating rows of surface atoms bonded to each other as dimers. The dimers buckle slightly from a perfectly symmetric configuration, although they oscillate rapidly back and forth between the two mirror-image buckled configurations. The buckled configuration can be describing as having an “upper” atom residing slightly further above the surface than the “lower” atom. The Si(111) plane is the natural cleavage plane of Si, and it reconstructs below roughly 1,000 K into a complicated (7×7) structure shown in Fig. 5.23. The surface was originally described as possessing a “dimer adatom stacking-faulted” structure. It can be broken into three regimes: bulk, reconstruction, and adatom

Fig. 5.23 Top view of the Si(111)-(7×7) reconstruction. The four non-equivalent types of adatoms are labeled 1) unfaulted corner 2) unfaulted edge 3) faulted corner and 4) faulted edge. Reprinted figure with permission from Lim H, Cho K, Capaz RB et al. (1996) Phys Rev B: Condens Matter 53: 15422. Copyright (1996) by the American Physical Society.

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Fig. 5.24 Diagram of the Si(111)-(1×1) slab including a vacancy before relaxation. Only the first two surface layers and the adatom layer are shown for clarity. Reprinted from Dev K, Seebauer EG, “Vacancy charging on Si(111)-(1×1) investigated by density functional theory,” (2004) Surf Sci 572: 486. Copyright (2004) with permission from Elsevier.

(Takayanagi and Tanishiro 1986). The bulk unreconstructed (111)-(1×1) phase exists beneath the second reconstruction layer. Above roughly 1,100 K, the Si(111)-(7×7) reconstruction undergoes a reversible phase transition to the (1×1) phase. Experiments employing second harmonic generation (SHM) (Hofer et al. 1995), scanning tunneling microscopy (STM) (Yang and Williams 1994), and reflection high-energy electron diffraction (RHEED) (Kohmoto and Ichimiya 1989; Fukaya and Shigeta 2000) have deduced that the (1×1) surface has a relaxed bulk-like structure with an adatom coverage of about 0.25 monolayers. At temperatures where the (1×1) reconstruction is observed, the adatoms move quickly and therefore do not reside in single sites for very long. However, a useful static model of this dynamic surface is one in which adatoms are placed in a (2×2) periodicity, thereby reproducing the 0.25 monolayers coverage, as shown in Fig. 5.24 (Dev and Seebauer 2004). Quantum calculations by Meade and Vanderbilt have indicated that this configuration is second only to the (7×7) in stability at room temperature (1989). 5.2.1.1 Point Defects Due to the buckled configuration of Si(100), two distinct types of monovacancy can form: the upper monovacancy in response to removal of the upper atom, and analogously for the lower monovacancy. For the most part, when a single-atom vacancy forms, the atom that was previously dimerized with the missing atom tends to leave the dimer row fairly rapidly, transforming the missing-atom vacancy into a missing-dimer vacancy. The discussion of divacancy defects on Si(100), however, will be postponed until the section on associates and clusters.

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The dominant defect on Si(111)-(7×7) is the monovacancy. Four vacancy locations on the (111) surface have been explored in the literature: the unfaulted corner (UFC), unfaulted edge (UFE), faulted corner (FC), and faulted edge (FE) (Lim et al. 1996; Dev and Seebauer 2003). The different vacancy types (which were also depicted in Fig. 5.24), arise from the unusual adatom surface arrangement, where the local geometry of Si(111)-(7×7) is described by unfaulted and faulted triangular units of six adatoms. However, the effects of charging on these various vacancy locations are likely to be similar. In fact, the UFE vacancy can have dominant charge states of (–2), (–1), and (0), but structural rearrangements due to charging are negligible (Dev and Seebauer 2003). On Si(111)-(1×1), monovacancies can have charge states of (–1), (0) or (+1) (Dev and Seebauer 2004). Notice the stable donor state that does not exist for the UFE vacancy on the (7×7) reconstruction. As with the (7×7) reconstruction, structural relaxation effects due to vacancy charging are negligible. 5.2.1.2 Associates and Clusters Virtually the entire literature for associates and clusters on silicon surfaces focuses on the (100) orientation. Neutral and charged divacancies (or “dimer vacancies”) are the dominant defects on the Si(100)-(2×1) reconstruction near and below room temperature (Tromp et al. 1985; Hamers and Kohler 1989; Hamers et al. 1986; Kitamura et al. 1993; Owen et al. 1995; Pandey 1985; Roberts and Needs 1990; Wang et al. 1993). Larger charged defects such as double dimer vacancies, splitoff dimer defects, and dimer vacancy lines can form from single divacancies. At least one report utilizes Kelvin probe force microscopy and STM to obtain evidence for similar charge trapping on the atomic steps and disordered domains of the Si(111)-(7×7) surface (Jiang et al. 2006). The dimer vacancy defect was first observed on Si(100) using scanning tunneling microscopy in 1985 (Tromp et al. 1985; Hamers et al. 1986). By STM imaging the Si(100) surface, Hamers and Kohler observed the single dimer vacancy (“A”) as well as two other types of defects, a double dimer vacancy (“B”) and a defect that appears as two half-dimers (1989). The latter defect, the so-called “type C” vacancy, is best described as two adjacent Si atoms missing along a <1 −1 0> direction. In contrast, the dimer vacancy involves the removal of two Si atoms along a <110> symmetry, and has mirror symmetry with respect to reflection in both the <110> and <1 −1 0> planes. The terminology for these defects is not always consistent. For example, there is at least one mention in the literature where the single dimer vacancy is referred to as a “type B” defect (Brown et al. 2002). Divacancy formation is energetically favorable, as the defect stabilizes the oscillatory dimer buckling of the silicon surface by pinning the dimers into one or the other asymmetric buckling orientation (Cricenti et al. 1995). The surface divacancy induces the adjacent pairs of atoms in the underlying layer to relax toward and rebond to each other along the row to partially fill the void, as shown in Fig. 5.25. Although the distances between these pairs are identical in the undefected

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Fig. 5.25 Ball and stick models of single dimer vacancy defects on Si(100) including the (a) nonbonded and (b) rebonded geometries. Silicon atoms that have a dangling bond are shaded in black. Reprinted figure with permission from Schofield SR, Curson NJ, O’Brien JL et al. (2004) Phys Rev B: Condens Matter 69: 085312-2. Copyright (2004) by the American Physical Society.

structure, upon divacancy formation the extent of rebonding differs. The stability of the defect in three equilibrium structures, nonbonding, rebonding, and weak bonding, has been examined (Jeong et al. 1995). Upon relaxation, Chan et al. have found that the distance under the lower atom of the original dimer contracts nearly 0.5 Å more than the corresponding distance under the upper atom (2003). The degree of relaxation of the neighboring atoms does not change significantly with charge state – less than about 0.1 Å This change is smaller than comparable charge-induced changes for the bulk divacancy, which exceed 0.5 Å. At least for Si(100), the single dimer vacancy (1-DV) joins with other surface defects to form larger defect clusters, much as monovacancies do in bulk silicon. For instance, the double dimer vacancy (2-DV) is formed by removing two adjacent dimers from a perfectly dimerized Si(100) surface. Brown et al. have observed both 2-DV and 1+2-DV (also called split-off dimer defect) in neutral and positive charge states on the Si(100) surface using STM imaging (2002; 2003). Koo et al. also observed appreciable amounts of 2-DV and 1+2-DV defects in their STM studies of the Si(100) surface (1995). Other clusters such as 1+1-DV, which consists of a rebonded 1-DV and a nonbonded 1-DV separated by separated by a split-off dimer (Schofield et al. 2004), and 1+3+1-DV, which consists of a cluster of three dimer vacancies with a single dimer vacancy on each side (Wang et al. 1993), have also been reported experimentally. Several computational studies have explored the geometry of 2-DV on Si(100), although none address the symmetry and rebonding of the positive charge state of the defect, which is visible via STM. Using ab initio calculations, Wang et al. (1993) and Chang and Stott (1998) predicted a rebonded structure for (2-DV)0, where the exposed atoms at the center of the defect rebond to exposed atoms on either side of the defect. This configuration is necessarily asymmetric and the rebonded side has a bond length of 2.49 Å, which is 9% longer than that of the surface dimer, but much shorter than the 2.79 Å bond length in 1-DV. This is shown schematically in Fig. 5.26. This asymmetry is also observed experimentally in STM imaging where one of the two dimers adjacent to 2-DV is depressed by more than 0.5 Å (Koo et al. 1995).

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Fig. 5.26 Ball and stick model of the rebonded 2-DV cluster where the white circles represent dimerized surface atoms and the shaded circles represent bulk atoms. Reprinted figure with permission from Wang J, Arias TA, Joannopoulos JD (1993) Phys Rev B: Condens Matter 47: 10499. Copyright (1993) by the American Physical Society.

As both 1-DV and 2-DV can exist in multiple charge states, it is possible that clusters comprised of these defects may also have ionization levels within the band gap. Schofield et al. have considered the geometry of the 1+1-DV cluster, which consists of a rebonded 1-DV and nonbonded 1-DV separated by a split-off dimer (Fig. 5.27) (2004). Another common cluster, 1+2-DV, consists of a rebonded 1-DV, a split-off dimer and a 2-DV with a rebonding atom (also shown in Fig. 5.27) (Schofield et al. 2004; Wang et al. 1993). The most significant atomic displacements are exhibited by the exposed second layer atoms of 1-DV, which are directly adjacent to the non-rebonded side of 2-DV. The 0.07 Å displacement of these two

Fig. 5.27 Ball and stick models of the 1+1-DV and 1+2-DV defects on the Si(001)-(2×1) surface. Reprinted figure with permission from Schofield SR, Curson NJ, O’Brien JL et al. (2004) Phys Rev B: Condens Matter 69: 085312-2. Copyright (2004) by the American Physical Society.

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atoms results in a shortening of the bond length in the rebonded 1-DV from 2.79 Å to 2.71 Å (Wang et al. 1993). When observing these defects via STM, 1+1-DV can be distinguished by its enhanced brightness (in comparison to 1+2-DV), a result of the larger surface strain that its presence induces. Regarding adatom dimers (or “surface di-interstitials”), Ihara et al. initially suggested a model involving interstitial dimers recessed into the Si(100) surface (1990). Three years later, however, Wang et al. readdressed this hypothesis to show that the interstitial dimer leads to a density of states at odds with experimental observations; in contrast to the bulk, the self-interstitial dimer is not a dominant defect on the Si(100) surface (1993). Density functional theory calculations suggest that they are significantly more stable even a few atomic layers below the surface due to the reduction of dangling bonds accompanying surface rearrangement and the delocalization of electrons on the local surface (Kirichenko et al. 2004). Most recently, surface di-interstitials have been readdressed in the context of mobile surface defects, thermal oxidation, or diffusion of implanted dopants. The mobility of these species, which is discussed in several computational studies, may explain why the self-interstitial is not often regarding as an important defect on Si(100) (Martin-Bragado et al. 2003; Law et al. 1998; Hane et al. 2000). Little else has been mentioned about the geometry or structure of these defects, however.

5.2.2 Germanium As germanium naturally cleaves along the (111) plane, its surface geometry and electronic structure has been studied extensively. Cleaving in vacuum at temperatures less than 40 K results in Ge(111)-(1×1), whereas fracturing the crystal at 40–300 K results in the (2×1) superstructure (Popik et al. 2001). The face phase transitions to the stable Ge(111)-c(2×8) upon heating above approximately 300ºC (Selloni et al. 1995). In this configuration, the topmost atoms of the surface saturate ¾ of the dangling bonds of the ideal first layer and donate their extra electron to the remaining first-layer atoms. Becker et al. proposed a model for the surface consistent with that of Si(111) comprised of alternating rows of (2×2) and c(4×2) adatoms on second-layer atoms (T4 sites) on a (1×1) substrate (1989). The Ge(100) surface is characterized by a strong short-range reconstruction with a weaker long-range ordering across the domains. It is the long-range interactions between dimers on the surface of Ge(100) that lead to higher-order surface reconstructions (Loscutoff and Bent 2006). Similar to the Si(100) surface, the Ge(100) surface exhibits a reversible transformation to a “paramagnetic-like” (2×1) reconstruction at room temperature, and a c(4×2) arrangement when cooled to below 220 K (Ferrari et al. 2001). The c(4×2) reconstruction consists of an alternate arrangement of buckled dimers along the [110] and [1 –1 0] surface directions; the buckled dimers help to minimize the surface free energy that results from the dangling bonds of the surface atoms (Zeng and Elsayed-Ali 2002). For

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Ge(100)-c(4×2), Ferrrari et al. calculated a Ge dimer bond length of about 2.6 Å and buckling angle between the atoms of the dimer of 17.5º with respect to the surface plane. 5.2.2.1 Point Defects Isolated point defects are observed on Ge(111)-c(2×8), although their formation is far less favorable than that of similar species on Si(111). In this case, the evidence for defect charging is exclusively experimental. When Ge divides along its natural (111) cleavage plane, single atom vacancies, as well as defects due to antiphase shift, are the dominant point defects (Lee et al. 2000a, b). Lee et al. suggest that surface monovacancies are negatively charged based on the delocalized depressions (indicative of upward band bending) they induce in empty state STM images. No additional information is given about the specific charge states of the defect. Once could hypothesize, however, that the Ge surface monovacancy takes on more than one charge state such as VSi−2, VSi−1, VSi0 on Si(111)-(7×7) or VSi−1, VSi0, VSi+1 on Si(111)-(1×1). 5.2.2.2 Associates and Clusters While experiments reveal that single and double dimer vacancy associates and clusters also exist on the Ge(100)-(2×1) surface, it appears that their charging has yet to be explored on this or any other Ge surface. The most prevalent cluster on the Ge(100) surface is the split-off dimer 1+2-DV, first reported by Yang et al. using STM imaging (Kubby et al. 1987; Yang et al. 1994). The structure of 1+2-DV on Ge(100) deduced by these authors is essentially same as 1+2-DV on Si(100). Interestingly, the formation energy of 1-DV on Ge(100) is three times larger than that of 1-DV on Si(100) (Ciobanu et al. 2004). Single and double dimer vacancies also exist. Dimer vacancy defects on the Ge surface are often studied in the presence of metal impurities (Fischer et al. 2007; Wang et al. 2005, 2006; Gurlu et al. 2004); the in-diffusion of metal atoms pops germanium atoms out of the crystal lattice and leads to the spontaneous generation of 1+1-DV, 1+2+1-DV, and 1+2-DV defects. Zandvliet has written a review article that serves as a comprehensive introduction to the Ge(001) surface including surface dimerization and missing dimer defects (2003). Extended defects such as vacancy and adatom islands and DVLs also form on Ge(100).

5.2.3 Gallium Arsenide GaAs naturally cleaves along the non-polar (110) plane, yielding surfaces that are similar to an ideal truncated bulk (110) plane; a (1×1) unit cell contains one anion

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and one cation, each with a broken bond (Feenstra and Fein 1985). In its unrelaxed form, the GaAs(110) surface consists of “chains” of alternating Ga and As atoms with C1h point-group symmetry directed in the [110] direction (Schwarz et al. 2000). The reconstructed surface consists of gallium and arsenic atoms that protrude downwards and upwards, respectively, relative to the zincblende geometry, in a manner that conserves the nearest-neighbor bond lengths (Duke et al. 1981). This geometry was originally described as “wavelike” or “rippled,” as it contains planar or nearly planar As-Ga-As wave fronts parallel to the (110) direction in the surface plane (Lubinsky et al. 1976). Several reconstructions on the (001) face of GaAs are also stable. The GaAs(001) surface shows a c(4×4) symmetry for As-rich conditions, but changes its periodicity to (2×4)/c(2×8) and finally (4×2)/c(8×2) as the surface gets more cation rich (Schmidt 2002). The (2×4) phase occurs in several forms, called α, β, and γ, depending on the temperature of the substrate (LaBella et al. 2005). For example, α(2×4), which occurs at the highest substrate temperature, has been suggested to correspond to anion dimers orientated along the [–110] direction with cation-cation bonds parallel to [110]. The β phase, which is stable for more anionrich conditions, was described by Chadi as possessing three As dimers in the top level (1987). A modified structure, β2, identified seven years later by Northrup and Froyen has been accepted as the more energetically favorable form of the (2×4) reconstruction; β2 has two As dimers in the top layer and a third As dimer in the third layer (1994). 5.2.3.1 Point Defects Vacancies on GaAs have been studied in considerable detail by both experimental and computational means. GaAs(110) is decorated with charged arsenic vacancies under gallium-rich gas-phase conditions, and with charged gallium vacancies and adatoms under arsenic-rich conditions. The configurations of the gallium vacancies on GaAs(110) have not been explored in depth, however, although it is suggested based on DFT calculations employing the LDA that the defect is stable in the (+1), (0), and (−1) charge states and is characterized by a comparatively large relaxation (0.5 Å) of the surface nearest-neighbor atoms into the surface (Schwarz et al. 2000). In contrast, VAs exhibits negative-U behavior with only a (+1/−1) ionization level within the band gap. This vacancy also induces similarly large nearest-neighbor relaxations (0.3 Å), however (Schwarz et al. 2000). Arsenic vacancies on the GaAs surface behave differently from those in the bulk of the semiconductor; the charge state of the surface species can vary from (+1) to (−1), and their geometry depends strongly on the Fermi energy (Ebert et al. 1994; Zhang and Zunger 1996). When relaxed, the GaAs(110) surface maintains its (1×1) periodicity, but the electrons in the Ga dangling bonds transfer to As, leading to the formation of fivefold-coordinated As and threefold-coordinated Ga atoms. Kim and Chelikowsky have suggested that the three non-rebonded Ga atoms around VAs+1 relax symmetrically inward with respect to the (110) plane, as

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Fig. 5.28 The relaxed geometry of the singly positive As vacancy on the GaAs(110) including (a) top view and (b) cross-sectional view of the top four layers. The solid and open circles denote the As and Ga ions, respectively. The nearest and the next nearest neighbors to the vacancy site are indexed from 1 to 10. Reprinted from Kim H, Chelikowsky JR, “Electronic and structural properties of the As vacancy on the (110) surface of GaAs,” (1998) Surf Sci 409: 438. Copyright (1998) with permission from Elsevier.

shown in Fig. 5.28 (1996; 1998). Zhang and Zunger found, however, that a nonsymmetric rebonded structure reduces the total energy by an additional 0.16 eV for VAs+1 (1996). The same nonsymmetric, rebonded configuration was found to be 0.17 eV more stable by Schwarz et al. (2000). Neither of these static equilibrium vacancy configurations exactly match experimental STM observations, and it is suggested that, at room temperature, the vacancy rapidly flips between the two energetically favorable configurations with an associated energy barrier of ~0.08 eV (Ebert 2002; Ebert et al. 2001). In the neutral and (−1) charge states, DFT investigations indicate that a symmetric rebonded configuration is always preferred (Cox et al. 1990; Zhang and Zunger 1996; Domke et al. 1998). In contrast to bulk gallium arsenide, in which charged AsGa and GaAs defects are well characterized, antisite defects on the GaAs surface have received considerably less attention. It is suggested, however, that they are most stable in the neutral charge state and have no ionization levels within the band gap, in contrast to those in the bulk, which are known to have a charge of (–2) (GaAs) or (+2) (AsGa) (Schwarz et al. 2000; Iguchi et al. 2005). There are two possible configurations for the anion adatom: one in which the adatom is bonded to the surface anions (As1+1, As10, As1−1), and one in which it is bonded to the surface cations (As20 and As2−1). The former is located 1.1 Å above the surface, while the latter is at 0.8 Å above the surface (Schwarz et al. 2000). The suggested geometries of both the antisite and adatom defects on GaAs(110) are depicted in Fig. 5.29.

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Fig. 5.29 Atomic relaxations of surface point defects on GaAs(110) including antisites (a, b), vacancies (c, d), and adatoms (e, f). Reprinted figure with permission from G. Schwarz, J. Neugebauer, and M. Scheffler, Point defects on III–V semiconductor surfaces. In: Proc. 25th Int. Conf. Phys. Semicond. (Eds.) N. Miura, T. Ando. Springer Proc. in Physics, Vol. 87, Springer, Berlin/Heidelberg 2001, p. 1379.

5.2.3.2 Associates and Clusters Associates and clusters comprised of arsenic and gallium vacancies arise on both GaAs(110) and GaAs(100). One notorious “associate,” the bulk EL2 defect, has been discussed in a previous section. The reader should be aware that older articles on the topic of the EL2 defect may discuss charged antisite-vacancy associates on the GaAs surface; the EL2 defect is now identified as the isolated point defect AsGa in the bulk. Defect cluster and associates including VAsVAs (Kanasaki et al. 2007), VGaVGa, and VGaVGaVAsVAs (Gwo et al. 1993) have been observed on GaAs(110) using STM. Green’s function (Ren and Allen 1984) and ab-intio total-energy (Yi et al. 1995) calculations have been employed to study neutral antisite-vacancy (AsGaVAs and GaAsVGa) and dimer vacancy associates on the GaAs(110) surface. Dimer vacancies (of unspecified charge state) have also been observed on the GaAs(100) surface, which is commonly used for the growth of electronic and photonic devices via molecular beam epitaxy (Xu et al. 1993). According to STM images, the (2×4) reconstruction of the Ge(100) surface consists of three As dimers and one missing As dimer (Pashley et al. 1988; Biegelsen et al. 1990). A second

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phase (also observable via STM) consists of two As dimers and two missing As dimers (Biegelsen et al. 1989; Hashizume et al. 1994; Bakhtizin et al. 1997). There is a Fermi-level dependence to the As vacancy clustering that occurs on the GaAs(110) surface according to STM images taken before and after 460 nm laser irradiation with pulse fluences ranging from 0.5 to 4 mJ/cm2 (Kanasaki et al. 2007). The laser fluence is low enough to allow examination of non-thermally stimulated kinetic effects on the surface. Arsenic vacancy clusters form from negatively charged monovacancies on n-type surfaces, while p-type surfaces are described by high concentrations of positively charged As monovacancies. The number and size of the As vacancy clusters on GaAs(110)-(1×1) increases with the number of laser pulses. For example, after 2,000 laser pulses, vacancy clusters containing up to six As vacancies are observed. According to Kanasaki et al., charged vacancies on surfaces can strongly affect subsequent carrier localization to induce local bond rupture on adjacent sites. The behavior cannot be attributed to the effects of vacancy redistribution, as the net charge of monovacancies is unity for both n- and p-type surfaces; as a consequence their repulsive interaction is basically the same. On n-type surfaces, upward band bending increases the rate of electronic bond rupture adjacent to existing negatively charged monovacancies, which raises the probability of vacancy cluster formation. On p-type surfaces, in contrast, downward band bending reduces the rate of bond rupture near positively charged vacancies; valence holes are prevented from localizing around existing vacancies and destabilizing the monovacancies. Fundamentally, the localization of a hole is equivalent to an electron being stripped out of a chemical bond at a localized region. Viewed in this context, the concept of holes mediating bond breakage is almost a given. This argument may not hold for higher processing temperatures, at which the distribution of defects will differ from that artificial freezing-in of a specific configuration at low temperature.

5.2.4 Other III–V Semiconductors The most stable cleavage faces of the zincblende and wurtzite forms of the group III–V semiconductors are the (110) and (1010) surfaces, respectively (Jaffe et al. 1996; Filippetti et al. 1999). Lastly, the (001) and (0001) surfaces of the III–V compounds, those which are relevant for single-crystal growth, have also been explored, particularly with respect to diffusion (Schmidt 2002). The zincblende-structure compound semiconductors containing Al, Ga, or In as the group III atom and As, P, or Sb as the group V atom cleave along their nonpolar (110) planes. On all of these materials explored to date, a (1×1) reconstruction is observed (Ebert 2002). The surface consists of an equal number of anions and cations, and the atoms are displaced relative to the ideal truncated bulk plane.

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Similar to GaAs, the (001) faces of GaP, InP, InAs, etc. undergo reconstructions as a function of anion and cation composition. Surface structures reconstruct to (2×4) periodicity, with either a Ga dimer or mixed-dimer on the top layer, occuring for Ga-rich preparation conditions of GaP. The (2×4)-α2 geometry is relevant for surfaces with balanced stoichiometry, whereas for more P-rich conditions, (2×4)-β2 forms. The InP(001) surface is also (2×4) reconstructed; for In-rich and stoichiometric conditions it is described by the formation of mixed In-P dimers on the top of a cation terminated surface and β2 geometry, respectively (Schmidt 2002). The geometry of the stable cleavage plane of wurtzite group III-nitrides, (1010), and the plane that forms when the crystal is grown by MBE, (0001), have been explored in detail. Both GaN(1010) and GaN(0001) are terminated by metallic layers containing about one bilayer of Ga atoms (Feenstra et al. 2005). GaN(1010) consists of an equal number of threefold-coordinated Ga and N in the surface layer of atoms, which ensures charge neutrality without the need for reconstructions or surface defects (Northrup and Neugebauer 1996). Terraces extend as strips in the [11-2-0] direction, and a (1×1) reconstruction is observed on the surface under Ga-rich conditions (Feenstra et al. 2005). For wurtzite GaN(1010), in contrast to zincblende GaN(110), symmetry allows dimers to rotate only orthogonally to the surface plane. When grown by MBE under Ga-rich conditions, GaN(0001) is known to be terminated by slightly more than a bilayer of Ga atoms. The structure can be visualized as seven unit cells of Ga sitting atop six unit cells of GaN in an incommensurate arrangement (Smith et al. 1998). A (2×2) N adatom model is energetically favored under extremely N-rich conditions, while a (2×2) Ga adatom structure is more stable for larger Ga:N ratios; for both reconstructions, the adatoms rest on a GaN-bilayer terminated GaN(0001)-(1×1) surface (Takeuchi et al. 2005). 5.2.4.1 Point Defects Published work for the indium-containing semiconductors InP, InAs, and InSb focuses exclusively on vacancies. Much of the discussion related to the geometry of the charged vacancy on InP, InAs, and InSb centers around the existence (or lack) of symmetry around the defect site. Similar to GaAs, the stable atomic structure depends critically on the vacancy charge state (Ebert et al. 1994; Engels et al. 1998). For InP and InAs, the surface anion vacancy is stable in the (+1), (0), and (−1) charge states; for InSb, on the other hand, the (−1) charge state is stable for all Fermi energies within the band gap (Qian et al. 2002). For VP, VAs, or VSb, the general trend is for the surrounding In atoms to relax inward. In analogy to the symmetry associated with VAs on GaAs(110), the positively charged anion vacancy has a nonsymmetric configuration with one rebonded dimer, while both the neutral and (−1) charged vacancies show a symmetric configuration with one loosely rebonded trimer (Qian et al. 2002; Cox et al. 1990; Ebert et al. 2000; Zhang and Zunger 1996; Schwarz et al. 2000; Domke et al. 1998).

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Fig. 5.30 Simulated STM images above the surface of a (+1) charged P vacancy on InP(110) surface for different atomic configurations. The simulation shows images for the nonsymmetric and symmetric rebonded configurations, as well as a time average of the two. Reprinted figure with permission from Ebert P, Urban K, Aballe L et al. (2000) Phys Rev Lett 84: 5818. Copyright (2000) by the American Physical Society.

These classifications were initially considered to be at odds with early STM investigations; VP+1 on p-type InP(110) was observed to have a symmetric image, whereas VP−1 on n-type InP(110) and VS−1 on n-type InSb(110) had asymmetric geometries (Ebert et al. 1994; Ebert et al. 2000; Heinrich et al. 1995; Whitman et al. 1991; Qian et al. 2002). But the STM observations can be explained in terms of thermal flipping between two degenerate defects configurations (Fig. 5.30) (Ebert et al. 2000). For the (+1) charge state, thermal flipping prompts the reaction path to pass through a saddle point with a symmetric configuration. For the (−1) charge state, the anion vacancy actually has two local minima, one corresponding to symmetric geometry and the other to a nonsymmetric geometry, although the former has a lower energy (Qian et al. 2002). Qian et al. have also looked at the relaxations associated with the stable anion surface vacancies on InAs, InP, and InSb. In the nonsymmetric configuration of VAnion+1, one of the surface In ions forms a dimer with the subsurface In ion. The three nearest-neighbor In atoms relax substantially inwards by different amounts. On InAs, the surface indium ion, the subsurface indium ion with which it dimerizes, and the surface indium ion that becomes twofold coordinated relax by 11.2, 20.9, and 41.9% of the bulk bond length, respectively. Around the symmetric VAnion0 and VAnion−1 sites with rebonded trimer, the two surface In atoms neighboring the defect move inward, while the subsurface In atom shifts toward the vacancy. For VAnion0, relaxations on the order of 15.9–27.1% of the bulk bond length now take place, with the relaxation of the surface atom typically being slightly larger than that of the subsurface atom, i.e., 21.1% vs. 17.8% in InAs. In going from the (0) to (−1) charge state, the surface indium atoms relax further towards the vacancy site along both the [001] and [110] directions. For example, the surface and subsurface atoms in InAs experience net displacements of 25.2% and 22.1%, respectively.

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The relaxations of the (110) surface of zincblende III–V semiconductors containing nitrogen differ greatly from those of GaAs. For GaAs both the anion and cation are displaced along the z-axis, but for GaN, only the surface cation experiences large shifts (Filippetti et al. 1999). The main relaxation consists of a contraction of the GaN bond in the surface layer and a slight buckling rehybridization with N atoms tending to adopt p3 coordination and Ga atoms adopting an sp2 configuration (Northrup and Neugebauer 1996). The rotation angles associated with atomic relaxation in InN, GaN, and AlN are nearly half those for GaAs, while the bond contractions (CB) for the same compounds are appreciable, in contrast to those in GaAs (Filippetti et al. 1999; Miotto et al. 1999; Pandey et al. 1997; Zapol et al. 1997). Filippetti et al. predicted that materials with small cohesive energies and ionicities will tend to have large rotations, whereas very ionic and strongly bound solids will tend towards small rotations. Rotation angles of between 11.7 and 21.7º are calculated for the nitride semiconductors, in contrast to the approximately 30º found for GaAs (Filippetti et al. 1999; Grossner et al. 1998; Miotto et al. 1999; Swarts et al. 1981). The bond contractions in AlN, InN, and GaN, which are about five times larger than those in GaAs, correlate well with cation size: CB, InN > CB, GaN > CB, AlN (Filippetti et al. 1999; Jaffe et al. 1996; Ooi and Adams 2005; Grossner et al. 1998). Using Hartree–Fock ab initio calculations, Jaffe et al. found that the relatively large surface bond contractions in the nitrides correlate with hybridization effects of cation d states with anion-derived valence states. 5.2.4.2 Associates and Clusters Defect clusters on III–V semiconductor surfaces other than GaAs have been described primarily for InP and secondarily for GaP and GaN. Ohtsuka et al. used photoluminescence to study the reactive ion etching of InP with ethane and hydrogen (1991). They attributed a shift in the peak energy of defect-related bands to a complex of In and P vacancies located close to the surface (supposedly the (100) plane (Yamamoto et al. 1998), although this detail is unclear in the original article). From STM studies of InP(110) and GaP(110) on n-type material, Ebert and Urban reported divacancy clusters on terraces (1993). On both surfaces, the scanning tip prompted P monovacancies to combine to form P divacancies. Furthermore, continued scanning reversibly converted the divacancies into four-vacancy clusters with a triangular structure. InP(110) has also been studied by STM imaging before and after room temperature photoexcitation at wavelengths ranging from 420 to 600 nm (Kanasaki et al. 2007). Laser pulse fluences of 2.7–2.8 mJ/cm2 stimulate formation of phosphorus vacancy clusters containing up to 6 P monovacancies on n-type material. Images of similarly treated p-type InP(110) surfaces reveal only monovacancies (Fig. 5.31). Kanasaki et al. provided an explanation for this behavior based on hole localization and electronic bond rupture which has already been discussed in the context of GaAs(110).

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Fig. 5.31 STM images of surface P-sites on the InP(110)-(1×1) surface before (a) and after excitation of n-type (b) and p-type (c) samples with 5,000 shots of 460 nm laser pulses of 2.7 mJ/cm2. Reprinted from Kanasaki J, Inami E, Tanimura K, “Fermi-level dependent morphology in photoinduced bond breaking on (110) surface of III–V semiconductors,” (2007) Surf Sci 601: 2368. Copyright (2007) with permission from Elsevier.

While the defect clustering behavior on the InP(110) surface resembles that on GaAs(110), InP(100)-(2×4) does not resemble its GaAs counterpart. Arsenic dimer rows are prevalent on GaAs(100) and InAs(100), and As dimer vacancies (of unspecified charge state) form on both of these surfaces (Huff et al. 1998). On InP(100), In-In, P-P and P-In dimers can co-exist (Chao et al. 2002; Schmidt et al. 1999). Chao and co-workers studied the evolution of In-terminated surfaces upon annealing using photoemission spectroscopy (PES) and observed clusters of up to 30 In atoms. Positron annihilation spectroscopy experiments performed on GaN bulk crystal associate a lifetime component at 470±50 ps with vacancy clusters involving up to 20 Ga vacancies (Tuomisto et al. 2005). The clusters are present near the N(0001) face of bulk high pressure grown GaN and have been identified as hollow pyramidal defects via TEM of Mg-doped GaN (LilientalWeber et al. 1999).

5.2.5 Titanium Dioxide The bulk-truncated rutile (110)-(1×1) surface is the most stable and best characterized for TiO2, although two other low-index planes, (100) and (001), have also been explored. The rutile (110)-(1×1) surface contains two different types of titanium atoms; rows of sixfold coordinated Ti atoms (as in the bulk) lie along the [001] direction, whereas fivefold coordinated Ti atoms with one dangling bond are situated perpendicular to the surface (Diebold 2003). Bridging oxygen atoms are absent one bond to the Ti atom in the missing layer, and are twofold coordinated; note should be taken of this unsaturated coordination, as it is found to have a large impact on the occurrence of point defects on the rutile surface. The main relaxations in the rutile structure occur perpendicularly to the surface; only the in-plane oxygens move laterally towards the fivefold coordinated Ti atoms. Bridging oxygen atoms relax downwards by –0.27 Å, while sixfold coordinated Ti atoms are

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displaced vertically upwards by 0.89±0.13 Å. The fivefold coordinated Ti atoms move downwards by approximately 1.16±0.05 Å (Diebold 2003). Several low index surface planes are observed for anatase TiO2 including (101), (100), (010), and (001). The (101) surface is corrugated, with a characteristic sawtooth profile perpendicular to the [010] direction (Lazzeri et al. 2001). The surface comprises both fivefold- and sixfold-coordinated Ti atoms, as well as twofold and threefold oxygen atoms. Upon relaxation, the twofold- and threefold-coordinated oxygen atoms experience small inwards (0.21 Å) and outwards (0.06 Å) displacements. The slightly buckled geometry of the surface occurs due to the 0.17 Å inward relaxation of the fivefold-coordinated Ti atoms. Similar to the reconstruction that rutile undergoes following annealing to high temperatures, anatase is known to undergo a (1×4) reconstruction upon sputtering and annealing in ultrahigh vacuum (Hengerer et al. 2000). 5.2.5.1 Point Defects The twofold coordinated oxygen atoms on the TiO2(110)-(1×1) surface are extremely susceptible to removal during thermal annealing. As originally suggested by Henrich and Kurtz (1981), this process generates a defect in the row of bridging O atoms on the crystal surface (Diebold 2003; Kuyanov et al. 2003). Oxygen atoms in the same plane as the Ti atoms on TiO2(110) can also be removed from the surface, leading to the formation of in-plane oxygen vacancies (POV) rather than bridging oxygen vacancies (BOV) (Fukui et al. 1997). For both the BOV and the POV, the coordination of the cations adjacent to the O vacancy is reduced, as is the screening between the cations (Henrich 1983). A significant number of papers discuss the defect structure of TiO2(110), yet few make mention of defect charging (Mutombo et al. 2008; Morgan and Watson 2007). While some researchers have investigated the charging of specific surface defects, other suggest that the defect charge of reduced TiO2(110) is shared by several surface and subsurface Ti sites (Kruger et al. 2008). For all BOV charge states, BOV0, BOV+1, and BOV+2, there is a slight (~0.01 Å) outward relaxation of the bond between the fivefold-coordinated titanium atom and the in-plane oxygen atom. The Ti-bridging oxygen bond changes from 2.023 Å for the undefected surface to 1.990, 1.991, and 1.968 Å in the presence of BOV0, BOV+1, and BOV+2 (Wang et al. 2005). The Ti-Ti distance across the vacancy increases from 2.959 to 3.305 Å after the BOV formation; it is lowered to 3.200 and 3.239 Å when the vacancy gains a (+1) or (+2) charge, respectively. The 0.14 Å expansion of the sixfold-coordinated Ti atoms away from BOV+2 can be compared to the 0.28–0.30 Å outward displacement of titanium atoms from VO+2 in bulk TiO2 (Wang et al. 2005; Cho et al. 2006). The changes in bond lengths associated with the formation of an in-plane oxygen vacancy on the TiO2(110) surface, POV0, POV+1, and POV+2, as well as those of the BOV, are illustrated in Fig. 5.32.

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Fig. 5.32 Top view of the perfect (a) bridging-oxygen and (b) in-plane-oxygen defect sites on the rutile TiO2(110) two-layer model surfaces. Reprinted from Wang SG, Wen XD, Cao DB et al., “Formation of oxygen vacancies on the TiO2(110) surfaces,” (2005) Surf Sci 577: 72. Copyright (2005) with permission from Elsevier.

5.2.6 Other Oxide Semiconductors There are four low Miller index surfaces for wurtzite ZnO: the nonpolar (1010) and (1120), the polar zinc-terminated (0001)-Zn, and the polar oxygen terminated (0001)-O surfaces. According to low-energy electron diffraction experiments, all exhibit a (1×1) reconstruction (Duke et al. 1977). Meyer and Marx have determined the relaxations of all four surface planes using density-functional theory (2003). Other experimental (Jedrecy et al. 2000) and computational (Whitmore et al. 2002) studies yield comparable results. Among all the low index UO2 surfaces, the (111) surface, the natural cleavage plane of the fluorite-type structure, is the most stable. The atomic structures of two other low-index planes of uranium dioxide, (110) and (001), have also been investigated (Muggelberg et al. 1998; Hedhili et al. 2000; Muggelberg et al. 1999). For UO2(111), the top layer is composed of threefold coordinated oxygen atoms while the second layer is made of sevenfold coordinated uranium atoms. Between each oxygen atom, and 0.7 Å further into the bulk for uranium atoms, (1×1) lattice periodicity is observed (Castell et al. 1998). The spacing between O (or U) atoms along the (111) surface is equal to 3.87 Å, while that between U and O atoms is 2.37 Å (Senanayake et al. 2005). On the (110) surface of CoO, each cation possesses five anion ligands, and the cations are well shielded from each other by the large, polarizable intervening O2− ions. All of the ions below the topmost layer are sixfold coordinated, as in the bulk (Mackay and Henrich 1989). A reconstruction of the surface, consisting of pyramidal surface depressions, appears upon annealing to 1,100 K (Weichel and Moller

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1998). Weichel and Moller cite the driving force of the reconstruction as the defect structure of the CoO surface. The formation of microfacets, or pyramidal surface cavities, reduces the energy of the highly cobalt and oxygen deficient surface. 5.2.6.1 Point Defects Oxygen vacancies are the predominant surface defects on other oxide semiconductors such as ZnO, UO2, and CoO. For example, several oxygen-derived surface defects on ZnO(1010) have been considered: an oxygen vacancy within the surface Zn-O dimer, an oxygen vacancy in the second layer of the material, and a whole dimer vacancy (Wander and Harrison 2000; Whitmore et al. 2002). According to Whitmore et al., the formation energy of the latter is lowest when linear vacancies form in the [001] direction. The existence of charged cation vacancies on zinc oxide is mentioned briefly in the literature. For example, the compensation of ionic excess charge on polar ZnO(1000) may occur, in part, through the formation of doubly positively charged zinc vacancies at the Zn-terminated surface (Kresse et al. 2003). Dulab et al. have found that the removal of Zn atoms happens through the formation of triangular shaped reconstructions (2003). Oxygen vacancies on UO2(111) dramatically affect the surface energy of the semiconductor; ab initio computation of its surface properties is confounded by the complicated hybridization and spin-orbit coupling of UO2 (King et al. 2007). On ionic cobaltous oxide, it has been shown that 500 eV Ar+-ion bombardment leads to a removal of about 4% of the monolayer of surface oxygen ions (Mackay and Henrich 1989; Jeng et al. 1991). For the unrelaxed vacancy defect on the surface, the shielding between cations adjacent to the defect is greatly reduced compared to the stoichiometric surface (Jeng et al. 1991).

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Chapter 6

Intrinsic Defects: Ionization Thermodynamics

6.1 Bulk Defects As discussed in Chap. 2, the concentration of charged defects in the bulk is a function of Fermi energy and ambient conditions such as temperature and pressure. The ionization levels of bulk defects will be discussed in light of the experimental and computational methods discussed in Chap. 3 and defect geometries introduced in Chap. 5. Although a definitive understanding of defect charging is difficult to synthesize from the incomplete and contradictory literature on the subject, an assortment of trends and dissimilarities are clearly discernable. Included in this section, along with numerous literature reports of charged defect ionization levels, are suggested defect ionization levels calculated using maximum likelihood approximation. In many instances, the calculated confidence intervals for these average values are exceedingly small (often 0.01 eV). Intuitively, this small confidence interval appears to be at odds with the wide spread of ionization level values frequently found in the literature. As is mentioned in Sect. 3.2.3, however, the error bars on the average values are statistically defined and influenced by numerous factors. Despite this fact, it is still meaningful and useful to derive an “average” ionization level from a varied assortment of experimental and theoretical reports.

6.1.1 Silicon Silicon has a band gap of about 1.1 eV at 300 K. Most experimental determinations of defect ionization levels in silicon have come from deep-level transient spectroscopy, electron paramagnetic resonance, and diffusion measurements. Numerous calculations of ionization levels by density functional theory have also been published, although many conflict with each other and with experimentally E.G. Seebauer, M.C. Kratzer, Charged Semiconductor Defects, © Springer 2009

131

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obtained values. Some of the deviation may be due to the fact that DFT numbers are all at 0 K, whereas transient enhanced diffusion measurements, for example, are carried out above 1,200 K. Acceptor levels, which track the temperature variation of the conduction band minimum, move from their 0 K positions by more than 0.4 eV at 1,200 K. Neutral and charged defects form readily in single crystal silicon at typical microelectronics processing conditions. Self-interstitial agglomerates are involved in phenomena such as transient enhanced diffusion and extended defect formation. Larger defect clusters and extended defects have been studied in the context of defect recombination, which is highly relevant to implant damage annealing and diffusion. 6.1.1.1 Point Defects Experimental and computational studies have considered the stability of the Si vacancy in the (+2), (+1), (0), (–1), (–2), and (–4) charge states. The ionization levels for these reactions have been examined in detail over several decades; for some levels there exist over twenty reported values. EPR investigations performed by Watkins and co-workers in the 1960s and 1970s led to the first identification of vacancy ionization levels (1963; 1975; 1964; 1979). The formation reactions for the single and double donor and acceptor vacancies in silicon are presented in Eqs. 6.1–6.4.

VSi0 ↔ VSi+1 + e −

(6.1)

VSi+1 ↔ VSi+2 + e−

(6.2)

VSi0 ↔ VSi−1 + h +

(6.3)

VSi−1 ↔ VSi−2 + h + .

(6.4)

and

The ionization levels for vacancy donors are still not fully established in bulk silicon. The biggest issue of contention in the literature has been the existence of negative-U behavior, or the instability of VSi+1 in comparison to VSi+2 and VSi0. Such a situation occurs when, upon capture of the first hole, the energy barrier of the vacancy for capturing a second hole is lowered instead of raised. This behavior has been observed experimentally, yet has been both confirmed and refuted in theoretical reports. If negative-U behavior does exist, Eqs. 6.1 and 6.2 should instead be replaced by

VSi0 → VSi+2 + 2e− .

(6.5)

There exist several reports detailing (+2/0) negative-U behavior for the silicon vacancy. Watkins and Troxell (1980) and Newton et al. (1983) obtained the first

6.1 Bulk Defects

133

direct evidence via EPR experiments. They identified (+2/+1) and (+1/0) vacancy ionization levels at Ev + 0.13 eV and Ev + 0.05, respectively. As the (+1/0) level is located below the (+2/+1) level, or inverted from the expected ionization level ordering, this implies a direct transition between the (+2) and (0) charge states. Upon examining the equations that relate defect charge state and formation energy, it is clear that the ionization level of the direct (+2/0) transition is midway between that of (+2/+1) and (+1/0) at Ev + 0.09 eV. Baraff and co-workers had already predicted such behavior from self-consistent calculations for single-particle states (1979; 1980a; 1980b). Hall-effect measurements on gamma-irradiated samples yield similar double and single donor levels of Ev + 0.128 eV and Ev + 0.028 eV, in that order (Emtsev et al. 1987). Recently, Zangenberg et al. identified a negative-U ionization level at Ev + 0.118 eV using DLTS (2002). Comparable ionization levels have been found from an assortment of theoretical methods. In early Green’s-function total energy calculations, Car et al. found VSi+2 to be stable below EV + 0.12 eV, with a switch to VSi0 for 0.12 < EF < 1.06 eV (1984). Supercell calculations with 216- and 256 atomic sites place the (+2/0) ionization level at Ev + 0.15 eV (Puska et al. 1998) and Ev + 0.09 eV (Lento and Nieminen 2003), respectively. Mueller et al. also observed negative-U behavior for the defect using a 216-atom supercell and a plane-wave-basis set with a kinetic-energy cutoff of 12 Ry; the corrected (+2/+1) and (+1/0) ionization levels are at 0.30 and 0.08 eV above the valence band maximum, respectively (2003). In some instances, the silicon vacancy exhibits negative-U behavior when the formation energy is calculated using the GGA, in others it does not (Shim et al. 2005). On a plot of defect formation energy versus electron chemical potential, negative-U behavior manifests as a direct transition between the (+2) and (0) charge states of the defect, as shown in Fig. 6.1 (right). Also, the figure, based on the DFT calculations of Wright, illustrates how using the LDA versus the GGA alters the position of the (+1/0) ionization level relative to the (+2/+1) level (2006). Some recent reports, both of which use the LDA form of exchange and correlation, show a stable (+1) charge state (Centoni et al. 2005; Schultz 2006). Two separate studies utilizing similar supercell sizes and k-point sampling have failed to reproduce the observed negative-U behavior in the positive charge states of the vacancy. The findings indicate that the difference between the (+2/+1) and (+1/0) ionization levels is only ~0.1 eV. Centoni et al. obtained only a single donor level (+1/0) at Ev + 0.09 eV, while Schultz, who used a novel modification method in his computation, placed (+1/0) and (+2/+1) at 0.19 and 0.07 eV above the valence band maximum, respectively. Much discrepancy exists over the locations of the acceptor ionization levels of the silicon vacancy. In short, several authors have proposed the existence of a second negative-U system for VSi0, VSi–1, and VSi–2, a feature that has not been clearly observed experimentally (Puska et al. 1998; Boyarkina 2000). While some have claimed that the (–1/–2) ionization level lies within the conduction band, others have posited the stability of VSi–2, and even VSi–4, for Fermi energies within the band gap. One can at least be fairly certain that VSi0 is stable for most Fermi energies in the middle third of the band gap.

134

6 Intrinsic Defects: Ionization Thermodynamics

Fig. 6.1 Formation energy of the Si vacancy in charge states from (+2) to (–2) as a function of the Fermi energy based on the data of (Wright 2006). Three vacancy ionization levels arise when the charged defect formation energy is calculated using the LDA (left). In contrast, two ionization levels and negative-U behavior are observed when the GGA is employed (right).

Regarding the (0/–1) vacancy level, irradiation methods (Matsui and Hasiguti 1965, Naber et al. 1973), tracer diffusion (Fairfield and Masters 1967), DLTS, PAS, and other unspecified methods (Van Vechten 1975, Fair 1977) have been used to obtained experimental estimates. Troxell and Watkins concluded that the acceptor level had to be less than 0.9 eV above the valence band maximum in an early DLTS investigation (1979). Subsequent DLTS (Watkins 1986) and positron annihilation (Makinen et al. 1989) experiments suggested that the (0/–1) level was around the middle of the band gap. Green’s function calculations revealed VSi–1 to be stable for all Fermi energies from Ev + 1.06 eV to the conduction band minimum, i.e., the defect has no (–1/–2) level within the band gap (Car et al. 1984). More recent theoretical estimates of the (0/–1) ionization level have been obtained from LDA calculations. Lento and Nieminen (2003), Mueller et al. (2003), Centoni et al. (2005), and Schultz (2006) suggest values of Ev + 0.84 eV, Ev + 0.66 eV, Ev + 0.65 eV, and Ec – 0.39 eV (meaning Ev + 0.70 at 300 K). There is relatively close agreement regarding the location of the (–1/–2) ionization level amongst those who believe it lies below the CBM. Kimerling and coworkers (1977, 1979) and Fair (1977) used DLTS and an unspecified method to identify a level at 1.03 eV and 1.01 eV above the valence band maximum, respectively. LDA computations with a 256-atomic supercell yield a (–1/–2) ionization level at Ev + 0.94 eV (Lento and Nieminen 2003). Using the plane-wave pseudopotential code VASP with a supercell of 216 atoms, Mueller et al. calculated a slightly lower value of Ev + 0.77 eV (2003). Schultz, using a novel method modification, obtained a double vacancy acceptor level of Ec – 0.27 eV (meaning ~Ev + 0.83 eV at 300 K) (2006). Latham et al. calculated a (–1/–2) level at Ec – 0.43 eV or Ec – 0.24 eV (Ev + 0.67 eV or Ev + 0.86 eV at 300 K) depending on whether the vacancy defect was allowed to relax to the C3v or C2v configuration, in that order (Latham et al.

6.1 Bulk Defects

135

Fig. 6.2 Formation enthalpies of vacancies in Si vs. Fermi. The heavy solid line indicates the lowest formation enthalpy at a given Fermi level. Both the lattice vacancy and the split vacancy are shown. The latter is more energetically stable only for the (–2) state. Reprinted figure with permission from Centoni SA, Sadigh B, Gilmer GH et al. (2005) Phys Rev B: Condens Matter 72: 195206-3. Copyright (2005) by the American Physical Society.

2006). Wright computed values of Ev + 0.721 eV and Ev + 0.590 using extrapolated LDA and GGA formation energies, respectively; the latter value is likely more relevant as the GGA reproduces the observed symmetry for VSi–1 and negative-U behavior for the VSi donor states (2006). Centoni et al. identified a similar transition at Ev + 0.74 eV but to the split, rather than lattice, vacancy configuration (2005). Fig. 6.2 illustrates the proposed preferable formation of VB–2 over VL–2. Additional exploration of the charge-state dependence of the split and lattice vacancy formation energies is clearly needed. There exist some anomalous reports in the literature regarding negative-U behavior and the stability of VSi–4. Three studies have posited the existence of negative-U behavior for the negatively charged silicon vacancy. Г-point 64-atomic-site and 216-atomic-site supercell calculations indicate the existence of a (0/–2) ionization level at 0.65 and 0.49 eV above the valence band maximum (Puska et al. 1998). Boyarkina and Vasil’ev interpreted experimental results to obtain (0/–1) and (–1/–2) ionization levels at Ec – 0.09 eV and Ec – 0.39 eV (meaning Ev + 1.01 eV and Ev + 0.71 eV at 300 K) (2000). When Latham et al. allowed the vacancy to relax to

136

6 Intrinsic Defects: Ionization Thermodynamics

C3v symmetry, they obtained a double acceptor negative-U level at Ec – 0.37 eV (Ev + 0.73 eV) (2006). Mueller et al. have also suggested a second negative-U phenomenon for the hitherto unconsidered (–3) and (–4) charge states of the lattice vacancy defect (2003). For Fermi energies close to the conduction band minimum, VSi–3 was not thermodynamically stable. The uncorrected and corrected estimates of the (–2/–4) ionization level are Ev + 0.66 eV and Ev + 0.87 eV, respectively. Table 6.1 summarizes the experimentally and theoretically determined silicon vacancy ionization levels. Based on these values, the maximum likelihood method yields ionization levels of Ev + 0.12 ± 0.01 eV (+2/+1), Ev + 0.07 ± 0.01 eV (+1/0), Ev + 0.09 ± 0.01 eV (+2/0), Ev + 0.71 ± 0.01 eV (0/–1), Ev + 0.88 ± 0.01 eV (–1/–2), and Ev + 0.71 ± 0.02 eV (0/–2). As the maximum likelihood value of the (+1/0) ionization level is closer to the valence band maximum than that of the (+2/+1) level, the ML method supports negative-U behavior and the existence of a direct (+2/0) charge state transition. It is also worthwhile to note that the value of the (+2/0) ionization level given directly by ML is almost identical to the value extracted from the (+2/+1) and (+1/0) ML ionization levels (which would be Ev + 0.095 ± 0.01 eV). It has been suggested that the singly positive charge state of the self-interstitial is not stable. An equation analogous to Eq. 6.5 can then be written to describe the corresponding negative-U behavior:

Sii0 → Sii+2 + 2e− .

(6.6)

Silicon self-interstitials become negatively charged in a manner similar to silicon vacancies:

Sii0 → Sii−1 + h +

(6.7)

Sii−1 → Sii−2 + h + .

(6.8)

Unfortunately, the self-interstitial defect has never been experimentally observed by a direct method; the charge states and ionization levels have been inferred from thermally-stimulated capacitance measurements and irradiation studies. Most additional information regarding ionization levels comes from DFT calculations. There is considerable debate over the best method to use because the combination of strong and weak bonds in many interstitial configurations poses significant challenges for electronic structure calculations. It is apparent that the defect formation energy depends strongly upon local geometric structure. Figure 6.3 exemplifies the manner in which the formation energy of Si self-interstitials, including their geometric configuration, correlates to the Fermi energy within the semiconductor. Different calculation methods yield highly variable formation energies; the generalized gradient approximation is known to yield higher values than the local density approximation (Bernstein and Kaxiras 1997; Jeongnim et al. 2000; Jeongnim et al. 1999). Consequently, the charge states for Si interstitials in p- and n-type silicon are still not clearly established.

6.1 Bulk Defects

137

Table 6.1 Experimentally and computationally determined ionization levels for the vacancy defect in bulk silicon (+2/+1) (+1/0) (+2/0)

(0/–1) (–1/–2) (0/–2) Method

Reference This work

0.12 ± 0.01 –

0.07 ± 0.09 ± 0.01 0.01 – –

0.71 ± 0.88 ± 0.01 0.01 0.2 –

–

–

–

0.78

–

–

0.44

–

0.73

–

–

0.37

–

0.55

1.0

0.71 ± Maximum likelihood 0.02 – Defect formation during gamma irradiation – Tracer diffusion under extrinsic conditions – Carrier removal after electron irradiation – Unspecified

–

0.11

–

–

1.03

–

–

0.12

–

–

–

–

– –

0.35 –

– –

0.68 1.01 ≤ 0.95 ≤ 0.95

– –

–

0.14

–

–

1.03

–

Kinetics after electron irradiation measured by DLTS

0.13

0.05

0.09

–

–

–

EPR

0.112

–

–

0.13

0.0567 0.084 ± – 0.004 0.05 Yes –

–

–

0.128

0.028

0.078 ± – 0.002

–

–

–

–

–

0.19

0.88

–

–

–

–

0.39

0.85

–

–

–

0.078

–

–

–

–

0.2– 0.88 0.5 0.58– – 0.65

–

–

0.13– 0.56 1.03 0.73

–

0.13

–

–

– –

–

–

–

Kinetics after electron irradiation measured by DLTS Kinetics after electron irradiation measured by junction capacitance Unspecified DLTS

(Matsui and Hasiguti 1965) (Fairfield and Masters 1967) (Naber et al. 1973) (Van Vechten 1975) (Kimerling 1977)

(Brabant et al. 1977) (Fair 1977) (Troxell and Watkins 1980) (Kimerling et al. 1979)

(Watkins and Troxell 1980) Hall effect (Mukashev et al. 1982) DLTS (Newton et al. 1983) Hall effect (Emtsev et al. 1984; Emtsev et al. 1987) Defect introduction (Gubskaya et al. by electron irradiation 1984) Unspecified (Van Vechten 1986) Defect formation during (Emtsev et al. gamma irradiation 1989) Formation of radiation (Lugakov and defects Lukashevich 1989) PAS (Makinen et al. 1989) Annealing of Si-E (Boyarkina 2000) centers EPR (Watkins 2000)

138

6 Intrinsic Defects: Ionization Thermodynamics

Table 6.1 (continued) (+2/+1) (+1/0) (+2/0)

(0/–1) (–1/–2) (0/–2) Method

Reference

–

–

0.118

–

–

–

DLTS

–

–

0.12

1.06

–

–

0.45

0.55

–

0.66

0.92

–

–

–

0.15

–

–

0.53

Green’s function calculations Green’s function calculations DFT-LDA

(Zangenberg et al. 2002) (Car et al. 1984)

–

–

0.09

0.84

0.94

–

DFT-LDA

0.3

0.08

–

0.66

0.77

–

DFT-GGA

–

0.09

–

0.65

0.74

–

DFT-GGA

0.418

0.453

0.435

0.531 0.834

0.682 DFT-GGA

0.07 0.13 0.13 0.05

0.19 0.237 0.06 –0.04

– – –

0.85 0.845 0.714 0.81

– – – 0.73

0.73 0.721 0.59 0.86

DFT-LDA DFT-LDA DFT-GGA DFT-AIMPRO

(Puska 1989) (Puska et al. 1998) (Lento and Nieminen 2003) (Mueller et al. 2003) (Centoni et al. 2005) (Shim et al. 2005) (Schultz 2006) (Wright 2006) (Wright 2006) (Latham et al. 2006)

DLTS deep level transient spectroscopy, EPR electron paramagnetic resonance, PAS positron annihilation spectroscopy, DFT density-functional theory, LDA local density approximation, GGA generalized-gradient approximation, AIMPRO ab initio modeling program. All values are in eV and referenced to the valence band maximum.

The silicon self-interstitial is thought to be stable in four charge states – Sii+2, Sii–1, and Sii–2. Recent evidence based on diffusion experiments suggests that tetrahedral Sii+2 dominates in p-type silicon, with a switch to neutral Sii in a <110>split configuration for weakly n-type material. The formation energies and ionization levels of the negatively charged defect in strongly n-doped Si are still poorly characterized. Numerous reports suggest that Sii+1 is unstable and that Sii+2 ionizes directly to 0 Sii in a manner characteristic of negative-U behavior. Most early experimental reports, as seen in Table 6.2, made mention only of a (+1/0) ionization level, not a (+2/0) ionization level. One experimental radiation defect formation investigation suggested a negative-U ionization level at Ev + 0.67 – 0.69 eV (Lugakov and Lukashevich 1989). Abdullin et al. assigned (+2/+1) and (+1/0) Sii ionization levels at Ec – 0.39 eV and Ec – 0.26 eV (meaning ~Ev + 0.73 eV and Ev + 0.86 eV at 300 K), respectively, based on the annealing properties of two traps observed with DLTS (1992a). They later reversed the ordering of these levels to suggest negative-U behavior (1992b). Subsequent thermally-stimulated capacitance measurements performed by Abdullin and Mukashev confirmed a level at Ec – 0.36 eV (Ev + 0.74 eV) associated with a positively charged defect, presumably the unstable

Sii0,

6.1 Bulk Defects

139

Fig. 6.3 Formation enthalpies of interstitials in Si vs. Fermi level. The heavy solid line indicates the lowest formation enthalpy at a given Fermi level. The tetrahedral, hexagonal, and split-<110> configuration are shown. The split-<110) is most stable except for the (+2) charge state. Reprinted figure with permission from Centoni SA, Sadigh B, Gilmer GH et al. (2005) Phys Rev B: Condens Matter 72: 195206-3. Copyright (2005) by the American Physical Society.

Sii+1 (1994). Based on the analysis of the secondary processes of radiation-induced defect formation in Si crystals, Lukjanitsa obtained a significantly lower (+2/0) ionization level of Ev + 0.43 eV (2003). Fair’s TED profile simulation has been revisited by Jung et al., with the finding that experimental data can be accurately described in terms of just Sii+2 and Sii0 defects, with an effective negative-U ionization level at Ev + 0.12 ± 0.05 eV (2005). The LDA calculations of Harrison yielded a significantly higher value of Ev + 0.6 eV (1998). DFT and first-principles studies by Zhu et al. (1998), Hakala et al. (2000), and Centoni et al. (2005) found a transition from tetrahedral Sii+2 to <110>-split Sii0 at Ev + 0.45 eV, Ev + 0.62 eV (this value was calculated from the data in their Table II), and ~Ev + 0.5 eV, respectively. Considerably fewer (and more divergent) estimates of the (0/–1) and (–1/–2) ionization levels exist. The experimentally determined values lie considerably below the computationally determined ones. DLTS, diffusion and OED measurements suggest a single acceptor level anywhere between 0.63 and 1.03 eV above

140

6 Intrinsic Defects: Ionization Thermodynamics

Table 6.2 Experimentally and computationally determined ionization levels for the self-interstitial defect in bulk silicon (+2/+1) (+1/0) (+2/0)

(0/–1) (–1/–2) Method

Reference

Maximum likelihood

This work

Self-diffusion Self-diffusion Quenching of defects Annealing of irradiation induced defects Unspecified DLTS OED IED OED OED

(Chik 1970) (Seeger and Frank 1973) (Leskoschek et al. 1973) (Naber et al. 1973)

0.76 ± 0.01 – – – –

0.64 ± 0.01 0.315 0.4 0.4 0.46

0.65 ± 0.01 – – – –

0.77 ± 0.01 0.695 0.67 0.8 0.73

0.81 ± 0.03 – – – –

– – – – – –

0.4

– – – – – –

0.72 0.63 1.03 1.03 1.01 –

– – – – – –

–

–

–

–

0.73 0.86 –

0.86 0.73 0.74

0.67– 0.69 – – –

– –

– – –

–

0.5

–

0.72

0.82

0.24

–

0.43

0.66

–

– –

– –

– –

– 1.04

Formation of radiation defects DFT-LDA DFT-LDA

1.01 – 0.78 –

– – – –

DFT-LDA DFT-LSDA DFT-GGA Fitting of TED profiles

(Zhu et al. 1998) (Hakala et al. 2000) (Centoni et al. 2005) (Jung et al. 2005)

0.66 0.93

0.727 1.1

DFT-GGA DFT-LDA

(Shim et al. 2005) (Schultz 2006)

– – – – 0.558 0.57

0.34 0.39 0.17 0.33

0.6 0.73 (+2/–1) – 0.45 – 0.62 – 0.5 – 0.12 ± 0.05 0.587 0.573 0.68 –

Formation of radiation defects DLTS DLTS Capacitance measurements OED

(Frank 1975) (Lefevre 1980) (Giles 1989) (Giles 1991) (John and Law 1993) (Tsoukalas and Chenevier 1987) (Lugakov and Lukashevich 1989) (Abdullin et al. 1992a) (Abdullin et al. 1992b) (Abdullin and Mukashev 1994) (Roth and Plummer 1994) (Lukjanitsa 2003) (Harrison 1998) (Lee et al. 1998b)

DLTS deep level transient spectroscopy, OED oxidation enhanced diffusion, IED inverse error diffusion, TED transient enhanced diffusion, DFT density-functional theory, LDA local density approximation, GGA generalized-gradient approximation, AIMPRO ab initio modeling program. All values are in eV and referenced to the valence band maximum.

the valence band maximum (Chik 1970; Seeger and Frank 1973; Leskoscheck et al. 1973; Naber et al. 1973; Frank 1975; Lefevre 1980; Giles 1989, 1990; John and Law 1993). Lukjanitsa obtained a value of Ev + 0.66 eV for the (0/–1) ionization level by analyzing radiation-induced defect formation (2003). Lee et al. (1998b) reported a switch to from Sii0 to Sii–2 in the [110]-split configuration

6.1 Bulk Defects

141

above Ev + 1.04 eV, while Zhu et al. (1998) found a switch to Sii–1 at Ev + 1.01 eV. First-principles calculations predict the ionization of Sii,<110> to a (–1) charge state for Fermi energies above Ev + 0.78 eV (Centoni et al. 2005). Using DFT with a modified supercell method, Schultz obtained values of Ev + 0.93 eV and Ev + 1.1 eV for the (0/–1) and (–1/–2) ionization levels, in that order (2006). The only (indirect) experimentally determined (–1/–2) ionization level is located at Ev + 0.82 (Roth and Plummer 1994). Table 6.2 summarizes the computational and experimental values proposed for the ionization levels of self-interstitials in silicon. Using maximum likelihood estimation, the (+2/+1), (+1/0), (+2/0), (0/–1), and (–1/–2) ionization levels are Ev + 0.76 ± 0.01 eV, Ev + 0.64 ± 0.01 eV, Ev + 0.65 ± 0.01 eV, Ev + 0.77 ± 0.01 eV, and Ev + 0.81 ± 0.03 eV.

6.1.1.2 Associates and Clusters Of all the defect associates and clusters in silicon, the divacancy is the best characterized in terms of ionization levels. It is fairly well established that VSiVSi can take on charge states of (+1), (0), (–1) and (–2), and that the most favorable state changes progressively in this order as the Fermi energy moves up from the valence band to the conduction band. Multiple authors have reported values around Ev + 0.25 ± 0.01 eV for the single donor (+1/0) ionization level, including Ev + 0.25 eV from EPR (Watkins and Corbett 1965), Ev + 0.27 eV from IR absorption and measurements of Hall effect (Konozenko et al. 1969), Ev + 0.28 eV from IR spectroscopy (Svensson et al. 1989), and Ev + 0.32 eV from photoconductivity (Lappo and Tkachev 1970). Curiously, however, DLTS measurements all converge upon a somewhat lower value between 0.18 and 0.25 eV above the valence band maximum (Asghar et al. 1993; Khan et al. 1999; Kimerling 1977; Lee et al. 1980; Londos 1987; Mooney et al. 1977; Trauwaert et al. 1996; Kauppinen et al. 1998). The origin of the discrepancy is unclear. One sole computational report has identified the (+1/0) divacancy ionization level at Ev + 0.24 eV (Mueller et al. 2003). Early experiments placed the (0/–1) level of VSiVSi around the middle of the band gap, while more recent experimental and computational papers report a slightly higher value. Photoconductivity experiments identified an ionization level at Ev + 0.53 eV (Lappo and Tkachev 1970) and Ec – 0.54 eV (Ev + 0.56 eV) (Kalma and Corelli 1968). Similar values were obtained by Konozenko et al. using IR absorption and Hall effect (1969) and Gregory and Barnes using lifetime measurements (1968). Similar to the (+1/0) level, measurements of the single acceptor level using DLTS tend to be slightly different. A level associated with (VSiVSi)–1 has been identified at Ev + 0.65 – 0.68 eV (Chung et al. 2003), Ev + 0.67 eV (Zangenberg et al. 2002; Fleming et al. 2007), Ec – 0.41 eV (Ev + 0.69 eV) (Svensson and Willander 1987), Ev + 0.7 eV (Evwaraye and Sun 1976), and Ev + 0.71 eV (Kimerling

142

6 Intrinsic Defects: Ionization Thermodynamics

1977; Tokuda et al. 1979; Kuchinskii and Lomako 1987). Recent DFT calculations (Wixom and Wright 2006) yield different values depending on the chosen format of exchange and correlation (LDA or PBE); the single acceptor level occurs at Ev + 0.75 eV using the LDA and Ev + 0.71 eV using the PBE. The ionization level value computed using the GGA (Ev + 0.47 eV) is far smaller than both of these estimates (Mueller et al. 2003). EPR and photoconductivity experiments also tend to underestimate the (–1/–2) ionization level of the divacancy in comparison to DLTS and DFT-LDA calculations. For example, values such as Ec – 0.42 eV (Ev + 0.68 eV) (Lappo and Tkachev 1970), Ec – 0.4 eV (Ev + 0.7 eV) (Watkins and Corbett 1965), and Ec – 0.39 eV (Ev + 0.71 eV) (Kalma and Corelli 1968) are almost identical to those recently proposed for the (0/–1) ionization level. Conversely, DLTS (Chung et al. 2003; Chung et al. 2002; Evwaraye and Sun 1976; Fleming et al. 2007; Kimerling 1977; Kuchinskii and Lomako 1987; Svensson and Willander 1987; Tokuda et al. 1979) and annealing (Cheng et al. 1966) experiments suggest that (VSiVSi)–2 is stable for Fermi energies 0.84 – 0.89 eV above the valence band maximum. As with the single acceptor level, the use of the GGA seems to yield an unreasonably low value of Ev + 0.56 eV (Mueller et al. 2003). Wixom and Wright obtained two different values for the double acceptor (–1/–2) level; Ev + 0.80 eV using the LDA and Ev + 0.77 eV using the PBE, respectively (2006). Table 6.3 summarizes the computational and experimental values proposed for the ionization levels of divacancies in silicon. On this basis, the maximum likelihood method yields ionization levels of Ev + 0.25 ± 0.01 eV (+1/0), Ev + 0.65 ± 0.01 eV (0/–1), and Ev + 0.87 ± 0.01 eV (–1/–2). The literature concerning the ionization levels for vacancy clusters larger than the divacancy is essentially non-existent. EPR studies of neutron-irradiated material reveal negative charge states for V3 and V4 (Lee and Corbett 1974) as well as V5 (Lee and Corbett 1973), but do not report ionization levels. Far less information exists concerning the ionization levels of the silicon diinterstitial. As the di-interstitial is highly mobile, experimental characterization of the defect is difficult; computations indicate that SiiSii is stable in the (+1), (0), and (–1) charge states. Ab initio calculations performed using both the local-density and generalized gradient approximation yield values of Ev + 0.1 eV for the single donor (+1/0) level and Ev + 0.6 eV for the single acceptor (0/–1) level of SiiSii with C1h point-symmetry (Kim et al. 1999). Kim et al. also discovered almost degenerate donor and acceptor levels for the metastable C2v structure at 0.4 eV above the valence band maximum. Density functional calculations reveal a (+1/0) donor ionization level of SiiSii corresponding to the “form C” structure with C1h symmetry at Ev + 0.32 eV (Eberlein et al. 2001). This value is close to the donor level of Ev + 0.4 eV associated with the C2 structure by Lee and coworkers based on EPR data (1976).

6.1 Bulk Defects

143

Table 6.3 Experimentally and computationally determined ionization levels for the divacancy defect in bulk silicon (+1/0)

(0/–1)

(–1/–2)

Method

0.25 ± 0.01 0.65 ± 0.01 0.87 ± 0.01 Maximum likelihood 0.25 – 0.7 EPR – – 0.89 Annealing of irradiation induced defects – 0.56 0.71 Photoconductivity 0.35 0.6 – Lifetime measurements 0.27 0.56 0.90 ± 0.01 IR Absorption and Hall effect 0.32 0.53 0.68 Photoconductivity 0.25 0.7 0.87 Annealing of irradiation induced defects 0.21 0.71 0.87 DLTS 0.23 – – DLTS – 0.71 0.89 DLTS 0.21 0.71 0.89 DLTS 0.19 – – DLTS – 0.69 0.87 DLTS 0.28 – – IR spectroscopy 0.21 – – DLTS 0.19 – – DLTS 0.23 – – PAS 0.18–0.20 – – DLTS – 0.65–0.68 0.843 DLTS 0.18 0.67 0.86 DLTS – 0.65–0.68 0.843 DLTS – 0.67 0.86 DLTS 0.24 0.47 0.56 DFT-GGA – 0.75 0.8 DFT-LDA – 0.71 0.77 DFT-PBE

Reference This work (Watkins and Corbett 1965) (Cheng et al. 1966) (Kalma and Corelli 1968) (Gregory and Barnes 1968) (Konozenko et al. 1969) (Lappo and Tkachev 1970) (Evwaraye and Sun 1976) (Kimerling 1977) (Mooney et al. 1977) (Tokuda et al. 1979) (Kuchinskii and Lomako 1987) (Londos 1987) (Svensson and Willander 1987) (Svensson et al. 1989) (Asghar et al. 1993) (Trauwaert et al. 1996) (Kauppinen et al. 1998) (Khan et al. 1999) (Chung et al. 2002) (Zangenberg et al. 2002) (Chung et al. 2003) (Fleming et al. 2007) (Mueller et al. 2003) (Wixom and Wright 2006) (Wixom and Wright 2006)

DLTS deep level transient spectroscopy, EPR electron paramagnetic resonance, IR infrared, PAS positron annihilation spectroscopy, DFT density-functional theory, LDA local density approximation, GGA generalized-gradient approximation, PBE Perdew–Burke–Ernzerhof. All values are in eV and referenced to the valence band maximum.

With regard to charging behavior of interstitial clusters larger than I2, only the single donor level (+1/0) of the tri-interstitial and the tetra-interstitial appear to have been studied. For the tri-interstitial, DFT computations place this ionization level at Ev + 0.19 eV (Carvalho et al. 2005) and ~Ev + 0.1 eV (Coomer et al. 1999), or close to the value of Ev + 0.15 eV associated with the W photoluminescence center. Mukashev et al. assigned a donor level of Ev + 0.29 eV to I4 based on the Si-B3 spectra (1982), which was later supported using ESR methods (Pierreux and Stesmans 2003). Similar values of Ev + 0.16 – 0.27 eV (Coomer et al. 2001), Ev + 0.37 eV (Arai et al. 1997), and Ev + 0.29 eV (Carvalho et al. 2005) have been

144

6 Intrinsic Defects: Ionization Thermodynamics

obtained computationally using density functional theory and semi-empirical tight binding methods. For the Frenkel pair (VSiSii associate), support exists for a single donor (+1/0) and double acceptor (–1/–2) ionization level. Optical absorption and electrical measurements of irradiated silicon indicate that VSiSii may have a donor level at Ev + 1.03 eV (Vook and Stein 1968). A similar number was reported by Gregory and Barnes (1968). Based on the kinetics of defect formation in gamma-irradiated Si, Emtsev et al. have since suggested that Ev + 1.03 eV corresponds, instead, to a Frenkel pair double acceptor level (1989).

6.1.2 Germanium Germanium has a band gap of about 0.67 eV at 300 K. As a semiconductor, it has a low effective mass and higher mobility of charge carriers in comparison to silicon, which makes it a promising material for high-performance logic devices. Some of the defects in germanium behave qualitatively in the same way as defects in silicon. Take, for instance, the vacancy, which exists in several charge states in the band gap. Experimental characterization of charged defects in Ge is difficult due to the rapid mobility (and consequent recombination) of the defects at temperatures below room temperature (Haesslein et al. 1998). 6.1.2.1 Point Defects Experimental and computational evidence exists for three germanium vacancy charge states within the band gap: VGe0, VGe–1, and VGe–2. In contrast to silicon, the vacancy defect in Ge is never a single or double donor; it likely has no (+1/0) or (+2/+1) ionization levels in the band gap. The energy of correlation between the first and second donor states is not sufficiently negative as to produce a (0/+2) level within the band gap (Janotti et al. 1999); the existence of a (0/–2) level has not been ruled out, however (Coutinho et al. 2005). Van Vechten and Thurmond were among the first to invoke Ge vacancy charging in order to accurately predict the results of quenching experiments (1976). Using perturbed angular correlation spectroscopy (PACS), Haesslein et al. deduced a deep acceptor at 0.2 ± 0.04 eV above the valence band maximum, which they attributed to VGe–1 (1997). Zistl et al. identified the same level at Ev + 0.33 eV using DLTS and PAC (1997). Ionization level values obtained from early Green’s function calculations (Puska 1989) show mixed agreement with more recent values. The DFT-LDA calculations of Janotti et al. (1999) and Fazzio et al. (2000) agree well with each other, but differ from a DFT study performed by Coutinho et al. (2005) using the local spin density approximation and Carvalho et al. (2008) using AIMPRO, which explored the Ge vacancy in the (–2) and (–3) charge states. According to Coutinho et al., the (–1/–2) and (–2/–3) ionization levels are located

6.1 Bulk Defects

145

at Ev + 0.17 – 0.20 eV and Ev + 0.37 eV, respectively. The values of Carvalho et al. are only a few eV lower. Due to the relative proximity of the ionization levels for VGe–1 and VGe–2, Coutinho et al. pointed out that the Ev + 0.2 eV level inferred from PACS measurements could, in fact, correspond to a (0/–2) ionization level. The last reported DFT-LDA calculations place the (0/–1) and (–1/–2) ionization levels at Ev + 0.02 eV and Ev + 0.26 eV, in that order (Spiewak et al. 2007). The existence of a negative-U correlation energy arising from the strong structural relaxations VGe0 and VGe–2 has not been ruled out, although the findings of Spiewak et al. suggest that the PACS level at Ev + 0.2 eV may simply correspond to the (–1/–2) rather than (0/–1) level. The proposed ionization levels of the germanium vacancy are summarized in Table 6.4. Using maximum likelihood estimation, the positions of these levels are suggested to lie at Ev + 0.05 ± 0.05 eV (+2/+1), Ev + 0.14 ± 0.05 eV (+1/0), Ev + 0.19 ± 0.01 eV (0/–1), Ev + 0.27 ± 0.02 eV (–1/–2), and Ev + 0.43 ± 0.07 eV (–2/–3). The germanium self-interstitial is definitely stable within the band gap in the neutral charge state; conflict exists in the literature as to the stability of the (+2), (+1), and (–1) states, likely due to the assortment of geometries that have been considered for the defect. Additionally, experimental evidence for charged germanium self-interstitials is scant. The defect has been observed via electron

Table 6.4 Experimentally and computationally determined ionization levels for the vacancy defect in bulk germanium (+2/+1) (+1/0) (0/–1) (–1/–2) (–2/–3) Method 0.05 ± 0.05 –––

0.14 ± 0.19 ± 0.27 ± 0.05 0.01 0.02 ––– ––– 0.29

0.43 ± 0.07 –––

– –

– –

– –

– –

0.42 0.399 0.395 0.2 0.26 0.17 ± 0.1

0.49 – – 0.37 – –

0.16 0.001 0 – – –

0.33 0.20 ± 0.04 0.21 0.27 0.209 0.373 0.21 0.366 –0.11 0.2 – 0.02 – 0.19 ± 0.1

Maximum likelihood

Reference This work

Quenching experiments

(Van Vechten and Thurmond 1976) DLTS & PAC (Zistl et al. 1997) PAC (Haesslein et al. 1997; Haesslein et al. 1998) Green’s function calculations (Puska 1989) DFT-LDA (Janotti et al. 1999) DFT-LDA (Fazzio et al. 2000) DFT-LSDA (Coutinho et al. 2005) DFT-LDA (Spiewak et al. 2007) DFT-AIMPRO (Carvalho et al. 2008)

DLTS deep level transient spectroscopy, PAS positron annihilation correlation, DFT densityfunctional theory, LDA local density approximation, LSDA local spin density approximation, GGA generalized-gradient approximation, AIMPRO ab initio modeling program. All values are in eV and referenced to the valence band maximum.

146

6 Intrinsic Defects: Ionization Thermodynamics

paramagnetic resonance (Trueblood 1967), DLTS (Bourgoin et al. 1980), and irradiation experiments (Saito et al. 1971; Shimotomai and Hasiguti 1971). An early quenching study invoked Gei+2 to explain an unidentified energy level located between the vacancy acceptor level and the middle of the band gap (Hobstetter and Renton 1962). While experiments indicate that Gei possesses donor ionization levels close to the conduction band minimum, theoretical investigations have suggested that the (+1/0) level lies within the valence band (Moreira et al. 2004) or slightly above the valence band maximum (Janotti et al. 1999; da Silva et al. 2000). For example, Haesslein et al. identified the (+1/0) ionization level at Ec – 0.04 eV (Ev + 0.63 eV at 300 K) using PACS (1997). Moreira et al. suggested that the PACS level introduced by the self-interstitial near the conduction band minimum might actually be an acceptor. From the work of Haesslein et al., da Silva and co-workers estimated a value of Ev + 0.11 – 0.16 for the (+1/0) level. Carvalho et al. have shed light upon much of this confusion by calculating self-interstitial ionization levels for three different geometric configurations: tetrahedral, hexagonal, and “caged” (2007a). For example, they obtain values of Ec – 0.08 eV and Ec – 0.24 eV for the (+2/+1) and (+1/0) ionization levels of the caged self-interstitial, respectively (2008). An alternate work provides details on the suspected negative-U behavior that destabilizes Gei+1 (Carvalho et al. 2007b). It should be noted that the band gap was taken to be 1.17 eV (0 K) for all these calculations, however, Carvalho et al. cite the accuracy of their density functional pseudopotential calculations as only about 0.1 eV. Consequently, negative-U behavior is still substantially unconfirmed. The existence of Gei–1 and Gei–2 was first inferred by analyzing the low temperature recovery stages associated with the thermally activated migration of selfinterstitials (Frank and Thomas 1960). Two estimates of the single acceptor ionization level exist. DFT calculations suggest that ionization level is located at Ev + 0.31 eV (da Silva et al. 2000) or Ev + 0.37 eV (Moreira et al. 2004) above the valence band maximum. The authors point out, however, these calculations are based upon a theoretical band gap of 0.43 eV that is much lower than the experimental value of 0.67 eV. More recent DFT investigations fail to find any germanium self-interstitial acceptor states in the band gap (Carvalho et al. 2007b). The proposed ionization levels of the germanium self-interstitial are summarized in Table 6.5. On the basis on these values, the (+2/+1), (+1/0), and (0/–1) ionization levels are predicted to lie 0.55 ± 0.02 eV, 0.38 ± 0.01 eV, and 0.37 ± 0.02 eV above the valence band minimum. It is unlikely that the Gei defect exhibits negative-U behavior; the proximity between the (+1/0) and (0/–1) ionization levels obtained using maximum likelihood estimation stems from the values obtained via DFT using the LDA. Future calculations of the (0/–1) ionization level utilizing a different computational scheme would presumably find values much closer to the conduction band minimum.

6.1 Bulk Defects

147

Table 6.5 Experimentally and computationally determined ionization levels for the self-interstitial defect in bulk germanium (+2/+1)

(+1/0)

(0/–1)

Method

Reference

0.55 ± 0.02 0.47–0.57 0.55 – – – – 0.59 0.47 0.59

0.38 ± 0.01 – 0.62 0.63 ± 0.02 0.15 0.07 –0.08 0.43 – 0.43

0.37 ± 0.02 – – – – 0.31 0.37 – – –

Maximum likelihood EPR DLTS PAC DFT-LDA DFT-LDA DFT-LDA DFT-AIMPRO DFT-AIMPRO DFT-AIMPRO

This work (Trueblood 1967) (Bourgoin et al. 1980) (Haesslein et al. 1998) (Janotti et al. 1999) (da Silva et al. 2000) (Moreira et al. 2004) (Carvalho et al. 2007b) (Carvalho et al. 2007a) (Carvalho et al. 2008)

DLTS deep level transient spectroscopy, EPR electron paramagnetic resonance, PAC positron annihilation correlation, DFT density-functional theory, LDA local density approximation, AIMPRO ab initio modeling program. All values are in eV and referenced to the valence band maximum.

6.1.2.2 Associates and Clusters Experimental and computational evidence exists for four germanium divacancy charge states, (+1), (0), (–1), and (–2), within the band gap. In contrast, the isolated vacancy in germanium has no (+1/0) state in the band gap. An acceptor ionization level located a few tenths of an eV above the valence band, potentially (0/–1), has long been observed via DLTS (Kolkovsky et al. 2007; Mooney et al. 1983; Marie and Levalois 1994; Fourches et al. 1991; Christian Petersen et al. 2006; Fage-Pedersen et al. 2000; Fukuoka and Saito 1982; Marie et al. 1993; Nagesh and Farmer 1988; Kovacevic et al. 2006). Emtsev et al. have also obtained an estimate of the (0/–1) ionization level via annealing of irradiation induced defects (1991). EPR is not typically used for studying associates in Ge, as large spin-orbit coupling leads to extremely weak signals (Nagesh and Farmer 1988). One computational investigation has explored the single donor (+1/0) ionization level of the Ge divacancy (Coutinho et al. 2006). Using a spin-density-functional code together with a Pade form for the LDA, Coutinho et al. looked at the electrical levels of VGeVGe by comparing its ionization energy and electron affinity with that of the SbV “marker.” They obtained values of Ev + 0.08 eV or Ev + 0.03 eV depending on whether the theoretical or experimental lattice constant was used. Regarding the acceptor levels of VGeVGe, Emtsev et al. tentatively identified acceptor-type divacancies with an ionization level at Ev + 0.2 eV by investigating the annealing behavior of irradiated Ge (1991). Comparable levels at Ev + 0.22 eV (Marie and Levalois 1994), Ev + 0.29 eV (Fage-Pedersen et al. 2000), Ev + 0.29 eV (Kovacevic et al. 2006), and Ev + 0.31 (Fukuoka and Saito 1982) have been obtained via DLTS. The value obtained by Nagesh and Farmer, who also studied gamma- and neutron-irradiated Ge with DLTS and identified a VGeVGe acceptor

148

6 Intrinsic Defects: Ionization Thermodynamics

Table 6.6 Experimentally and computationally determined ionization levels for the divacancy defect in bulk germanium (+1/0)

(0/–1)

(–1/–2)

Method

Reference

0.06 ± 0.03 – – –

0.27 ± 0.01 0.31 – 0.2

0.40 ± 0.01 – 0.50 –

This work (Fukuoka and Saito 1982) (Nagesh and Farmer 1988) (Emtsev et al. 1991)

– – – 0.03–0.08

0.22 0.29 0.29 0.27–0.34

– – – 0.39–0.40

Maximum likelihood DLTS DLTS Annealing of irradiation induced defects DLTS DLTS DLTS DFT-LDA

(Marie and Levalois 1994) (Fage-Pedersen et al. 2000) (Kovacevic et al. 2006) (Coutinho et al. 2006)

DLTS deep level transient spectroscopy, DFT density-functional theory, LDA local density approximation. All values are in eV and referenced to the valence band maximum.

level at Ec – 0.17 eV (Ev + 0.50 eV), stands out as an outlier (1988). Coutinho and co-workers also explored these acceptor ionization levels with their ab initio calculations (2006). They obtained values of Ev + 0.27 (0/–1) and Ev + 0.40 (–1/–2) using the experimental lattice constant and Ev + 0.34 (0/–1) and Ev + 0.39 (–1/–2) using the theoretical lattice constant, respectively. Based on the computational assignment of the (0/–1) ionization level to 0.27 eV above the VBM, one can assume that most of the experimentally determined ionization levels relate to the (0/–1) transition. The acceptor level determined by Nagesh and Farmer may, instead, pertain to the (–1/–2) ionization level. The proposed ionization levels of the germanium divacancy are summarized in Table 6.6. Using maximum likelihood estimation, the (0/–1) and (–1/–2) ionization levels are estimated to be Ev + 0.27 ± 0.01 eV and Ev + 0.40 ± 0.01 eV. A simple average of the (+1/0) ionization levels values of 0.03 and 0.08 eV obtained using the experimental and theoretical Ge lattice constant yields a recommended ionization level value of Ev + 0.06 ± 0.03 eV.

6.1.3 Gallium Arsenide Gallium arsenide has a band gap of 1.43 eV at 300 K (Adachi 2005). It is a direct band gap semiconductor and its high saturation electron velocity, high electron mobility, and high breakdown voltage make it an attractive choice for an assortment of devices. As a binary semiconductor, it supports intrinsic antisite defects in addition to vacancy and self-interstitial defects. Additionally, the relative contribution of different bulk defects often correlates to the deviation from exact 1:1 GaAs stoichiometry, where the amount of Ga and As in the sample is determined by electrochemical titration or the average mass of the unit cell (Hurle 1999).

6.1 Bulk Defects

149

6.1.3.1 Point Defects For crystals grown under Ga-rich, or p-type, conditions, the concentrations of Asi and AsGa are negligible. Similarly, for As-rich, or n-type, conditions, VGa plays a much more important role than VAs (Hurle 1999). The lowest formation energies are found for AsGa+2 in p-type and VGa–3 in n-type GaAs under As-rich conditions (El-Mellouhi and Mousseau 2005), and for Gai+3 in p-type and GaAs–2 in n-type GaAs under Ga-rich conditions, all respectively (Zhang and Northrup 1991; Schick et al. 2002). Arsenic atoms exit the lattice as interstitials, a process that can be described by both an elementary and overall gas-phase reactions

As 0As ↔ VAs0 + Asi0 As 0As ↔

1 As4 ( g ) + VAs0 . 4

(6.9)

(6.10)

Similarly, the formation of gallium vacancies entails 0 0 GaGa ↔ VGa + Gai0

(6.11)

1 0 As4 ( g ) ↔ As 0As + VGa . 4

(6.12)

and

Both types of vacancies can also be formed by the reaction of vacancies already present within the crystal with antisite defects. For example, a gallium vacancy can be formed from an arsenic vacancy and an arsenic antisite, 0 0 AsGa + VAs0 ↔ VGa .

(6.13)

All of these defects undergo ionization reactions to donor and acceptor states similar to those laid out for silicon. Note that the antisite defect, which was not considered for either silicon or germanium, may also be charged. The stable charge states for the arsenic vacancy are still a matter of debate. Experiments have suggested VAs+1, VAs0, VAs–1, and VAs–2 as candidates in the bulk, whereas recent theoretical calculations propose negative-U behavior and the subsequent destabilization of the (0) and (–1) charge states. Positron annihilation spectroscopy has been used to identify acceptor ionization levels close to the conduction band minimum. For example, Corbel et al. located the (0/–1) and (–1/–2) levels at Ev + 1.33 ± 0.02 eV and Ev + 1.395 ± 0.015 eV, respectively (where these values have been converted from the CBM with a band gap of 1.43 eV) (1988). Ambigapathy et al. (1994) and Kuisma et al. (1996) used the same technique to obtain a (+1/0) level at Ev + 1.29 eV and (0/–1) level at Ev + 1.4 eV. Although several early Green’s function (Baraff and Schlüter 1985; Puska 1989; Delerue 1991), DFT (Jansen and Sankey 1989), and tight-binding (Seong and Lewis 1995) studies predicted no negative-U behavior for the defect, more recent density-functional theory investigations have suggested otherwise. Cheong and Chang (1994) and

150

6 Intrinsic Defects: Ionization Thermodynamics

Fig. 6.4 Formation energies as a function of the Fermi energy of various charge states of As vacancies in GaAs at 0 K. The Fermi level is calculated with respect to the valence band maximum. Arrows point to the location of the a) (+1/–1) and b) (–1/–3) ionization levels. Reprinted figure with permission from El-Mellouhi F, Mousseau N (2005) Phys Rev B: Condens Matter 71: 125207-9. Copyright (2005) by the American Physical Society.

Pöykkö et al. (1996) were the first to report the negative-U effect for the (+1/–1) ionization level of VAs. This behavior was confirmed by Chadi, who found the electron capture VAs+1 + 2e– → VAs–1 to be favored by strong Jahn–Teller distortions, leading to the stability of the (–1) charge state for Fermi energies above midgap (2003). Recently, El-Mellouhi and Mousseau have suggested a shallower (+1/–1) ionization level and a second negative-U electron capture at Ev + 1.27 eV with the reaction VAs–1 + 2e– → VAs–3 (2005). In this model, illustrated in Fig. 6.4, the arsenic vacancy exhibits two successive negative-U transitions, meaning that neither the neutral nor the (–2) states are ever majority species. The ionization levels of the arsenic vacancy in bulk GaAs are summarized in Table 6.7. On the basis of these values, the maximum likelihood method yields Ev + 1.26 ± 0.02 eV (+1/0), Ev + 1.35 ± 0.01 eV (0/–1), and Ev + 1.39 ± 0.02 eV (–1/–2). The conclusive identification of the stable charge states of VGa in n-type GaAs is a relatively recent development. Initially, computational studies hinted at the predominance of VGa–3, while diffusion experiments indicated the existence of VGa–1 or VGa–2 (Bracht et al. 1999). First-principles calculations have now definitively shown that, for intrinsic and n-type GaAs, VGa–3 has a lower formation energy than VGa0, VGa–1, or VGa–2. Some authors have reported ionization levels lying well below the midgap with no negative-U effect (Jansen and Sankey 1989; Puska 1989;

6.1 Bulk Defects

151

Table 6.7 Experimentally and computationally determined ionization levels for the arsenic vacancy defect in bulk GaAs (+1/0)

(0/–1)

(–1/–2)

Other

Method

Reference

1.26 ± 0.02 –

1.39 ± 0.02 1.395 ± 0.015 –

–

Maximum likelihood

This work

–

PAS

(Corbel et al. 1988)

1.29

1.35 ± 0.01 1.33 ± 0.02 1.4

–

PAS

–

–

–

0.07 (+2/+1)

1.25 1.2

1.35 1.3

– 1.39

– –

0.83

1.04

1.15

1.26 (–2/–3)

0.93

0.64

–

0.785 (+1/–1)

1.3 1.29 –

1.428 0.43 –

– – –

– 0.86 (+1/–1) 0.27 (+1/–1), 1.27 (–1/–3)

Green’s function calculations DFT-LDA Green’s function calculations Green’s function calculations First-principles calculations TB Calculations DFT-LDA DFT-LDA

(Ambigapathy et al. 1994; Kuisma et al. 1996) (Baraff and Schlüter 1985) (Jansen and Sankey 1989) (Puska 1989) (Delerue 1991) (Cheong and Chang 1994) (Seong and Lewis 1995) (Pöykkö et al. 1996) (El-Mellouhi and Mousseau 2005)

PAS positron annihilation spectroscopy, DFT density-functional theory, LDA local density approximation. TB tight-binding. All values are in eV and referenced to the valence band maximum.

Delerue 1991; Zhang and Northrup 1991; Schick et al. 2002; Janotti et al. 2003), while others have identified defects with ionization levels closer to the center of the band gap (Baraff and Schlüter 1985; Cheong and Chang 1994; Gorczyca et al. 2002), and even negative-U behavior (Seong and Lewis 1995; Delerue 1991). Recently, in a fashion akin to the investigation of the temperature-dependence of defect ionization levels in silicon performed by Van Vechten, El-Mellouhi and Mousseau revisited the divergent experimental and theoretical findings for VGa in GaAs (2006). They used LDA results to explore the effects of temperature on the Gibbs free energy of formation and found that the thermal dependence of the Fermi level and ionization levels leads to a reversal of the preferred charge state as the temperature increases. Figure 6.5 depicts the minimal effect of temperature on vacancy ionization enthalpies, as well as the linearly increasing entropic term in the Gibbs free energy. With such an analysis, El-Mellouhi and Mousseau were able to reproduce the conflicting experimental results of Bracht et al. (1999) and Gebauer et al. (2003). To summarize, the calculated and experimentally determined ionization levels for the gallium vacancy defect are tabulated in Table 6.8. Using maximum likelihood estimation, the (0/–1), (–1/–2), and (–2/–3) ionization levels of VGa are 0.10 ± 0.01 eV, 0.38 ± 0.03 eV, and 0.55 ± 0.03 eV above the valence band maximum, respectively.

152

6 Intrinsic Defects: Ionization Thermodynamics

Fig. 6.5 Entropy and enthalpy (inset) contribution to the free energy of ionization for various charge transitions of VGa in GaAs. The enthalpy shows little temperature dependence, but the ionization entropy is significant, leading to a substantial entropy contribution to the free energy of ionization that increases linearly with temperature. Reprinted with permission from El-Mellouhi F, Mousseau N (2006) J Appl Phys 100: 083521-4. Copyright (2006), American Institute of Physics.

Table 6.8 Experimentally and computationally determined ionization levels for the gallium vacancy defect in bulk GaAs (0/–1) (–1/–2) (–2/–3) Other 0.10 ± 0.01 0.42 0.19

0.38 ± 0.03 0.6 0.52

0.55 ± 0.03 – – 0.72 –

0.08 0.11 0.25 0.19

0.27 0.22 0.39 0.2

0.49 0.33 0.57 0.32

– –

0.49 –

0.69 –

0.39 0.09 0.13 0.05

0.52 0.13 0.15 0.4

0.78 0.2 0.18 0.55

Method

Reference

Maximum likelihood

This work

Diffusion (Bracht 1999) Green’s function calculations (Baraff and Schlüter 1985) – DFT-LDA (Jansen and Sankey 1989) – Green’s function calculations (Puska 1989) 0.13 (+1/0) Green’s function calculations (Delerue 1991) – DFT-LDA (Zhang and Northrup 1991) – First-principles calculations (Cheong and Chang 1994) 0.035 (+1/–1), TB calculations (Seong and Lewis 1995) 0.078 (–1/–3) – DFT-LDA (Gorczyca et al. 2002) – DFT-LDA (Schick et al. 2002) – DFT-LDA (Janotti et al. 2003) – DFT-LDA (El-Mellouhi and Mousseau 2005)

DFT density-functional theory, LDA local density approximation. TB tight-binding. All values are in eV and referenced to the valence band maximum.

6.1 Bulk Defects

153

Gallium interstitials are likely stable in the (+1), (0), and (–1) charge states as the Fermi energy in the bulk rises from the valence band maximum to the conduction band minimum. In the early 1990s, investigators focused more heavily upon the (+2) gallium self-interstitial (Chadi 1992; Tan et al. 1991); for example, self-consistent pseudopotential calculations performed by Chadi placed the (+2/+1) level of the defect at the valence band maximum. These authors also calculated a (+1/0) ionization level of Ev + 0.6 – 0.7 eV for the <110> split-interstitial configuration. Northrup and Zhang found a low formation energy for the (+3) charge state of Gai in both intrinsic and p-type material using the LDA and first-principles pseudopotentials (1994). Using a similar calculation method, they obtained both (+3/+2) and (+2/+1) ionization levels slightly below midgap, yet did not cite their exact locations (1991). Hurle “arbitrarily assigned” levels of Ev + 0.3 eV and Ev + 0.5 eV to the (+2/+1) and (+1/0) charge state transitions, respectively (1999). Ab initio calculations found negative-U effects for (+3/+1) and (+1/–1) levels located at Ev + 0.29 eV and Ev + 1.23 eV (Zollo et al. 2004). Malouin et al. identified three stable charge states for the low formation energy tetrahedral interstitial configuration; the (+1/0) and (0/–1) ionization levels are located 1.02 eV and 1.20 eV above the valence band maximum, respectively (2007). Additionally, (+3) and (+2) charged gallium interstitials have prohibitively high formation energies for almost all possible geometries. Earlier diffusivity profiles have been revisited in light of these findings. Bracht and Brotzmann have shown that experimental data of Zn diffusion in gallium arsenide is better explained by the presence of Gai+1 than by Gai+2 and Gai+3 (2005). As with the gallium interstitial, the charge on the arsenic split-interstitial transitions from (+1) to (0) to (–1) for increasing Fermi energies within the band gap. Under equilibrium conditions, charged arsenic interstitials, which tend to adopt the splitinterstitial configuration over the ideal tetrahedral configuration, have formation energies that are much higher than those of antisites and vacancies in bulk GaAs (Zhang and Northrup 1991; Schick et al. 2002). Nevertheless, they are easily created by irradiation and are likely responsible for the deviation from stoichiometry (towards As-rich unless the concentration of Ga in the melt is considerably greater than 50%) observed in melt-grown GaAs (Hurle 1999; Bourgoin et al. 1988). Interstitial arsenic defects are difficult to observe experimentally, as even after irradiation or heat treatment, they are often EPR, electrically, and optically invisible (Delerue 1991). Consequently, most of the insight into their configuration and electronic structure arises from self-consistent semi-empirical tight-binding, pseudopotential, and first-principles molecular-dynamics calculations. Early work suggested negative-U behavior for the Asi defect with a (+1/–1) ionization level at about Ev + 0.35 eV (Chadi 1992). Instead, recent DFT results obtained by Schick et al. using the LDA with the Ceperley–Alder form for the exchange and correlation potentials as parameterized by Perdew and Zunger reveal (+1/0) and (0/–1) ionization levels for the As split-interstitial located at Ev + 0.3 eV and Ev + 0.5 eV, respectively (2002). The defect interaction that leads to the formation of the antisite defect, or the migration of a vacancy to a nearest neighbor, is expressed as 0 0 As 0As + VGa ↔ AsGa + VAs0 .

(6.14)

154

6 Intrinsic Defects: Ionization Thermodynamics

The ionization reactions for the antisite defect are 0 +1 AsGa ↔ AsGa + e−

(6.15)

+1 +2 AsGa ↔ AsGa + e− .

(6.16)

and

The arsenic antisite is stable in the (+2), (+1), and (0) charge states within the band gap. DLTS experiments have revealed a (+2/+1) ionization level at Ev + 0.61 eV (Lagowski et al. 1982) and (+1/0) ionization level at Ev + 0.61 eV (Martin et al. 1977) or Ev + 0.89 eV (Lagowski et al. 1982). Based on EPR measurements, Weber assigned ionization levels of Ev + 0.52 and Ec – 0.75 eV (or Ev + 0.68 eV at 300 K) to the (+2/+1) and (+1/0) ionization levels of the stable arsenic antisite in GaAs, respectively (1983). The levels calculated by Bachelet et al. using the local-density approximation were significantly higher, at Ev + 0.83 and Ev + 1.10 eV for the (+2/+1) and (+1/0) ionization levels, respectively (1981). The range of computational estimates for the same levels varies widely, however. No trends can be discerned by differentiating between Green’s function (Baraff and Schlüter 1985; Puska 1989; Delerue 1991) and DFT (Jansen and Sankey 1989; Gorczyca et al. 2002; Schick et al. 2002; Overhof and Spaeth 2005) calculations. The gallium antisite defect acts as a shallow acceptor in n-type GaAs but as a deep acceptor in p-type GaAs. It arises when a gallium interstitial reacts with an arsenic vacancy:

Gai0 + VAs0 ↔ Ga 0As .

(6.17)

Some reports suggest that the neutral gallium antisite ionizes to GaAs–1 according to −1 Ga 0As ↔ Ga As + h+ ,

(6.18)

while others indicate a direct ionization from GaAs0 to GaAs–2 (or negative-U behavior): −2 Ga 0As ↔ Ga As + 2h + .

(6.19)

The gallium antisite is stable in the (0), (–2), and possibly (–1) charge states. Values for the acceptor ionization levels of the gallium antisite defect vary widely; similar discrepancies exist amongst the donor levels of the arsenic antisite. For n-type samples, Bugajski et al. used photoluminescence to determine (0/–1) and (–1/–2) ionization levels for GaAs at Ev + 0.20 eV and Ev + 0.077 eV, respectively (1989). However, only GaAs–2 defects were observed, lending support to the suggested negative-U behavior of the defect, with a (0/–2) ionization level at about Ev + 0.17 eV. In p-type GaAs, Bugajski et al. found only a single acceptor level at Ev + 0.068 eV. These values agree well with those from an earlier PL and Hall effect study of Yu and Reynolds (1982) and Yu et al. (1982). Zhang and Chadi obtained similar ionization levels of Ev + 0.24 eV (0/–1) and Ev + 0.09 eV(–1/–2) using DFT (1990), which appear almost reversed from those obtained by Jansen

6.1 Bulk Defects

155

Table 6.9 Experimentally and computationally determined ionization levels for both arsenic and gallium antisites in bulk GaAs AsGa AsGa GaAs (+2/+1) (+1/0) (0/–1)

GaAs Method (–1/–2)

Reference

0.65 ± 0.02 – 0.61 0.52 0.83 1.25 0.57 0.88 1.2 0.31 1.0 0.98

0.79 ± 0.01 0.61 0.89 0.75 1.10 1.5 0.84 1.07 1.37 0.66 1.1 1.18

Maximum likelihood

This work

– –

– –

0.29 ± 0.01 – – – – 0.62 0.28 0.28 – – – – 0.23 0.199 0.7 0.2

DLTS DLTS EPR Green’s function calculations Green’s function calculations DFT-LDA Green’s function calculations Green’s function calculations DFT-LDA DFT-LDA DFT-LSDA PL Hall effect DLTS PL

(Martin et al. 1977) (Lagowski et al. 1982) (Weber 1983) (Bachelet et al. 1981) (Baraff and Schlüter 1985) (Jansen and Sankey 1989) (Puska 1989) (Delerue 1991) (Gorczyca et al. 2002) (Schick et al. 2002) (Overhof and Spaeth 2005) (Yu and Reynolds 1982) (Yu et al. 1982) (Wang et al. 1984) (Bugajski et al. 1989)

– –

– –

0.09 0.62

DFT-LDA First-principles calculations

(Zhang and Chadi 1990) (Cheong and Chang 1994)

0.13 ± 0.01 – – – – 0.3 0.07 0.11 – – – – 0.077 0.071 0.4 0.068– 0.077 0.24 0.77

DLTS deep level transient spectroscopy, EPR electron paramagnetic resonance, PL photoluminescence, DFT density-functional theory, LDA local density approximation, LSDA local spin density approximation. All values are in eV and referenced to the valence band maximum.

and Sankey (1989) and Puska (1989). On the other hand, Wang et al. (1984), Baraff and Schlüter (1985), and Cheong and Chang (1994) located ionization levels considerably above the valence band maximum, Ev + 0.30 to 0.77 eV for GaAs–1 and Ev + 0.62 to 0.70 eV for GaAs–2, all respectively. The ionization levels for both arsenic and gallium antisites are summarized in Table 6.9. The maximum likelihood estimates for the AsGa (+2/+1) and (+1/0) ionization levels are Ev + 0.65 ± 0.02 eV and Ev + 0.79 ± 0.01 eV. Those for GaAs (0/–1) and (–1/–2) are Ev + 0.13 ± 0.01 eV and Ev + 0.29 ± 0.01 eV.

6.1.3.2 Associates and Clusters

Divacancy defects, as well as an assortment of vacancy-interstitial pairs of unspecified charge state, have been identified using DLTS (Bourgoin et al. 1988). The two acceptor ionization levels of the mixed GaAs divacancy, (0/–1) and (–1/–2),

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6 Intrinsic Defects: Ionization Thermodynamics

have been explored via theoretical calculations. Lin-Chung and Reinecke studied the divacancy using a large cluster recursion approach and reported levels at 0.13, 0.48, 0.82 and 1.17 eV above the valence band maximum, although no attempt was made to assign levels to specific charge state transitions (1983). Ab-initio calculations provide the most comprehensive treatment, to date, of divacancy charging (Pöykkö et al. 1996). Pöykkö et al. reported single (0/–1) and double (–1/–2) ionization levels of Ev + 0.12 eV and Ev + 0.19 eV for VGaVAs, respectively.

6.1.4 Other III–V Semiconductors The band gaps of other III–V semiconductors are tabulated in Table 6.10. Many of them are used for high temperature applications and electro-optical devices that operate in the short-wavelength range of the visible spectrum. Boron containing compounds, in particular, have unique physical properties such as low densities, extremely high thermal conductivities, and low resistivities (Zaoui and El Haj Hassan 2001). When charged defects in these other group III–V semiconductors are compared to those in gallium arsenide, many similarities can be observed. This correspondence is most pronounced for compounds that contain no boron or nitrogen. For example, in gallium phosphide, which has a band gap of 2.26 eV at 300 K, the lowest formation energies are found for PGa+2 in p-type and VGa–3 in n-type material under P-rich and stoichiometric conditions. In other respects, however, charged defects in III–V compounds that contain elements from the second row of the Periodic Table behave differently from those in GaAs. For AlN, InN, and GaN,

Table 6.10 Band gap values at 300 K for the group III–V semiconductors Material

Band gap (eV)

Reference

BN BP BAs BSb AlN AlP AlAs AlSb GaN GaP GaSb InN InP InAs InSb

6.1–6.4 2.0 0.67 0.87 6.026 2.48 2.17 1.615 3.2 2.26 0.726 0.7 2.21 0.354 0.17

(Rummukainen et al. 2004) (Madelung et al. 2002a) (Madelung et al. 2002b) (Ferhat et al. 1998) (Guo and Yoshida 1994) (Adachi 2005) (Levinshtein et al. 1999) (Adachi 2005) (Bougrov et al. 2001) (Levinshtein et al. 1996) (Levinshtein et al. 1996) (Davydov et al. 2003) (Adachi 2005) (Levinshtein et al. 1996) (Levinshtein et al. 1996)

6.1 Bulk Defects

157

for the most part, only vacancies have low enough formation energies to affect the electronic properties of the bulk under equilibrium conditions. Exceptions to this rule occur in BN, where the mismatch between the radii of the boron and nitrogen atoms is comparatively small, and in zincblende AlN, where the lattice defects can interact with a greater number of nearest neighbor atoms; charged antisites and interstitials may occur in these materials (Orellana and Chacham 1999; Stampfl and Van de Walle 2002; Van de Walle and Neugebauer 2004). 6.1.4.1 Point Defects

The charging of defects in GaP, InAs, InP, and InSb has been studied using density functional theory (Hoglund et al. 2006); the behavior of charged defects in GaP will be discussed in most detail. The primarily difference between the defects in GaP versus GaAs is the existence of additional stable charge states not available to defects in GaAs, probably as a result of the almost 1 eV larger band gap. Gai+2 and GaP–2 have low formation energies in p-type and n-type material under Ga-rich conditions, respectively (Jansen and Sankey 1989; Hoglund et al. 2005). Note that the (+2) charge on Gai differs from the (+1) charge state predicted for Gai in GaAs; revisiting the GaP system in the manner of Malouin et al. may yield similar results (Malouin et al. 2007). For example, fitting of Zn diffusion profiles in GaSb indicates that Gai+1 and Gai0 mediate the changeover of interstitial zinc atoms to substitutional gallium sites (Sunder et al. 2007). According to LDA calculations (Hoglund et al. 2005) and ESR experiments (Kaufmann et al. 1981) performed for GaP, the (+2/+1) ionization level for the phosphorus vacancy lies at Ev + 0.97 eV and Ev + 1.25 eV, respectively. Hoglund et al. explored the energetics of several other charged defects in GaP. For charge states ranging from (+1) to (–5), the phosphorus vacancy exhibits negative-U behavior; only VP–1 and VP–3 are energetically favorable. The formation energy of VGa–2 is approximately 0.5 eV higher than that of VGa–3, and for Ga-rich conditions, the (–2/–3) ionization level is about 0.20 eV above the valence band maximum. Under stoichiometric conditions, there is a small region in the band gap between 0.65 and 0.94 eV where GaP–2 has a large concentration in the bulk. Negative-U behavior is observed for the gallium antisite defect, for which there is only one (0/–2) ionization level inside the band gap. The complexity of the defect chemistry in GaP, as well as a visual representation of how close some of the defect levels are, can be gleaned from Fig. 6.6. Different rules must be invoked to describe the defect behavior in III–V semiconductors containing B or N. For conditions rich in the group-III element, it has been suggested that the (–3) charge state of the group-III vacancy is the dominant point defect in n-type material, while the concentration of either (VGroup V)+1 or (VGroup V)+3 becomes significant in p-type material (Orellana and Chacham 1999; Stampfl et al. 2000; Boguslawski et al. 1995; Mattila and Nieminen 1997; Gorczyca et al. 2002; Fara et al. 1999; Tian et al. 2007). For all four group-III nitride compounds, according to this model, the nitrogen vacancy exhibits negative-U behavior. Recently, Ganchenkova and Nieminen have suggested that negatively

158

6 Intrinsic Defects: Ionization Thermodynamics

Fig. 6.6 Formation energies as a function of the Fermi level for relaxed native defects in GaP under P-rich conditions. The antisite defects are shown as dotted lines, vacancies as full lines, and interstitials as broken lines. The alternative y-axis on the right gives the defect concentration in orders of magnitude at T = 300 K. Reprinted figure with permission from Hoglund A, Castleton CWM, Mirbt S (2005) Phys Rev B: Condens Matter 72: 195213-5. Copyright (2005) by the American Physical Society.

charged nitrogen vacancies, which were not even considered in earlier works, dominate in n-type conditions (2006). Progressing from Fermi energies near the conduction band minimum to those at the valence band maximum, their model proposes that VN–3, VN–1, and VN+1 are the dominant defects in the bulk. The gallium vacancy in GaN is a triple acceptor, much like that in GaAs; in n-type material, VGa exists in the (–3) charge state. Positron annihilation measurements give direct experimental evidence for negatively charged Ga vacancies at temperatures less than 600 K (Saarinen et al. 2006). According to theoretical calculations, the (0/–1), (–1/–2), and (–2/–3) ionization levels are located at approximately 0.2, 0.7, and 1.1 eV above the valence band maximum (Limpijumnong and Van de Walle 2004; Neugebauer and Van de Walle 1996; Mattila and Nieminen 1997). VN+1 and VN+3 are shallow donors in GaN, and although they have high formation energies under n-type conditions, they can act as compensating species in the case of p-type doping (Van de Walle and Neugebauer 2004). Using a larger supercell that allows for better relaxation of VN+3, Limpijumnong and Van de Walle have placed the (+3/+1) ionization level for VN at EV + 0.59 eV, or considerably higher than previously published values (Neugebauer and Van de Walle 1996; Park and Chadi 1997; Wright and Mattsson 2004). Ganchenkova and Nieminen appear to be the first to have considered the possibility of negatively charged nitrogen vacancies in n-type material (2006). This relative neglect is surprising, considering as the acceptor ionization levels of VAs in GaAs and VP in GaP have been explored for a long time. The authors suggested that, for n-type

6.1 Bulk Defects

159

Fig. 6.7 Formation energies for gallium and nitrogen vacancies and mixed divacancy in GaN. Notice the turn-over in the slope of the VN line at about 2.5 eV, which suggests that negatively charged nitrogen vacancies may, in fact, contribute to the defect chemistry in bulk GaN. Reprinted figure with permission from Ganchenkova MG, Nieminen RM (2006) Phys Rev Lett 96: 196402-2. Copyright (2006) by the American Physical Society.

material, VN–1 and VN–3 will have smaller formation energies than VGa–3, and that VN exhibits negative-U behavior for the newly investigated ionization states. The vacancy is unstable in the neutral and (–2) charge states. Using the Zhang– Northrup formalism, Ganchenkova and Nieminen calculated nitrogen vacancy ionization levels at 2.43, 2.58, and 2.60 eV above the valence band maximum for the (+1/–1), (–1/–2), and (–2/–3) ionization levels, respectively. Figure 6.7 illustrates the turnover in defect formation energy that occurs at about 2.5 eV when the (–1) and (–3) charge states are considered. The authors acknowledged that PAS experiments have shown no evidence of nitrogen vacancies in n-type GaN, yet suggested that the signals of the negatively charged vacancies are similar to that of the bulk, and are thus PAS-invisible (Mattila and Nieminen 1997; Saarinen et al. 1997; Oila et al. 2001; Saarinen et al. 2001; Boguslawski et al. 1995). The positron lifetimes for VN–1 and VN–3 are about 131 and 132 ps, respectively, which is essentially the same as in the bulk (Ganchenkova and Nieminen 2006). Interstitial defects created in GaN upon irradiation can be observed by optically detected electron paramagnetic resonance (Chow et al. 2000). Density-functional theory and first-principles calculations have also been employed to predict their electronic structure and defect levels. The interstitial defects in GaN have been suggested to exist in considerably different charge states than those in GaAs. According to density-functional theory calculations performed by Limpijumnong and Van de Walle, Gai is stable only as Gai+3 and Gai+1 (2004). The (+3/+1) ionization level occurs at about 2.5 eV above the valence band minimum. Based on

160

6 Intrinsic Defects: Ionization Thermodynamics

experimental ODEPR measurements, Chow et al. predicted second and third ionization levels for the defect at Ev + 1.9 eV and Ev + 2.25 eV, respectively (2004). As a reminder, the gallium interstitial in GaAs is stable in the (+1), (0), and (–1) charge states; none of the donor states are destabilized by negative-U behavior. Nitrogen interstitials, which can occur in charge states ranging from (+3) to (–1), have high formation energies for all Fermi energies within the band gap, and can be produced in GaN only upon irradiation (Van de Walle and Neugebauer 2004). Nitrogen interstitials in GaN, in comparison to arsenic interstitials in GaAs, have more stable charge states within the band gap. The (+3/+2), (+2/+1), (+1/0), and (0/–1) ionization levels calculated by Limpijumnong et al. are located at Fermi energies approximately 0.75, 1.0, 1.5, and 1.0 eV above the VBM, respectively. These levels are far higher than those suggested for Asi in GaAs (assuming no negative-U behavior). Schick et al. calculated ionization levels for Asi+1/Asi0 at Ev + 0.30 eV and Asi0/Asi–1 at Ev + 0.50 eV (2002). 6.1.4.2 Associates and Clusters

Little is known about the ionization levels or stable charge states of associates and clusters in the III–V semiconductors other than GaAs; there is little point in separating out the materials that contain B and N atoms for purposes of comparison. Similar to his treatment of divacancies in GaAs, Xu has performed self-consistent semiempirical tight-binding calculations to describe the electronic structure of VInVP that considers defect charge states of (+6) to (–2) (1990). In a later publication, the same author applied a similar analytical treatment to charged divacancies in GaAs, GaP, and GaSb (1992). For each semiconductor, Xu provides values for energy level as a function of charge state, symmetry, level occupancy, and net charge; it is not readily apparent from these calculations which charge states are stable or the locations of charge state transitions within the band gap. Zhao et al. (2006c) associated the divacancy in InP with an experimentally observed deep level at Ec – 0.13 eV (or Ev + 1.37 eV based on a calculated band gap Eg of 1.50 eV), which compares quite well to the earlier calculated value of Ec – 0.15 eV (Ev + 1.35 eV) (Xu 1990). Ab initio calculations (Hakala et al. 2002) predict double (–1/–2) and triple (–2/–3) acceptor ionization levels at Ev + 0.28 eV and Ev + 0.52 eV for VGaGaSb in GaSb, which has a band gap of approximately 0.72 eV (Pichler 2004).

6.1.5 Titanium Dioxide Although ideal undoped titanium dioxide would be insulating and have an exact 2:1 ratio of oxygen to titanium, titania typically exists as an n-type semiconductor with a slight excess of metal. Titania has been studied extensively for its unique physical and chemical properties including high refractive index, excellent optical transmittance in the visible and near-infrared region, high dielectric constant, and

6.1 Bulk Defects

161

chemical stability (Matsumoto et al. 2001). Anatase and rutile TiO2, the two most studied phases, differ considerably in their electronic properties; at 300 K, rutile has a direct band gap of around 3.03 eV while anatase has an indirect band gap of approximately 3.18 eV (Jellison et al. 2003; Jin et al. 2001). 6.1.5.1 Point Defects

Defect formation in oxide semiconductors tends to be more complicated than in Group IV or III–V semiconductors, and the equations one typically finds to describe these processes often leave out quite a bit of mechanistic detail. One example involves the oxygen vacancy that appears as a product in the Schottky defect formation reaction, which entails the simultaneous creation of both an anion and a cation vacancy: M M + OO ↔ VM + VO + MO .

(6.20)

It is important to keep in mind that the production of an oxygen vacancy must be accompanied by the formation of a corresponding oxygen interstitial according to OO ↔ Oi + VO .

(6.21)

The overall Schottky reaction (to be discussed momentarily) often leaves out this detail. In anatase, the oxygen interstitial can form a substitutional diatomic molecule, (O2)O, a species similar to the split-interstitial in silicon. The oxygen species is presumably important in describing oxygen exchange with the surrounding gas. Vacancy and interstitial defects are formed by identical chemical and electronic reactions in rutile and anatase TiO2. Antisites do not occur, as the large cationanion size mismatch and strong ionicity cause them to have prohibitively large formation energies. TiO2, as well as non-stoichiometric TiO2–x, can undergo reversible reactions with oxygen in the ambient (DeFord and Johnson 1972) g TiO2 ↔ Ti +4 + O2( ) + 4e− .

(6.22)

The production of neutral oxygen vacancies in the bulk is nominally described by the overall reaction 1 g OO ↔ VO0 + O2( ) . 2

(6.23)

Oxygen is lost to the gas phase as a result of the spontaneous binding of interstitial oxygen to oxygen in lattice sites, and subsequent out-diffusion to yield gaseous oxygen. It is likely, though not certain, that the out-diffusing species is interstitial oxygen rather than the diatomic molecule:

OO ↔ Oi0 + VO0 OO ↔

Oi0

+ ( O2 )O

( O2 )O ↔ OO +

1 (g) O2 . 2

(6.24) (6.25) (6.26)

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6 Intrinsic Defects: Ionization Thermodynamics

In theory, all of the defects in TiO2 can ionize in a manner akin to silicon vacancies, interstitials, etc. Typically, however, the formation of charged defects in TiO2 is described via the formation of Schottky and Frenkel defects. For example, the process of Schottky defect formation involves the simultaneous creation of a charged anion and a cation vacancy

TiTi + 2OO ↔ VTi−4 + 2VO+2 + TiO2 .

(6.27)

The fundamental Schottky equilibrium expression is

Zero ↔ VTi0 + 2VO0 .

(6.28)

The analog to the Schottky reaction that forms an oxygen vacancy is the Frenkel reaction that forms a titanium interstitial. The reaction can be visualized as a titanium lattice ion moving to an interstitial site, leaving a titanium vacancy behind:

TiTi0 ↔ Tii+4 + VTi−4 .

(6.29)

The fundamental Frenkel equilibrium expression is

Zero ↔ Tii0 + VTi0 .

(6.30)

The formation of neutral titanium interstitials can also be represented by the overall reaction g TiTi + 2OO ↔ Tii0 + O2( )

(6.31)

that is obtained from the following quasi-chemical elementary reactions with the knowledge that oxygen in the TiO2 lattice is easily reduced

TiTi + 2OO ↔ Tii0 + VTi + 2OO

(6.32)

1 g OO ↔ VO + O2( ) 2

(6.33)

2VO + VTi ↔ Zero .

(6.34)

Also, according to Na-Phattalung et al., as titanium interstitials have low formation energies, they can also be formed when oxygen vacancies react with TiO2 (2006) according to the overall reaction

2TiO2 + 2VO0 ↔ TiO2 + Tii0 .

(6.35)

As the formation of charged defects in TiO2 is often described by Schottky and Frenkel reactions involving specific defect charge states, the term “ionization level” appears only in recent literature concerning TiO2 defects. For the past 60 years researchers have focused primarily upon identifying the charge states of the defects that arise in the bulk as a function of ambient temperature and pressure. For example, to explain the electronic behavior of reduced and stoichiometric rutile and anatase TiO2, defect models involving three main types of point defects, Ti3+ interstitials, Ti4+ interstitials, and O+2 vacancies have been proposed (Diebold

6.1 Bulk Defects

163

Fig. 6.8 Two-dimensional defect formation scheme as a function of oxygen partial pressure and temperature calculated at three different Fermi levels (0.5 eV in a), 1.5 eV in b), and 2.5 eV in c)). Reprinted from He J, Behera RK, Finnis MW et al, “Prediction of high-temperature point defect formation in TiO2 from combined ab initio and thermodynamic calculations,” (2007) Acta Mater 55: 4334. Copyright (2007) with permission from Elsevier.

2003). Few authors even consider the Ti interstitial in the (+2) charge state (Cho et al. 2006). Recent work indicates that defects of heretofore unconsidered charge states may play a role in the overall defect chemistry of TiO2 at certain pressures and temperatures (He et al. 2007). Figure 6.8, a defect formation scheme, illustrates the complex dependence of dominant defect type and charge state on these conditions, as well as Fermi energy. A clearer understanding of the ionization levels of Tii, VTi, and VO is clearly necessary. Oxygen vacancies exist in the (+2), (+1), or (0) charge states in bulk TiO2 and act as intrinsic n-type dopants that shift the Fermi-level toward the conductionband minimum (Batzill et al. 2006; Cho et al. 2006). Based on experimental findings at an assortment of ambient conditions, it seems reasonable to suppose that (+2) oxygen vacancies are the dominant defects at increasing partial pressures of oxygen; only at high oxygen partial pressure (105 Pa) would one be able to obtain exactly stoichiometric material (Bursill et al. 1983; Yahia 1963; Marucco et al. 1981; Picard and Gerdanian 1975; Millot et al. 1987). Alternatively, based on thermogravimetry and electrical conductivity measurements performed in the 1980s, it has been suggested that VO+1 prevails at very high oxygen partial pressure (TiO2–x where x ≤ 0.0001) (Marucco et al. 1981). This experimentally observed behavior, however, can also be explained in terms of trivalent impurities such as Al+3 and Fe+3 that combine with Tii+4 or VO+2 (Blumenthal et al. 1966; Tannhauser 1963). A recent DFT study performed by He et al. that considered oxygen vacancy formation energy as a function of Fermi energy, oxygen partial pressure, and temperature may resolve this confusion (2007). It appears that VO+1 has a low formation energy in TiO2 at 1400 K and 10–2 atm for Fermi energies between approximately 1.6 and 2.1 eV above the valence band maximum. Titanium vacancies in the (–4) charge state have low formation energies in p-type TiO2 at certain temperatures and pressures. Experimental evidence for VTi–4 is rare as most experimental defect measurements are carried out in n-type material. At 300 K for both low and ambient oxygen partial pressures (10–15 and 10–2 atm,

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6 Intrinsic Defects: Ionization Thermodynamics

respectively), VTi–4 is the predominant defect in bulk TiO2 for Fermi energies ranging from approximately Ev + 1.5 eV to the CBM (He et al. 2007). Debate has long simmered over the relative importance of (+3) and (+4) titanium interstitials in rutile TiO2. The existence of the (+3) (Frederikse 1961; Young et al. 1961; Chester 1961) and (+4) (Frederikse 1961; Kofstad 1962; Tannhauser 1963; Blumenthal et al. 1967) charge states of Tii in TiO2–x was first suggested by electron paramagnetic resonance experiments in the 1960s. Mass action laws can be applied to the defect formation reactions in TiO2 to determine the pressure, and thus stoichiometry, dependence that should be observed in electrical conductivity measurements when a given defect charge state dominates. By comparing the slopes of experimental quantities such as ln(PO2)/ln(x) (where x is the deviation from stoichiometry) or ln( PO2)/ln(e) (where e stands for electronic conductivity) with predictions from mass action laws, the existence of Tii+3 or Tii+4 can, in principle, be inferred (accounting for the degree of TiO2 non-stoichiometry) (Capron and Boureau 2004; Kofstad 1962; Millot et al. 1987; Knauth and Tuller 1999; Nowotny et al. 1997; Kevane et al. 1963; Bak et al. 2003b, a). Electrical conductivity results suggest that the complete ionization of Ti+3 ions to Ti+4 occurs in stoichiometric TiO2 (Bursill and Blanchin 1984; Millot and Picard 1988). Computational studies of the titanium interstitial indicate that five charge states – (+4), (+3), (+2), (+1), and (0) – may contribute to the defect chemistry in rutile TiO2 (Bak et al. 2003a, b; Capron and Boureau 2004; Hallil et al. 2006; He and Sinnott 2005). Cho et al. provided support for the existence of Tii+2 based on the delocalized character of doped electrons revealed by first-principles calculations (2006). They cited defect clustering or the inaccuracy of the LDA as possible causes for the discrepancy between their calculations and the “typical” defect picture, yet presented little explanation for why a single model involving Tii+2 should replace the many involving Tii+3 and Tii+4. The computational results of He et al., on the other hand, indicate that Tii+4 is a dominant defect in highly reduced n-type TiO2 at 300 K while Tii+3, Tii+2, Tii+1, and Tii0 have low formation energies in highly reduced material at 1,900 K for Fermi energies ranging from the valence band maximum to the conduction band minimum. A better understanding of the effect of temperature and oxygen partial pressure on defect charge states in rutile TiO2 is clearly necessary. There is reason to believe that these experimental conditions have a large impact on the formation energy of charged defects in the bulk. Table 6.11 details the recent findings of He et al. including the ionization levels of the four main point defects in TiO2 (2007). The defect chemistry of anatase TiO2 has received increasing attention in the last decade due to the novel photocatalytic properties associated with the crystal structure. While experimental work suggests that anatase behaves in a similar fashion to rutile, computational work has recently suggested that the concentration of VO+2 is not appreciable in anatase for any range of ambient oxygen partial pressure. However, experiments using impedance spectroscopy have suggested that VO+2 dominates at low T and high oxygen partial pressures (105 > PO2 > 10 Pa), whereas Tii+4 becomes important at increasing temperatures and low oxygen

6.1 Bulk Defects

165

Table 6.11 Ionization levels of point defects in rutile TiO2 valid over a wide range of temperatures and pressures (300–1,900 K, 10–2–10–15 atm) calculated by (He et al. 2007) Ionization Level

VTi

VO

Tii

Oi

(+4/+3) (+3/+2) (+2/+1) (+1/0) (0/–1) (–1/–2) (–2/–3) (–3/–4)

– – – – – 0.39 0.22 0.82 1.44

– – 2.11 2.53 – – – –

1.71 2.22 2.48 3.08 – – – –

– – – – 2.69 1.95 – –

All values are in eV and referenced to the valence band maximum.

partial pressure (10–9 > PO2 > 10–19 Pa) (Knauth and Tuller 1999). The experimental papers published by these groups do, however, consider nanocrystalline and ceramic TiO2, and also differ slightly in their pressure and temperature ranges. Weibel et al. found the transition from Schottky disorder (VO+2 and VTi–4) to Frenkel cation disorder (Tii+4 and VTi–4) to occur at 580ºC; the activation energies of the two types of defects are 1.3 ± 0.1 eV and 2.2 ± 0.2 eV, respectively (2006). The temperature at which the transition from oxygen vacancy to titanium interstitial dominance occurs is much lower in anatase than in rutile, where interstitial formation is observed only above 1,100ºC and under 10–1 Pa. The more favorable formation of interstitials in anatase can be attributed to the fact that it has a 10% lower density than rutile. Na-Phattalung et al. have used DFT with the LDA to obtain low formation energies for the fundamental native defects in anatase TiO2 (Tii, Oi, VTi, and VO) and high formation energies for the Ti- and O-antisite defects (2006). None of the four low-energy native defects were found to have ionization levels inside the DFT band gap, as shown in Fig. 6.9. According to these authors, VO+2 is not a dominant native defect near equilibrium growth conditions; even at high oxygen deficiencies, the formation of the oxygen vacancy is less favorable than that of the (+4) titanium interstitial Ti- and O-antisite defects. For Ti-rich material, VTi–4 becomes more energetically favorable than Tii+4 for Fermi energies about 2.8 eV above the valence band maximum. In O-rich material, the range over which Tii+4 is likely to exist in substantial concentrations is far smaller; instead, neutral (O2)O and VTi–4 have low formation energies for Fermi energies between Ev + 0.6 and Ev + 1.35 eV and from Ev + 1.35 eV to the conduction band minimum, respectively. As Tii+4 has a negative formation energy for EF < 2.4 eV and 0.5 eV under Ti-rich and O-rich conditions, respectively, the p-type doping of TiO2 under equilibrium growth conditions is unlikely. The small formation energy of Tii+4 makes the following reaction favorable and explains the insignificant concentration of VO+2 in the bulk: mTiO2 + 2nVO+2 ↔ ( m − n ) TiO2 + nTii+4 + 2ne − .

(6.36)

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6 Intrinsic Defects: Ionization Thermodynamics

Fig. 6.9 Defect formation energies as a function of Fermi level under Ti-rich (a) and O-rich (b) growth conditions. The Fermi energy, referenced to the valence band maximum, is traced all the way up to the experimental band gap. The vertical dotted line is the calculated band gap at the special k-point. Notice the low formation energy proposed for (– 4) titanium vacancies in strongly n-type material, especially under O-rich conditions. Reprinted figure with permission from NaPhattalung S, Smith MF, Kwiseon K et al. (2006) Phys Rev B: Condens Matter 73: 125205-4. Copyright (2006) by the American Physical Society.

6.1.5.2 Associates and Clusters

It is difficult to even speculate on the stable charge states, much less ionization levels, of VTiVTi and VOVO in rutile TiO2. The cation divacancy may act as a donor and introduce doubly degenerate levels at Ev + 0.5 eV (Ahn et al. 2007). These levels arise due to the bonding between the oxygen atoms left behind by the removal of the two titanium atoms. The oxygen divacancy in the apical configuration introduces a state in the band gap at Ev + 1.5 eV (assuming Eg ~ 1.7 eV), whereas no levels are found for the nearest-neighbor divacancy (Cho et al. 2006).

6.1.6 Other Oxide Semiconductors The band gaps of ZnO, UO2, and CoO are respectively 3.2 eV (Pearton et al. 2003), 1.3 eV (Samsonov 1982), and 6.0 eV (Jeng et al. 1991). Zinc oxide is an attractive

6.1 Bulk Defects

167

material for UV and photonic applications; it is well-suited to these applications due to its large exciton binding energy, high breakdown strength, and high saturation velocity. Uranium dioxide is of interest to the microelectronics, energy, and defense communities; it can withstand high operating temperatures and radiation damage. The conductivity of intrinsic UO2 is approximately the same as that of GaAs, yet its dielectric constant (~22) is nearly double that of both Si and GaAs. Cobaltous oxide has found use as a catalyst component in industrial hydrodesulfurization and air purification. Charged defects can affect the performance of these oxide semiconductors as well as countless others. For example, undesirable moisture absorption in La2O3, a high permittivity, large band gap semiconductor being studied as a gate insulator, is attributed to oxygen vacancies (Zhao et al. 2006b). The abnormally high p-type conductivity of SnO2 is thought to arise from charged Sn vacancies (Togo et al. 2006). Most other wide band gap oxides with high p-type conductivity are based on copper oxide compounds.

6.1.6.1 Point Defects

The ionized point defects in ZnO, UO2, and CoO have been investigated by the same pressure dependency measurements as those in TiO2, as well as DFT calculations. As ZnO tends towards a reduced state under equilibrium conditions, oxygen vacancies in the (+2) and (0) charge states are the dominant defects under p-type and n-type conditions, respectively. For O-rich conditions, or high oxygen partial pressures, discrepancy exists in the literature as to whether charged zinc vacancies or oxygen interstitials dominate (Kohan et al. 2000; Oba et al. 2001; Erhart et al. 2005; Zhao et al. 2006a). VO+2 is also the dominant defect in hypostoichiometric UO2–x, whereas Oi–2 forms to accommodate the hyperstoichiometry of UO2+x (Crocombette et al. 2001). In CoO, which prefers to exist as Co1–xO, VCo0 and VCo–1 are the predominant defects at high oxygen partial pressure, depending upon Fermi energy, while VCo–2 becomes important at low oxygen partial pressures (Dieckmann 1977; Hoshino et al. 1985). Implicit in this brief introduction is the implication that the defects in both ZnO and CoO have ionization levels within the band gap. For example, in a recent density-functional theory investigation using the GGA, Zhao et al. identified charge transfer levels for VO, VZn, Oi, and Zni, as shown in Fig. 6.10 (2006a). The defect chemistry of ZnO has been well-studied; ion-gas and electronic reactions and their rate constants have been explored in great detail by Kröger (1964), Hagemark (1976), and Mahan (1983). With respect to specific charge states, the following defect species have been considered: oxygen vacancies VO+2, VO+1, and VO0; zinc vacancies VZn0, VZn–1, VZn–2; and oxygen interstitials Oi+2, Oi+1, Oi0, Oi–1, Oi–2. Antisite oxygen, as well as neutral, singly, and doubly ionized zinc interstitials have also been considered in the literature, yet their formation energies are always higher than those of O and Zn vacancies and O interstitials (Zhao et al. 2006a; Bixia et al. 2001; Reynolds et al. 1997).

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6 Intrinsic Defects: Ionization Thermodynamics

Fig. 6.10 Calculated defect formation energies for selected vacancy, interstitial, and antisite defects in ZnO as a function of the Fermi level at low oxygen partial pressure. Reprinted figure with permission from Zhao J-L, Zhang W, Li X-M et al. (2006) J Phys: Condens Matter 18: 1504. Copyright (2006) by the Institute of Physics Publishing.

The reaction between ZnO and the ambient can be described by the solid-gas reaction 1 g g ZnO ↔ Zn( ) + O2( ) . 2

(6.37)

Oxygen vacancies and interstitials are created by the elementary reaction OO ↔ Oi + VO .

(6.38)

If ZnO behaves like TiO2, then interstitial oxygen binds to oxygen in lattice sites and outdiffuses to form gaseous oxygen (Na-Phattalung et al. 2006), according to the overall reaction 1 g OO ↔ VO + O2( ) , 2

(6.39)

which is formed by summing the following elementary reactions: OO ↔ Oi + VO

(6.40)

OO + Oi ↔ ( O2 )O

(6.41)

( O2 )O ↔ OO +

1 (g) O2 . 2

(6.42)

Zinc interstitials and vacancies are produced by the following reactions: ZnZn ↔ Zni + VZn

Zni ↔ Zn

(g)

.

(6.43) (6.44)

6.1 Bulk Defects

169

All of the defects in ZnO exist in multiple charge states. Their formation can be described by ionization reactions analogous to those shown earlier for TiO2. There exists significant experimental and computational evidence that oxygen vacancies in a neutral or (+2) state are the dominant defects in zinc-rich ZnO (Zhao et al. 2006a; Mahan 1983; Tomlins et al. 2000; Janotti and Van de Walle 2005). Under p-type conditions the oxygen vacancy is in the (+2) charge state, while under n-type conditions the oxygen vacancy is in the neutral charge state. The (+1) charge state is thus never thermodynamically stable in undoped material. Density-functional theory calculations using the GGA for the exchange-correlation potential have instead located this ionization level at 1.07 eV above the valence band maximum (Zhao et al. 2006a). These authors do point out, however, the tendency of the GGA to underestimate the bandgap and strongly alter the defect formation energies and ionization levels. This is confirmed by DLTS experiments (Pfisterer et al. 2006; Hofmann et al. 2007) and DFT-LDA calculations (Van de Walle 2001) that obtain values of Ev + 2.77 eV and Ev + 2.7 eV for the (+2/0) ionization level. While the literature agrees that VZn in the (–2) charge state is the dominant point defect in n-type, O-rich ZnO, its existence at high concentrations as either VZn0 or VZn–1 under p-type conditions is controversial. Calculations by Zhang et al. (2001), Lee et al. (2001), Erhart et al. (2005) suggest that VZn–2 is the defect with the lowest formation energy only for Fermi energies above midgap, and that the defect has no ionization levels in the band gap. On the other hand, other researchers find that the formation energy of the zinc vacancy is always lower (even under p-type conditions) than that of the other intrinsic defects in O-rich ZnO (Kohan et al. 2000; Oba et al. 2001; Zhao et al. 2006a). Also, these authors all find at least one charge state transition within the band gap. Kohan et al. and Zhao et al. computed (0/–1) and (–1/–2) ionization levels at 0.2–0.3 and 0.6–0.8 eV above the valence band maximum. In an earlier first-principles study also using the GGA, Oba et al. ruled out VZn0 as a stable defect within the band gap and cited a (–1/–2) level at about 0.05 eV above the VBM. The values published by Zhao et al. in 2006 may be more reliable, however, as they were obtained using a 72-atom supercell with four-k-point sampling and a plane-wave cutoff energy of 400 eV; Oba et al. used only one k-point with a cutoff energy of 380 eV. The dumbbell oxygen interstitial in the neutral charge state is the alternative defect that has been suggested to have a low formation energy in p-type ZnO under O-rich conditions (Erhart et al. 2005). Oi can serve as both a donor and acceptor in ZnO with charge states ranging from (+2) to (–2) (Zhao et al. 2006a). The debate over the formation energy of the oxygen interstitial in p-type O-rich ZnO is strongly related to the geometry of the defect. For Fermi energies within 0.5 eV of the valence band, Erhart et al. and Lee et al. (2001) claim that Oi, db0 is the dominant point defect in the bulk. Although both Oi, db+2 and Oi, db+1 are stable, the (+2/+1) and (+1/0) ionization levels for the two defects are right at the valence band maximum. This picture is not at odds with prior research, which had always revealed the octahedral oxygen interstitial to be a high formation energy defect in both Zn- and O-rich material. Oi,oct is an acceptor in ZnO; its (0/–1) and (–1/–2)

170

6 Intrinsic Defects: Ionization Thermodynamics

ionization levels are normally placed at Ev + 0.3–0.6 eV and Ev + 0.8–1.5 eV, respectively (Zhang et al. 2001; Zhao et al. 2006a; Kohan et al. 2000; Oba et al. 2001). Erhart et al. also calculated formation energies for Oi, oct, in the (0), (–1), and (–2) charge states in close agreement with these values. A recent density-functional theory investigation may not be justified in claiming that the formation of VZn0 is at least 1 eV more favorable than that of Oi0, as the work considers the octahedral instead of the dumbbell configuration of the defect (Zhao et al. 2006a). The proposed ionization levels for the oxygen interstitial defects, as well as those for VO, VZn, and Zni are summarized in Table 6.12. On this basis, the maximum likelihood estimation value for the (+2/0) ionization level of VO is 2.36 ± 0.02 eV above the valence band maximum. Individual maximum likelihood values of the (+2/+1) and (+1/0) oxygen vacancy ionization levels are not presented here. The (0/–1) and (–1/–2) ionization levels of VZn are at Ev + 0.11 ± 0.05 eV and Ev + 0.55 ± 0.05 eV. The (+1/0), (0/–1), and (–1/–2) ionization levels of Oi are at Ev + 0.07 ± 0.05 eV, Ev + 0.63 ± 0.04 eV and Ev + 0.97 ± 0.05 eV. Lastly, maximum likelihood predicts ionization levels of Ev + 1.0 ± 0.06 eV (+2/+1), Ev + 1.0 ± 0.06 eV (+1/0), and Ev + 0.8 ± 0.06 eV (+2/0) for Zni. Uranium and oxygen vacancies, as well as oxygen interstitials, occur in undoped UO2; they arise in the (–4), (+2), and (–2) charge states, respectively. Frenkel pairs, which consist of a vacancy and an interstitial of the same elemental type, VO+2 and Oi–2 in this case, are the most stable defect in stoichiometric UO2. Based on neutron diffraction studies of interstitials in near-stoichiometric UO2, Willis proposed an anion Frenkel disorder model for the crystal (VO+2 and Oi–2) (1964); the alternative models of Schottky (VO+2 and VU–4) or cation Frenkel disorder would have required a cation vacancy structure for the anion excess crystals. A later defect survey of UO2 initially considered both the singly and doubly charged Frenkel pair (Catlow 1977). The calculations revealed that the formation energy of the singly charged pair far exceeds that of the doubly charged pair. This result implies that the negatively charged oxygen interstitial provides a considerably deeper electron trap than the ion vacancy, whose effective charge is positive. Support for the predominance of divalent oxygen Frenkel disorder in UO2 has also been obtained from subsequent diffraction experiments and modeling work (Catlow 1977; Jackson et al. 1987; Hutchings 1987; Petit et al. 1998). Most researchers agree that only a small minority of Schottky defects will exist in the bulk (Catlow 1977; Tharmalingam 1971); recent ab initio calculations, for instance, have shown that the formation energies of the oxygen vacancy and interstitial are more than 10 eV lower than that of the uranium vacancy (Petit et al. 1998; Crocombette et al. 2001; Freyss et al. 2005). The principal point defects to be considered in Co1–xO are cation vacancies and charge compensating electrons or holes; the cation vacancies can exist in the neutral, (–1), or (–2) state (Carter and Richardson 1954; Shelykh et al. 1966; Eror and

6.1 Bulk Defects

171

Table 6.12 Experimentally and computationally determined ionization levels for oxygen vacancies, zinc vacancies, oxygen interstitials, and zinc interstitials in bulk ZnO Defect (+2/+1) (+1/0) (0/–1) (–1/–2) Other VO

–

–

–

–

3.16

–

–

–

–

–

–

–

– 3.3 2.9

– 2.0 1.94

– – –

– – –

VZn

2.26 – – –

0.94 – – –

Oi

– – – – – –

Zni

– – – – – 1.0 ± 0.06 0.41 – – 1.1 1.55

– – – 0.11 ± 0.05 – 0.35 – – – 0.05 – –0.5 – 0.19 0.07 ± 0.63 ± 0.05 0.04 0.03 0.38 0.12 1.4 – 0.48 – 0.7 0.08 0.38 1.0 ± – 0.06 0.45 – – – – – 0.8 – 1.73 –

– – – 0.55 ± 0.05 0.78 0.44 – 0 0.62 0.97 ± 0.05 0.95 – 0.89 1.7 0.78 – – – – – –

Method

2.36 ± 0.02 Maximum (+2/0) likelihood 2.77 (+2/0) DLTS 0.17 (+2/0) 1.1 (+2/0) 0.5 (+2/0) 2.7 (+2/0) 2.42 (+2/0) 1.6 (+2/0) 1.14 (+2/0) 0.15 (+2/0) –

DFT-LDA DFT-LDA DFT-GGA DFT-LDA DFT-LDA

DFT-LDA DFT-GGA DFT-GGA Maximum likelihood – DFT-LDA – DFT-LDA – DFT-GGA – DFT-LDA – DFT-GGA – Maximum likelihood – DFT-LDA – DFT-LDA – DFT-GGA DFT-LDA – DFT-GGA 0.8 ± 0.06 Maximum (+2/0) likelihood – DFT-LDA 0.4 (+2/0) DFT-LDA 1.05 (+2/0) DFT-GGA 1.0 (+2/0) DFT-LDA – DFT-GGA

Reference This work (Pfisterer et al. 2006; Hofmann et al. 2007) (Kohan et al. 2000) (Lee et al. 2001) (Oba et al. 2001) (Van de Walle 2001) (Janotti and Van de Walle 2005) (Lany and Zunger 2005) (Zhao et al. 2006a) (Yu et al. 2007) This work (Kohan et al. 2000) (Lee et al. 2001) (Oba et al. 2001) (Zhang et al. 2001) (Zhao et al. 2006a) This work (Kohan et al. 2000) (Lee et al. 2001) (Oba et al. 2001) (Zhang et al. 2001) (Zhao et al. 2006a) This work (Kohan et al. 2000) (Lee et al. 2001) (Oba et al. 2001) (Zhang et al. 2001) (Zhao et al. 2006a)

DLTS deep level transient spectroscopy, DFT density-functional theory, LDA local density approximation, GGA generalized-gradient approximation. All values are in eV and referenced to the valence band maximum.

Wagner 1968; Bransky and Wimmer 1972; le Brusq and Delmaire 1973; Fisher and Tannhauser 1966; Wagner and Koch 1936; Dusquesnoy and Marion 1963). While an ideal point-defect model was initially considered suitable to interpret the properties of CoO, more recently, models involving cobalt vacancies and interstitials, oxygen vacancies, antisite defects, and complexes have been proposed. The Debye–Huckel model of Nowotny and Rekas, for example, which neglects the

172

6 Intrinsic Defects: Ionization Thermodynamics

concentrations of neutral and singly ionized vacancies, as well as interstitials, accurately fits the experimental conduction data for Co1–xO for all values of x (1989). The large ionic size of the oxygen atom in CoO serves as the main prohibitive factor to the easy formation of oxygen interstitials in the bulk. Dieckmann (1977) and Koel and Gellings (1972) both showed that VCo0 or VCo–1 are important at oxygen pressures around 1 atm, while VCo–2 becomes the majority defect at lower pressure. As CoO under low PO2, when the concentration of defects is lower than about 0.1 atomic %, displays very little non-stoichiometry, it can be assumed that VCo–2 is important in nearly-stoichiometric material (Bak et al. 1997). The ionization level for the (–1) vacancy has been predicted to be between 0.45 and 0.65 eV above the valence band maximum by several authors (Fisher and Tannhauser 1966; Dieckmann 1977; Carter and Richardson 1954; Eror and Wagner 1968). The slightly higher values of 0.98 – 1.1 eV obtained by Fryt (1976) and Bransky and Wimmer (1972) probably considered additional intrinsic defects such as Schottky or Frenkel type defects. Values of between Ev + 0.65 eV to Ev + 0.77 eV have been found for the (–1/–2) ionization level of the cobalt vacancy (Bransky and Wimmer 1972; Fisher and Tannhauser 1966; le Brusq and Delmaire 1973; Dieckmann 1977). 6.1.6.2 Associates and Clusters

There have been some isolated studies on defect ionization levels in ZnO. For instance, using DLTS, Simpson and Cordaro associated a trap at 0.24 eV with a cluster-like oxygen vacancy-related defect in ZnO (1990), probably VO+1VZn–1 but possibly VZnZnOOZnZni. Seghier and Gislason have reported extremely shallow donor levels in ZnO at approximately Ev + 15 meV using electrical and optical measurements; they have tentatively assigned these levels to clusters involving Zni and VO, but make no mention of specific constituents or charge states (2007). No ionization levels have been determined experimentally for the defect clusters in cobaltous oxide and uranium dioxide. One study has, however, looked at the formation enthalpy of the charged Willis cluster in UO2. The formation of the 2:2:2 cluster in UO2+x is represented by the reaction:

(

2OO0 + O2 + nUU0 ↔ 2Oia 2Oib 2VO

)

−n

+ nUU+1 .

(6.45)

where n is the effective charge of the cluster. For small deviations from stoichiometry, or UO2+x where 0 < x < 0.07, Ruello et al. have calculated an enthalpy of formation for (2:2:2)–1 of –1.7 ± 0.6 eV based on chemical diffusion and electrical conductivity measurements (2004). One semi-empirical tight-binding study sheds light upon the ionization levels of the 2:2:2 and lesser known 2:1:2 cluster in UO2 (Lim et al. 2002). Lim et al. obtained defect levels of Ev + 0.3 eV and Ev + 0.5 eV for the 2:2:2 and 2:1:2 clusters, respectively; no mention was made of the charge states associated with these ionization levels.

6.2 Surface Defects

173

6.2 Surface Defects Vacancies, adatoms, and defect clusters on semiconductor surfaces can become charged. Charge state, in turn, affects concentration, mobility, and mass transport in the bulk. The use of an “active surface” that can selectively remove self-interstitials over dopant interstitials simultaneously improves profile spreading and sheet resistance in ultrashallow junctions. Point defect annihilation rates at surfaces can, in principle, become controlling factors of solid-state diffusion rates (Kwok 2007). An understanding of these phenomena is critical for integrated circuit manufacturing. The experimental and computational methods used to explore the charging of bulk defects do not translate directly to surface defects. The identification of charged defects with STM is a relatively recent development. Surface defect calculations vary depending on the selection of slab geometry and number of atomic layers that are allowed to relax. It has only recently become possible to draw direct comparisons between bulk and surface defect charging.

6.2.1 Silicon Non-thermal diffusion on the silicon surface is mediated by charged vacancies. As a consequence, many microelectronics processing steps that rely upon optical illumination (such as rapid thermal processing) may be affected. In an attempt to better understand this phenomenon, ionization levels have been calculated for the upper and lower monovacancy and divacancy on Si(100). The behavior of the unfaulted edge vacancy on the Si(111)-(7×7) reconstructions has also been probed. 6.2.1.1 Point Defects

An assortment of computational methods have been used to calculate the formation and ionization energies of the monovacancy on Si(100), which is not considered as frequently as the divacancy due to its high formation energy (0.87 – 1.50 eV) (Chan et al. 2003; Brown et al. 2002; Brown 2003). As mentioned earlier, two distinct types of monovacancy can form as a consequence of the buckled dimer structure on the silicon surface: the upper monovacancy in response to removal of the upper atom, and analogously for the lower monovacancy. The charge state-dependent stability of both the lower and upper monovacancy has been investigated. The lower monovacancy supports only the (0) and (–1) states. The upper monovacancy is very similar to that of the bulk silicon vacancy (Chan et al. 2003) in terms of stable charge states, however. The upper monovacancy exists in four charge states (+2), (0), (–1), and (–2), complete with negative-U defect behavior that leads to metastability of the (+1) charge state (Puska et al. 1998; Baraff et al. 1980a; Watkins and Troxell 1980; Lento and Nieminen 2003). So far this case

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6 Intrinsic Defects: Ionization Thermodynamics

Table 6.13 The ionization levels for the lower and upper monovacancy defects on Si(100) published by (Chan et al. 2003) Si surface defect

(+2/0)

(0/–1)

(–1/–2)

Lower monovacancy Upper monovacancy

– 0.07

0.82 0.62

– 1

represents the only known example of negative-U behavior for a surface defect. Table 6.13 summarizes the findings of Chan et al., who provide values for the ionization levels of the lower and upper monovacancy on Si(100). Dev and Seebauer estimated the entropy of ionization for the monovacancy on Si(001)-(2×1) to predict the behavior of its ionization levels at temperatures between 0 and 1,600 K (Dev and Seebauer 2003b). For both types of monovacancy, increasing the temperature causes a lowering of the ionization levels for negatively charged vacancies. The corresponding levels for positive vacancies remain unchanged, resulting in a decrease in the range of Fermi energies over which the neutral species is stable. Subsequent work examined the unfaulted edge (UFE) vacancy on Si(111)-(7×7). Depending on the position of the Fermi energy, the UFE vacancy is stable in the (0), (–1), and (–2) charge states; positive charge states are not stable for any value of EF (Dev and Seebauer 2003a). For comparison, the monovacancy in bulk Si is stable as V+2, V0, V–1, and V–2, while the lower and upper monovacancies on Si(100)-(2×1) support charges of (0) and (–1), and (+2), (0), (–1), and (–2), respectively. Clearly, there is only a modest correspondence in the number and type of stable charge states among the bulk and various surface crystallographic orientations. In considering the temperature dependence of the UFE vacancy, the (0/–1) level intersects the valence band at about 640 K, implying that UFE0 is not stable above this temperature for any value of Fermi energy. If, as reported by Himpsel et al. (1983), a high density of surface states sets the Fermi energy near midgap for undoped, room temperature Si(111), Dev and Seebauer suggest that the (–2) charge state of the vacancy dominates under virtually all temperatures and dopant concentrations (Fig. 6.11).

Fig. 6.11 Formation energies as a function of Fermi energy of various charge states of the UFE vacancies on Si(111)-(7×7) at 0 K. The formation energy is referenced to the neutral vacancy, while the Fermi energy is referenced to the valence band maximum. Reprinted from Dev K, Seebauer EG, “Vacancy charging on Si(111)-(7×7) investigated by density functional theory,” (2003) Surf Sci 538: L497. Copyright (2003), with permission from Elsevier.

6.2 Surface Defects

175

6.2.1.2 Associates and Clusters

DFT studies indicate that the divacancy on Si(100) is stable in the (0), (–1), and (–2) charge states (Chan et al. 2003), although STM imaging studies have observed only the positively charged species (Brown et al. 2002). It is possible that the failure to observe negatively charged dimer vacancies on Si(001) via STM or calculate positively charged dimer vacancies on the same surface is related to the rebonding configuration (Brown et al. 2003). Also, the experimentally prepared surface may not be sufficiently n-type for negatively charged defects to arise, or tip-induced band bending might obscure the observation of negatively charged dimer vacancies (Brown et al. 2003). The positively charged single dimer vacancy and split-off dimer defect imaged by Brown et al. are shown in Fig. 6.12. For the n-type material utilized in the experiment, perturbations associated with both positive and negative charge density are visible in empty-state images, whereas only those associated with negative charge density are visible in filled-state images (2002). An assortment of computational methods have been used to calculate the formation energies of 1-DV, 2-DV, 1+1-DV, and 1+2-DV on Si(100) (Wang et al. 1993; Roberts and Needs 1989, 1990; Nurminen et al. 2003; Schofield et al. 2004; Ciobanu et al. 2004). Ab initio calculations yield formation energies of 0.22 eV/dimer, 0.16 eV/dimer, and 0.14 eV/dimer for neutral 1-DV, 2-DV, and 1+2-DV, respectively (Wang et al. 1993). The formation energies for 2-DV and 1+2-DV obtained using empirical potentials are 0.505 eV and 0.618 eV (or 0.225 eV/dimer and 0.309 eV/dimer), respectively (Ciobanu et al. 2004). Schofield et al. reported much higher formation energies of ~1.13 eV/dimer and ~0.85 eV/dimer for the 1+1-DV and 1+2-DV clusters from first-principles calculations using the Car–Parrinello molecular dynamics program (2004). Chan et al. have employed density functional theory to obtain ionization levels for the divacancy on Si(001) (2003). For the divacancy, the formation energies of

Fig. 6.12 Empty-state image of defects on Si(100)-(2×1) including a) the positively charged divacancy and b) the positively charged split-off dimer defect. Reprinted with permission from Brown GW, Grube H, Hawley ME et al. (2002) J Appl Phys 92: 822. Copyright (2002), American Institute of Physics.

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6 Intrinsic Defects: Ionization Thermodynamics

the positive charge states increase as the Fermi energy moves away from the valence band, while those of the negative states decrease. The Fermi energy on the free Si(001) surface resides at Ev + 0.41 eV, meaning that divacancies should exist predominantly in the neutral charge state at temperatures near absolute zero. Only the neutral, (–1), and (–2) states are stable on the surface, in contrast to the bulk, where the singly positive charge state (+1) is stable for some values of the Fermi energy. The (0/–1) and (–1/–2) ionization levels of the surface dimer vacancy at 0 K have been calculated to be at 0.62 and 1.06 eV above the valence band maximum, respectively (Chan et al. 2003). As with the monovacancy on Si(100), increasing the temperature causes a lowering of the ionization levels for the negatively charged defects. The corresponding levels for positive vacancies remain unchanged, leading to a decrease in the range of Fermi energies over which the neutral species is stable.

6.2.2 Germanium Evidence suggests that charged vacancies on the germanium surface mediate lead to changes in activation energy and preexponential factor for mass transport. The charging of these defects is not as well understood as those on silicon. 6.2.2.1 Point Defects

No computational work exists concerning the ionization levels of Ge(100) and Ge(111) surface defects, although several ab initio molecular dynamics works have investigated the energetics, geometries, and band structures associated with the undefected germanium surface (Takeuchi et al. 1994; Pasquali et al. 1998; Bechstedt et al. 2001). Defects on the (100) crystal plane of germanium have not been explored experimentally to the extent of those on Si(100), in part due to the inherent stability of the (2×1) surface reconstruction (Kubby et al. 1987; Yang et al. 1994). Yang et al. have observed delocalized contrast surrounding these features on Ge(100)-(2×1), which potentially indicates that the defects are not charged relative to the neutral surface. However, when Lee et al. put forth this explanation in 2000, they cited a similar absence of charged vacancies on Si(111)-(7×7) and Si(100)-(2×1) (2000b); the evidence for charged defects on these surfaces is now very strong (Chan et al. 2003; Brown et al. 2002). For Ge(111)-c(2×8), variable voltage STM has been used to demonstrate the existence of charged surface defects and to provide insight into the local band bending that they cause (Stroscio et al. 1987; Yang et al. 1994; Lee et al. 2000a, b). Five years after Molinàs-Mata and Zegenhagen (1993) first used STM to investigate defects on the Ge(111)-c(2×8) reconstruction, Lee et al. revealed the presence of surface defects with a net charge (1998a; 2000a). The defects on Ge(111) can be classified into three categories: adsorbed H atoms, Ge atom vacancies, and

6.2 Surface Defects

177

Fig. 6.13 STM images of the defects labeled as C (adatom vacancy) and C' (rest atom vacancy) on the Ge(111) surface taken at various sample voltages (a) +2.5 V (b) +1.5 V (c) +0.5 V (d) –2.0 V (e) –1.0 V and (f) –0.5 V, respectively. Reprinted from Lee G, Mai H, Chizhov I et al, “Charged defects on Ge(111)-c(2×8) characterization using STM,” (2000) Surf Sci 463: 60. Copyright (2000), with permission from Elsevier.

defects due to antiphase shift. There is no spatial designation assigned to the vacancies, in contrast to those on Si(111)-(7×7), which supports four distinct adatom vacancy locations, the unfaulted corner (UFC), unfaulted edge (UFE), faulted corner (FC), and faulted edge (FE) (Lim et al. 1996; Dev and Seebauer 2003a). All of the possible Ge surface defects are voltage-dependent, indicating that they are either charged point defects or neutral composites of defects with opposite charges (Fig. 6.13). The germanium vacancy on the surface is associated with a bright spot due to the charge localized at the dangling bonds of the three nonbonded first-layer atoms. From the direction of surface band bending, Lee et al. inferred that the adatom vacancy defect is negatively charged (2000b; 2000a); the UFE vacancy on Si(111) investigated by Dev and Seebauer is also negatively charged (Dev and Seebauer 2003b). In some STM images, however, the vacancy appears to be neutral, a phenomenon the authors attribute to neutralization of the negative adatom vacancy by a positively charged rest-atom vacancy, although no individual positively charged rest-atom vacancies were observed (Lee et al. 2000a). The authors make no mention of the degree of band bending or the concentration of adatom vacancy defects that would be required to lead to Fermi energy pinning over the whole surface. The antiphase shift vacancy discussed by Lee et al. is a point defect generated by a shift of an adatom row by a half period in the row direction; it perturbs the c(2×8) reconstruction and causes a line stacking fault, or antiphase domain boundary. In some images the point defect is negatively charged, while in others it appears to be neutral relative to the background of the unperturbed surface. No comparable defect is observed on the Si(111)-(7×7) reconstruction.

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6 Intrinsic Defects: Ionization Thermodynamics

6.2.2.2 Associates and Clusters

The ionization levels of single and double dimer vacancy associates and clusters on Ge(100)-(2×1) have not been investigated. One report uses empirical potentials to compare the formation energies of neutral 2-DV and 1+2-DV on Ge(100) to the same defects on Si(100) (Ciobanu et al. 2004). It is worth noting that both of these defects are highly stable on Ge(100); they tend not to disaggregate into smaller dimer vacancy defects (1-DV, for instance). According to Ciobanu et al., 2-DV and 1+2-DV have formation energies of approximately 0.35 eV/dimer and 0.61 eV/dimer, in that order.

6.2.3 Gallium Arsenide Surface defects can be generated on GaAs by cleaving or segregation of bulk defects to the surface. Most studies have been carried out on the natural cleavage plane of GaAs, of which comparably defect-free surfaces can be obtained. A better understanding of the charged defects on GaAs is desired in order to understand the influence of cleaving and sputter-anneal cleaning of defected surfaces on Fermi level position. 6.2.3.1 Point Defects

GaAs(110) is decorated with charged gallium and arsenic vacancy defects; the observation of these charged defects on n- and p-type material is a result of localized defect states introduced in the band gap by the vacancies (Ebert 2002). STM has been used extensively to image the vacancies on GaAs, and to determine their charge states (Ebert 2002; de la Broise et al. 2000; Chao et al. 1996b, a; Domke et al. 1996; Ishikawa et al. 1998). Under gallium-rich conditions, VAs+1 and VAs–1 are the dominant defects for p-type and n-type GaAs(110), respectively, whereas under arsenic-rich conditions, charged gallium vacancies and arsenic adatoms become important (Schwarz et al. 2000b). Surface photovoltage measurements and density-functional theory and ab initio calculations have been employed to obtain values for defect ionization levels (Yi et al. 1995a; Zhang and Zunger 1996; Kim and Chelikowsky 1996), although the determined positions of these levels vary widely, as well as the degree of band bending they cause (Lang 1987; Stroscio et al. 1987; Lengel et al. 1994; Aloni et al. 1999, 2001). As opposed to bulk gallium arsenide, in which ionized AsGa and GaAs play an important role in the defect chemistry, antisite defects have yet to be observed on the GaAs surface. From DFT calculations, however, it is known that the surface and near-surface antisite defect has no charge-transfer level within the band gap, and thus, does not alter the electronic properties of the (110) surface (Schwarz et al. 2000b; Iguchi et al. 2005). Nevertheless, Schwarz et al. find surface antisite defects to be most

6.2 Surface Defects

179

Fig. 6.14 Topography (A) and surface photovoltage (B) images taken simultaneously at VS = –2.8 V around an arsenic vacancy on GaAs(110). Reprinted with permission from Aloni S, Nevo I, Haase G (2001) J Chem Phys 115: 1877. Copyright (2001), American Institute of Physics.

stable in the neutral charge state, in contrast to those in the bulk, which are known to act as either double acceptors (GaAs) or double donors (AsGa). Arsenic and gallium vacancies at the surface are formed by the following surface-gas reactions 1 As2 ( g ) + VAs 2

(6.46)

1 As2 ( g ) ↔ As As + VGa 2

(6.47)

As As ↔

where there is a noticeable difference in the number of gallium atoms surrounding the vacancy (Chiang and Pearson 1975). Scanning tunneling microscopy images such as those seen in Fig. 6.14 reveal the presence of missing arsenic atoms, or charged surface anion vacancies, on Ga(110) (Feenstra and Fein 1985; Lengel et al. 1994; Ebert et al. 1994). The (+1) and (–1) charge states of VAs are relevant to the GaAs(110) surface. Chao et al. used a method based on STM and compensation by ionized dopant atoms to determine an isolated arsenic vacancy charge of (+1) (1996b; 1996a). Several authors have considered the Fermi energy dependence of the formation energies of VAs–1, VAs0, and VAs+1. Ab initio calculations performed by Yi et al. found the (–1) arsenic vacancy to be the predominant surface defect for all Fermi energies within the band gap (1995a; 1995b). In contrast, the latter two groups identified VAs+1 and VAs–1 as the dominant defects for p-type and n-type GaAs(110), respectively. They also found that VAs0 is stable only over a narrow energy range, as the (+1/0) ionization level is fairly close to that of (0/–1); more recently, it has also been suggested that the arsenic vacancy behaves as a negative-U center (Zhang and Zunger 1996; Kim and Chelikowsky 1996; Schwarz et al.

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6 Intrinsic Defects: Ionization Thermodynamics

2000b). The energy levels calculated via DFT by Kim and Chelikowsky and Zhang and Zunger deviate considerably from one another. The former obtain (+1/0) and (0/–1) ionization levels of Ev + 0.10 eV and Ev + 0.24 eV while the latter predict values of Ev + 0.32 eV and Ev + 0.40 eV for the same levels. Schwarz et al., the authors who proposed a negative-U center for the arsenic vacancy, placed the (+1/–1) level at about 0.22 eV above the valence band maximum. Kim and Chelikowsky have cited the inaccuracy in evaluating the band-maximum line-up of different charge states as a possible reason for the varying ionization level values. Under arsenic-rich conditions, two different types of charged defects become important: gallium vacancies and arsenic adatoms. The surface gallium vacancy bears little resemblance to the bulk gallium vacancy; VGa–3 is the lowest energy defect in n-type Ga-rich GaAs. On the surface, the (–3) state does not exist, and for n-type Ga-rich GaAs(110), VGa–1 has a formation energy that is about 1 eV greater than that of VAs–1 (Schwarz et al. 2000a). Normally, without special treatment, gallium vacancies can be observed on the surface of n-type GaAs wafers (Lengel et al. 1996). They were first identified in filled-state STM images of n-type GaAs(110) by Lengel et al. (1993). Schwarz et al. calculated ionization levels for the gallium vacancy of Ev + 0.24 eV and Ev + 0.31 eV for the (+1) and (–1) states, respectively. For p-type material, VGa+1 is almost degenerate in formation energy with the positively charged arsenic adatom. The same authors calculated two ionization levels for the arsenic adatoms As1; (+1/0) at Ev + 0.36 eV and (–1/0) at Ev + 0.60 eV. The cation-bound adatom was predicted to be singly negatively charged for all positions of the Fermi energy except for extremely p-type material, in which the neutral charge state of the defect is stable, as shown in Fig. 6.15.

Fig. 6.15 Formation energies of surface point defects on GaAs(110) as a function of the surface Fermi level for arsenic-rich conditions. Reprinted figure with permission from G. Schwarz, J. Neugebauer, and M. Scheffler, Point defects on III–V semiconductor surfaces. In: Proc. 25th Int. Conf. Phys. Semicond. (Eds.) N. Miura, T. Ando. Springer Proc. in Physics, Vol. 87, Springer, Berlin/Heidelberg 2001, p. 1380.

6.2 Surface Defects

181

6.2.3.2 Associates and Clusters

The mixed divacancy on GaAs(110), VGaVAs, may assume a negative charge state. Using ab initio total-energy calculations with the Car–Parrinello method, Yi et al. found that the VGaVAs favors negatively charged states regardless of the position of the bulk Fermi level (1995a). They made no mention of the specific negative charge states under consideration, however. As a reminder, both gallium and arsenic monovacancies are stable in the (+1), (0), and (–1) charge states on GaAs(110). The similarity in electronic levels between the associates and isolated defects implies that the Fermi energy pinning location for surface vacancies are only slightly perturbed by the formation of antisite-vacancy associates on the GaAs(110) surface.

6.2.4 Other III–V Semiconductors Many III–V semiconductor surfaces are decorated with charged defects similar to those on GaAs(110). Surface vacancies, and anion vacancies in particular, are produced by the evaporation of surface atoms. Even on the cleavage surfaces of these materials, low-temperature desorption of atoms causes the appearance of monovacancy defects at room temperature. 6.2.4.1 Point Defects

Under indium-rich conditions, VP+1 and VP–1 have low formation energies on p-type and n-type InP(110), respectively (Semmler et al. 2000; Morita et al. 2000; Ebert 2002, 2001; Kanasaki 2006). Interestingly, antisite defects, rather than indium vacancies or phosphorus adatoms, are predicted to arise under anion-rich conditions (Ebert et al. 2001; Hoglund et al. 2006). While the morphology and electronic structure of the surfaces of the boron and nitride-containing semiconductors have been investigated, little information exists concerning their charged defects (Filippetti et al. 1999; Ooi and Adams 2005; Miotto et al. 1999). Typically, the (1010) ZnO surface is used as a reference system for AlN, GaN, and InN (Filippetti et al. 1999). For a more thorough discussion of point defects on compound semiconductor surfaces, Ebert has published over thirty journal articles on group III–V surfaces, including a summary of defects observable using STM, shown in Fig. 6.16 (2001; 2002). The stable charge states and ionization levels of defects on the indium-group V compounds can be compared to similar defects in the bulk. VP and VAs on InP and InAs possess two ionization levels ((+1/0) and (0/–1)) within the band gap while VSb is only stable in the (–1) charge state. For purposes of comparison, the anion vacancy in InP and InAs is stable in considerably different charge states ((+1), (–1), (–2), (–3), and (–5) for the former and (+1) and (–1) for the latter) (Hoglund et al.

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Fig. 6.16 Anion and cation vacancies on different materials. The upper and lower frames of each vacancy show the occupied and empty state STM images, respectively. Reprinted from Ebert P, “Atomic structure of point defects in compound semiconductor surfaces,” (2001) Current Opinion in Solid State & Materials Science 5: 218. Copyright (2001), with permission from Elsevier.

2006); the antimony vacancy in bulk InSb is also only stable in the (–1) charge state. Additional correspondences based on the computational work of Hoglund et al. are tabulated in Table 6.14. Some of these charge states have been observed experimentally. Using combined STM and photoelectron spectroscopy, Ebert et al.

6.2 Surface Defects

183

Table 6.14 Ionization levels of bulk and surface defects for InP, InAs, InSb as determined by (Hoglund et al. 2006) using density-functional theory within the local-density approximation Defect

(+1/0)

(0/–1)

(+1/–1)

(–1/–2)

(–2/–3)

Other

No IL

VP in InP VP on InP(110) VIn in InP VIn on InP(110) VAs in InAs VAs on InAs(110) VIn in InAs VIn on InAs(110) VSb in InSb VSb on InSb(110) VIn in InSb VIn on InSb(110)

– 0.40 – 0.10 – 0.21 – – – – – –

– 0.58 – 0.36 – 0.27 – 0.11 – – – –

0.57 – – – 0.30 – – – – – – –

0.63 – – – – – – – – – – –

0.70 – – – – – – – – – – –

1.06 (–3/–5) – 1.38 (–3/–4) – – – – – – – 0.005 (–1/–3) –

– – – – – – –3 – –1 –1 – –1

All values are in eV and referenced to the valence band maximum. For those defects with no ionization level within the band gap, the stable charge state for all Fermi energies within the band gap is listed in the “No IL” column.

obtained a (+1/0) ionization level for VP in InP of Ev + 0.52 eV and Ev + 0.45 eV corresponding to the nonsymmetric and less stable symmetric vacancy, respectively (2000). Qian et al. used density-functional theory to obtain ionization levels of 0.388 and 0.576 eV above the valence band maximum for (+1/0) and (0/–1) in InP, respectively, with a calculated band gap of 1.11 eV (2002). The same authors also determined the levels for VAs on InAs(110); (+1/0) and (0/–1) are 0.20 and 0.277 eV above the valence band maximum. For InSb(110), Qian et al. found no ionization level for the Sb vacancy within the band gap. The anomalous behavior of the antimony vacancy can be explained in terms of the electronic configuration of the defect. Of the three In-based compounds, InSb shows the strongest hybridization, with the localized surface state derived from the valence and conduction states. In order to reduce the total energy when the Fermi energy is shifted from the valence band maximum to the conduction band minimum, the vacancy state of InSb, located within the valence band, is always occupied (Qian et al. 2002).

6.2.5 Titanium Dioxide The defect chemistry of the TiO2 surface may have a profound impact on its use for catalytic applications, in particular. For example, the charging of surface bridging oxygen vacancies may affect desorption and reaction rates of water, oxygen, and organic molecules on TiO2(110) (Kim et al. 2008). Research to understand the charging of both the rutile and anatase surfaces is currently underway.

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6 Intrinsic Defects: Ionization Thermodynamics

6.2.5.1 Point Defects

The bridging (BOV) and in-plane (POV) oxygen vacancies that occur on the rutile TiO2(110) surface are also easily ionized. It has recently been suggested that these defects can occur as BOV+2, BOV+1, POV+2, and POV+1, in addition to BOV0 and POV0 (Fukui et al. 1997; Diebold et al. 1998; Chen et al. 2001; Wang et al. 2005). This phenomenon makes sense, as the existence of positively charged oxygen vacancies in bulk TiO2 is well known. The low surface free energy of anatase TiO2 leads to a different tendency toward reduction and/or the formation of oxygen vacancies compared to rutile TiO2. Despite the fact that several studies have considered the effect of surface defects on adsorbate reactions, the existence and possible structure of the surface oxygen vacancy are still uncertain (Tilocca and Selloni 2003, 2004). Using the embedded-cluster numerical discrete variational method, Chen et al. suggested the existence of charged F-type centers, i.e. positively charged oxygen vacancies, on TiO2 (110) (2001). Positively charged bridging and in-plane oxygen vacancies can form when negatively charged, rather than neutral, bridging-oxygen ions (O–1 and O–2) are removed (Wang et al. 2005). Most computational studies of the reduced (110)TiO2-(1×1) surface predict gap states, although the published reports do not come to an agreement as to the surface defect structure responsible for these states (Lindan et al. 1997). This is in contrast to VO+2 in bulk TiO2, which does not give rise to any defect levels within the band gap (Cho et al. 2006). Wang et al. recently compared the energetics behind the formation of in-plane and bridging oxygen vacancies in the neutral, (+1), and (+2) states (2005). The DFT-derived formation energies of all three states of both defects are tabulated in Table 6.15. The ionization potential for losing one electron to form BOV+1 is –0.9 eV, while the second ionization potential to form BOV+2 is 1.3 eV; these values indicate the stability of BOV+1. The first and second ionization potential of the POV are –0.7 and 1.1 eV, respectively. These numbers can be compared to the formation energies detailed above for the BOV, revealing a stability order of the oxygen vacancies on the TiO2 (110) surface as BOV+ ≈ POV+ > BOV ≈ BOV+2 > POV ≈ POV+2. In 2000, Heibenstreit and co-workers reported the first scanning tunneling microscopy study of single-crystal anatase. STM images of anatase surfaces must be interpreted differently from those of rutile surfaces. In the former, the corrugated nature of the surface causes the twofold-coordinated oxygen atoms at the highest position to be imaged bright, whereas in the latter, the 3d Ti-derived empty defect

Table 6.15 The formation energies (in eV) of the neutral and positively charged vacancies on TiO2(110) according to (Wang et al. 2005) Defect

(+2)

(+1)

(0)

BOV POV

8.1 8.6

4.2 4.5

7.9 8.3

6.2 Surface Defects

185

Fig. 6.17 STM images of an anatase (101) surface. Four features could possibly represent oxygen vacancies including (A) single black spots, (B) double black spots, (C) bright spots, and (D) half black spots. Reprinted figure with permission from Hebenstreit W, Ruzycki N, Herman GS et al. (2000) Phys. Rev. B: Condens Matter 62: R16336. Copyright (2000) by the American Physical Society.

states appear brighter. Hebenstreit et al. observed four types of imperfections on anatase (101), yet were unable to easily associate any of them with the oxygen vacancy that is easily identified in studies of rutile (110) (Fig. 6.17) (2000). The authors suggested that either the anatase (101) surface is very stable against the loss of twofold-coordinated oxygen atoms, or STM is not capable of imaging such vacancies. It is still undetermined as to whether these atomic-scale imperfections are oxygen vacancies. If the imperfections are point defects, they would be present in far lower number densities than are observed on rutile (110). This low concentration would not, however, be at odds with the general surface science picture of the material; not only is the surface energy of anatase (101) known to be low, but also Woning and van Santen have predicted the easier reduction of rutile versus anatase titanium dioxide surfaces (1983). Synchrotron photoemission spectroscopy experiments performed by Thomas et al. have indicated that the defects may be surface O vacancies in the form of surface Ti+3 (2003). The authors observed the defects to give rise to a state below EF similar to that seen on the surface of rutile TiO2. From the band-gap state at 1.1 eV, and associated resonance in the spectrum, they suggested that Ti 3d contributes to the defect state, probably as a result of surface O vacancies.

6.2.6 Other Oxide Semiconductors The intrinsic surface defects on ZnO, UO2, and CoO have not been explored to the same extent as those on titanium dioxide. Their effect on mass transport in the bulk is expected to be of critical importance for applications employing these materials, just as it is for silicon.

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6.2.6.1 Point Defects

When the rare natural form of ZnO, zincite, is cleaved, nonpolar (1010) surfaces result; for this surface, it is known that the most commonly occurring defects are oxygen vacancies (Gopel et al. 1980a; Gopel et al. 1980b; Wander and Harrison 2003). Papers published between 1957 and 1979 by Heiland and co-workers indicated the existence of donor-type ZnO defects near the surface, with ionization levels at about 0.2 eV below the conduction band minimum (1978). The same picture is not true for the (0001)-Zn surface, where one of two processes can occur: either zinc atoms are removed from the lattice, leading to the formation of (+2) zinc vacancies, or oxygen atoms are adsorbed to the surface (Wander and Harrison 2003; Jedrecy et al. 2000). No information exists concerning the ionization levels of surface zinc vacancies. Using embedded cluster calculations Fink ascertained (+2/+1) and (+1/0) ionization levels for VO on (0001)-Zn of Ev + 0.05 eV and Ev + 1.8 eV, respectively (2006). Defects can be artificially created on UO2 by ion sputtering; the removal of surface oxygen atoms creates charged oxygen vacancies on the surface (Stanek et al. 2004). In STM images taken by Castell et al. (Fig. 6.18), bright lattice positions correspond to uranium sites; point defects in the image simply appear as vacancies, without causing a detectable brightening or darkening of the uranium sites (1998). Although Muggelberg et al. observed the stabilization of the UO2(110) surface through the creation of vacancy defects, they make no mention of their charge state (1998). Oxygen vacancy defects are also the predominant surface defects on ionic cobaltous oxide (McKay et al. 1987). When a surface O2– ion is removed from the lattice, two electrons must be trapped at the defect site in order to maintain local charge neutrality. Investigating the CoO(100) surface with photoemission spectroscopy reveals that the oxygen vacancy is associated with an emission in the bulk band gap just above the valence band maximum (Jeng et al. 1991). Unusual conductivity behavior arises from the creation of isolated oxygen vacancies on CoO(111), in comparison to other transition-metal oxides. Mackay and Henrich found that the defective surface was less conducting than the ascleaved surface, despite the fact that electrons must be trapped at the oxygen vacancies (1989; 1985). Fig. 6.18 Empty state STM image of the UO2(111) surface with a sample bias of 1.9 V from (Castell et al. 1998). Missing uranium ions do not create an observable perturbation on their neighbors. Reused with permission from M. R. Castell, S. L. Dudarev, C. Muggelberg, A. P. Sutton, G. A. D. Briggs, and D. T. Goddard, Journal of Vacuum Science & Technology A, 16, 1055 (1998). Copyright 1998, AVS The Science & Technology Society.

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Watkins GD, Troxell JR (1980) Phys Rev Lett 44: 593–6 Weber ER (1983) Physica 116B: 398–403 Weibel A, Bouchet R, Knauth P (2006) Solid State Ionics 177: 229–236 Willis BTM (1964) J Phys Radiat 25: 431–9 Wixom RR, Wright AF (2006) Phys Rev B: Condens Matter 74: 205208 Woning J, Van Santen RA (1983) Chem Phys Lett 101: 541–7 Wright AF (2006) Phys Rev B: Condens Matter 74: 165116–1 Wright AF, Mattsson TR (2004) J Appl Phys 96: 2015–22 Xu H (1990) Phys Rev B: Condens Matter 42: 11295–11302 Xu H (1992) Phys Rev B: Condens Matter 46: 12251–12260 Xu H, Lindefelt U (1990) Phys Rev B: Condens Matter 41: 5979–90 Yahia J (1963) Phys Rev 130: 1711–1719 Yang WS, Wang XD, Cho K et al. (1994) Phys Rev B: Condens Matter 50: 2406–8 Yi JY, Ha JS, Park SJ et al. (1995a) Phys Rev B: Condens Matter 51: 11198–11200 Yi JY, Koo JY, Lee S et al. (1995b) Phys Rev B: Condens Matter 52: 10733–10736 Young CG, Shuskus AJ, Gilliam OR (1961) Bull Amer Phys Soc 6: 248 Yu PW, Reynolds DC (1982) J Appl Phys 53: 1263–1265 Yu PW, Mitchel WC, Mier MG et al. (1982) Appl Phys Lett 41: 532–534 Yu ZG, Wu P, Gong H (2007) Physica B 401–402: 417–420 Zangenberg N, Goubet JJ, Nylandsted Larsen A (2002) Nucl Instrum Methods Phys Res, Sect B 186: 71–7 Zaoui A, El Haj Hassan F (2001) J Phys:Condens Matter 13: 253–262 Zhang SB, Chadi DJ (1990) Phys Rev Lett 64: 1789–92 Zhang SB, Northrup JE (1991) Phys Rev Lett 67: 2339–42 Zhang SB, Zunger A (1996) Phys Rev Lett 77: 119–122 Zhang SB, Wei SH, Zunger A (2001) Phys Rev B: Condens Matter 63: 075205 Zhao J-L, Zhang W, Li X-M et al. (2006a) J Phys: Condens Matter 18: 1495–508 Zhao Y, Kita K, Kyuno K et al. (2006b) Appl Phys Lett 89: 252905 Zhao Y, Dong Z, Miao S et al. (2006c) J Appl Phys 100: 123519 Zhu ZY, Hou MD, Jin YF et al. (1998) Nucl Instrum Methods Phys Res, Sect B 135: 260–264 Zistl C, Sielemann R, Haesslein H et al. (1997) Mater Sci Forum 258–263: 53–58 Zollo G, Lee YJ, Nieminen RM (2004) J Phys: Condens Matter 16: 8991–9000

Chapter 7

Intrinsic Defects: Diffusion

7.1 Bulk Defects Bulk self-diffusion in semiconductors is mediated by point defects such as vacancies and interstitials, as well as defect associates and clusters. Multiple defects can contribute to overall motion in the bulk; the relative contribution of each defect (or defect charge state) depends upon crystal structure, stoichiometry, Fermi level, and temperature. The diffusion of native defect associates and clusters has been studied to a far lesser extent than that of intrinsic point defects. Quite a few of the results presented here originate from computations, which currently still manifest significant weaknesses. For example, for semiconductor surface diffusion both ab initio (DFT) and classical approaches generally predict higher activation energies for diffusion than seen experimentally; ab initio approaches, however, generally get closer. Semi-empirical and classical methods often wrongly predict the positions of potential minima and saddle points of the potential energy surface, leading to incorrect ground and transitional states for adsorbates (1996). Classical potentials such as the Stillinger–Weber and Tersoff cannot describe the vacancy ionization phenomena (Ditchfield et al. 1998) or certain surface reconstruction effects. As an example of the latter problem, surfaces like the dimerized Si(100)-(2×1) and GaAs(100)-(4×2) contain highly-distorted, non-tetrahedral covalent bonds. Diffusion on such surfaces sometimes involves a concerted exchange mechanism with a complex bonding scheme that requires quantum methods to model accurately. Thus, methods like MD that employ classical potentials can offer quantitatively reliable results only in special cases where these effects play no important role (Allen et al. 1997). Otherwise, one should probably expect to reproduce experiment with only qualitative accuracy. For this purpose no classical potential appears to be superior to the others under all circumstances; each has its peculiar balance of strengths and limitations (Garrison and Srivastava 1995).

E.G. Seebauer, M.C. Kratzer, Charged Semiconductor Defects, © Springer 2009

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7.1.1 Point Defects The energetics of point defect migration have been examined experimentally and computationally. Although activation energies and pre-exponential factors for diffusion are frequently reported in the literature, they typically display a large variance. It has recently come to light that some experimental diffusion measurements in the bulk may have been affected by the chemical state of nearby surfaces in ways that are often uncontrolled. 7.1.1.1 Silicon Remarkably, considerable debate still surrounds the mechanism of native defect diffusion in silicon (Bracht et al. 1998; Bracht et al. 1995; Ural et al. 1999; Masters and Fairfield 1966; Frank et al. 1984; Fair 1981; Sharma 1990; Pandey 1986; Blochl et al. 1993), the question of whether interstitial atoms are the prime mediators (especially at temperatures below about 900°C) (Ural et al. 1999), and the value of the interstitial formation energy. Computational approaches have not proven to be sufficiently reliable to resolve these questions, with calculated formation energies ranging from 2.2 eV to 4.5 eV (Blochl et al. 1993; Clark and Ackland 1997; Van Vechten 1986; Sinno et al. 1996; Needs 1999; Leung et al. 1999; Goedecker et al. 2002; Marques et al. 2005). Recent experiments by Seebauer et al. have shown that surface chemical bonding state affects the self-diffusion rate in silicon by influencing the concentration of point defects within the solid (2006). Diffusion measurements have typically been made in the presence of surfaces whose dangling bonds are largely saturated with adsorbates of various kinds. However, maintaining an atomically clean surface opens a pathway for native point defect formation at the surface that is much more facile than corresponding pathways within the solid. The surface pathway fosters a much larger solid defect concentration on a several-hour laboratory time scale than in conventional approaches, with correspondingly larger self-diffusion rates, as shown in Fig. 7.1. Data taken under clean conditions therefore yields substantially lower values for the defect formation energy. Also, these experiments employed a kinetic short-time limit (Vaidyanathan et al. 2006). This short-time limit circumvents a problem of data interpretation that has plagued most experimental work on Si self-diffusion. The energetics and mechanisms of vacancy and self-interstitial diffusion in silicon have been explored extensively (Tang et al. 1997; Munro and Wales 1999; ElMellouhi et al. 2004). Many reports have suggested that self-interstitials mediate diffusion at high temperatures, while vacancies dominate at lower ones (Shimizu et al. 2007). More recent results, including those obtained in this laboratory, argue for interstitial-dominated diffusion at all temperatures (Bracht et al. 1998; Vaidyanathan et al. 2006). Several mechanisms of self-interstitial migration in silicon have been posited: direct-interstitial, interstitial-interchange or “kick-out” (where interstitial atoms exchange with the lattice), and pair-diffusion (where Si interstitials

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migrate along with Si vacancies) (Kato 1993; Jung et al. 2004). Vaidyanathan et al. have recently shown conclusively that interstitials are the primary mediators of self-diffusion for temperatures from 650–1,000ºC (2007). The work also shows that interstitials mediate self-diffusion through a kick-out mechanism of exchange with the crystalline lattice, rather than a Frank–Turnbull mechanism of interstitialvacancy creation. Experimental measurements of defect diffusivities, which often yield both a pre-exponential factor and activation energy for site-to-site hopping, have likely been significantly affected by chemical adsorbate on the sample surface. Even computational investigations of silicon self-interstitial diffusion by DFT, tight-binding, and quantum-based molecular dynamics methods all exhibit an enormous variance (Jung et al. 2004). The theoretical investigation of selfinterstitial diffusion is complicated by the assortment of configurations available to the defect, the stability of which is known to depend upon charge state. For example, some authors believe that the saddle point for the migration of Sii0, Sii−1, and Sii−2 (all of which prefer the <110>-split configuration) is the tetrahedral site, while others believe that it is the hexagonal site (Lee et al. 1998). To further complicate matters, the fraction of time spent in different self-interstitial configurations during migration is a strong function of temperature (Sahli and Fichtner 2005). In addition, studies differ in the details of the computational method, a result of the debate over which method is most applicable to the combination of strong and weak bonds in many interstitial configurations (Leung et al. 1999; Marques et al. 2005; Schultz 2006).

Fig. 7.1 Measured self-diffusion coefficients in n-doped Si for the atomically clean (100) surface compared with other literature reports with various methods and doping levels. “HJ” refers to “heterojunction method.” The numbers for the clean surface lie much higher than the others, and imply a correspondingly larger defect concentration that must be caused by the surface. Reprinted figure with permission from Seebauer EG, Dev K, Jung MYL et al. (2006) Phys Rev Lett 97: 055503-2. Copyright (2006) by the American Physical Society.

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As the energetics of self-interstitial diffusion can be quantified only indirectly, the experimental results reveal a wide variance. The effect of doping on the diffusion coefficients of charged self-interstitials in Si is only just starting to be understood (Bracht 2006). Also, the distinction between overall diffusion activation energies, defect formation energies, and migration barriers for site-to-site hopping is also often unclear (Kato 1993). In early quenching experiments, Seeger and Frank showed that self-interstitial defects had activation energies of migration of 1.5 eV or more (1973). Taniguchi and Antoniadis inferred an activation energy of 4 eV for the effective mesoscale diffusion of self-interstitials in silicon from phosphorus-enhanced diffusion (1985). From the in-diffusion of Pt in silicon, Mantovani et al. estimated an activation energy of the self-interstitial diffusion coefficient of 5.1 eV (1986). Stolwijk et al. studied the diffusion of gold in silicon with the aid of a neutron activation analysis to obtain a comparable activation energy of 4.8 eV (1984). Bronner and Plummer, however, obtained a smaller overall activation energy of 2.4 eV from the gettering of gold in silicon (1987). This is closer to the recent 1.86 eV value of Wijaranakula based on experiments where oxygen donors were used to trace Si interstitial motion (1990). Tan and Gosele determined a 0.4 eV activation energy for the diffusion of self-interstitials based on the assumption that interstitial-type dislocation loops are formed in the bulk, which they acknowledge as highly controversial (1985). The energy barrier for the site-to-site hopping of the silicon self-interstitial is always considerably smaller. Panteleev et al. characterized the diffusion of self-interstitials with photostimulated electron emission, obtaining a migration energy barrier of 0.12 ± 0.04 eV (1976). By monitoring the disappearance of proton-beam-generated point defects below room temperature, Hallen et al. assigned an migration energy barrier of 0.065 ± 0.015 eV to Sii (1999). In the early 1980s, Baraff and Schluter (1984), Car et al. (1984), and Bar-Yam and Joannopoulos (1984) used Green’s function techniques to consider the isolated diffusion of both Sii+2 and Sii0. For example, the latter authors found a low energy barrier, 1.1 ± 0.3 eV, for the exchange of Sii+2 with a lattice atom; the migration of the tetrahedral self-interstitial to the hexagonal site has an identical energy barrier. The hopping mechanism of the neutral defect is less obvious due to the near degeneracy of the hexagonal, bond-centered, and lattice sites. Exchange of Sii0 with a lattice atom is associated with an energy barrier of 1.7 ± 0.4 eV. In early DFT-LDA work, Nichols et al. obtained a barrier of 0.4 eV for hopping of Sii0 (1989). Kato et al. proposed a combined mechanism of direct-interstitial and interstitial-interchange for the diffusion of Sii0, with an associated energy barrier for migration of 1.2–1.7 eV (1993). Their corresponding value for Sii+2, which is thought to diffuse solely via site-to-site hopping, is 1.3–2.3 eV. Gilmer et al. employed a classical MD simulation to obtain a result of 0.9 eV (1995). Zhu et al. reported a barrier of 1.4 eV for Sii0, which decreases to 0.9 and 0.7 eV for Sii+1 and Sii+2, respectively (1996). A high barrier of 1.37 eV for Sii0 has been computed by Tang et al. from tight-binding molecular dynamics simulations (1997). Lee et al. reported a range of 0.15 to 0.18 eV in LDA for Sii0 depending on path, with corresponding values of 0.47 to 0.59 eV for Sii+1 and a lower bound of

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1.0 eV for Sii+2 (1998). These authors also postulated a charge-assisted hopping mechanism in which the interstitial is neutral in the initial and final states, but converts to Sii−2 in the transition state. The barrier for such motion was calculated to be less than 0.05 eV. Leung et al. used DFT to obtain a range of barriers of 0.03 to 0.15 eV in LDA and 0.18 to 0.20 eV in the generalized gradient approximation (GGA) for Sii0, with the results depending on diffusion path (1999; 2001). Needs et al. determined energy barriers for the diffusive jump of the interstitial defect between the hexagonal and split-<110> sites of 0.15 eV (LDA) and 0.20 eV (GGA) (1999). By studying the diffusion of the self-interstitial in Si using an ab initio molecular dynamics package, Sahli and Fichtner obtained a migration energy of 0.45 eV (2005). Using maximum likelihood methods and a wide variety of literature reports, this group has estimated a migration energy barrier of 0.72 ± 0.03 eV for Sii. Experimentally and computationally determined vacancy diffusion parameters suffer from the same uncertainties as their interstitial counterparts. Watkins was the first to use EPR and DLTS to report diffusion coefficients and migration energies for VSi+2, VSi0, and VSi−2; these charge states were associated with barriers of 0.32 eV (1963; 1979), 0.33 eV (1979), and 0.18 ± 0.02 eV (1975), respectively. Neutron irradiation studies in p-type and n-type silicon yield similar values of 0.3–0.4 eV and 0.17, in that order (Gregory and Barnes 1968). Ershov et al. investigated the Fermi-level dependence of defect annealing kinetics after photo-stimulated electron emission (1977). These authors associated migration energies of 0.48 ± 0.05 eV, 0.33 ± 0.03 eV, and 0.18 ± 0.02 eV with VSi+1, VSi0, and VSi−1. It can be inferred that they are, in fact, referring to the doubly charged defect species in comparing their values to those from other studies. For example, Panteleev et al. measured a value of 0.18 eV for the migration energy of VSi−2 and one of 0.25 eV for the single negatively charged defect, VSi−1 (1976). Early simulations based on the supercell approximation, Car–Parinello approach, Stillinger–Weber interatomic potential, and empirical tight-binding method all determined migration energies of between 0.3 and 0.43 eV for VSi0 (Nichols et al. 1989; Maroudas and Brown 1993; Gilmer et al. 1995; Rasband et al. 1996; Shao et al. 2003; Car et al. 1984). Kelly et al. (1986) and Sugino and Oshiyama (1992) calculated energy barriers for the (+2) vacancy of 0.42 and 0.40 eV, respectively. The value that Sugino and Oshiyama proposed for the migration energy of VSi−1, 0.1 eV, differs considerably from the experimental value of Panteleev et al. (1976). Using molecular dynamics simulations, Tang et al. obtained a migration energy barrier of only 0.1 eV for VSi0, which they attribute to an error in the calculation method, as the overall mesoscale activation energy of 4.07 eV compares well with that from experiments (1997). More recent theoretical predictions are in good agreement with almost all of the experimental values, although few authors have considered the diffusion of VSi−2 and VSi+2. Bernstein et al. calculated a migration energy barrier of 0.3 eV for the neutral vacancy with a nonorthogonal tight-binding model Hamiltonian based on the extended Huckel approach (2000). Using the Car–Parinello molecular dynamics code with GGA functional, Kumeda et al. found a migration barrier for VSi0 of 0.58 eV (2001). El-Mellouhi et al.

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associated a migration energy barrier of 0.40 ± 0.02 eV with the movement of VSi0 along the <111> direction; the defect passes through the metastable split-vacancy site before reassuming the more stable tetrahedral configuration (2004). Voronkov and Falster inferred an activation energy for vacancy diffusion (with no charge state specified) of 0.38 eV from simulated RTA vacancy profiles and the lifetime of radiation-induced vacancies (2006). Using maximum likelihood estimation, the energy barrier for site-to-site hopping of VSi is 0.30 ± 0.01 eV. While numerous reports concentrate on activation energies for the diffusion of charged defects, it is also important to consider the variation in pre-exponential factors as a function of charge state for site-to-site and mesoscale diffusion. Theoretical treatments of diffusion often pay scant attention to prefactors. A few reports containing prefactors for Sii and VSi diffusion have been found; no clear trends emerge. A review of numerous experimental studies summarized the prefactors for interstitial and vacancy diffusion as 914 cm2/s and 0.6 cm2/s, respectively (Gosele et al. 1996). Considerably different values were obtained by Tang et al., who used tight-binding molecular dynamics simulations to study diffusivity data (1997). They obtained prefactors of 1.18 × 10−4 cm2/s and 1.58 × 10−1 cm2/s for vacancies and self-interstitials in silicon. Bracht examined the contributions to the self-diffusivity from self-interstitials and vacancies and associated prefactors of 2,980 cm2/s and 0.92 cm2/s with the mesoscale diffusion of the two, in that order. Silvestri and co-workers presented experimental results of dopant- and self-diffusion in extrinsic silicon doped with As (2002). They also reexamined the data of Bracht (1998) to obtain prefactors specifically for the mesoscale diffusion of Sii+1 and Sii0 of 3.33 ± 2.81 × 102 cm2/s and 9.29 ± 8.17 cm2/s. Sahli and Fichtner obtained a value of 5.18 × 10−3 cm2/s for neutral Sii using molecular dynamics simulations (2005). 7.1.1.2 Germanium The mechanisms and energetics of self-diffusion in germanium, particularly as they relate to charge state effects, have received less attention than those in silicon. In contrast to silicon, however, it is commonly believed that self-diffusion in germanium occurs via a vacancy mechanism (Valenta and Ramasastry 1957; Campbell 1975; Janotti et al. 1999; Fuchs et al. 1995; Letaw et al. 1956). Seeger and Chick (1968)and Van Vechten (1974) both found the formation energy of the germanium interstitial to be prohibitively high; the interstitial is thus unable to influence diffusion at equilibrium. At least one recent study indicates that the metastable Gei+1 may contribute to diffusion at temperatures around 220 K (Carvalho et al. 2007). Based on the known ionization levels of the defect, it can be inferred that VGe0 contributes greatly to self-diffusion in p-type material whereas VGe−2 plays a greater role in n-type material. Logically, for intermediate Fermi levels within the band gap, VGe−1 should also contribute to self-diffusion unless its migration rate is sufficiently low so that most mass flux is carried by a minority defect.

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The vacancy diffusion process has been studied and analyzed by an assortment of techniques including radioactive tracers, sectioning by grinding or sputtering, and Steigmann’s or Gruzin’s absorption method; many of these methods yield similar values for the activation energy of mesoscale diffusion (about 3 eV) but do not discuss the effect of defect charging (Sharma 1990). Mayburg (1954) and Vanhellemont et al. (2005), for example, obtained vacancy formation energies on the order of 2.0 eV and migration energies of approximately 1.1 eV, which sum to approximately 3 eV. Ershov et al. found migration energies for the neutral and singly charged vacancy in Ge of 0.52 ± 0.05 eV and 0.42 ± 0.04 eV, respectively (1977). By measuring the effect of dopants on self-diffusion and modeling the dependence of the negatively charged vacancy concentration on the Fermi energy, Werner et al. determined that VGe−1 and VGe0 are responsible for 77% and 23% of the transport for self-diffusion in intrinsic material at 700ºC, respectively (1985). No experimental data exist for the diffusion of self-interstitials in germanium. As with silicon, the variation in calculated germanium defect formation and migration energies is quite large. Some of the computational values do, however, closely match those obtained experimentally (Bailly 1968; Bennemann 1965; Scholz and Seeger 1963; Phillips and Van Vechten 1973; Swalin 1961). Tightbinding calculations yield a formation energy of 3.6 eV for the neutral gallium vacancy (Bernstein et al. 2002); to compare, a LDA/DFT study produces a value of 1.9 eV (Fazzio et al. 2000). Lauwaert et al., using a molecular dynamics code with a Stillinger–Weber potential, obtained a neutral vacancy formation energy of 3.38 eV and migration energy of 0.17 eV (2006). These authors also calculated considerably higher values for self-interstitials, 5.81 eV and 0.42 eV, respectively, confirming that such defects do not contribute to diffusion in germanium. The local density functional study of Pinto et al. yielded a neutral vacancy formation energy of 2.6 eV with migration barriers of 0.4, 0.1, and 0.04 eV for the (0), (−1), and (−2) charge states of the defect (2006). Figure 7.2 illustrates the potential energy along the migration path of the Ge vacancy described by their calculations, including its dependence on charge state. Lastly, Vanhellemont et al. have performed ab initio calculations to estimate the formation energies of VGe0, VGe−1, and VGe−2 as 2.35 ± 0.11 eV, 1.98 ± 0.11 eV, and 2.19 ± 0.11 eV, in that order (Vanhellemont et al. 2007). The migration energy of the uncharged vacancy in Ge is between 0.4 and 0.7 eV according to these authors. 7.1.1.3 Gallium Arsenide For gallium arsenide and the other III–V semiconductors whose crystal structure consists of two elemental sublattices, the basic mechanism of self-diffusion is one of migration within a specific sublattice (Goldstein 1960). As illustrated in Fig. 7.3, an assortment of migration mechanisms is thought to occur within the bulk of the semiconductor. These include (a) interstitials and vacancies diffusing through the lattice (b) direct exchange between nearest neighbors on opposite sublattices leading to the generation of two antisite defects (c) indirect exchange

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Fig. 7.2 Potential energy along the migration path of a vacancy in Ge obtained with a cluster NEB calculation. Neutral (□), negatively charged (Δ), and double negatively charged (○) vacancies were considered. Reprinted from Pinto HM, Coutinho J, Torres VJB et al., Formation energy and migration barrier of a Ge vacancy from ab initio studies,” (2006) Mater Sci Semicond Process 9: 501. Copyright (2006), with permission from Elsevier.

through a ring on the same sublattice (d) exchange with the nearest neighbor vacancy on the opposite sublattice and (e) exchange with the next-nearest neighbor vacancy (Cohen 1997). In undoped, n-type, and p-type GaAs, gallium self-diffusion is mediated by gallium vacancies, though various charge states have been suggested to dominate (Tan and Gosele 1988; Zhang and Northrup 1991; Morrow 1990; Bracht 1999). This assignment arose from studying micrographs of quantum well interdiffusion versus arsenic partial pressure (Kaliski et al. 1987). By analyzing doping enhanced AlAs/GaAs superlattice disordering data, Tan proposed a mesoscale activation energy of 6.0 eV for the diffusion of VGa−3 in n-type GaAs under an As-rich ambient (Tan 1995). Wang et al. used a Ga tracer isotope technique to explore this defect, which they associated with a mesoscale activation energy of 4.24 eV (1996), in good agreement with the values calculated by Zhang and Northrup (1991) and Walukiewicz (1990) of 4.0 ± 0.5 and 4.6 ± 0.3 eV, respectively. The pre-exponential factors for mesoscale diffusion of gallium vacancies in GaAs are also very large compared to the typical values of 10−3 cm2/s. For instance, Tan obtained a value of 7 cm2/s along with his activation energy of 4 eV. Distinct theories have been invoked to explain simultaneous high activation energies and prefactors. For example, Van Vechten and coworkers (1975; 1973) proposed a ballistic model for diffusion in which the movements of atoms near a diffusing defect occasionally conspire to produce an open pathway for migration

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Fig. 7.3 Schematic illustration of several diffusion mechanisms in a III–V semiconductor. Impurity atoms are shown as filled circles. a) Interstitial and vacancy generation-recombination via a Frenkel reaction b) direct exchange between nearest neighbors on opposite sublattice generating two antisite defects c) indirect, concerted exchange through a simple ring on the same sublattice d) exchange with nearest-neighbor vacancy on opposite sublattice e) exchange with next-nearest-neighbor vacancy (or nearest-neighbor on same sublattice). The shaded diamond shows one face of the unit cell. Possible jumps to the vacancy from three (of the nearest 12) group III sites are also shown. Reprinted from Cohen RM, Point defects and diffusion in thin films of GaAs,” (1997) Materials Science & Engineering, R 20: 193. Copyright (1997), with permission from Elsevier.

having essentially no activation barrier. An atom with sufficiently high translational energy can then simply squirt through the opening, with diffusion parameters characteristic of a freely translating particle. For charged defects, however, one can also rationalize the high pre-exponential factor in terms of a high formation entropy. As mentioned in Sect. 2.1.2, the ionization entropy can be sizeable for the formation of certain bulk defects. VGa in the (−1) and (−2) charge states has also been discussed in the context of Ga self-diffusion (Mei et al. 1988; Li et al. 1997; Muraki and Horikoshi 1997). A clearer picture of the conditions under which these different charge states mediate gallium self-diffusion in GaAs now exists. For instance, both Bracht (1999) and El-Mellouhi and Mousseau (2006b) have suggested that neutral and

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Table 7.1. Calculated diffusion barriers (in eV) for VGa in GaAs for all possible migration paths identified by (El-Mellouhi and Mousseau 2007) using SIEST-A-RT Migration Path

VGa0

VGa−1

VGa−2

VGa−3

First neighbor Plane-passing Cluster-assisted Fourth neighbor

0.84 1.7 – –

0.90 1.7 2.44 4.24

1.86 1.85 2.89 4.24

– 2.0 3.24 4.3

negatively charged Ga vacancies mediate self-diffusion in undoped, p-type, and n-type material, with relative contributions that depend on temperature and doping; this concept is depicted in Fig. 7.4. Under p-type conditions, negatively charged vacancies contribute far less to diffusion than neutral vacancies; the concentrations of VGa−1 and VGa−2 decrease even more with decreasing temperature. Under intrinsic conditions, Bracht claims that either VGa−1 or VGa−2, depending on the temperature, mediate diffusion, with a mesoscale activation enthalpy of 3.71 eV. El-Mellouhi and Mousseau agree that VGa−3 does not contribute significantly to diffusion under intrinsic conditions; VGa−3 affects diffusion in n-type material at temperatures less than 1,150 K. Using SIEST-A-RT, a local-basis density functional package with activation relaxation technique, El-Mellouhi and Mousseau also calculated the diffusion barriers of VGa0, VGa−1, VGa−2, and VGa−3 for four different migration paths – first neighbor, plane-passing, cluster-assisted, and fourth neighbor (2007b). The reader is referred to (El-Mellouhi and Mousseau 2006a) for a thorough discussion of these paths (including figures). Their results, which illustrate a clear correlation between defect charge state and migration path, are detailed in Table 7.1. By examining the relationship between VGa charge state and site-tosite diffusion barrier, it is clear that the “first-neighbor” path is favorable for VGa0, VGa−1, and VGa−2, while VGa−3 may also diffuse according to a “plane-passing” path. All but the “fourth neighbor” migration barriers are compatible with the overall mesoscale activation enthalpy of 3.71 eV obtained by Bracht. Arsenic diffusion in GaAs is governed by a substitutional-interstitial diffusion mechanism, most likely that of kick-in and kick-out (Schultz et al. 1998; Noack et al. 1997). It has not been explored in as much detail as gallium diffusion, as the

Fig. 7.4 Temperature dependence of the thermal equilibrium concentrations of VGa0, VGa−1, and VGa−2 in a) intrinsic b) n-type and c) p-type GaAs. Reprinted with permission from El-Mellouhi F, Mousseau N (2006) J Appl Phys 100: 083521-6. Copyright (2006) American Institute of Physics.

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formation of arsenic interstitials in the bulk at equilibrium conditions is unfavorable in comparison to that of vacancies and antisites. The activation barrier for diffusion of VAs is high (~2.4 eV), however, and highly charge state-independent, in contrast to that of VGa (El-Mellouhi and Mousseau 2007b, a). Based on the DFT results of Schick et al., one can infer that Asi+1, Asi0, and Asi−1 are the primary contributors to As diffusion for Fermi energies from Ev to 0.3 eV, 0.3 eV to 0.5 eV, and 0.5 eV to Ec, all respectively (2002). The first estimates of self-diffusion of arsenic in GaAs were obtained by Palfrey et al. (1983) and Goldstein (1960) via the in-diffusion of radioactive arsenic isotopes. The instability of these isotopes led to widely differing activation energies and indicated that self-diffusion was governed by a vacancy mechanism (Schultz et al. 1998; Egger et al. 1997). More recently, strained GaAsP/GaAs and GaAsSb/GaAs superlattices have proven more useful in studying the self-diffusion coefficient of arsenic (Schultz et al. 1998; Egger et al. 1997). Interdiffusion coefficients determined by Egger et al. for these compounds are higher under arsenic-rich conditions than under gallium-rich conditions, pointing to an interstitial-substitutional type of diffusion mechanism (1997). Schultz et al. found an increase in the effective diffusion coefficient of an arsenic tracer isotope in GaAs for high arsenic vapor pressures, also an indication that interstitial-substitutional migration dominates over vacancy migration (1998). 7.1.1.4 Other III–V Semiconductors In Chap. 4, a distinction was made between the III–V semiconductors containing elements from the second row of the periodic table, such as GaN and BN, and compounds such as GaAs and GaP. That delineation came into play when discussing the likelihood of interstitial and antisite formation. It also helps to shed light upon the different mechanisms of point defect diffusion that have been observed in several III–V semiconductors. For example, Group III vacancy-type defects do not dominate diffusion in GaN for all Fermi energies, as they do in GaAs. Also, as excess nitrogen atoms in the crystal lattice induce significant strains to form the low-symmetry bonding configuration, it is not surprising that nitrogen vacancies, rather than interstitials, mediate nitrogen diffusion in GaN. For the III–V semiconductors such as GaSb and GaP in which defect diffusion has been explored, many similarities to GaAs exist. According to the model of diffusion in GaSb proposed by Bracht et al., Ga migrates via vacancies whereas Sb migrates via interstitials, in analogy to the diffusion of VGa and Asi in gallium arsenide (2000; 2001). Several mentions of charged gallium interstitials contributing to self-diffusion has been found in the literature. Hakala et al. suggested a diffusion model involving both VGa and Gai in order to account for diffusion measurements performed in Ga- and Sb-rich ambient conditions (2002). Sunder and co-workers identified Gai0 and Gai+1 as key mediators in the diffusion of Zn through GaSb; it is now known that charged interstitials contribute much less to the total Ga diffusion coefficient than do vacancies, even under Ga-rich conditions (Sunder et al. 2007; Sunder and Bracht 2007). Experiments have conclusively

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revealed that diffusion in GaSb is mediated by a next-nearest-neighbor diffusion mechanism where migration occurs in two distinct sublattices (Bracht et al. 2000; 2001). Migration via a nearest-neighbor diffusion mechanism is associated with strongly asymmetric reaction energies that render the mechanism ineffective in GaSb (Hakala et al. 2002). Only one direct experimental study of gallium diffusion in GaP exists (Wang et al. 1996). The authors felt it premature, without substantial evidence, to attribute Ga self-diffusion to the charged gallium vacancy, although they concluded that diffusion was primarily mediated through one type of simple native defect. They measured an activation energy for mesoscale diffusion of 4.5 eV in GaP, or about 0.25 eV greater than their value for GaAs, a difference attributed to the stronger Ga-P bond in comparison to the Ga-As bond. Defect diffusion in semiconductors containing elements from the second row of the periodic table, such as B and N, is quite different. Among these materials, gallium nitride has received the most consideration. In p-type GaN, gallium interstitials mediate self-diffusion, as the formation energy of gallium vacancies are prohibitively high (Limpijumnong and Van de Walle 2004). In this regime, the energy barrier for the migration of Gai+3 is at least 1.0 eV lower than that of VGa0, which is counterintuitive, as one might expect the large radius of the Ga atom to inhibit its movement in the small crystal lattice of GaN. It is likely that the interstitial defect migrates via a pair-diffusion mechanism. In n-type GaN, conversely, the migration of gallium interstitials is energetically unfavorable and diffusion is mediated by gallium vacancies. According to Limpijumnong and Van de Walle, the activation energy for self-diffusion and formation energy of VGa−3 are 3–4 eV and 1–2 eV, respectively. Tuomisto et al. obtained a similar migration barrier of 1.8 ± 0.1 eV for the gallium vacancy from positron annihilation measurements (2007). Both vacancies and interstitials have been studied in the context of nitrogen diffusion in GaN; as with GaAs, temperature may greatly affect which defect mediates N self-diffusion. For instance, nitrogen vacancies in the (+3) charge state may mediate self-diffusion in p-type material at temperatures greater than 500ºC (Myers et al. 2006). Limijumnong and Van de Walle calculated migration barriers of 2.6 eV for VN+3 and 4.3 eV for VN+1 (2004). They attributed the strongly charge state-dependent energies to the saddle-point associates used to determine the migration barriers. Wright and Mattsson identified two potential diffusion paths, a perpendicular one producing movement perpendicular to the c axis, and a diagonal one producing movement both perpendicular and parallel to the c axis (2004). Lower activation energies are associated with the perpendicular path; values of 2.49, 2.80, and 3.55 eV correspond to the migration of VN+3, VN+2, and VN+1, respectively. Ganchenkova and Nieminen calculated site-to-site hopping migration energies for charged nitrogen vacancies (2006). Figure 7.5 illustrates how the energy barrier decreases from about 4.0 eV for the (+1) charge state of the defect to ~2.0 eV for the (−3) charge state, as well as the effect of the migration barrier on the overall activation energy of VN. When it comes to nitrogen self-interstitials, Limpijumnong and Van de Walle calculated the Ni diffusion barrier as 2.4 eV for the (0) charge state and 1.6 for the (−1) charge state (2004). Wixom and Wright

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Fig. 7.5 Activation energies for self-diffusion of gallium and nitrogen vacancies in the GaN sublattice. Solid lines represent the activation energies for the lowest energy charge state of a defect; the dashed line is the effective activation energy for vacancy-mediated self-diffusion. The inset demonstrates the charge dependence of vacancy migration energies. Reprinted figure with permission from Ganchenkova MG, Nieminen RM (2006) Phys Rev Lett 96: 196402-2. Copyright (2006) by the American Physical Society.

used DFT to obtain migration barriers for Ni+3, Ni+2, and Ni+1 of 1.79, 2.12, and 1.98 eV, in that order (2006). These correspond well to the experimental value of 1.99 eV obtained by Fleming and Myers by monitoring infrared absorption following room temperature irradiation (2006). 7.1.1.5 Titanium Dioxide The self-diffusion of oxygen and titanium in rutile TiO2 has been examined extensively; oxygen migrates via a vacancy diffusion mechanism, whereas excess Ti atoms diffuse through the crystal as interstitials (Diebold 2003). The migration barrier for diffusion of VO+2 is significantly higher than that of Tii+2, so Ti cations are the major diffusive species in the bulk. This fact was less apparent in early studies, which yielded similar overall activation energies for the two defects; values ranged from 2.0–2.9 for titanium (Hoshino et al. 1985; Kitazawa et al. 1977; Akse and Whitehurst 1978; Venkatu and Poteat 1970; Lundy et al. 1973; Sawatari et al. 1982) and 2.4–2.8 eV for oxygen (Arita et al. 1979; Haul and Dumbgen 1965; Iguchi and Yajima 1972; Derry et al. 1981). In the semiconductors discussed up until this point, especially silicon, the effect of defect charge-state upon the migration energy for site-to-site hopping has been discussed. TiO2 is different for two main reasons: the energy barriers for the site-to-site migration of Tii and VO, as well as the effect of charge state on the energetics of diffusion, have only recently begun to be discussed in the literature.

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The self-diffusion of oxygen in non-stoichiometric rutile titanium dioxide has been studied experimentally by radioactive tracer and gaseous exchange methods (Barbanel et al. 1971; Millot and Picard 1988; Derry et al. 1981; Bagshaw and Hyde 1976; Arita et al. 1979). Haul and Dumbgen were the first to propose that oxygen diffusion occurs via (+2) vacancies, yet the possibility of cooperative motion of both anion and cation defects has also been considered (1965). The diffusion coefficients of oxygen vacancies in TiO2 grow with increasing nonstoichiometry; oxygen self-diffusion is faster in highly reduced titania (Millot and Picard 1988; Gruenwald and Gordon 1971). Ab initio pseudopotential total-energy calculations have been utilized to examine the diffusion of VO+2 through the crystal lattice (Iddir et al. 2007). The direct diffusion of the charged oxygen vacancy along the [1–10] direction is associated with a migration barrier of 0.69 eV, in contrast to that of 1.77 eV associated with diffusion along the [001] channel. Titanium diffuses through titanium dioxide by an interstitial-type mechanism; Tii+3 and Tii+4 are likely to contribute more to diffusion at reduced and stoichiometric conditions, respectively (Hoshino et al. 1985; Bak et al. 2003a, b). This mechanism of self-diffusion was suggested by the tracer diffusion experiments of Sasaki et al. (1985). He and Sinnott considered an assortment of structural Frenkel models to shed light upon the barrier to neutral Ti interstitial diffusion in rutile TiO2 (2005). They found defect motion through the open channel in the [001] direction to be more favorable than that involving movement from a lattice site in the [100] or [010] direction. In a recent publication, Iddir et al.

Fig. 7.6 (a) Diffusion profile of Tii+4 along [001]. (b–d) Snapshots of the diffusing Tii+4 and a portion of the surrounding bulk at the positions 1, 2, and 3 as given in (a). Reprinted figure with permission from Iddir H, Ogut S, Zapol P et al. (2007) Phys Rev B: Condens Matter 75: 073203-2. Copyright (2007) by the American Physical Society.

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examined Tii+4 diffusion along both the [001] and [110] directions; they also found that interstitials diffuse along open [001] channels and obtained migration barriers of 0.37 eV and 0.7 eV for Tii+4 and Tii0, respectively (Iddir et al. 2007). The diffusion profile of Tii+4 is depicted in Fig. 7.6. 7.1.1.6 Other Oxide Semiconductors The stoichiometry of a metal oxide semiconductor can, in part, provide insight into the mechanisms of defect diffusion at play within the bulk of the crystal. A brief review of terminology is useful, as the terms utilized in the literature tend to vary from report to report. While TiO2–x is typically referred to as “reduced” titanium dioxide, ZnO1–x is described as “Zn-rich” zinc oxide and UO2–x as “hypostoichiometric” uranium dioxide. Similarly, the terms “oxidized,” “O-rich,” and “hyperstoichiometric” can be used interchangeably. It is valuable to become familiar with these conventions, as defect diffusion in the remaining metal oxide semiconductors is often discussed in conjunction with regimes of stoichiometry or chemical potential. Under extreme conditions, e.g., reducing and oxidizing environments, charged vacancies comprised of the deficient species and charged interstitials (and antisites) comprising the overabundant species tend to preferentially form. Consequently, it is not surprising that in stoichiometric and hypostoichiometric oxides, oxygen always diffuses via a vacancy mechanism. Only under oxygen-rich conditions, typically with hyperstoichiometric material, does an interstitial-mediated mechanism of anion diffusion come into effect. On the other hand, in the metal oxides that will be discussed shortly, metal cations diffuse as vacancies, rather than interstitials. This naturally prompts the question as to why the diffusion of titanium atoms in TiO2 has historically been suggested to occur via the migration of metal interstitials; this paradox will be readdressed at the end of this section. Oxygen self-diffusion occurs through a vacancy mechanism in Zn-rich ZnO, in which VO0 and VO+2 are the dominant point defects (Hoffman and Lauder 1970; Mackrodt et al. 1980; Binks 1994; Tomlins et al. 1998; Erhart and Albe 2006b). Under oxygen-rich conditions, oxygen diffusion is mediated by interstitials in the (0) or (−2) charge state (Sabioni 2004; Haneda et al. 1999; Erhart and Albe 2006b; Erhart et al. 2005). The latter mechanism has been observed experimentally, as undoped zinc oxide normally exhibits n-type behavior, and all experiments have been performed under oxygen-rich conditions. Early experimental studies were somewhat unreliable, as they relied on gaseous-exchange techniques (Hoffman and Lauder 1970; Moore and Williams 1959), whereas modern investigators have used secondary ion mass spectroscopy to obtain diffusivities from depth profiles in undoped and doped ZnO samples (Tomlins et al. 1998; Hallwig et al. 1976). For example, Haneda et al. (1999) and Sabioni (2004) saw that extrinsic doping, which leads to the formation of additional Oi−2, increased the oxygen diffusion coefficient. Additionally, the diffusion prefactors and activation energies from these studies vary widely. A recent first-principles investigation by Erhart and

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Fig. 7.7 Dependence of self-diffusion of oxygen in ZnO on chemical potential and Fermi level at 1,300 K, illustrating the competition between vacancy and interstitial mechanisms. The dark gray areas indicate the experimental data range around 1,300 K. Reprinted figure with permission from Erhart P, Albe K (2006) Phys Rev B: Condens Matter 73: 115206-8. Copyright (2006) by the American Physical Society.

Albe has helped to shed light upon the complex interplay between vacancy and interstitial-mediated diffusion (2006a); these authors also explored the effect of defect charge state on the preferred path of Oi through the crystal, as shown in Fig. 7.7. The lowest computed energy barrier for migration of Oi0 (0.81 eV) corresponds to the movement of one of the atoms in the dumbbell to one of the first nearest out-of-plane neighbor positions, and the formation of a new dumbbell interstitial. For intermediary chemical potential, a balance, dependent upon both stoichiometry and Fermi level, occurs between the two migration mechanisms. This occurs as oxygen interstitials have greater formation enthalpies, yet smaller site-to-site migration enthalpies, in comparison to oxygen vacancies. In contrast to TiO2, experiments and computations suggest that cation vacancies, e.g., VZn0, VZn−1, and VZn−2, are responsible for zinc diffusion in ZnO. This fact is counterintuitive, as ZnO falls into the category of a metal-excess, or oxygen-deficient, oxide semiconductor. Zinc interstitials do not mediate self-diffusion within the bulk, however, as their formation energy is prohibitively high under most conditions. According to experiments using the method of thin sections performed by Moore and Williams (1959) and Kim et al. (1971), however, zinc diffuses isotropically in ZnO; it is known from computation that zinc interstitials diffuse in an anisotropic manner through the crystal (Binks 1994). These results imply that zinc vacancies, rather than interstitials, mediate zinc diffusion in ZnO. Using 70Zn as a tracer isotope and SIMS for data collection, Tomlins et al. also recently confirmed that zinc self-diffusion is controlled by a vacancy mechanism (2000). They briefly discuss the energetics of defect migration; the lowest activation energy for the migration of zinc interstitials (the sum of the formation energy and migration energy of Zni−2) is still 0.8 eV higher than of VZn−2. This calculation uses the migration energy of the zinc vacancy obtained by Binks, 0.91 eV, who

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proposed a “double-jump” model, a combination of two c-axis jumps, for the diffusion of the defect (1994). Zinc interstitials likely diffuse via a kick-out mechanism, where the interstitial zinc atom moves in the direction of a substitutional zinc atom and replaces it (Janotti and Van De Walle 2006). The migration energy of the defect may be as low as 0.55–0.57 eV despite the fact that their formation energy is prohibitively high under most conditions. Oxygen self-diffusion in urania occurs in a similar manner to that in ZnO; in UO2+x, anion migration is mediated by interstitials (Matthews 1974), while in UO2 and UO2–x, a vacancy mechanism of anion diffusion dominates (Matthews 1974; Catlow et al. 1975; Catlow 1977; Murch and Catlow 1987; Senanayake et al. 2007). The hyperstoichiometric and stoichiometric regimes can be likened to O-rich and Zn-rich ZnO, where oxygen interstitials and oxygen vacancies mediate diffusion, respectively. Such behavior has also been found in other alkaline earth fluorites (Catlow 1977; Jackson et al. 1985). Matthews obtained an activation energy for the migration of Oi−2 in UO2+x of 1.3 eV, in comparison to the higher value of 2.8 eV obtained for the diffusion of the same defect in stoichiometric UO2 (1974). Catlow and coworkers directly compared the energetics of interstitial versus vacancy diffusion in UO2; they obtained a migration enthalpy of ~0.52 eV for VO+2, or about 0.5 eV lower than that of Oi−1 (Catlow et al. 1975; Murch and Catlow 1987). They also found that the “kick-out” or “interstitialcy” mechanism, where diffusing atoms interchange between substitutional and interstitial sites in the crystal lattice, of interstitial migration is more energetically favorable than the direct migration of anion interstitials (Catlow 1977). Similar to ZnO, the literature indicates that the cation vacancy in the (−4) charge state, VU−4, mediates the diffusion of metal atoms in UO2+x and UO2. Catlow (1977), Matthews (1974), and Matzke (1987) discovered that the rate of uranium vacancy diffusion increases along with the deviation from stoichiometry in UO2+x. In UO2+x the overall diffusion coefficient of uranium increases in approximate proportionality with x2 by roughly five orders of magnitude between UO2 and UO2.20 at 1,400–1,600ºC (Matzke 1987). This fast diffusion is attributed to increased VU–4 concentration, since U5+ions diffuse more slowly than U4+ ions. Jackson et al. proposed four different pathways for cation migration; for stoichiometric UO2, a mechanism whereby a cation moves into a cation vacancy in the presence of an adjacent oxygen vacancy dominates (1986). Kupryazhkin et al. obtained an effective activation energy of uranium ion diffusion of 5.0 ± 0.3 eV (Kupryazhkin et al. 2008), in good agreement with the experimental value of 5.6 eV (Matzke 1987). In cobaltous oxide, which tends towards Co1–xO, both cobalt and oxygen selfdiffuse via a vacancy mechanism (Carter and Richardson 1954; Koel and Gellings 1972; Chen and Jackson 1969). In 1969, Chen and Jackson calculated a favorable translational energy at the saddle point associated with the motion of cation vacancies (1969). Tracer diffusion studies place the activation energy for site-to-site hopping of the cation vacancy VCo−2 at approximately 1.6 eV (Chen and Jackson 1969; Carter and Richardson 1954; Fryt et al. 1973). The self-diffusion coefficient of cobalt has a strong dependence on oxygen partial pressure, particularly at low

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pressures, where the (−2) charge state of VCo dominates. Chen and Jackson also suggested a vacancy-mediated mechanism for oxygen diffusion. Assuming that oxygen diffuses via (+1) vacancies, they calculated an activation energy for migration of 4.12 eV at 1,429ºC and 0.21 atm. For purposes of comparison, in the temperature range 1,000–1,600ºC, the activation energy for cation vacancy self-diffusion in CoO is much smaller than that for anion diffusion (1.65 eV for cobalt and 4.12 eV for oxygen) (Chen and Jackson 1969; Carter and Richardson 1954; Fryt 1976; Fryt et al. 1973). It is now logical to return to the subject of the predominant cation diffusion mechanism in TiO2. In n-type ZnO, UO2, and CoO, vacancies in the (−2), (−4), and (−2) charge states mediate the migration of metal atoms within the bulk. Similar behavior has been observed and calculated in BaO, CaO, MgO, NiO, SrO (Peterson and Wiley 1985; Mackrodt and Stewart 1979). For example, in nickel oxide, the rapid cation self-diffusion relative to anion self-diffusion suggests that excess oxygen ions are accommodated by the formation of cation vacancies in the (−1) and (−2) charge state (Peterson and Wiley 1985). As a reminder, Na-Phattalung et al. identified VTi−4 as the defect present in highest concentrations in O-rich anatase TiO2 (2006). These authors were the first to explore the Fermi energy dependence of the formation energy of the metal vacancy, and not just that of the metal interstitial. Oxidation experiments indicate, however, that the diffusion rate of titanium vacancies in the bulk is very low and limited by their formation at the gas/solid interface (Nowotny et al. 2005). Thus, it remains likely that the migration of titanium in TiO2 proceeds via an interstitially-mediated mechanism.

7.1.2 Associates and Clusters The diffusion of native defect associates and clusters has been studied to a far lesser extent than that of charged impurity-native defect pairs. In fact, charging is rarely discussed in the context of intrinsic di-interstitial and divacancy diffusion. The silicon literature focuses primarily on the migration of neutral self-interstitial associates, which are highly mobile and affect process steps such as post-implantation anneals. Silicon divacancy and vacancy cluster diffusion has been examined in far less detail. Considering as the ionization levels of VSiVSi are fairly well established, and the diffusion of charged monovacancies has been studied in depth, this lacuna is somewhat surprising. What little knowledge exists regarding the diffusion of comparable charged associates and clusters in Ge, III–V, and oxide semiconductors will be discussed in brief. 7.1.2.1 Silicon Diffusion in silicon is mediated by interstitial defects that migrate through the crystal lattice; interstitial-dominated diffusion may include contributions from

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Si di-interstitials and tri-interstitials (Du et al. 2005; Gharaibeh et al. 2001) in addition to mono-interstitials, however. Under certain processing conditions such as during post-implantation annealing, where the concentration of interstitial associates and clusters approaches that of single self-interstitial defects, these contributions may be important. The high mobility of di-interstitials may also be invoked to reconcile divergent dopant profiles in oxidation enhanced diffusion (OED) and transient enhanced diffusion (TED) studies (Hane et al. 2000; Martin-Bragado et al. 2003). The diffusion energetics of the silicon tri-interstitial will not be discussed in any detail, as the literature fails to mention charging in conjunction with I3 migration. Simulations suggest, however, that the migration of I3 could be even faster than that of Sii and SiiSii (Gilmer et al. 1995). Despite the fact that SiiSii is stable in the (+1), (0), and (−1) charge states, only the diffusion of the neutral defect has been considered. The most energetically favorable diffusion pathway for the defect has not yet been established, but the experimental and theoretical values for activation energy are in fairly close agreement. The activation energy of diffusion for the di-interstitial has been determined experimentally via electron paramagnetic resonance experiments (Lee 1998; Eberlein et al. 2001). The motion of the P6 center attributed to the silicon di-interstitial is associated with an energy barrier of 0.6 ± 0.2 eV at temperatures ranging from 344 to 370 K. Kim et al. determined possible migration processes for di-interstitial diffusion by constant temperature molecular dynamics simulations using a tightbinding potential (1999). They observed only defect reorientation at 600 K, but diinterstitial hopping to nearest-neighbor sites with an energy barrier of 0.7 ± 0.1 eV at around 1,200 K. Eberlein et al. obtained a 0.5 eV activation barrier of diffusion for SiiSii0 using DFT with a migration pathway involving several of the structures presented in Section 5.1.1.2 (2001). Tight-binding molecular dynamics simulations performed by Cogoni et al. have shown that SiiSii0 diffuses almost as fast as the dumbbell Sii at room temperature (2005). These authors obtained a migration energy of 0.89 eV for SiiSii, and identify a relatively high-energy metastable state for the intermediate configuration of the defect associate during migration. Du et al. (2006), however, calculated a diffusion barrier of only 0.3 eV for the ground state di-interstitial structure with C1h point-symmetry (Richie et al. 2004). The lowest energy migration pathway for the defect consists of two steps along the <111> direction, a translation/rotation step and a reorientation step. These authors hypothesized that the diffusion barrier for SiiSii might be even lower (by about 0.1 eV) for the (+1) charge state of the defect. It is fairly well established that VSiVSi can take on charge states of (+1), (0), (−1) and (−2), yet most reports only discuss the migration of neutral VSiVSi. Van Vechten was one of the first to discuss the role of the silicon divacancy in diffusion near the melting temperature of the semiconductor (1,685 K) (1986). Kinetic lattice Monte Carlo (Caliste and Pochet 2006) and molecular dynamics (Prasad and Sinno 2003) simulations have since shed more light upon the temperature dependence of divacancy and vacancy cluster mediated diffusion. Several mechanisms of divacancy diffusion have been put forth; earlier reports considered a twostep detachment and recombination process (La Magna et al. 1999), whereas more

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recent studies detail a one-step hopping migration path (Hwang and Goddard 2002). An EPR study performed by Watkins and Corbett yielded an activation energy of 1.3 eV associated with the two-step model described above (1965). Mikelsen et al. performed isothermal annealing studies of VSiVSi in electron irradiated Si to arrive at an identical value of 1.30 ± 0.02 eV and a diffusion coefficient of 4 × 10−3 cm2/s (2005). Hwang and Goddard obtained an activation energy of 1.35 eV for the one-step hopping of the neutral silicon divacancy using the LDA with a 216-atom supercell (Hwang and Goddard 2002). Prasad and Sinno used molecular dynamics simulations to calculate the temperature dependent activation energy for VSiVSi; the temperature range of 1,300 < T < 1,650 K was well described by an energy barrier of 1.45 eV (2003). Lastly, one report has considered the impact of divacancy charging on diffusion in the bulk (Zhang et al. 2005). Zhang and co-authors used DFT calculations within the GGA to calculate charge state-dependent values for migration barrier based on Hwang’s one-step hopping model of diffusion; the activation barriers are 1.36, 1.38, and 1.44 eV for the (−1), (0), and (+1) charge states of the silicon divacancy, respectively. 7.1.2.2 Other Semiconductors Only a handful of reports discuss the diffusion of charged defect clusters in Ge and III–V semiconductors; no mention has been found of charged defect cluster diffusion in TiO2, ZnO, CoO, and UO2. For the most part, the equilibrium concentrations of di-interstitial and divacancy defects in these materials are so low that they cannot have a large effect on self-diffusion (Pöykkö et al. 1996). Under specific temperature conditions, such as after a long high-temperature anneal followed by a quenching step, charged defect clusters can diffuse and alter the dopant profile in the bulk (Morrow 1991). For certain temperature regimes, vacancies contribute to diffusion germanium much as they do to diffusion in silicon (Hashimoto and Kamiura 1974). Janke et al. have performed supercell and cluster DFT calculations to obtain migration energies for three of the four possible charge states of the germanium divacancy ((−2), (−1), and (0) but not (+1)) (2007). The authors considered a diffusion pathway wherein the end points were relaxed divacancies separated by one atomic jump. The activation energies for the migration of (VGeVGe)−2, (VGeVGe)−1, (VGeVGe)0 are 1.3, 1.2, and 1.1 eV, respectively. The magnitude of the energy barrier is close to that of VSiVSi and the experimental estimate of 1.0 eV (Hashimoto and Kamiura 1974). The charge state dependence of the migration barrier, however, is opposite to that of both the silicon divacancy (Zhang et al. 2005) and the germanium monovacancy (Pinto et al. 2006). Janke and co-workers attribute this trend to the reduction in space through which the mobile atoms can move as the charge state becomes more negative; the relaxations of (VSiVSi)−2 necessitate that the defect distort more so than (VSiVSi)−1 during the migration process. The mixed divacancy in GaAs, VGaVAs, has been invoked to explain the drop in net carrier concentration and conversion of GaAs from semi-insulating to p-type

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following a high-temperature anneal (Morrow 1991). The defect is thought to be stable in the (−2), (−1), and (0) charge states (Pöykkö et al. 1996). Dubonos and Koveshnikov studied the in-diffusion of a defect thought to be the divacancy at temperatures up to 350ºC and obtained a diffusion coefficient and activation energy for diffusion of 0.12 cm2/s and 1.14 eV, respectively (1990). A simple model obtained by fitting experimental data indicates that the diffusivity of the acceptortype defect VGaVAs (in the (−1) charge state) over the temperature range of 250–950ºC can be described by a prefactor of 3 × 10–3 cm2/s and activation energy of 0.94 eV (Morrow 1991). There is reason to believe that the mixed divacancy in GaN, VGaVN, would be well-suited to a similar treatment. The defect is stable in the (−3) charge state for Fermi levels above approximately Ev + 1.5 eV and exists in higher concentration than gallium vacancies, which are known to contribute to diffusion in the bulk (Ganchenkova and Nieminen 2006).

7.2 Surface Defects Semiconductor surface diffusion in general, and of defects in particular, has been review extensively by Seebauer and co-workers (Seebauer and Allen 1995; Seebauer and Jung 2001). Surface defect diffusion is mediated by vacancies and adatoms; there is reason to believe that defect associates and clusters migrate on the surface much as they do in the bulk, although few researchers have studied the phenomenon. Crystal structure, stoichiometry, Fermi level, and temperature dictate which defects mediate migration on the surface, much as they do in the bulk. Charge state effects do exist, but have been studied rather rarely, and mostly in the context of heterodiffusion. Such work will be detailed in the present chapter instead of Chap. 8 on extrinsic defects because the effects connect so closely with intrinsic surface defects and should also apply to them.

7.2.1 Point Defects Self-diffusion and many kinds of heterodiffusion on metals are typically mediated by native point defects such as vacancies or adatoms, though small adatom clusters sometimes play a role. The rate of mesoscale mass transport (i.e., over many atomic diameters) depends upon the number of mobile defects on the surface, and also upon the rate at which these defects move from site to site. In the case of a vacancy, the defect moves when an atom within the surface plane shifts into the original vacancy position. In the case of an adatom, site-to-site motion can take place by simple hopping of the atom or by one of various surface exchange mechanisms (Chang et al. 1996; Jun and Lei 1999; Allen et al. 1997). In simple exchange, for example, the atom “dives” into the surface, simultaneously “pushing” a substrate atom into an adatom position. Atomic motion is much easier to image directly on surfaces than in

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the bulk. Thus, both site-to-site hopping and mesoscale diffusion (by continuum methods) have been examined experimentally in considerable detail. 7.2.1.1 Silicon Up to now, hopping diffusion of defects has been explored only for neutral defects and only on the Si(001) surface. For example, Kirichenko et al. discussed possible mechanisms for the disappearance of neutral monovacancies on Si(001) (2004). They proposed four models for this phenomenon, including escape of the remaining atom from the “defect” dimer, diffusion of monovacancies along a dimer row and formation of dimer vacancies, rapid hopping of the remaining atom between two possible energy minima in the “defect” dimer, and diffusion of the surface monovacancy into the subsurface layer. Kitamura et al. measured the migration of single-dimer vacancies on Si(001) using high-T real-time STM, revealing that diffusion is predominantly one dimensional along the dimer row with a corresponding activation barrier of 1.7 ± 0.4 eV (1993). Using first-principles density functional calculations, Zhang et al. (1993) and Wang et al. (1993) have proposed a barrier to dimer vacancy migration as high as 2.5 eV. The former authors also proposed a displacement for the single dimer vacancy consisting of a wavelike displacement with the concerted motion of two second layer and two top layer atoms, as seen in Fig. 7.8. The diffusion of isolated adatoms on Si(100) has also been investigated. Mo et al. obtained an energy barrier of 0.67 eV for their diffusion by comparing island number densities measured with STM with those from simulations (1991). Brocks et al. calculated a comparable activation energy of 0.6–1.0 eV, depending on the path, from first-principles total-energy calculations; diffusion along the dimer rows is more favorable than diffusion perpendicular to the rows (1991). Activation energies for mesoscale diffusion are available for Si/Si(100), In/Si(111), Ge/Si(111), and Sb/Si(111). Webb et al. (1991) and Keeffe et al. (1994) both associated an activation energy of ~2.25 with the diffusion of silicon on the (001) surface. Mo et al. reported the diffusion of Sb dimers on a Si(001) surface using a STM whose tip approached the same region before and after sample annealing, giving an activation energy and a prefactor of diffusion of 1.2 eV and 10–4 cm2/s (1991). Measurements of mesoscale surface diffusion on both Si(100) and Si(111) have shown a dependence on native defect charge state (Allen et al. 1996, 1997; Allen and Seebauer 1995; Webb et al. 1991; Keeffe et al. 1994; Mo et al. 1991; Brocks et al. 1991; Hibino and Ogino 1996; Mo 1993; Kitamura et al. 1993; Zhang et al. 1993; Wang et al. 1993). In principle, the ionization of either surface adatoms or vacancies can be involved (Allen et al. 1996). Activation energies obtained for In, Ge, and Sb on the Si(111) surface for mesoscale diffusion using second harmonic microscopy range from 1.83 eV to 2.48 eV. The activation energies for diffusion and diffusion coefficients associated with these surface and adsorbed species are tabulated in Table 7.2.

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Fig. 7.8 Schematic diagram of a) a portion of a dimer row in Si(100) containing a single-dimer vacancy (top view). The large, medium, and small circles represent atoms in the top, second, and third layers, respectively. A solid line between any two atoms shows a Si-Si bond. Also shown are the b) initial c) transition state and d) final configurations along the diffusion pathway (side view). The arrows indicate the directions of atom motion leading to the next configuration, with longer arrows indicating larger displacements. Reprinted figure with permission from Zhang ZY, Chen H, Bolding BC et al. (1993) Phys Rev Lett 71: 3677. Copyright (1993) by the American Physical Society.

Table 7.2 Summary of activation energies for diffusion (in eV) and diffusion coefficients (in cm2/s) for defects and adsorbed species on Si(111) and Si(100) Species

Surface EA (eV)

D (cm2/s)

Sb In Ge Si Si Si Si Pb Sb Single dimer vacancy Single dimer vacancy Single dimer vacancy

Si(111) Si(111) Si(111) Si(001) Si(001) Si(001) Si(001) Si(111) Si(100) Si(100) Si(100) Si(100)

6 × 103 ± 0.7 3 × 103 ± 0.3 6 × 102 ± 0.5 –– ––

2.6 ± 0.13 1.82 ± 0.02 2.48 ± 0.09 2.25 ± 0.04 2.25 ± 0.04 0.67 ± 0.08 0.6 – 1.0 1.2 1.2 ± 0.1 1.7 ± 0.4 2.5 ≥ 2.2

Reference

(Mo 1993) (Allen et al. 1996) (Allen and Seebauer 1995) (Keeffe et al. 1994) (Hibino and Ogino 1996) (Brocks et al. 1991) –– (Webb et al. 1991) 2 × 1,010 to 8 × 1,011 (Allen et al. 1997) 10 – 4 ± 1 cm2/s (Lannoo and Allan 1982) –– (Kitamura et al. 1993) –– (Mo et al. 1991) –– (Wang et al. 1993)

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Fig. 7.9 Comparison of preexponential factors for mesoscale surface diffusion of groups III and V adsorbates on Si(111). The V shape provides evidence for the ionization of surface defects, since the ionization entropy of such defects is positive (and increases the diffusional pre-exponential factor) for both positive and negative defects. Reprinted figure with permission from Allen CE, Ditchfield R, Seebauer EG (1996) J Vac Sci Technol, A 14: 28. Copyright (1996) by the American Vacuum Society.

Since Sb and In have the same valence (three) and roughly the same atomic size, the primary differences in Ediff,M should arise from ionization effects associated with either electronegativity or acceptor/donor effects. The diffusional pre-exponential factor comprises the product of the intrinsic prefactor D0,I together with an entropic formation term. The entropic formation term comprises several components analogous to those discussed for ΔHf. There is a positive chemical entropy of formation ΔSfc associated with lattice mode softening near a terrace vacancy, which decreases local vibrational frequencies (Zener 1951). ΔSfc for a terrace vacancy should not be too far from that in the bulk as a first approximation, and both analytical (Lannoo and Allan 1982, 1986) and MD (Suni and Seebauer 1994, 1996) calculations suggest that this term can be 6R to 11R or higher. Adatoms themselves do not appear to contribute to ΔSfc appreciably (by hardening the lattice, for example). An ionization entropy ΔSfi may also exist because of the band gap. As in the case of bulk diffusion, the effect arises from local mode softening, this time due to a charge carrier confined near a charged vacancy or ion core (Van Vechten and Thurmond 1976a). If the carrier is associated with a hydrogenic donor or acceptor in a semiconductor, the carrier’s charge is generally screened by the large dielectric constant of the material and becomes highly delocalized. Hence little contribution to ΔSfi is made. This effect can be seen in the diffusion prefactor for In, Si, and Sb on Si; the quantity is larger for both In and Sb than for Si (Fig. 7.9). 7.2.1.2 Germanium Although some researchers have explored the diffusion of adatoms and vacancies on the Ge(111) surface with STM, their behavior is not as well characterized as that of similar defects on Si(111). Also, no computational work exists concerning the ionization levels of surface defects, so the effect of charge state on the energetics of diffusion has not been discussed in any theoretical investigations.

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Fig. 7.10 Schematic diagrams of diffusion hopping of a single vacancy on Ge(111)-c(2×8). Diagram a) shows a two dimensional view with the vacancy surrounded by rest atoms (R) and adatoms (A) together with metastable sites (M) to which it can hop. In b), the 1D simplification is shown with the approximate hopping rates. Reprinted from Mayne AJ, Rose F, Bolis C et al., “A scanning tunnelling microscopy study of the diffusion of a single or a pair of atomic vacancies,” (2001) Surf Sci 486: 232. Copyright (2001), with permission from Elsevier.

From the direction of surface band bending in STM images, Lee et al. inferred that the adatom vacancy defect on the Ge(111)-c(2×8) occurs in both the neutral and negative charge state (2000). Molinas-Mata et al. (1998), Mayne et al. (2001),and Brihuega et al. (2004) observed the thermal diffusion of vacancies and vacancy associates with scanning tunneling microscopy, yet did not mention defect charge state. One group of researchers has shown that, by the thermally activated hopping of neighboring adatoms to a vacancy site, the vacancy, presumably either in the neutral or (−1) charge state, can diffuse on the surface (Molinas-Mata et al. 1998). Artificially generated single vacancies can either switch to a different lattice site or split into two so-called “semivacancies.” The semivacancies are separated by a variable number of Ge adatoms in metastable T4 positions and can diffuse or eventually merge back into a single vacancy. Mayne et al. considered the mechanisms of both single vacancy hopping and the hopping of two vacancies occupying adjacent surface sites, although without regard to vacancy charge state (Fig. 7.10). Brihuega et al. also observed isotropic diffusion of the single vacancy to take place via a typical two-dimensional random walk pattern. The process is thermally activated with an effective energy barrier for the migration of the neutral defect of Ed = 0.89 ± 0.01 eV and 0.88 ± 0.02 eV according to Mayne et al. and Brihuega et al., respectively. At room temperature, the diffusion coefficient along [110] is more than twice the value obtained along [112], indicating that the diffusion processes leading to single vacancy migration on the surface are slightly anisotropic (Brihuega et al. 2004). 7.2.1.3 Gallium Arsenide Scanning tunneling microscopy has also been used to investigate the diffusion mechanisms of anion and cation vacancies on the surface of GaAs(110) (Lengel et al. 1996). The injection of minority carriers by the STM tip serves as a catalyst

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for vacancy motion; under gallium-rich conditions, VAs+1 and VAs−1 are the dominant defects for p-type and n-type GaAs(110), respectively, whereas under arsenic-rich conditions, charged gallium vacancies and arsenic adatoms become important (Schwarz et al. 2000). Gwo et al. (1993) and Lengel et al. (1993; 1996) both investigated the directional movement of gallium and arsenic on the GaAs surface, yet made no mention of the effect of defect charge state on migration mechanism. Ab initio calculations on the energetics and dynamics of surface vacancies on GaAs(110) performed by Yi et al. found that the diffusion of gallium and arsenic vacancies on the surface is more likely to occur via motion along the zigzag chains, rather than between them (1995). The diffusional barrier heights for neutral VGa and VAs are 2.5 and 1.5 eV, respectively, while those for the same singly negatively charged vacancies are 1.9 and 2.5 eV. These activation energies imply that VGa−1 diffuses more easily than the neutral vacancy, whereas charging impedes the motion of VAs−1 compared to the neutral defect. The considerable barrier height for the diffusion of VAs−1 is due to the high stability of the vacancy when it is charged. No mention is made in the literature of the diffusion of the singly positively charged VAs+1 and VGa+1 that are hypothesized to exist in high concentrations on p-type GaAs(110) under Ga-rich and As-rich conditions, respectively. Some experimental and computational information exists on the diffusion of cations on the GaAs(001)-β2 surface (Kley et al. 1997; Kawabe and Sugaya 1989; Nishinaga and Kyoung-Ik 1988; Neave et al. 1985; Shitara et al. 1993; Ohta et al. 1989). The experimentally deduced migration barriers for Ga adatoms vary over a wide range from 1.1 to 4.0 eV (Nishinaga and Kyoung-Ik 1988; Neave et al. 1985; Shitara et al. 1993; Ohta et al. 1989). These studies were undertaken mainly to provide a model for surface growth during molecular beam epitaxy, and thus neglect to mention the relative contribution of the arsenic adatom in the (+1), (0), and (−1) states. Kley et al. identified two diffusion channels that affect the migration of adatoms: in one channel the adatom jumps across the surface dimers and leaves the dimer bonds intact, while in the other, the dimer bond is broken (1997). 7.2.1.4 Other III–V Semiconductors Few studies exist regarding the diffusion of charged defects on the remaining III–V semiconductor surfaces. In particular, little is known about the migration of electrically active defects on semiconductors containing elements from the second row of the periodic table, such as BN and GaN. In the literature that does exist, Ebert et al. explored the diffusion of phosphorus vacancies on InP(110) and GaP(110) with scanning tunneling microscopy. The migration is most likely triggered by the STM tip, as the probability of diffusion is significantly higher when the tip contacts the surface. Under indium-rich conditions, the formation energies of VP+1 and VP−1 on p- and n-type InP(110), in that order, are less than 1.0 eV (Semmler et al. 2000; Morita et al. 2000; Ebert 2002; Ebert et al. 2001; Kanasaki 2006). Jumps of vacancies in both the [1 −1 0] and [001] directions are observed; jumps in the former direction occur three times more frequently than those in the latter direction. The

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authors estimate the diffusion coefficient of VP+1 as 10–18 cm2/s at room temperature (Ebert et al. 1992). On GaP, diffusion can lead to the recombination of neutral adatoms with charged vacancies and the formation of adatom clusters. In contrast to isolated P vacancies, however, these defect clusters show no local variations in height in STM images. As a consequence, they induce no band bending and are uncharged (Ebert 2002). 7.2.1.5 Titanium Dioxide Experimental and computational evidence has revealed a mechanism of adsorbatemediated vacancy diffusion on rutile (110). Two molecular configurations and one dissociated (atomic) configuration of O2 adsorption can occur on defected surfaces, whereas no adsorption takes place on stoichiometric TiO2(110) (Wu et al. 2003). In 2003, Schaub et al. found experimental STM evidence for O vacancy diffusion along the [110] direction, or perpendicular to the bridging O rows of the TiO2 (110) surface (2003). The phenomenon was observed for surfaces with 0.1 to 1 monolayer of coverage at temperatures ranging from 180 to 300 K. The authors proposed a diffusion mechanism whereby adsorbed O2 molecules mediate vacancy mobility through a cyclic process involving the loss of an oxygen atom to a vacancy and sequential capture of a bridging oxygen atom. These results were later discounted, as the background pressure in the vacuum chamber was high enough that oxygen vacancies were replaced by bridging hydroxyl pairs (Wendt et al. 2005). Additionally, the diffusing species was likely water rather than O2 (Wendt et al. 2006). DFT calculations (Wang et al. 2004) and STM images (Zhang et al. 2007) have subsequently revealed that bridging oxygen vacancies do, indeed, diffuse exclusively along bridging oxygen rows, but with a substantial energy barrier for migration. Oxygen vacancies are immobile on the adsorbate-free surface, as indicated by the 4 eV barrier calculated by Rasmussen et al. for the unassisted diffusion of a bridging oxygen atom from a bridging row to a titanium row (2004). Wu et al. proposed that oxygen-assisted O vacancy diffusion results from transformations among several different O2 adsorption states, which have comparable energies and are separated by modest energy barriers (1.1–3.4 eV); the process is especially favorable when the O vacancy concentration exceeds that of adsorbed O2 (2003). DFT calculations by Wang et al. predict that O2 dissociation at the defect site is the rate-limiting step for O2-mediated vacancy diffusion, with a barrier of 1.39 eV (2004). Their computational results do, however, explain vacancy diffusion across bridging oxygen rows in terms of the ease of O-O recombination, vacancy density increase upon atomic O deposition, and the temporal and spatial correlation of vacancy diffusion. Arrhenius analysis of isothermal STM images yields a diffusion barrier of 1.15 ± 0.05 eV and preexponential factor of 1 × 1012.2±0.6 s−1 for bridge-bonded oxygen vacancies; additionally, it was found that the hoping rate of vacancies increases exponentially with temperature (Zhang et al. 2007). The energy barrier for migration obtained experimentally accords with the value of 1.03 eV calculated by the same authors using slab DFT calculations.

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7.2.2 Associates and Clusters Although the surfaces of group IV, III–V, and oxide surfaces are decorated with charged defect associates and clusters, little attention has been given to the migration of these defects. The Hwang laboratory has investigated the migration of special Si clusters dubbed “magic” clusters on Si(111) (2002; Ho et al. 2005). STM images have been utilized to determine activation energies and pre-exponential factors for the hopping motion of these defects, which are observable and mobile for temperatures greater than 400ºC. Values for activation energy range from 1.96 to 2.64 eV, depending upon whether the cluster hops in the faulted or unfaulted half of the Si(111)-(7×7) reconstruction. Hu et al. have investigated the cluster-cluster interactions that arise during epitaxial deposition using a phasefield model for monolayer atomic clusters that includes substrate-mediated interactions (2007). Neither or these reports mention defect charging, however. Reports examining the formation and migration of metal adatom clusters on Si and TiO2 surfaces are more plentiful. The atomistic process of metal-cluster growth on Si(111)-(7×7) involving the diffusion of adatoms between half-unit cells has been considered (Wang et al. 2008). Bimetallic Pt-Rh clusters containing up to 100 atoms are mobile at room temperature on rutile TiO2(110) according to scanning tunneling microscopy images (Park et al. 2006). One-dimensional modeling indicates that Pt cluster diffusion on anatase TiO2(001) depends upon cluster size and exhibits an Arrhenius temperature dependence, with the prefactor and activation energy varying with cluster size (El-Azab et al. 2002). None of these studies have attempted to correlate thermal surface cluster diffusion to defect charge state in the manner discussed in Section 7.2.1.1.

7.3 Photostimulated Diffusion Several laboratories have reported that photostimulation may non-thermally influence defect diffusion in the semiconductor bulk (Wieser et al. 1984; Ishikawa and Maruyama 1997; Gyulai et al. 1994; Noel et al. 1998; Fair and Li 1998; Lojek et al. 2001; Ravi et al. 1995) and on the surface (Ditchfield et al. 2000, 1998). Photostimulated enhancement and retardation have both been observed. Yet experimental interpretation has often proven difficult because heating by the probe light or changes in heating configuration as probe intensity varied cast doubt on the results. Definitive measurements have awaited experimental configurations in which heating and illumination could be decoupled, which has been accomplished only in recent years (Ditchfield et al. 2000; Jung 2004). Photostimulated diffusion offers a different perspective from other experimental methods into understanding the behavior of charged defects in semiconductors. There are technological implications as well. In microelectronic fabrication, for

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example, current technology for making shallow pn junctions employs heating by incandescent lamps, higher-intensity flash lamps, or lasers (Lindsay et al. 2003). The optical stimulation is typically thought to supply heating alone. The existence of photostimulated dopant diffusion or activation might require modification of widely-used process models (Zechner et al. 2002), and would also provide a new means to kinetically distinguish among rapid heating technologies, which employ vastly differing illumination intensities. There is physical reason to suppose that illumination can affect bulk diffusion in semiconductors non-thermally. Both vacancies and interstitials (of Si and dopants) are capable of existing in multiple charge states. The formation energy and entropy (Van Vechten and Thurmond 1976b, a) for the forward reaction and the migration energies (Fair 1981) of the point defects thus formed can vary with charge state. Since illumination affects the availability of charge carriers and therefore the most favored charge state of the defects, illumination can, in principle, change both the formation and migration energies (and entropies) of point defects (Jung et al. 2000). The modification of defect concentration via photostimulation was discussed in Chap. 2. Here, the impact of photostimulation on both site-to-site hopping and mesoscale diffusion rates will be presented in more detail.

7.3.1 Photostimulated Diffusion in the Bulk In general, several distinct mechanisms can influence the rate of photoinfluenced defect diffusion, including changes in defect charge and various electron-hole recombination mechanisms. Chapter 2 described how photostimulation can change the average charge state of semiconductor defects. The fact that such changes can influence site-to-site hopping rates was recognized in the 1970s and 1980s in connection with Fermi level variation through forward biasing of diode structures (Landsberg 1991; Lang 1982). That body of literature did not treat photostimulation explicitly, but many of the principles outlined there form a basis for interpreting photostimulated diffusion data. Recent experimental work has unambiguously demonstrated photostimulated enhancement of self-diffusion within silicon due to changes in the average charge state interstitials (Jung et al. in preparation; Vaidyanathan 2007; Jung 2003). Figure 7.11 compares diffusion profiles for illuminated and unilluminated specimens of n-type material. Illumination increases the diffusivity by a factor of up to 25 in response to optical fluxes near 1.5 W/cm2. Figure 7.12 shows the Arrhenius dependence of photostimulated diffusivity, compared to dark, at the maximum intensity. The degree of illumination enhancement varies with temperature and intensity. By contrast, no photostimulation effects could be observed for p-type material under similar experimental conditions. The difference in behavior between n- and p-type material gives strong evidence that the observed enhancement in n-Si is genuine, and not an artifact of some unknown heating or similar spurious

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Fig. 7.11 SIMS profile of 30Si that has diffused in a 28Si matrix, showing non-thermal illumination enhancement of diffusion for two different intensities. Annealing conditions are 800ºC for 1 hr in n-type material.

effects. The exponential profile shapes represent the signature of diffusion by a quick-moving intermediate species, which has been shown to be the Si interstitial under the conditions of these experiments (Vaidyanathan et al. 2006). The exact charge states involved are not known with certainty, as the most likely charge states for Si interstitials in n-type Si are not definitively established. But the primary candidates are Sii+2, Sii0, and Sii–1. More detailed experimental analysis showed that both the site-to-site hopping rate and the concentration of mobile interstitials changed under illumination. There is similar unambiguous evidence for photostimulated diffusion of boron and arsenic implanted into silicon (Vaidyanathan et al. in preparation; Vaidyanathan 2007). Both the diffusion and activation of these dopants vary significantly with illumination at the 1 W/cm2 level. Although mechanisms for photostimulated diffusion based upon carrier recombination have been postulated (Ravi et al. 1995) such mechanisms have not yet been observed unambiguously. Almost all of the literature on diffusional effects

Fig. 7.12 Illumination enhancement factor versus illumination intensity for n-type Si.

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from carrier recombination is based on experiments with electrically biased diode structures. Starting in the 1970s (Lang and Kimerling 1974), several laboratories have reported greatly enhanced bulk diffusion of Si self-interstitial atoms in such configurations. Indeed, sometimes the diffusion is nearly athermal (independent of temperature) in the presence of excess carriers. The literature has been reviewed extensively (Landsberg 1991; Kimerling 1978; Watkins et al. 1983; Lang 1982), and several possible mechanisms have been identified for this defect and several others in Si, GaAs and GaP. In the Bourgoin–Corbett mechanism, the potential energy surface for diffusion is such that the defect switches charge state as it moves from site to site by alternately capturing electrons and holes (Stievenard and Bourgoin 1986). In the “phonon-kick” mechanism, coupling between charge carriers and atom cores in the bulk is takes place via dipoles (or higher-order moments) caused by lattice distortions near a defect. Stronger dipoles lead to stronger coupling (Weeks et al. 1975) and more efficient recombination. Once recombination takes place, the electronic energy dumped into lattice vibrations sometimes remains localized long enough to induce vacancies or interstitial atoms to hop, especially if the excess energy can channel into the appropriate diffusive mode (Weeks et al. 1975; Van Vechten 1988). In the electronic excitation mechanism, recombination creates an excited state that is electronic rather than vibrational, and the excited state is postulated to have a lower barrier to migration than the ground state. Thus, if the electronic excitation has a sufficiently long lifetime, diffusion can proceed by the lower energy pathway.

7.3.2 Photostimulated Diffusion on the Surface In principle, all the photostimulation mechanisms described for bulk diffusion can also apply to surface diffusion. However, only the charge state mechanism has been identified explicitly. On the Si(111) surface, low-level optical illumination (< 2 W/cm2) was shown to either enhance or inhibit diffusion of indium, germanium and antimony on Si by close to an order of magnitude, depending on the doping type (n or p) of the underlying substrate (Ditchfield et al. 2000). Figure 7.13 shows an example case. All systems exhibited a similar convergence of the Arrhenius plots for n and p-type material. The effects arise primarily from illumination-induced changes in the formation energy for surface vacancies, which was the only defect common to all the adsorption systems. Also, correspondence between the convergence temperature in the Ge/Si(111) system and the disappearance of the (7×7) reconstruction (with its large charge separations) offered additional evidence for the importance of ionization on surface diffusion. Illumination of a p-type material makes the surface more n-type, presumably increasing the average negative charge of the vacancies. Adatoms on the Si(111)-(7×7) surface, however, have a positive charge, which would result in electrostatic attraction between the negatively charged vacancies and positive adatoms and thereby inhibiting mass transport.

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Fig. 7.13 Arrhenius plots of In diffusion n-type and p-type Si(111) at about 1018 cm–3 doping under dark and illuminated conditions. Error bars derive from standard error analysis of the diffusion profiles, while lines represent least-squares fits. For diffusion in the dark, n-type and p-type material yields identical fits. For n-type illuminated material the least squares fit includes only data about 390 K. The drop-off in D below 390 K appears to represent a true change in slope. Reprinted figure with permission from Ditchfield R, Llera-Rodriguez D, Seebauer EG (2000) Phys Rev B: Condens Matter 61: 13711. Copyright (2000) by the American Physical Society.

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Chapter 8

Extrinsic Defects

8.1 Bulk Defects Semiconductors often contain intentionally introduced dopant atoms or accidentally introduced impurity species that alter the type and amount of charged defects in the bulk. Doping can change the predominant charge state of intrinsic bulk defects or introduce new extrinsic defects. The location of these new defects within the semiconductor depends on dopant or impurity atomic radius, charge, and bulk crystal structure. While some atoms prefer to reside in substitutional sites, replacing atoms on the crystal lattice, others assume interstitial configurations. Dopant and impurity atoms can remain as isolated point defects or join with each other (or intrinsic defects) to form defect complexes. Dopant complexes can trap charge and lead to larger extended defects such as planar faults and voids. Under some circumstances, doping-related defects may be present in the crystal in large enough concentrations that they outnumber native point defects. Dopants are frequently characterized as either “shallow” or “deep.” A shallow donor implies a (+1/0) ionization level that lies very close to the valence band. A deep donor implies a (+1/0) ionization level lying within the band gap much further from the valence band maximum. These expressions translate for acceptor defects except that shallow and deep, instead, refer to the distance between the defect ionization level and the conduction band minimum. A standard notation will be used to refer to extrinsic defects. Substitutional defects will be written “XSite” where impurity species X may be further specified as an acceptor atom “A”, donor atom “D”, metal atom “M”, or actual element such as N, S, etc. The subscript “Site” on the substitutional defect indicates the lattice site on which the defect resides. For example, DSi represents a substitutional donor atom on a silicon lattice site and AGa a substitutional acceptor atom on the Ga sublattice of GaAs. Generic interstitial defects may also be written as Ai, Di, or Mi. The formation of isolated substitutional defects can occur via a reaction involving native vacancies. Take, for instance, the formation of substitutional silicon in E.G. Seebauer, M.C. Kratzer, Charged Semiconductor Defects, © Springer 2009

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gallium arsenide. As there are two sublattices (and thus two types of native vacancies), the substitutional atom can reside on either a gallium lattice position 0 0 Si + VGa ↔ SiGa

(8.1)

0 . Si + VAs0 ↔ SiAs

(8.2)

or an arsenic lattice position

Defect pairs comprising native defects and dopants or impurities form much in the same way as native divacancies, di-interstitials, etc. Take, for instance, the formation of a neutral donor defect-vacancy pair from N, P, As, or Sb: D +1 + VSi−1 ↔ ( DVSi ) . 0

(8.3)

A pair formed from VSi–2 would, instead, have an overall charge of (–1). A comparable reaction is observed for the formation of a defect pair from a charged interstitial such as oxygen: Oi+ q + VSi−1 ↔ ( OiVSi )

q −1

(8.4)

where the unit charge q on the oxygen interstitial contributes to the charge state of the oxygen interstitial-silicon vacancy pair. Many of these defect formation reactions are competitive; in silicon the capture energy of a neutral interstitial by VSi–1 is lower than that of D+1 by VSi–1 (Peaker et al. 2005).

8.1.1 Silicon The presence of foreign atoms in the silicon crystal lattice, even at concentrations as low as 1010 cm–3, dramatically affects the electrical behavior and transport properties of the bulk. Dopants can be introduced into silicon via an assortment of methods, including during Czochralski growth of large ingots, or by diffusion or ion implantation into wafers. Charged defects created by intentional p- and n-type doping are the primary mediators of thermal diffusion. The unintentional incorporation of metal, oxygen, or hydrogen atoms during wafer processing can create electron-hole recombination centers that substantially lower device efficiency. Silicon is doped “p-type” via the addition of B, Al, Ga, or In. While boron is the most important acceptor dopant for IC device fabrication, aluminum is frequently used to manufacture power semiconductors with deep p-n junction depths ranging from a few to a hundred microns (Krause et al. 2002). Gallium and indium are better suited to Czochralski-grown solar cells and infrared detectors. The addition of N, P, As, and Sb to intrinsic silicon creates an abundance of carrier electrons in the bulk or “n-type” doping. Nitrogen atoms in silicon can lock dislocations to increase mechanical strength, enhance oxygen precipitation, and suppress thermal donors (Yang et al. 1996). Arsenic is often diffused into silicon

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235

to form p-n junctions, while phosphorus is the species of choice for bulk doping of silicon wafers. Metallic impurities, particularly the transition elements from the 3d series, serve as efficient electron-hole recombination centers and strongly reduce carrier lifetimes in bulk Si. They can easily creep into silicon wafers during heat treatment, and their effects are so detrimental that their behavior has been considered extensively (Beeler et al. 1990; DeLeo et al. 1981; Heiser et al. 2003; Istratov and Weber 1998; Tavendale and Pearton 1983; Weber 1983). Copper and nickel may cause a breakdown in silicon oxides (Honda et al. 1984; Hiramota et al. 1989) and iron-boron pairs have a damaging effect on solar cell efficiency even at the 1011 cm–3 level (Reehal et al. 1996; Reiss et al. 1996). The inclusion of platinum, palladium, and gold in silicon has been studied in an attempt to control the minority-carrier lifetime in semiconductor devices (Bollmann et al. 2006; Sachse et al. 1997a; Sachse et al. 1997b; Watkins et al. 1991). Czochralski-grown silicon has a high concentration of oxygen (typically 1018 oxygen atoms/cm3) that is distributed homogeneously throughout the bulk and becomes supersaturated for a wide range of temperatures (Lee and Nieminen 2001). The variable retention time phenomenon of dynamic random access memory (DRAM), which is observed universally in 64 Kbit–16 Mbit DRAMs of various manufacturers, has been attributed to the VSi-oxygen complex (Umeda et al. 2006). Sulfur is also stable in the Si crystal lattice. Below-band gap light absorption and photocurrent generation have been attributed to sulfur impurities in silicon microstructures (Wu et al. 2001). Lastly, hydrogen may be incorporated into silicon during electronic device processing steps including plasma treatment (Ulyashin et al. 2001) and high-energy proton implantation (Lévêque et al. 2001). The charged extrinsic defects in silicon are well characterized. Consequently, their properties, charging, and diffusion behavior will be discussed in brief and the reader referred to several more comprehensive resources on the topic. 8.1.1.1 Structure Typically, B, Al, Ga, and In atoms occupy substitutional positions in the Si lattice, although electron irradiation easily displaces the atoms into interstitial configurations. The relaxations around a substitutional defect can be predicted based on the atomic radius of the impurity atom. For example, B has a smaller ionic radius than Si, which explains the inward relaxations around BSi. Substitutional defect selfinterstitial complexes such as ASiSii are more favorable in B- and In-doped Si (Windl et al. 1999), while ASiAi or isolated Ai have lower formation energies in Al- and Ga-doped material (Schirra et al. 2004; Melis et al. 2004). First-principles total-energy and DFT calculations have been performed to identify the stable and metastable configurations of charged Bi and BSiSii in B-doped Si (Tarnow 1991; Zhu et al. 1996; Hakala et al. 2000). As with the native point defects in silicon, different configurations are favored depending on defect charge state (Fig. 8.1). The stable charge state-dependent point symmetry of the defect also varies depending

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Fig. 8.1 Calculated atomic structure of the BSiSii defect in its two possible configurations. For the (+1) charge state only the C3v configuration is found, whereas for the (0) and (–1) charge states both the C3v and C1h configurations are found. The bond lengths d1–d5 are also charge state dependent. Reprinted figure with permission from Hakala M, Puska MJ, Nieminen RM (2000) Phys Rev B: Condens Matter 61: 8156. Copyright (2000) by the American Physical Society.

on dopant species (Al versus B, for instance). Substitutional defect vacancy clusters such as ASiV and ASiV2, which play a role in vacancy-mediated diffusion, have also been explored. B and Ga are capable of forming stable acceptor-silicon vacancy pairs (Melis et al. 2004), while similar defects comprised of In and Al are often unstable (Alippi et al. 2004). An assortment of configurations for BSiVSi and BSiVSiVSi have been investigated using density functional theory (Adey et al. 2005). For the most part, n-type dopants prefer to form defect complexes. In nitrogendoped material, NSi is rare and Ni only exists in discernable concentrations in implanted material annealed to 600ºC, not in as-grown material. The donor defectvacancy complex known as the E center is discussed frequently in the literature (Adam et al. 2001; Rummukainen et al. 2005), as is the donor defect-interstitial complex. At high doping levels (> 1020 cm–3), vacancy clusters decorated with multiple dopant atoms or dopant-vacancy-pair-interstitial complexes occur in the bulk (Voyles et al. 2003; Sawada and Kawakami 2000). The charge-state dependent configurations and relaxations of the E center formed from N, P, As, and Sb have been investigated computationally (Ganchenkova et al. 2004; Ögüt and Chelikowsky 2003). For instance, Ganchenkova and co-workers observed an inward relaxation of the first nearest-neighbors of the PVSi complex for all charge states ((+1), (0), (–1)), while the degree of relaxation increased from ~4 to 12% as the Fermi energy was raised from the VBM to the CBM. Sawada and Kawakami have explored a variety of charged N defect configurations in silicon using calculations based on the local-density approximation. Four low energy structures are typically considered for the donor defect-interstitial defect: a [110] dumbbell interstitial where a donor atom and a Si atom share a lattice site orientated in the [110] direction, a pair with approximate C2 symmetry, a structure with the interstitial donor atom in the hexagonal site, and a dumbbell-like interstitial oriented in the [100] direction (Liu et al. 2003). Isolated donor interstitials can also join with impurities such as oxygen introduced during the Cz-Si growth process to produce both electrically inactive defects and shallow thermal donors (Ewels et al. 1996;

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237

Fig. 8.2 The configuration of the CuiSii defect that strongly resembles a CuSiSiiSii defect with the Cu atom (black sphere) closer to the substitutional site than the two Si atoms. The complex has Cs symmetry. Reprinted from Estreicher SK, “First principles theory of copper in silicon,” (2004) Mater Sci Semicond Process 7: 106. Copyright (2004) with permission from Elsevier.

Voronkov et al. 2001). A common example is the NiOiOi defect that occurs when one Ni atom bonds with two Oi atoms, forcing the O atoms to move slightly out of the bond-centered sites. The geometries of interstitial and substitutional copper point defects and complexes are better understood than those of the other 3d transition metals. In general, however, slow diffusers such as titanium prefer to remain as isolated atoms, whereas faster diffusers such as Cu, Ni, and Co form aggregates and precipitates (Estreicher 2004). In p-type and intrinsic silicon, copper exists as the tetrahedrallysituated interstitial Cui+1 ion. Substitutional copper defects, CuSi, form when interstitial copper atoms encounter isolated silicon vacancies. Only a small fraction of copper atoms, typically less than 0.1% of the equilibrium concentration, are situated at these substitutional sites (Heiser et al. 2003). According to Latham et al., Td symmetry is preserved for the (+1) charge state of the defect, while Jahn–Teller distortions lead to tetragonal D2d point-symmetry for CuSi0 and CuSi–1 (2005). Another computational report claims that neutral Co, Ni, and Cu adopt D3d symmetry while Ti, V, Cr, Mn, and Fe assume Td symmetry (Kamon et al. 2001). Mention of defect complexes including CuSiCuSi, CuiCui, CuSiCui, and CuiSii also appears in the literature; the charging of these defects has not been explored in any detail (Estreicher et al. 2003; Estreicher et al. 2005). The latter is depicted in Fig. 8.2. Isolated oxygen atoms occupy a bond-centered interstitial position in silicon; in this position the defect is infrared active, and is known to improve the high temperature mechanical strength of the bulk (Sassella 2001). The small energy differences between C1, C1h, C2, or D3d point symmetry for the defect has led researchers to propose a model wherein the O atom tunnels between equivalent sites around the bond center, effectively assuming D3d symmetry with an Si-O-Si angle of 180º (Hao et al. 2004; Coutinho et al. 2000). At lower temperatures, on the order of 350–500ºC, shallow and deep donor and acceptor complexes are formed from isolated oxygen interstitials (Lee and Nieminen 2001). The well-studied A-center, also known as the vacancy-oxygen pair, contributes to thermal diffusion

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at high temperatures (Casali et al. 2001). For temperatures above 800ºC, the majority of intrinsic vacancy defects are bound in the form of OVSi complexes. In the most stable configuration of the complex, the oxygen atom moves away from the vacancy site in the [100] direction and forms bonds with two silicon atoms (Pesola et al. 1999). VSiO has orthorhombic C2v symmetry for all three stable charge states ((0), (–1), and (–2)). The multi-vacancy-oxygen defect, formed from the intrinsic silicon divacancy, has also been studied (Alfieri et al. 2003). Pesola et al. also discuss the geometry and relaxations of this defect; in VSiVSiO, the oxygen atom forms bonds with the two silicon atoms at one end of the divacancy. Jahn–Teller distortions induce symmetry lowering relaxation to C1h symmetry for all three charge states of V2O ((0), (–1), and (–2)). The defect behavior of sulfur, another chalcogen or group-IV atom, in silicon, has been compared to that of oxygen using first-principles total energy calculations by Mo et al. (2004). When protons are implanted into Si at cryogenic temperatures, they remain as isolated interstitial defects where their specific configuration depends on the position of the Fermi energy in the bulk (Holm et al. 1991). The (+1) and (–1) charge states of Hi favor bond-centered and tetrahedral configurations, respectively (Bonde Nielsen et al. 2002; Bonde Nielsen et al. 1999). When diffusing H atoms interact with each other they form metastable H2 dimers (Holbech et al. 1993), H2 molecules at interstitial tetrahedral sites (Leitch et al. 1998; Hourahine et al. 1998), and hydrogen-VSi pairs (Coomer et al. 1999; Bech Nielsen et al. 1997). According to Bech Nielsen et al., HVSi0 has C1h point symmetry with the Si-H bond bent a few degrees away from a <111> axis in the mirror plane; the structure corresponds closely to that of the PVSi0, the E center comprising a phosphorus atom located next to a monovacancy. Pairs consisting of multiple hydrogen atoms or silicon vacancies also arise (Coutinho et al. 2003). 8.1.1.2 Ionization Levels The point defects and defect complexes arising from the p-type doping of crystalline silicon assume an assortment of charge states. Substitutional B exists in the (–1) charge state for all Fermi energies within the band gap, although it prefers to bond with Sii to form (BSiSii)+1 or (BSiSii)–1. Note that the neutral charge state of BSiSii is metastable; the defect exhibits negative-effective-U behavior (Hakala et al. 2000). DFT calculations performed by Hakala et al. suggest that the (+1/–1) ionization level of the defect is located at Ev + 0.60 eV. As evidenced by these same authors and DLTS experiments (Troxell and Watkins 1980; Watkins 1975; Harris et al. 1982), the isolated B interstitial behaves similarly; for example, Harris et al. identified (+1/0) and (–1/0) levels at Ev + 0.97 ± 0.01 eV and Ev + 0.73 ± 0.08 eV, respectively (1987). While investigating the transient enhanced diffusion of boron in silicon, Jung et al. calculated an effective (+1/–1) ionization level of Ev + 0.33 ± 0.05 eV using maximum likelihood estimation (and relying upon numerous experimental and computational values found in the literature) (2005). For comparison, Ali in Al-doped Si has one ionization level – (+1/0) – in the tetrahedral configuration and

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239

two – (+2/+1) and (+1/0) – in the hexagonal configuration (Schirra et al. 2004). Also, the Al interstitial prefers to form a different complex, AlSiAli, that exists in two charge states with a (+1/0) ionization level at approximately Ev + 0.36 eV. Concerning vacancy complexes, BSiVSi is stable in the (+1), (0), and (–1) charge states, while BSiVSiVSi is stable in those charge states as well as (–2). The single donor level have been identified at Ev + 0.11 eV, Ev + 0.31 eV, and Ev + 0.37 eV via DLTS (Bains and Banbury 1985) and capacitance-transient spectroscopy (Londos 1986); these values likely correspond to transitions between the third-nearest neighbor, C1, or C1h configuration of the defect, respectively (Adey et al. 2005). Also, according to Adey et al., the acceptor level predicted to lie at Ev + 1.04 eV is difficult to observe using DLTS due to the high temperature at which the defect dissociates. Lastly, the (+1/0), (0/–1), and (–1/–2) ionization levels of BSiVSiVSi are located at Ev + 0.15 eV, Ev + 0.44 eV, and Ev + 0.90 eV, or close to those of VSiVSi. Donor or “n-type” doping induces similar point defects and defect complexes formed from substitutional and interstitial dopant atoms. NSi is a rare defect in silicon, although it has been detected in the (+1), (0), and (–1) charge states by EPR (Brower 1980; Murakami et al. 1988). While early investigators identified a (+1/0) ionization level for the defect (Pantelides and Sah 1974; Itoh et al. 1987), more recent DFT calculations disagree over the stability of NSi+1 (Goss et al. 2003). For instance, Sawada and Kawakami predict a direct switch from NSi0 to NSi+2 close to the VBM (< 0.1 eV) and a single acceptor level at approximately Ev + 0.5 eV (Sawada and Kawakami 2000). The nitrogen interstitial, like the boron interstitial, is stable in three charge states with (+1/0) and (0/–1) ionization levels at Ev + 0.5 eV and Ev + 0.9 eV (Goss et al. 2003). Isolated Pi and Asi have received little treatment in the literature, as their contribution to diffusion in the bulk in the form of donor-vacancy and donor-interstitial pairs is of greater significance. The PSii defect migrates in the (0) and (–1) charge states (Fig. 8.3) (Liu et al. 2003), while the donor-vacancy pair (or E center) in silicon adopts the (0), (–1), and (–2) charge states (Adam et al. 2001; Evwaraye 1977). The ionization levels of the E center are hard to resolve using conventional DLTS, as they lie close to those of the silicon divacancy (Peaker et al. 2005). High-resolution DLTS experiments suggest that the single acceptor levels of PVSi, SbVSi, and AsVSi are all about 0.4 eV below the CBM (or 0.7 eV above the VBM) (Auret et al. 2006). The charge states of defects induced by transition metal doping are more predictable; metals from Group 8 or less on the periodic table (e.g., Ti, V, Cr, Mo, Fe) tend to exist as interstitial donor defects, while those from Group 9 or higher (e.g., Au, Zn, Co, Ni, Cu) generally form substitutional acceptor defects (Graff 2000; Lemke 1994; Istratov and Weber 1998). Following high temperature processing, the diffusivities of the latter are large enough that interstitial atoms precipitate, which leaves only occupied substitutional sites (Macdonald and Geerligs 2004). Computational investigation of these defects is complicated by the need to account for a wide range of spin multiplicities and complex chemical behavior (Estreicher 2004). The copper interstitial in silicon likely has a (+1/0) ionization level within the band gap, as suggested by DLTS experiments that identify a donor level at Ev + 0.95 ± 0.01 eV associated with either Cui or a complex thereof (Istratov et al.

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Fig. 8.3 (a–d) Lowest energy structure of PSii pair in various charge states. In (a), (b), and (d), the small ball is the P atom and the large balls are Si atoms. In (c), the ball is the P atom. All other Si atoms are shown as a stick-only network. (e) Fermi level dependence of the PSii pair formation energetics. Reprinted with permission from Liu X-Y, Windl W, Beardmore KM et al. (2003) Appl Phys Lett 82: 1840. Copyright (2003), American Institute of Physics.

1997). Substitutional copper may exist in four charge states – (+1), (0), (–1), and (–2) – although the existence of the double acceptor is debated in the literature. Experimental studies indicate that the (+1/0) and (0/–1) ionization levels of the defect are located at Ev + 0.215 ± 0.015 eV and Ev + 0.44 ± 0.02 eV, respectively (Istratov and Weber 1998). Knack and co-workers identified a (–1/–2) ionization level for CuSi at Ec – 0.167 eV (or Ev + 0.933 eV at 300 K) close to the (+1/0) level of Cui, as well as slightly lower values for the single donor and acceptor ionization levels (1999). Similar charging behavior is predicted for other transition metal dopants (Sachse et al. 1997a). For example, using total energy calculations with AIMPRO, Latham et al. have compared the ionization levels of different substitutional metal defects in silicon (2005); their values are presented in Table 8.1. The ionization levels of metal vacancy and interstitial defect complexes are not well characterized. The CuSiCui defect likely exists in the (+1) and (0) charge states with a defect level within 0.1–0.2 eV of the VBM (Estreicher 2004; Estreicher et al. 2005; Estreicher et al. 2003). While Oi in silicon is electrically neutral (Ballo and Harmatha 2003), complexes containing oxygen atoms as well as vacancies and self-interstitials can assume an assortment of charge states. For instance, the simple complex formed

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241

Table 8.1 Ionization levels of several transition metal substitutional defects in Si as calculated by (Latham et al. 2005). Note that the defects possess fewer donor ionization levels and more acceptor ionization levels as the periodic table group increases from 10 to 12. Defect

(+1/+2)

(0/+1)

(0/–1)

(–1/–2)

PdSi PtSi CuSi AgSi AuSi ZnSi

0.140 0.09 – – – –

0.31 0.33 0.207 0.37 0.35 –

0.88* 0.87* 0.478 0.55* 0.542* 0.28

– – 0.933* – – 0.58

All values are in eV and referenced to the VBM. The values with an asterisk were originally referenced to the CBM but have been altered with a value of 1.1 eV for Eg of Si at 300 K.

from Oi and Sii gives rise to a (+1/0) ionization level at Ev + 0.40 eV (Pinho et al. 2003). DLTS experiments reveal that the OiOiSii complex exhibits negative-U behavior with a (+2/0) ionization level at Ev + 0.255 eV (Markevich et al. 2005) and a single acceptor or (0/–1) ionization level at Ev + 0.99 eV (Lindstrom et al. 2001). The “A center” or OVSi defect is stable in the (0), (–1), and (–2) charge states, although experimental and computational work disagrees over the locations of the ionization levels for the defect. Electron spin resonance (Watkins and Corbett 1961) and gamma irradiation (Makarenko 2001) experiments identify a (0/–1) ionization level at Ec – 0.17 ± 0.01 (or Ev + 0.93 ± 0.01 eV at 300 K) and Ec – 0.155 ± 0.005 eV (or Ev + 0.945 ± 0.005 eV at 300 K), respectively. Using density functional theory, however, Pesola et al. obtain values of Ev + 0.40 eV and Ev + 0.53 eV for the (0/–1) and (–1/–2) ionization levels, in that order (1999). The large discrepancy is attributed to the underestimation of the band gap when using the LDA; the authors utilized a different “local mass” approximation (LMA) for purposes of comparison and saw the (0/–1) level raise from 0.4 to 0.7 eV above the VBM. The similar defect formed from two, instead of one, oxygen vacancy possesses similar charge states. The single acceptor ionization level of OVSiVSi is placed at Ev + 0.63 eV using DLTS (Alfieri et al. 2003) and Ev + 0.34 eV using DFT (Pesola et al. 1999). These same authors obtained values of Ev + 0.39 and Ev + 0.87 eV for the (–1/–2) ionization level using the identical techniques. Once again, large discrepancies are observed between the experimental values and those calculated using the LDA. The hydrogen interstitial in silicon exhibits negative-U behavior; it experiences a direct transition from the (+1) to the (–1) charge state (Chang and Chadi 1989). Van de Walle et al. addressed the stability of the various charge states of the isolated hydrogen defect using first-principles calculations (1989). According to their work, the (+1/0) and (0/–1) ionization levels are at Ev + 0.9 eV and Ev + 0.5 eV, yielding a negative-U correlation energy of –0.4 eV. The levels of Hi have also been determined experimentally; Herring et al. obtained similar single donor and acceptor ionization levels of Ev + 0.94 eV and Ev + 0.48 eV via time-resolved capacitancetransient measurements (2001). Although HVSi has been investigated using EPR (Stallinga et al. 1998), only the neutral charge state of the defect appears to have

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been discussed and no mention is made of ionization levels within the band gap. On the other hand, HVSiVSi is stable in three charge states ((+1), (0), and (–1)) and H2VSiVSi in two charge states (only (0) and (–1)) (Coutinho et al. 2003; Lévêque et al. 2001), which suggests the existence of multiple charge states for HVSi. 8.1.1.3 Diffusion Transient enhanced diffusion (TED) results from the migration and dissolution of point defects and defect complexes and, in turn, critically affect device dimensions during integrated circuit fabrication via ion implantation and annealing (Fair and Weber 1973; Yu et al. 2004). The diffusion of many common n- and p-type dopants is faster than silicon self-diffusion, as evidenced in Fig. 8.4. Silicon is often doped p-type with boron, a relatively light species that suffers from channeling during implantation, TED, and clustering through complex defect interactions

Fig. 8.4 Temperature dependence of the diffusion coefficient of foreign atoms in silicon compared with self-diffusion. Solid lines represent diffusion data of elements that are mainly dissolved substitutionally and diffuse via the vacancy or pair diffusion mechanism. Long dashed lines illustrate diffusion data for hybrid elements, which are mainly dissolved on the substitutional lattice site, but their diffusion proceeds via a minor fraction in an interstitial configuration. The short dashed lines indicate the elements that diffusion via the direct interstitial mechanism. Reprinted figure with permission from Bracht H, “Diffusion mechanisms and intrinsic pointdefect properties in silicon,” MRS Bulletin Vol. 25, No. 1 (2000) MRS Bull p. 24, Fig. 2. Reproduced by permission of the MRS Bulletin.

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243

during thermal annealing (Duffy et al. 2005). For heavily p-type material, boron exists as Bi+1, interstitial silicon as Sii+2, and the substitutional boron-interstitial complex as (BSiSii)+1. Much controversy has surrounded the mechanism by which these defects diffuse; recently, this laboratory has employed maximum likelihood and Monte Carlo techniques to conclude that the “kick-out” mechanism dominates over the pair diffusion mechanism (Jung et al. 2004). The diffusion of group V atoms such as phosphorus, arsenic, and antimony in silicon has also been a subject of much controversy (Richardson and Mulvaney 1989; Fahey et al. 1989). Phosphorus has a high diffusivity in silicon and is often employed in conjunction with boron doping to produce p-n junctions for microelectronics device applications. For n-type material, depending on the degree of n-type doping, the isolated vacancy is stable as VSi0, VSi–1, or VSi–2, interstitial phosphorus as Pi0 or Pi–1, and the substitutional phosphorus-vacancy complex, or E center, as (PSiVSi)–1. Phosphorus, an atom with a small atomic radius, migrates in Si via an interstitial mechanism, while arsenic prefers to diffuse by a vacancy-mediated mechanism. The diffusion mechanisms associated with impurities in silicon are best understood in terms of the atomic radius, valence, and electronegativity of the dopant atom. In order to compensate lattice strain, small dopants (B, P) attract self-interstitials and repel vacancies, whereas bigger dopants (As, Sb) are more likely to attract vacancies than self-interstitials (Bracht 2000). Group III elements typically diffuse by an interstitial-mediated mechanism, whereas Group V elements more commonly diffuse via vacancies. Impurities can diffuse via vacancies regardless of whether or not a bonding interaction exists between the two species. If no impurityvacancy bonding occurs, then the impurity moves whenever it exchanges places with the vacancy due to random statistical motions. When the vacancy and impurity are tethered to each other (by either long or short range forces), then the impurity and vacancy have a greater probability of exchanging with each other so that the impurity can move. For instance, the binding energy of the silicon vacancy with a first-nearest As or Sb atom is 1.17 or 1.45 eV; similar values for B and P are only 0.17 and 1.05 eV (Nelson et al. 1998; Bunea and Dunham 2000). When the binding energy is greater than ~1 eV, pair or E-center diffusion may occur, where the vacancy moves out to a third neighbor site and returns along a different path, overcoming an exchange barrier (EB) between lattice sites, as illustrated in Fig. 8.5. Nelson et al. focused on the role of dopant-vacancy pair migration energy barriers in deciding the dominant mechanism of diffusion. These energies vary widely in the literature, although Nelson et al. suggest that they are proportional to the valence and the atomic size. The Pauling electronegativity, the measure of an atom’s tendency to transfer charge during bonding, may also affect the propensity for an impurity to tether to a vacancy and thus play a role in vacancy-dopant atom pair diffusion. Arsenic and antimony have large electronegativities, in addition to large atomic radii and small exchange barriers, and thus prefer to diffuse by a vacancy-mediated mechanism. The interplay between these quantities plays out even between the two species; it is suggested that interstitials mediate 40% of As diffusion versus a mere 2% of Sb diffusion in as-grown material (Bracht 2000). The proportions would probably change in implanted material due to the excess of

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8 Extrinsic Defects

Fig. 8.5 Schematic illustration of simple and pair (E center) vacancy-mediated diffusion mechanisms in silicon. The position of the dopant atom (D) and vacancy are highlighted by the letter D and the numbered atoms, respectively. In the case of E-center diffusion, the vacancy can move out to a third neighbor site and return along a different path, overcoming an exchange barrier between lattice sites. For simple diffusion, the vacancy exchanges with the dopant and then continues through the lattice to exchange with other dopants. Reprinted with permission from Nelson JS, Schultz PA, Wright AF (1998) Appl Phys Lett 73: 247. Copyright (1998), American Institute of Physics.

interstitial atoms. It is not surprising that nitrogen, with its comparatively huge electronegativity, behaves in an unusual manner. In fact, nitrogen prefers to form the donor defect-silicon vacancy complexes that were discussed in Sect. 8.1.1.1. Most early work on boron diffusion explained the phenomenon in terms of a “pair diffusion” mechanism, in which a boron and a silicon atom diffuse together as a bound complex (Orlowski 1988; Mulvaney and Richardson 1987; Morehead and Lever 1986). This hypothesis arose from surface oxidation experiments in which Si interstitials were found to enhance boron diffusion (Agarwal et al. 2000). It is well known that Sii mediates boron motion however, until recently, the precise nature of the relevant complex and its diffusion path were not agreed upon. The evidence available in the literature did not exclude other possible mechanisms, such as the “kick-out” mechanism that often mediates diffusion in solids. For B-doped Si, kick-out envisions boron motion beginning when a free Si interstitial encounters a substitutional B atom and exchanges with it, leaving the boron in an interstitial position. The boron then moves rapidly in the interstices until it exchanges with another Si atom in the host lattice, thereby becoming substitutional and regenerating interstitial Si. At about the same time, experimental work by Cowern and co-workers (1991; 1992), bolstered by the DFT calculations of Nichols et al. (1989), revealed that kick-out provided a better description of pair diffusion than was originally conceived. This view persisted throughout most of the 1990s, until Windl et al. (1999) and Sadigh et al. (1999) simultaneously published two DFT-based reports that led to an explicit debate regarding the dominant mechanism. The relative importance of pair diffusion and kick-out has been addressed by Seebauer and co-workers using mathematical techniques drawn from systems engineering; they conclude that kick-out very likely dominates pair diffusion in both implanted and unimplanted Si (Jung et al. 2004). The experimental and computational work that supports the kick-out mechanism of boron motion over that of pair diffusion is described as follows. Watkins employed annealing studies with electron paramagnetic resonance (EPR) to obtain a value of 58 kJ/mol (0.6 eV) (1975). However, this number is actually transposed from other EPR measurements concerning spin alignment reported in the article.

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245

Table 8.2 Summary of atomic radii, electronegativity, binding energy, exchange barrier, and dominant diffusion mechanism for common n- and p-type dopants in silicon Ele- Atomic ElectroBinding Exchange Mecha- Reference ment Radius (pm) negativity Energy (eV) Barrier (eV) nism Si B Al

117 81 125

1.9 2.04 1.61

– 0.17 1.42

– 2.49 –0.30

– I I

Ga In N P As Sb

125 150 70 110 121 141

1.81 1.78 3.04 2.19 2.18 2.05

2.2(a) 1.73 1.05 1.17 1.45

3.72 3.6 4.44 1.05 0.65 –0.05

I I N2V I V V

– (Jung et al. 2004) (Krause et al. 2002; Schirra et al. 2004) (Melis et al. 2004) (Alippi et al. 2004) (Stoddard et al. 2005) (Bracht 2000) (Bracht 2000) (Gossmann et al. 1997)

The values for binding energy in the table correspond to the VSi-dopant first-neighbor configuration as calculated by (Nelson et al. 1998). The values for “exchange barrier” come from the same report.

A least-squares fit of the data shown in Fig. 10 of Watkins’ article computed by Jung et al. yields a value of 43 kJ/mol (0.45 eV) (2004). More recently, Collart et al. studied room-temperature diffusion of B after low-energy implants in Si (1998). By combining the results of their work with those of Cowern et al. (1991; 1992), Collart et al. derived a value of 39 kJ/mol (0.4 eV). Using DFT calculations, Zhu et al. offered an estimate of 29 kJ/mol (0.3 eV) (1996). However, this number represents only a difference in formation energy between the initial and final states of hopping, which is not necessarily the same as a true transition-state barrier. Also, the calculation concerns neutral Bi, while in p-doped material the boron interstitial is likely to be positively charged. Zhu reported 19 kJ/mol (0.2 eV) in a different set of calculations (1998). This result, however, also pertains to a formation energy difference rather than a true barrier. The text of the article does not clearly specify the charge state of Bi. Uematsu simulated B diffusion profiles according to the kick-out reactions, yet assumed that only neutral B interstitials contribute to B diffusion (1997). Also, they only accounted for Sii+1 and Sii0, choosing to ignore the concentration of Sii+2 that may become non-negligible for heavy p-type doping. As early as 1991, evidence for the dominance of the pair diffusion mechanism of boron migration in silicon began to reappear (Hane and Matsumoto 1993; Heinrich et al. 1991; Vandenbossche and Baccus 1993). Then, in 1999, two simultaneously published DFT works directly opposed the earlier findings in support of a “kick-out” mechanism. Windl et al. used two variants of the nudged elastic band method (NEBM) in concert with a monopole correction for charged systems to obtain barriers between 39 and 68 kJ/mol (0.4 and 0.7 eV) for pair diffusion (1999). Sadigh et al. used two somewhat different methods to calculate barriers of 66 and 71 kJ/mol (0.68 and 0.73 eV) (1999). Since that time, Allipi et al. have reported a barrier of 68 kJ/mol (0.7 eV), in substantial agreement with the 1999

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work (2001a; 2001b). This estimate is sufficiently close to that for producing interstitial boron by kick-out (~97 kJ/mol or 1 eV) that, depending upon the preexponential factors and other aspects of the kinetic network, the rates for the two mechanisms could be of the same order of magnitude at the high temperatures (~1,000°C) characteristic of device processing. Recently, Jung et al. readdressed the conflict over kick-out versus pair-diffusion (2004). They provide the best comprehensive assessment of the two mechanisms to date, where rate parameters for the relevant steps, calculated based on literature reports and physical arguments, are coupled with a Monte Carlo technique. Using the maximum likelihood method, they obtain an activation energy barrier of 0.37 ± 0.04 eV for Bi+1, 0.72 ± 0.03 eV for Sii+2, and 1.05 ± 0.07 eV and 0.5 ± 0.1 eV for the formation of Bi+1 from (BSiSii)+1 and vice versa, respectively. Jung et al. determined that for both intrinsic and doped material, interstitial diffusion contributes to the overall diffusive flux to a far greater extent than pair diffusion. Martin-Bragado et al. reveal how a physical “kick-out” mechanism can be merged with experimental parameters to accurately describe diffusion in doped-Si (2005). The migration energies used in the model for Bi–1 and Bi+1, 0.36 eV and 1.1 eV, are derived from the energy required to form the charged defect as a function of Fermi level and the activation energy of its contribution to the effective boron distribution; the values are in agreement with the ab initio calculations of Windl et al. Traditionally, the diffusion of phosphorus in silicon has been explained in terms of a P-vacancy complex in an assortment of charge states; the model proposed by Fair and Tsai is one such example (1977). It explained high-concentration phosphorus doping profiles in terms of diffusion with VSi0, VSi–1, and VSi–2 and describes the kink and tail regions in SIMS profiles as the result of (PVSi)–1 dissociation. Experimental support for this mechanism comes from investigating the effect of nitridation, which induces vacancy supersaturation, on P diffusion. P diffusion is retarded in intrinsic material, while the nitridation retardation effect is much less significant in strongly n-type material (4 × 1020 cm–3). Fahey et al. took this to indicate that the vacancy mechanism starts to contribute to P diffusion under heavily n-type conditions (1985). The prevalence of excess self-interstitials during the diffusion of phosphorus (a phenomenon that is observable experimentally) led to the rejection of this diffusion model. It is far more likely that phosphorus atoms couple to silicon interstitials, which leads to a supersaturation of Sii at the front and back of the diffusion zone (Gosele and Strunk 1979). Using the point defect injection method, Bracht suggested that the fractional contribution of interstitials to the migration mechanism under intrinsic conditions is in the range of 86–100% in the worst case scenario (2000). The PSii complex in the (+1) charge state serves the primary mediator of interstitial diffusion; its formation and activation energy for diffusion has been determined by an assortment of methods (Liu et al. 2003; Christensen et al. 2003; Ural et al. 1999; Haddara et al. 2000). In heavily n-doped material, the charged E-center, (PVSi)–1, contributes to diffusion in the bulk (Liu et al. 2003). In contrast to phosphorus diffusion, the diffusion of arsenic and antimony appears to be well described by a model in which E-centers dissociate instantaneously

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as they migration from regions of higher to lower dopant concentration. Significant mention has been made in the literature of the enhancement and retardation of diffusivity that has been calculated and observed to occur as a function of doping level. For very high doping levels (> 2 – 3 × 1020 cm–3), experimental work by Nylandsted Larsen et al. (1993) and Fair and Weber (1973), as well as KLMC simulations (Dunham and Wu 1995; Bunea and Dunham 2000; Haley and Gronbech-Jensen 2005), reveal high diffusivity. This behavior is thought to be related to the interaction of vacancies with more than one dopant atom. For example, Pankratov et al. used a kinetic lattice Monte Carlo model with a single arsenicvacancy pair and interactions extending from the 3rd to 20th nearest neighbor showed an increase in arsenic diffusivity with temperature (1997). On the other hand, Haley and Gronbech-Jensen obtained a maximum As diffusivity for concentrations in the range of 1018 to 1019 m–3, while concentrations greater and less yielded a lower diffusivity. By looking at the time-dependence of clustering effects, they suggest that, especially at higher temperatures, where mobile AsVSi pairs attract free vacancies, a diffusion model based on more than merely the AsVSi pair might be necessary. Transition metals have very small solubilities at room temperature; they tend to diffuse to the silicon surface during the cooling period following high temperature processing. They undergo one of two different diffusion mechanisms. Co, Ni, and Cu, for example, behave as fast diffusers and migrate via site-to-site hopping from interstitial site to interstitial site (Fig. 8.6) (Coffa et al. 1993). They have high diffusion coefficients (approximately 10–5 cm2/sec) and low activation energies for diffusion (< 1 eV). The position of the Fermi level in the bulk affects more than just the charge state of the diffusing species. Istratov et al. have shown that Fermi level position determines the sign and magnitude of the electrostatic charge on copper precipitates and affects whether copper precipitates in the bulk or diffuses to the surface (2000). In p-type Si, positively charged copper precipitates repel Cui+1 whereas, in n-type Si, negatively charged precipitates trap Cui+1. In contrast

Fig. 8.6 Interstitial diffusion model through the path Td → D3d → Td. Reprinted from Kamon Y, Harima H, Yanase A et al., “Diffusion mechanism of the late 3d transition metal impurities in silicon,” (2001) Physica B: Condensed Matter 308–310: 392. Copyright (2001), with permission from Elsevier.

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Table 8.3 Experimental and theoretical migration barriers for diffusion of neutral transition metal impurities in Si 3d TM Element

Sc

Experiment (eV)

3.20 1.5– 1.55– 0.79– 0.63– 0.43– 0.37 0.13– 0.18– 2.05 2.8 1.1 1.3 1.1 0.76 0.86 5.48 4.97 3.96 2.78 1.74 0.95 0.49 0.75 0.35

Calculation without lattice relaxation (eV) Calculation with lattice relaxation (eV) Most stable atomic site

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

3.80 3.76 3.06

2.07

1.17

0.30

0.25 0.31

0.35

Td

Td

Td

Td

D3d

D3d

Td

Td

D3d

Calculated migration barriers and stable atomic site of impurity are from (Kamon et al. 2001) while experimentally observed migration barriers are from (Weber 1983).

to the late 3d metal impurities, Zn, Pt, and Au undergo fast long-range diffusion as interstitials until they assume a substitutional configuration in the silicon crystal lattice via a kick-out mechanism. The experimental and theoretical migration barriers for all of these defects are compared and contrasted in Table 8.3. The large differences and chemical trend in the diffusion coefficients of the 3d transition metals are attributed to enhanced stability of certain atomic sites and lattice relaxations (Kamon et al. 2001). Abnormally rapid diffusion in Si during low temperature processing (T < 700ºC) has been attributed to the interaction of Oi with lattice vacancies, self-interstitials, metallic elements, carbon, and hydrogen, as well as dislocation locking (Senkader et al. 2001). Newman has published an extensive review on the diffusion and precipitation of oxygen in Czhochralski silicon (2000). The diffusion of isolated oxygen interstitials occurs by atomic jumps from bond-centered sites to one of the six equivalent adjacent sites. The activation energy for diffusion of neutral Oi is estimated to be 2.73 eV (Dzelme et al. 1999). Defects formed from oxygen interstitials, such as the oxygen di-interstitial, can be extremely fast diffusers in silicon (Murin et al. 1998). The diffusivity of OiOi is several orders of magnitude higher than that of Oi (Lee et al. 2001b); the defect is the main contributor to the transport of oxygen at temperatures below 700ºC. There appears to be no variation in the activation energy of this defect as a function of Fermi level in the bulk (and consequently, defect charge state) (Glunz et al. 2003). The migration of the A center, or OVSi pair, likely occurs by single atomic jumps that lead to the local rearrangement of the defect. The high rate of single diffusion jumps of oxygen atoms in electron irradiated Si above 300ºC is attributed to the sequential trapping and de-trapping of VSi (Oates and Newman 1986). The diffusion coefficient of hydrogen in silicon is high even at low temperatures (Pearton et al. 1987); additionally, hydrogen diffusion depends heavily upon interactions with native and impurity defects (McQuaid et al. 1991). Saad et al. have undertaken to model hydrogen diffusion in single crystal silicon including the formation of bound hydrogen near the Si surface, the presence of “slow” and “fast” contributions to diffusion, and the interaction of hydrogen with other defects in the bulk (2006). Hydrogen concentration profiles after processing (as measured

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by SIMS) are wellfit by a model where interstitial hydrogen in the (+1), (0), or (–1) charge state serves as the “fast” diffusing species in the bulk. “Slow” diffusion is governed by the metastable hydrogen dimer discussed in Sect. 8.1.1.1.

8.1.2 Germanium Similar to silicon, germanium can be doped p- and n-type by the addition of trivalent and pentavalent impurities; the defects induced by intentional doping are not as well characterized as those in silicon, however. Doping with metal atoms, with application to infrared radiation detectors and spintronics, has also been explored. In brief, spintronics, a new field of device technology that considers the spin, rather than just the charge, of an electron, may revolutionize the storage, processing, and transmission of digital information. The vacancy-oxygen complex, also referred to as the A center, can be induced by irradiation with high-energy particles in oxygen-rich Ge. Interstitial hydrogen is probably the most common impurity in nominally pure crystals of Ge, as hydrogen is the only atmosphere suitable for the growth of high purity material (Hiller et al. 2005). Carbon doping is known to suppress the fast thermally activated phosphorus diffusion that prevents the formation of shallow source/drain p-n junctions (Luo et al. 2005). Vacancy complexes involving oxygen, hydrogen, carbon, and group V impurity atoms such as B, P, As, Sb, and Bi have been identified and studied. 8.1.2.1 Structure In germanium, almost all dopants occupy substitutional lattice sites. For example, in boron-doped Ge, the (0/–1) ionization level of BGe is right at the valence band maximum; the formation energy of BGe–1 is 0.72 eV at Ev and continues to decrease for rising Fermi energies within the band gap (Delugas and Fiorentini 2004). The higher formation energies (on the order of 4 eV) observed for boron interstitials depend upon defect geometry and Fermi level. The defect prefers to adopt a tetrahedral configuration for the (+1) and (0) charge states and a hexagonal configuration for the (–1) charge state. The structure of the BGeGei pair has been studied in the context of boron diffusion paths in Ge. The lowest energy configuration of (BGeGei)+1 has C3v symmetry with BGe coupled to Gei sitting on the trigonal axis towards the tetrahedral site. In the neutral charge state, the boron atom moves closer to the substitutional site. Jahn–Teller distortions to C1h symmetry occur for the (–1) charge state of the defect, which adopts a true <110> split-interstitial configuration (Janke et al. 2007). A pair comprising a p-type dopant atom and a germanium vacancy, InVGe, has also been identified experimentally. According to one PAC study, the defect may consist of an In atom on a tetrahedral interstitial site with a Ge vacancy trapped on the substitutional position along the <111> direction at the double nearest-neighbor distance (Feuser et al. 1990).

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In n-type Ge, little mention is found in the literature of isolated interstitial and substitutional defects. Even though the charging of the donor-VGe pair or E center is discussed frequently, its charge state-dependent geometry and relaxations appear not to have been explored. A few investigators have looked at the binding energy of the defect (Chroneos et al. 2007b; Chroneos et al. 2007a). The defects induced by the doping of Ge with transition metals such as Co, Ni, V, Fe, Mn, and Cr have been investigated. It is energetically favorable for these transition metal species to occupy substitutional sites in the crystal lattice. The low vacancy formation energy in Ge has been invoked to explain this phenomenon (Luo et al. 2004). As mobile Mn reacts with VGe to form MnGe, the formation energy of the vacancy directly affects the concentration of MnGe in the bulk. When the formation energy of vacancies in the bulk is higher (as it is for VSi, for instance), Mn prefers to occupy an interstitial site in the crystal lattice. The substitutional configuration leads to small (2–3%) bond length contractions with respect to the ideal Ge-Ge distance (Continenza et al. 2006). Although isolated interstitial sites are energetically unfavorable, substitutional-interstitial and substitutional pair complexes may occur in the bulk. Little information exists about the charge-state dependent geometries of any of these defects. Oxygen impurities exist as both isolated interstitials and oxygen OiVGe pairs in germanium. The latter have a comparatively high formation energy (3.05 eV versus 1.19 eV), and are only expected to exist in high concentrations at high temperatures or in the presence of a non-equilibrium concentration of vacancies in the bulk (Coutinho et al. 2000). According to these same authors, interstitial oxygen has comparable formation energies for structures with C2, C1, and C1h point-symmetry; the D3d structure has energy 0.1 eV higher than the others. Di-interstitial oxygen defects have not been identified in Ge, although defect clustering is expected to occur, much as it does in silicon. The OiVGe pair is formed when the oxygen atom moves away from a tetrahedral site and bridges to host atoms; it has C2v point-symmetry in its two stable charge states, (0) and (–1). Excess hydrogen in the germanium crystal lattice results in isolated point defects as well as HiGei and HiVGe pairs. Khoo and Ong considered the relative stability of interstitial atomic hydrogen, protons, and molecular hydrogen in Ge (1987). The isolated hydrogen impurity prefers a bond-centered site in the (–1) charge state rather than the tetrahedral site that Hi in Si adopts (Dobaczewski et al. 2004). The hydrogen-Gei complex has been investigated via infrared adsorption measurements and ab initio calculations (Budde et al. 1998). The split-interstitial with a hydrogen attached to one of the two equivalent core atoms is equivalent in energy for both the <110> and <100> configuration of Gei. After relaxation, both forms have C1h symmetry. HiHiGei adopts C2v and C2 symmetry when created from the <110> and <100> split-interstitial, respectively. Hydrogen can also saturate the dangling bonds of the vacancy defect in germanium, forming a HiVGe complex. Jahn–Teller distortions lead to symmetry-lowering relaxations for (HiVGe)0 and consequent C1h point-symmetry (Coomer et al. 1999). Hydrogen can also form more complicated defect complexes in Ge; it can bind to an already present substitutional atom in the lattice such as carbon or oxygen (Oliva 1984;

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Wang and C. 1973). In fact, the (OH)+1 pair is more stable than either Oi or Hi in germanium (Deak et al. 1993). 8.1.2.2 Ionization Levels Investigation of the charged defects in p-type Ge using DLTS or minority carrier transient spectroscopy is not easy, as it is difficult to form good rectifying junctions to p-type material of resistivity < 10 Ω·cm (Lindberg et al. 2005). Delugas and Fiorentini utilized DFT with the LDA to determine ionization levels of BGe, Bi, and BGeGei (Delugas and Fiorentini 2004). Substitutional boron has a (0/–1) ionization level lying very near the valence band maximum (with a suggested error of ± 0.05 eV); the defect exists in the (–1) charge state for all Fermi energies within the band gap. In contrast, Bi has three stable charge states in the tetrahedral configuration ((+1), (0), and (–1)) and two for the hexagonal ((0) and (–1)) and bond-centered ((+1) and (0)) configurations, respectively. The (+1/0) and (–1/0) ionization levels of defect are located at Ev + 0.22 eV and Ev + 0.27 eV. BGeGei exists in the (+1) charge state for Fermi energies from the VBM to Ev + 0.27 eV, and in the neutral charge state from Ev + 0.27 eV to the CBM, with the slight possibility that that (–1) is stable for Fermi energies very close to the CBM. Zistl et al. were able to identify a level at Ev + 0.33 eV in Ge implanted with 111In atoms using DLTS by fabricating a Schottky diode structure (Zistl et al. 1997). This level was attributed to the InVGe pair. DLTS experiments on Ga-doped Ge yield no conclusive extrinsic defect ionization levels (Christian Petersen et al. 2006). Positively charged donors bind to negatively charged germanium vacancies, increasing their concentration by raising the position of the Fermi level (Mitha et al. 1996). The donor-vacancy pair in Ge is stable in an assortment of charge states. The existence of an ionized defect comprising a phosphorus, arsenic, or antimony atom has been considered. The defect definitely exists in the (0), (–1), and (–2) charge states within the band gap; recently, a single donor charge state has also been suggested (Lindberg et al. 2005). Early Hall effect measurements on irradiated n-type Ge crystals revealed an energy level at Ec – 0.20 eV (or Ev + 0.47 eV at 300 K) that was associated with the donor-vacancy pair (Tkachev and Urenev 1971). Most DLTS studies agree roughly upon the positions of the single and double acceptor ionization levels, as evidenced in Table 8.4. The high-resolution Laplace DLTS investigation of Lindberg et al. finds an anomalous ionization level close to the valence band maximum, which the authors attribute to the single donor level of the SbVGe pair. They justify this assignment based on the observed dependence of the emission rate on the electric field, as well as the relatively small capture cross section of the defect. Recent ab initio density-functional calculations identify a comparable (+1/0) level for Sb3VGe at Ev + 0.1 eV, as well as acceptor levels for Sb2VGe and Sb3VGe (Coutinho et al. 2008). The charging and ionization levels of copper defects in germanium will be discussed in greater detail, as comparable information for other transition metal defects is scarce. CuGe has four stable charge states within the band gap, (0), (–1), (–2),

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Table 8.4 Experimentally determined ionization levels of the E center in P-, As-, and Sb-doped germanium Defect

(+1/0)

(0/–1)

(–1/–2)

Reference

PVGe

– –

AsVGe

– – –

0.348 0.32* 0.348 0.334 0.47* 0.307

– – 0.377 – – 0.46–0.48*

0.095 ± 0.006 – – – – –

0.309 ± 0.007 0.32* 0.30* 0.30* 0.27* 0.307

– – – 0.29* – 0.293

(Markevich et al. 2004b) (Nagesh and Farmer 1988) (Peaker et al. 2005) (Markevich et al. 2004b) (Fukuoka and Saito 1982) (Markevich et al. 2004a; Markevich et al. 2004b) (Lindberg et al. 2005) (Nagesh and Farmer 1988) (Fage-Pedersen et al. 2000) (Nyamhere et al. 2007) (Fukuoka and Saito 1982) (Peaker et al. 2005)

SbVGe

All values are in eV and referenced to the valence band maximum. The values accompanied by an asterisk were originally referenced to the CBM and corrected with a value of 0.67 eV for Eg of Ge at 300 K.

and (–3), while Cui has two, (+1) and (0). Early Hall effect measurements revealed single, double, and triple acceptor ionization levels at 0.04 eV, 0.33 eV, and 0.41 eV above the valence band maximum, respectively (Woodbury and Tyler 1957). Using DLTS, the (–2/–3) ionization level was identified at an identical position of Ec – 0.259 eV (or Ev + 0.401 eV at 300 K) (Clauws et al. 1989). A photoinduced current transient spectroscopy study performed by Blondeel and Claws placed the (0/–1) and (–1/–2) ionization levels at Ev + 0.037 eV and Ev + 0.322 eV, in that order (1999). The ionization levels of other transition metal impurities have been studied using DLTS and other experimental methods. For instance, the single and double acceptor levels for FeGe have been identified at Ev + 0.134 eV (Glinchuk et al. 1959) and Ev + 0.40 eV (Tyler and Woodbury 1954). Clauws et al. (2007) and Forment et al. (2006) report an assortment of acceptor ionization levels for Ti, Cr, Fe, Co, Ni, and Hf-doped Ge. Quenching studies by Kamiura et al. revealed a deep acceptor energy level of Ev + 0.08 eV in copper-contaminated material, and observed a disappearance of the defect following annealing at 260ºC (1984). The authors attributed the level to a pair consisting of a substitutional copper atom and mobile defect (C, Si, O), rather than a vacancy-copper interstitial or vacancy-selfinterstitial pair. The oxygen-VGe pair or “A center” has three relevant charge states within the band gap: (0), (–1), and (–2). Ito et al. (1986) and Markevich et al. (2003) utilized DLTS to obtain values of Ev + 0.22 eV and Ev + 0.27 eV for the single acceptor ionization level of the defect. Litvinov et al. studied the absorption spectra of Ge crystal enriched with oxygen isotopes and placed the same (0/–1) ionization level at Ev + 0.25 ± 0.03 eV (2002). Experimental studies indicate that the (–1/–2) ionization level is 0.20–0.27 eV below the CBM, or 0.39 to 0.47 eV above the VBM at

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300 K (Fukuoka et al. 1991; Fukuoka et al. 1983; Goncharov et al. 1972; Markevich et al. 2002; Markevich et al. 2006; Nagesh and Farmer 1988). The charge states associated with hydrogen impurities in Ge are not well established. Computational work suggests that the (+1/–1) level of the tetrahedral hydrogen interstitial lies at the valence band (Van de Walle 2003; Denteneer et al. 1989; Van de Walle and Neugebauer 2003). Consequently, for all Fermi energies within the band gap, Hi exists in the (–1) charge state. A metastable trigonal configuration assigned to bond-centered hydrogen has a single donor level at Ev + 0.101 ± 0.002 eV identifiable via Laplace DLTS; the single acceptor level of this defect is predicted to lie close to the VBM (Dobaczewski et al. 2004). Earlier self-consistent calculations suggested that interstitial H possessed a single donor-like state in the valence-band region (Pickett et al. 1979). Using local-density-functional pseudopotential theory, Coomer et al. identified acceptor (0/–1) levels for HVGe, H2VGe, H3VGe all within 0.35 eV of the valence band edge (1999). HVGe and H3VGe also possess single donor (+1/0) levels close to the valence band. Pairs formed from hydrogen atoms and other impurities such as carbon and oxygen can also assume multiple charge states (Oliva 1984; Navarro et al. 1987; Kahn et al. 1987). 8.1.2.3 Diffusion Understanding the diffusion mechanisms of extrinsic dopants in germanium is critical to the microelectronics industry; low-leakage junctions are crucial for successful device applications (Satta et al. 2005). In many cases, the diffusion of common dopants in germanium differs greatly from that in silicon; for example, boron diffuses much slower in Ge than in Si. The migration of transition metals (especially copper) in Ge has been investigated in the context of interconnect metals. As with silicon, the behavior of native impurities such as oxygen and hydrogen introduced by Czochralski growth have been studied. Although the predominant diffusion mechanism of group III and V impurities in Ge has been called into question by several authors (Mitha et al. 1996), substantial support now exists for a vacancy-mediated mechanism for all dopants and impurities other than boron and phosphorus. At typical diffusion temperatures, group III impurity atoms exist in the (–1) charge state and group V atoms in the (+1) charge state. The diffusivities of the latter (including P, As, Sb, and Bi) are about two orders of magnitude higher. Group III elements do not diffuse via neutral or positively charged complexes; as an example consequence, gallium diffusion in Ge is unaffected by doping (Riihimaki et al. 2007). The anomalously slow diffusion of B in Ge (Uppal et al. 2001) can be attributed to the high exchange barrier between B and VGe. This behavior resembles boron diffusion in silicon (Nelson et al. 1998), for which an alternate interstitial pathway therefore dominates diffusion. A SIMS investigation of B-implanted material yielded an activation energy of diffusion of 4.65 ± 0.3 eV in the temperature range of 800–900ºC (Uppal et al. 2004). Phosphorus exhibits a strong elastic interaction with native

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Fig. 8.7 Numerical fits to the As concentration profile obtained using SIMS depth profiling after annealing at 600ºC for 5 min from (Vainonen-Ahlgren et al. 2000). Fits are calculated for different combination of charge states for Ge vacancies. Reprinted with permission from Vainonen-Ahlgren E, Ahlgren T, Likonen J et al. (2000) Appl Phys Lett 77: 691. Copyright (2000), American Institute of Physics.

interstitials and migrates via an interstitial mechanism (Luo et al. 2005). The diffusion of P in bulk Ge can be slowed by the addition of carbon, which indicates that phosphorus and substitutional atoms compete to “capture” Ge self-interstitials. The diffusion of As, Sb, and Bi is mediated by dopant-vacancy pairs or E-centers and occurs much faster (Riihimaki et al. 2007). The attractive long range Coulomb attraction between the dopant and the vacancy defect is great enough that the complex resists dissociation while diffusing through the crystal lattice. For example, as illustrated in Fig. 8.7, arsenic atoms diffuse by binding to Ge vacancies in the (0) and (–2) charge states (Vainonen-Ahlgren et al. 2000). The fitting of experimental diffusion profiles provides activation energies of diffusion for P, As, and Sb of 2.07, 3.32, and 2.28 eV (Chui et al. 2003). The diffusion of transition metals in Ge has also been investigated. Copper diffuses in Ge via two different migration modes; one is VGe-controlled and the other is Cui-controlled (Bracht 2004). When the concentration of Ge vacancies in the bulk is small (in virtually dislocation-free material, for instance), the conversion of Cui to CuGe is controlled by the supply of vacancies from the surface. This phenomenon is interesting, as the surface pathway, rather than spontaneous creation of vacancies within the bulk, governs the conversion of Cui to CuGe. If the concentration of vacancies in the bulk is high (when process steps are not carried out to minimize the number structural defects, such as dislocations, in the bulk), the conversion of Cui is controlled by the supply of Cui. The interstitial-controlled mode of diffusion via the dissociative Frank–Turnbull mechanism also governs the migration of Ag, Au, and Ni. A SIMS study of the diffusion of tin in doped Ge indicates that the metal atom diffuses via a vacancy-mediated mechanism in the temperature range of 555–930ºC (Friesel et al. 1995). The retardation of tin diffusion as the Ga (acceptor) doping concentration increases indicates that migration is likely mediated by negatively charged SnVGe complexes (Riihimaki et al. 2007).

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Fig. 8.8 The diffusion path of interstitial oxygen in the germanium lattice. The position of the two nearest neighboring germanium atoms is added. It can be seen that the oxygen jumps from one interstitial position to the next. Reprinted from Lauwaert J, Hens S, Spiewak P et al., “Simulation of point defect diffusion in germanium,” (2006) Physica B 376–377: 260, Fig. 4, Copyright (2006), with permission from Elsevier.

The oxygen interstitial present in high purity single crystal germanium diffuses quickly compared to most other impurities. The migration path can be described by a succession of hops between interstitial positions; as illustrated in Fig. 8.8, the saddle point of the diffusion barrier is not always on a fixed position, and the diffusion path is extended (Lauwaert et al. 2006). The values of the activation energy and preexponential factor for diffusion calculated using a quantumchemical totalenergy method are 2.05 eV and 0.39 cm2/s, respectively (Gusakov 2005). The diffusion of H in silicon has been explored, but not with much reference to defect charge state (Frank and Thomas 1960; Gusakov 2006). Experiments indicate that the diffusion of atomic hydrogen is limited by trapping due to impurity oxygen atoms and dimers (Pokotilo et al. 2003).

8.1.3 Gallium Arsenide Gallium arsenide is frequently doped with group IV and group VI impurity atoms, as well as transition metals, in order to alter its electronic behavior. For concentrations greater than 1018 cm–3, silicon doping causes extrinsic point defects and segregation that can adversely influence device performance (Hubik et al. 2000). Group VI elements such as tellurium, oxygen, and sulfur compensate shallow

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acceptors and donors and act as non-radiative recombination centers that reduce the photoluminescence efficiency in the bulk (Huang et al. 1996). Magnesium and beryllium have been investigated as possible p-type dopants, although the relatively large ionization potential of Mg makes it difficult to achieve p-type conductivity at room temperature; information about charged Mg- and Be-related defects is almost non-existent. The diffusion of zinc from the vapor phase into n-doped GaAs is used to produce abrupt p-n junctions and heterojunctions; it is one of the most commonly used p-type dopants due to its high solubility, fast diffusion, and low ionization energy (Luysberg et al. 1992; Bracht and Brotzmann 2005). Doping with other transition metals such as Mn, V, Cr, Fe, Co, and Ni is undertaken to form dilute-ferromagnetic semiconducting devices (Goss and Briddon 2005; Mahadevan and Zunger 2004). For spintronics applications, GaxMn1–xAs is known to have the highest Curie transition temperature (100–160 K) of all the magnetic semiconductors and holds particular interest (Frymark and Kowalski 2005). 8.1.3.1 Structure When GaAs is doped with group IV atoms such as Si, Sn, and Ge, isolated substitutional defects form, as well as VGa complexes. Foreign atoms that occupy Ga lattice sites become shallow donors, while those that occupy As lattice sites become shallow acceptors. This “amphoteric” behavior, where a dopant can reside on either an anion or a cation lattice site, reduces the doping efficiency of silicon (Domke et al. 1998). For instance, an isolated donor such as SiGa+1 partially compensates for acceptor species in the bulk. Complexes of dopant atoms and native gallium vacancies are typically deep acceptors. These species are also invoked to explain the electrical deactivation that occurs in Si-, Sn-, and Ge-doped GaAs. Little information exists about the charge state-dependent geometries of extrinsic defects in bulk GaAs. One computational study suggests that (SiGaVGa)–2 consists of a SiGa+1 defect and a VGa–3 defect at the second-nearest-neighbor separation of 4 Å (Northrup and Zhang 1993). Larger defects clusters have also been discovered in the bulk. For example, Ashwin et al. found three localized vibrational modes in heavily Si-doped GaAs, corresponding to a more complicated, perturbed SiGaVGa electron trap involving a second Si atom or Si vacancy (1997). Newman et al. had previously attributed these modes to a deep electron trap with a planar structure such as VGaSiAsAsGa (1994). Group VI atoms such as Te, Se, S, and O, also act as amphoteric dopants in gallium arsenide; isolated dopant atoms on As sites behave as shallow donors, whereas VGa-dopant complexes serve as acceptors (Hurle 1999). Tellurium is a somewhat novel donor, as it causes “superdilation” of the lattice, or a dramatic increase in the lattice parameter with increased doping, at levels above 2 × 1018 cm–3 (Kuznetsov et al. 1973b; Mullin et al. 1976). Similar superdilation occurs in GaAs doped with Se (Kuznetsov et al. 1973a) and S (Venkatasubramanian et al. 1989). Transition metals form interstitial defects, as well as substitutional point defects in bulk GaAs. For instance, zinc can incorporate into GaAs as either a substitutional

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acceptor or an interstitial donor. Goss and Briddon have used local-spin-densityfunctional techniques to understand the structure of acceptor antisite defects in GaAs (Goss and Briddon 2005). ZnGa exhibits Td-point symmetry and experiences a 7% contraction in volume upon relaxation. Although MnGa also has Td symmetry, it experiences a far larger volumetric expansion of 18%. The Mn ion occurs in three different electronic configurations in doped-GaAs: (i) neutral Mn acceptor, Mn 3d5 inner-shell electrons with a weakly bound valence band hole (ii) ionized Mn acceptor, 3d5 state, and (iii) neutral Mn in 3d4 configuration (Sepega et al. 2002). According to Sapega et al., the latter of those states contributes most to the unique ferromagnetic behavior of the compound. Interstitial donors formed from transition metals may play an important role in diffusion in GaAs (van Gisbergen et al. 1991a, b). The interstitial zinc donor, which exists in the (+1) charge state, forms from Gai+2 and ZnGa–1. Hwang also predicted the formation of complexes of the type (ZnAsVAs)+1 in Zn-, Cd-, or Mn-doped material (Hwang 1968). This complex may form at high doping levels, when the concentration of Gai–2 (a participant in the formation of the complex) in the bulk increases dramatically (Shamirzaev et al. 1998). 8.1.3.2 Ionization Levels As the charged defects that arise in GaAs doped with Si, Sn, and Ge are all quite similar (Hurle 1979; Krivov et al. 1983), Si-doped material will be discussed in detail, as the lower atomic weight of the dopant allows for the characterization of charged defects using additional experimental techniques including local vibrational mode and Raman spectra; the reader is referred to a comprehensive review article on GaAs for more detail about Sn- and Ge-induced defects (Hurle 1999). SiGa exists in the (+1) charge state for all Fermi energies within the band gap, while SiAs takes on the (–1) charge state. SiAs–1 is observable using a combination of positron lifetime spectroscopy and scanning tunneling microscopy (Gebauer et al. 1997). Complexes formed from isolated substitutional defects are also charged. The complex formed from the charged arsenic vacancy, SiAsVAs, adopts a (+1) charge. SiGaVGa, on the other hand, is predicted to occur in both the (–1) and (–2) charge states. The charge compensation mechanism that restricts n-type doping with Si to 5 × 1018 cm–3 can be explained in terms of the formation of these defect complexes from point defects (Laine et al. 1996; Domke et al. 1996). Evidence for the former comes from photoluminescence experiments (Ky and Reinhart 1998). The behavior of SiGaVGa has been predicted computationally using DFT within the local density approximation (Northrup and Zhang 1993). (Si–1 is relevant under Ga-rich conditions, while (SiGaVGa)–2 occurs in high GaVGa) concentrations under As-rich conditions. The (–1/–2) ionization level of SiAsVGa has been identified at Ev + 0.54 eV via localized vibrational mode infrared absorption measurements of heavily Si-doped GaAs (Kung and Spitzer 1974). Figure 8.9 shows the equilibrium defect concentrations as a function of the Si concentration up to the Si solubility limit.

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Fig. 8.9 Concentrations of charged Si-induced defects in GaAs as a function of the total silicon concentration for the As-rich limit at T = 940ºC. Reprinted figure with permission from Northrup PA, Zhang SB (1993) Phys Rev B: Condens Matter 47: 6793. Copyright (1993) by the American Physical Society.

At least three different charged defects are known to exist in Te-doped GaAs: TeAs+1, TeAsVGa, and (TeAsVGaVAs)0. A thermodynamic model incorporating these defects explains both the compensation and superdilation in the material (Hurle 1999). Spectroscopy (Williams 1968) and positron annihilation (Krause-Rehberg et al. 1994) experiments provide direct evidence for the TeAsVGa complex. As the concentration of Te in the bulk increases, the probability of TeAsVGa formation increases, as does the concentration of TeAs+1 and VGa (Frigeri et al. 1997). Gallium vacancies exist in the (0), (–1), (–2), and (–3) charge states within the bulk; consequently, it is likely that the TeAsVGa complex has one, if not more, ionization levels within the band gap. Nishizawa et al. assigned a value of Ev + 0.18 eV to the (0/–1) ionization level of the defect complex based on photoluminescence data (1974). As discussed in Chap. 4, even though the principle of charge additivity may be invoked, there is no correspondence between the (0/–1) ionization level of TeAsVGa and that of VGa, which lies at Ev + 0.10 eV according to maximum likelihood estimation. The stress dependence of the photoluminescence band at 0.95 eV observed in Te-doped GaAs led to the identification of (TeAsVGaVAs)0 (Reshchikov et al. 1995). The carrier concentration in selenium- and sulfur-doped GaAs also saturates at a value slightly below 1 × 1019 cm–3; it is proposed that a similar process of compensation by dopant-vacancy complexes occurs in these materials (Wolfe and Stillman 1975; Vieland and Kudman 1963; Williams 1968; Gutkin et al. 1995). In sulfur-doped GaAs, an EL3 defect related to SAs+1 associated with an energy level at Ev + 0.56 eV is detectable using DLTS (Yokota et al. 2000). Substitutional defects such as ZnGa and MnGa have a (0/–1) ionization level close to the valence band maximum. The formation energy of ZnGa in the (0) charge state becomes higher than that of the (–1) charge state at Ev + 0.024 eV according to Hall effect measurements (Su et al. 1971), although a substantially different value of Ev + 0.42 eV has been put forth based on photoluminescence experiments (Shamirzaev et al. 1998). The comparable ionization level for MnGa is Ev + 0.11 eV (Zhang and Northrup 1991; Goss and Briddon 2005). The ionization levels of other transition metals including Cr, Mn, Fe, Co, Ni, and Cu are

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discussed in an early theoretical paper by Hemstreet (1980). It seems that interstitials formed from transition metals only exist in the (+1) charge state. Similarly, no mention has been found of ionization levels within the GaAs band gap for metal-VAs complexes. 8.1.3.3 Diffusion Diffusion of silicon in Ga has been explained in terms of isolated gallium vacancies as well as VGa-complexes. Models incorporating SiGa+1 and SiAs–1 are incapable of explaining the diffusion of silicon in Sb-doped GaAs (Yu et al. 1989). The charge state of VGa that mediates diffusion in the bulk depends upon the silicon doping level. An early study attributed diffusion of Si in the intrinsic regime to VGa0 or VGa–1 (Lee et al. 1990). This picture was revised to account for the newly identified VGa–3 defect, proposed to be the dominant vacancy defect for all doping levels (Tan et al. 1993). More recent simulations of Si-doped GaAs indicate that silicon diffusion is only governed by VGa–3 for dopant concentrations near or exceeding 10ni, where ni is the intrinsic carrier concentration in the bulk or 1.79 × 106 cm–3; for concentrations in the range of 5ni, diffusion is governed by VGa0 (Saad and Velichko 2004). Lei et al. built the (SiGaVGa)–2 complex into their computer simulations in order to better model the charged defects that actually govern diffusion behavior in the bulk (2002). They were able to explain the enhanced aggregation of VGa–3 and (SiGaVGa)–2 around experimentally observed dislocations in terms of silicon’s role in reducing the formation energy of the aforementioned defects. Dopants such as Te, S, and Se are introduced into bulk GaAs via ion implantation. Consequently, their tendency to diffuse during the high-temperature annealing used to repair implantation damage has been explored. Karelina et al. looked at the dependence of the Te diffusion coefficient on temperatures between 1,000–1,150°C, as well as the depth of the p-n junction on the vapor pressure of arsenic during diffusion (1974). They proposed that the diffusion of Te in GaAs takes place via arsenic sites. High temperature implants and annealing steps force additional Te atoms into As lattice sites (Pearton et al. 1989). The applicable reaction has been outlined by Nishizawa and Kurabayashi in their study of heavily Se-doped GaAs (1999). Arsenic vacancies serve as traps for selenium interstitials which, in turn, leads to the formation of SeAs+1 defects. Sulfur diffusion profiles indicate that the species migrates via a kick-out mechanism mediated by Asi (Tuck and Powell 1981; Uematsu et al. 1995). The kick-out reaction described by Uematsu et al. accounts for the conversion of Si+1 to SAs+1 and a neutral arsenic interstitial, Asi0. The diffusion of sulfur has been investigated as a means of better understanding arsenic self-diffusion (Engler et al. 2001; Scholz et al. 2000). The migration of Zn, Mn, and Cr in GaAs has been studied to shed light on the substitional-interstitial mechanism of self-diffusion in the bulk (Deal and Stevenson 1986) and explain the changes in ferromagnetic behavior after annealing (Edmonds et al. 2004). Much like Te and Se, it is suggested that Zn atoms move

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interstitially in the crystal until they arrive at vacancy Ga lattice sites; ZnGa–1 is formed as a consequence (Cohen 1997). A similar Frank–Turnbull mechanism with interchange occurring between Zni+1 and ZnGa–1 had already been proposed by Kahen et al. (1991); according to their model, the gallium vacancies that mediate diffusion are either in the (0) or (–1) charge state. Also, these researchers observed a slowing of the diffusivity of positively charged zinc interstitials by the formation of defect pairs comprised of ZnGa–1. A kick-out model of zinc diffusion, where highly mobile Zn interstitials become immobile substitutional atoms by displacing lattice Ga atoms, also appears in the literature (Gosele and Morehead 1981). Several authors have made assumptions as to the charge states of the relevant defects (Cohen 1997). Luysberg et al. explained the formation of extended defects such as interstitial loops and dislocation networks in terms of excess Gai formation by the interstitial-substitutional zinc exchange and the resulting supersaturation of VAs (1992).

8.1.4 Other III–V Semiconductors The other III–V semiconductors including GaSb, GaP, GaN, and AlN are doped with many of the same elements as GaAs. For instance, in GaSb, p-type conductivity is achieved by Ge or Sn doping and n-type conductivity is achieved by Te, Se, or S doping. Transition metals including Cr, V, and Ru may form either deep donor or acceptor levels that trap charge carriers and give rise to high conductivity (Hidalgo et al. 1998). Copper is introduced into GaSb to create a semi-insulating material with low carrier concentration (< 1014 cm–3) (Warren et al. 1990). The geometries of these extrinsic defects and their ionization levels are not well characterized. Consequently, Sect. 8.1.4.2 will contain only ionization level estimates for charged defects in GaN and AlN. Many of the extrinsic point defects in III–V nitride semiconductors have been discussed by Gorczyca et al. (1999) and Sheu and Chi (2002). Similar trends in the charge states of defects in GaN, AlN, and BN are observed; the absolute values of the corresponding ionization levels differ primarily due to the different band gaps of the three materials. For n-type doping of these materials, a number of elements including Si and Ge can be used to routinely achieve carrier concentrations exceeding 5 × 1020 cm–3 (Gotz et al. 1999). As with GaAs, magnesium and beryllium have been investigated as potential p-type dopants in GaN; more information exists on the charged defects that occur in Mg- and Be-doped GaN, however. Carbon has also been cited as a potential means of obtaining p-type conductivity (Kohler et al. 2001). Hydrogen can passivate both acceptors and donors in GaN (Van de Walle 2003; Neugebauer and Van de Walle 1996a). Transition metal dopants including Zn, Mn, and Cr have been investigated in the context of spindependent electronic devices requiring ferromagnetic GaN (Dietl et al. 2000). Also, Ni, Ti, and Au are often used to make metal contacts to GaN-based LED and FET devices, and may diffuse into the bulk during high temperature annealing.

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8.1.4.1 Structure Group IV atoms such as Si, Ge, and Sn behave as p-type dopants in GaSb, even though they act as n-type dopants in other III–V compounds such as GaAs. Whether they occupy primarily group III or group V lattice sites should depend on the difference between the covalent radius of the impurity and the host atom, as well as the concentration of vacancies in the bulk. As the radius of Si is considerably less than that of Ga and Sb, it has little difficulty occupying either lattice site. In practice, Si prefers the larger Sb site over the smaller Ga site and acts as a net acceptor (Subbanna et al. 1988); Ge and Sn exhibit similar behavior (Longenbach et al. 1991). Tellurium and selenium are typical n-type dopants for GaSb and GaP. The addition of Te to GaSb can lower the carrier concentration of undoped material, which is about 1.7 × 1017 cm–3, to less than 1015 cm–3 (Stepanek et al. 1999). The 765 meV cathodoluminescence band observed in Se-doped crystals is attributed to the acceptor VGaGaSbSeSb (Diaz-Guerra et al. 2007). Donor vacancy pairs such as SeSbVGa may be related to the formation of dislocation loops (Doerschel 1994). Transition metals such as Cr, V, and Ru can suppress the native acceptor concentration in GaSb, which in undoped form is always p-type due to native VGs and GaSb defects. Reports that discuss doping of GaSb with Cu and Cr focus mainly on the solubility of the dopant in the bulk (Sestakova and Stepanek 1994) and resulting ferromagnetic behavior (Abe et al. 2004). Interstitial metal defects in the (+1) charge state may mediate diffusion in the bulk (Mimkes et al. 1998; Sunder et al. 2007). Also, transition metal impurities bind with native antimony vacancies to form metal-VSb complexes. In GaN, n-type dopants such as silicon and germanium substitute on gallium lattice sites to form stable, shallow donors. This behavior cannot be generalized for all of the nitride semiconductors, as Ge becomes a deep donor in AlN. When the dopants occupy N lattice sites, they act as deep acceptors. Boguslawski and Bernholc have used quantum molecular dynamics calculations to obtain estimates of the atomic relaxations around substitutional impurities, which affects both the impurity levels and defect formation energies (1997). In GaN, SiGa0, GeGa0, SiN0, and GeN0 experience changes in bond length of –5.6%, –1.4%, 13.6%, and 13.5%, in that order. All of these relaxations preserve the local hexagonal symmetry of the defect. Comparable values are obtained for the defects in InN, except for GeAl0, which exhibits large outwards relaxations of 17.2% of the Ge-N bond length. The large relaxations (with respect to those in GaAs) appear in instances where the mismatch between the atomic radii of the impurity and the host atom are large. The Mg- and Be-related defects in GaN are discussed in the context of compensation of p-type doping by intrinsic point defects and defect complexes. Substitutional point defects including MgGa and BeGa (Fig. 8.10), and interstitial point defects such as Mgi and Bei, form in the bulk. Similar defects occur in Mg-doped InN (Stampfl et al. 2000). Under gallium-rich growth conditions, the compensation of MgGa–1 probably occurs via the formation of substitutional-interstitial pairs with a configuration (MgGaNMgi)+1 (Reboredo and Pantelides 1999). The inability to easily achieve p-type doping in Mg-doped GaN has also been attributed to other

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Fig. 8.10 Schematic representation of atomic positions in the (1120) plane for (a) BeGa–1 and (b) BeGa0 in wurtzite GaN. Large circles represent Ga atoms, medium circles N atoms, and the hatched circles represent Be. Dashed circles indicate ideal atomic positions, dashed lines bonds in the ideal lattice. The numbers denote the percentage of change in the bond lengths, referenced to the bulk Ga-N bond length. Reprinted figure with permission from Van de Walle CG, Limpijumnong S, Neugebauer J (2001) Phys Rev B: Condens Matter 63: 245205-6. Copyright (2001) by the American Physical Society.

defects, including VN+1, Mgi+2, Gai+3, (MgGaNGai)+2, and MgGaVN. Positron annihilation spectroscopy experiments indicate that the presence of MgGaVN explains both the electrical compensation in the bulk and the activation of p-type conductivity upon annealing (Hautakangas et al. 2003). Several different configurations have been considered for interstitial Be in GaN (Bernardini et al. 1997; Van de Walle et al. 2001). Donor complexes, namely (BeGaBei)+1, exist in substantial concentrations in p-type Be-doped GaN. Also, Be may not act as an efficient acceptor in GaN due to the formation of a charged BeiVGa complex that acts as a compensating donor (Naranjo et al. 2000). Transition metals such as Zn, Ti, and Ni occupy cation sites in the GaN and AlN crystal lattices. The formation energies of similar defects on nitrogen sites are energetically unfavorable, as are those of tetrahedral and octahedral interstitials (Xiong et al. 2006). For gold impurities, the formation energy for substitution on the Ga and N sites are comparable (Chisholm and Bristowe 2001a). Zinc doping leads to the formation of ZnGa impurities, which exhibit small outward relaxations on the order of 2% and 10% of the Ga-N and Al-N bond lengths, respectively (Gorczyca et al. 1999). Ti, Ni, and Au retain the four-fold coordination of the GaN lattice and induce outwards relaxations of varying magnitudes (Chisholm and Bristowe 2001b). No mention has been found in the literature of larger defects pairs or complexes. 8.1.4.2 Ionization Levels For most values of the Fermi energy, SiGa and GeGa prefer the (+1) charge state. In InN, which has a far smaller band gap than GaN, SiIn has a (+1) charge for the

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Fig. 8.11 Calculated formation energies of Si, Ge, Sb, P, As, and C on the N site of GaN and AlN under N-rich and cation-rich preparation conditions versus Fermi level. The range of EF corresponds to the DFT-LDA fundamental energy gaps of AlN and GaN, 3.29 eV and 2.03 eV, versus the experimental band gaps of 3.2 eV and 6.026 eV. Reprinted figure with permission from Ramos LE, Furthmuller J, Leite JR et al. (2003) Phys Rev B: Condens Matter 68: 085209-8. Copyright (2003) by the American Physical Society.

entire band gap (which is 1.9 eV in the original report, not the experimental value of 0.72 eV) (Stampfl et al. 2000). In contrast, SiN and GeN may adopt up to six stable charge states within the band gap. A computational study employing density-functional theory, ab initio pseudopotentials, and a supercell approach predicted a SiGa (+1/0) donor level 0.1 eV below the conduction band minimum (Neugebauer and Van de Walle 1996b). Ramos et al. obtained identical (+1/0) and (0/–1) ionization levels for SiGa and GeGa at Ev + 1.91 eV and Ev + 1.96 eV using DFT with the LDA fundamental band gap of 2.03 eV for GaN (versus the experimental band gap of 3.2 eV) (Ramos et al. 2003). The same authors found ionization levels for SiN and GeN; according to their results, the charge on each defect changes from (+4) to (–1) as the Fermi level progressively rises from the valence band maximum to the conduction band minimum. The ionization levels of these defects, as well as those induced by Sb, As, P, and C doping in GaN and AlN are illustrated in Fig. 8.11. The ionization levels of Mg and Be substitutional and interstitial defects have been obtained experimentally and computationally. Using density-functional theory, these same authors and Van de Walle et al. (2001) identified a MgGa (0/–1) ionization level at Ev + 0.06 ± 0.05 eV and Ev + 0.17 eV, respectively. Total energy DFT calculations yield a comparable value of Ev + 0.18 eV (Gorczyca et al. 1999).

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All of these ionization levels correspond to MgGa in wurtzite GaN; Van de Walle et al. obtained a value 0.03 eV higher for the same defect in zincblende GaN. Teisseyre et al. performed high-pressure measurements of the donor-acceptor pair recombination related to the shallow Be acceptor and compared their results to ab initio calculations using the linear muffin-tin orbital (2005). Their experiments indicated the presence of a (0/–1) ionization level for MgGa and BeGa at Ev + 0.247 eV and Ev + 0.104 ± 0.03 eV, respectively; values of Ev + 0.175 eV and Ev + 0.075 eV for the same defects were obtained computationally. Two computational reports of a Bei (+2/+1) ionization level at Ev + 2.3 eV (Bernardini et al. 1997) and approximately Ev + 1.9 eV (Van de Walle et al. 2001) have been found. Transition metals act as either acceptors or donors depending on where they reside in the nitride semiconductor crystal lattice. Substitutional Zn on a cation site in GaN and AlN forms a shallow acceptor state. ZnGa–1 in GaN and ZnAl–1 in AlN will have negative formation energies in n-type material, meaning that the defects form spontaneously. Bergman et al. used photoluminescence to derive a single acceptor ionization energy of Ev + 0.34 eV for ZnGa (1987); first-principles totalenergy calculations yield a similar value for the level (Neugebauer and Van de Walle 1999). Ab initio calculations using a supercell approach identify (0/–1) ionization levels for ZnGa and ZnAl at Ev + 0.3 eV and Ev + 0.4 eV (Gorczyca et al. 1999). According to LDA calculations performed by Chisholm and Bristowe, in GaN doped with Ti, Ni, and Au, TiGa, NiN, and Aui act as single donors and AuN and TiN as double donors (2001a). Several of their defect ionization levels compared well to those estimated experimentally with photoluminescence and electron paramagnetic resonance (Baur et al. 1995). 8.1.4.3 Diffusion The dynamics of charged defect diffusion in III–V semiconductors such as GaSb and GaP are not well understood. Mimkes et al. have studied the diffusion of several transition metal impurities in GaSb using Hall effect and conductivity measurements (1998). While silver migrates according to a substitutional mechanism mediated by VGa, a simple vacancy mechanism of diffusion does not apply for gold due to its large atomic radius. The high diffusion coefficient of Cu in GaSb indicates that copper diffuses via a kick-out mechanism; interstitial copper atoms force gallium atoms to jump to adjacent lattice sites. Zinc diffuses according to a similar kick-out mechanism in GaP, where zinc incorporates into gallium sublattice sites and displaces Gai agglomerates to form dislocation loops (Jager and Jager 2002). The charge states of the specific defects involved in this kick-out reaction have been extracted from experimental SIMS profiles using continuum theoretical calculations; the changeover from Zni+1 to ZnGa–1 is mediated by Gai0 and Gai+1 (Sunder et al. 2007). The (0) charge state of Gei is pertinent to diffusion at doping levels of 1 – 2 × 1019 cm–3 and the (–1) charge state for doping concentrations of 3 – 10 × 1019 cm–3 (Nicols et al. 2001). At higher dopant surface concentrations,

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Fig. 8.12 Schematics of the neutral Mg interstitial diffusion model for (a) horizontal diffusion and (b) vertical diffusion. Reprinted figure with permission from Harafuji K, Tsuchiya T, Kawamura K (2004) Jpn J Appl Phys, Part 1 43: 528. Copyright (2004) by the Institute of Pure and Applied Physics.

Zn diffusion in GaP may also be mediated by VP or VGaVP (Stolwijk and Popping 2003). Few reports exist regarding the diffusion of extrinsic dopants in group III-nitride semiconductors. Although some authors discuss the speed and redistribution of dopants in the bulk (Wilson et al. 1995), they provide no insight into the defects that mediate migration. Harafuji et al. have examined the diffusion of Mg in GaN using a Hartree–Fock ab initio method (2004). Neutral Mg interstitials diffuse by hopping from an initial hexagonal crystal lattice cage center to an adjacent cage center. Two different pathways (shown in Fig. 8.12) exist for this site-to-site hopping: a) by way of the position between Ga and N atoms aligned along the [0001] direction leading to horizontal diffusion on the (0001) plane and b) by displacing an adjacent Ga atom and residing at the lattice for a brief period of time before moving to the adjacent cage center, which drives vertical diffusion along the [0001] direction.

8.1.5 Titanium Dioxide Doping can increase the suitability of bulk TiO2 for photo- and heterogeneous catalysis or spintronics. It can also potentially increase the lifetime of charge carriers and narrow the band gap of the material, both of which increase photoreactivity (Li et al. 2003b). In some cases, however, the chemical doping of TiO2 with Cr, Mo, V, and other trivalent and pentavalent metal atoms has been observed to lead to instability and the overly rapid recombination of electrons and holes (Choi et al. 1994). Upon supra-band gap irradiation of TiO2 particles, positive holes are

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generated in the valence band and negative electrons in the conduction band according to TiO2 + hν (UV) → TiO2 + e– + h+.

(8.9)

After the charge carriers are produced, they can become trapped or recombine (Berger et al. 2005). Dopants can act as carrier scattering centers and traps, reducing the probability of carriers reaching the substrate surface and participating in the desired reactions (Na-Phattalung et al. 2006). For photocatalytic systems in which the rate-limiting step is interfacial charge transfer, the overall quantum efficiency of the photocatalytic process, a variable directly proportional to the rate of charge transfer and inversely proportional to the electron hole recombination rate in the bulk, can be enhanced via improved charge separation and inhibition of charge carrier recombination (Linsebigler et al. 1995). Several approaches have been taken to obtain increased photocatalytic efficiency in TiO2 including doping with transition metals, lanthanide ions, and nonmetals. The goal of doping with transition metals is to shift the absorption spectrum to longer wavelengths through the formation of induced states within the fundamental band gap (Glassford and Chelikowsky 1993). This strategy should be clearly distinguished from that undertaken in the doping of Si; for photocatalytic TiO2, the aim is to dope to such an extent that the fundamental electronic structure, not simply the Fermi level, is altered. TiO2 can be doped p- and n-type with transition metals such as Nb, Fe, Al, Cr, V, and Co. Based on XRD measurements of metal-doped TiO2 grown via molecular beam epitaxy, it is clear from Fig. 8.13 that species exhibit different solubility in rutile versus anatase TiO2 (Matsumoto et al. 2002). Oxygen sensors fabricated from Nb-doped TiO2 exhibit higher sensitivity and lower working temperatures (Arbiol et al. 2002). Nb-doped thin films can also be used to monitor methanol selectivity at ppm levels with negligible sensitivity to interfering gases, such as benzene and NO2 (Sberveglieri et al. 2000). Iron has been considered as a viable transition metal dopant for TiO2 epitaxial substrates, thin films, nanoparticles, and nanorods (Andersson et al. 1974; Li et al. 2003b; Ding et al. 2007). Doping with cobalt has been shown to affect the fundamental electronic structure, rather than just the Fermi level, of TiO2; the band-gap energy of Co-doped rutile TiO2 is found to increase with increasing Co composition. Room-temperature ferromagnetic behavior has been observed in cobalt-doped titanium dioxide, CoxTi1–xO2–x–y, with values of x up to ~0.4 (Jaffe et al. 2005; Lee et al. 2006a). Neodymium, a lanthanide metal ion, is capable of increasing the reactivity of TiO2 in applications where metal doping has been insufficient (Burns et al. 2004). The enhanced effectiveness of the dopant is partially attributed to the ionic radius and oxygen affinity of the ion species. It is commonly selected as an emission agent in optical applications because of its high efficiency and durable emission properties (Li et al. 2003a). Nd3+ has been also observed to remarkably enhance the photocatalytic efficiency of 2-chlorophenol degradation on doped-TiO2 (Fig. 8.14) (Shah et al. 2002).

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Fig. 8.13 The solubility limits of different transition metal ions in anatase (front row) and rutile thin films grown via MBE. Reprinted from Matsumoto Y, Murakami M, Hasegawa T et al., “Structural control and combinatorial doping of titanium dioxide thin films by laser molecular beam epitaxy,” (2002) Appl Surf Sci 189: 346. Copyright (2002), with permission from Elsevier.

Fig. 8.14 Photodegradation of 2-CP with undoped TiO2 and transition metal ion-doped TiO2 under a UV light source. Initial concentration was 50 mg in 1,000 mL at pH 9.5. Reprinted figure with permission from Shah SI, Li W, Huang C-P et al. (2002) PNAS 99: 6485. Copyright (2002) National Academy of Sciences, U.S.A.

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TiO2 doped with non-metal atoms, typically in the range of 1–10 atomic %, is predicted to be more photocatalytically active and stable than transition metaldoped TiO2 (Di Valentin et al. 2004). Several elements including N, S, C (Sakthivel and Kisch 2003), F (Yu et al. 2002), and P (Shi et al. 2006) have been considered in the literature; nitrogen has received more attention than the others (Asahi et al. 2001; Batzill et al. 2006). The addition of nitrogen atoms to films, powders, and nanoparticles of TiO2 has been seen to successfully improve absorption in the visible region, and leads to a corresponding increase in photochemical activity. S doping shifts the absorption edge of TiO2 into the lower-energy region, as mixing of the S 3p states with the valence band causes band gap narrowing. Nitrogen is better suited to band gap narrowing however, as the large ionic radius of S makes it difficult to incorporate into the TiO2 crystal lattice (Asahi et al. 2001). 8.1.5.1 Structure Most transition metal dopants occupy Ti sites in the TiO2 crystal lattice. For instance, trivalent Cr, Al, Ga, V, and Fe dissolve preferentially at substitutional sites to form CrTi–1, AlTi–1, GaTi–1, VTi–1, and FeTi–1 (Sayle et al. 1995). Positively charged defects including VO+2, Tii+3, and Tii+4 may form in the bulk in an attempt to compensate the negatively charged substitutional defects. In some instances, the simultaneous formation of extrinsic substitutional defects and intrinsic compensating defects can alter the crystal phase of the semiconductor (Wang et al. 2005). The rutile TiO2 octahedral unit cell has two fewer shared edges than the anatase unit cell; shared edges are known to cause cation-cation repulsion and structural destabilization. As a consequence, the additional cations induced by trivalent metal doping lead to the preferable formation of rutile TiO2. Conversely, the cation vacancies induced by tetravalent doping are better tolerated in anatase TiO2. Similar defects are observed in aluminum- and iron-doped TiO2. A complex in the (+2) or (+1) charge state comprising substitutional aluminum and oxygen on a lattice site, AlTiOO, was invoked based on early experiments to explain the pressure and temperature dependence of electrical conductivity in Al-doped TiO2 (Yahia 1963). Under moderate doping conditions, Al3+ occupies a cation lattice site to form AlTi–1 (Gesenhues and Rentschler 1999). Fe substitutes for Ti4+ in a similar manner (Andersson et al. 1974; Bally et al. 1998). At high doping levels (0.285 mol % concentration of Al2O3, for instance), Ali+3 increases in concentration, as does (AlTiAli)+2 and (AlTiAliVO)+4. Doping with niobium, an n-type dopant in titanium dioxide, leads to the formation of singly positively charged NbTi defects. A model involving these defects explains the temperature-dependent mobility of electrons in the bulk, as well as the equilibrium constant for intrinsic electronic ionization (Sheppard et al. 2006). Numerous researchers have explored the carrier compensation that occurs in bulk TiO2 following Nb addition (Frederikse 1961; Baumard and Tani 1977; Eror 1981; Morris et al. 2000). The enhanced solubility of Nb in anatase versus rutile TiO2 is attributed to the differences in VTi concentration or stress induced by the presence of Ti3+ in the two materials (Arbiol et al. 2002).

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In cobalt-doped anatase TiO2, CoTi, Coi, and complexes such as CoTiVO form. There are two distinct types of CoTiVO complexes; one pair is orientated along the c-axis while the other is nearly in the ab plane (Sullivan and Erwin 2003). A different defect model is necessary to explain the ferromagnetic behavior of Co-doped TiO2; high concentrations of both substitutional cobalt and oxygen vacancies are prerequisites for magnetic behavior (Jaffe et al. 2005). Under these conditions, cobalt atoms tend to form pairs or complexes on Ti sites, either with each other or oxygen vacancies. Neodymium atoms can enter the TiO2 crystal lattice on both substitutional (as NdTi–1) and interstitial sites. Nd3+ substitutes for Ti4+ on the body-centered and face-centered lattice sites in the anatase structure; as the effective ionic radius of Nd3+ is approximately 0.4 Å larger than that of Ti4+, distortions occur (Burns et al. 2004). The substitutional Nd3+ ions cause an expansion of the anatase lattice along the c direction with a maximum value of 0.15 Å at a Nd doping level of 1.5 atomic % Nd (Li et al. 2005a; Zhao et al. 2007). Above this doping concentration, Nd atoms likely incorporate into the lattice as interstitial atoms. Non-metal atoms can be doped into the TiO2 matrix as anions and substitute for native oxygen atoms, or cations and substitute for titanium atoms. For instance, in N-doped TiO2, N3– ions substitute for lattice O2– to form NO–1 defects (Batzill et al. 2006). The excess negative charge in the lattice (and concurrent reduction of the oxide semiconductor) induces the formation of anion vacancies and Ti3+ states (Emeline et al. 2007). Nitrogen interstitials also induce localized electronic states in the band gap (Di Valentin et al. 2005; Di Valentin et al. 2007). Yang et al. have calculated the formation energies of N-doped anatase supercells with N concentrations ranging from 0–4.17 atomic % using DFT within the GGA (2007). In chemically modified TiO2 photocatalysts, sulfur (S4+) can substitute for lattice titanium and oxygen atoms, as well as adopt an interstitial configuration in the crystal lattice. Using first-principles total energy calculations, Smith et al. have explored charged SO, STi, and Si in anatase TiO2 (2007). Especially under O-rich growth conditions, interstitial defects form readily wherein an S atom bonds strongly with one of the lattice O atoms. The defect adopts a split-interstitial configuration with a S-O bond distance of 1.8 Å, as illustrated in Fig. 8.15. 8.1.5.2 Ionization Levels Of all the defects induced by transition metal doping, only the extrinsic defects in Co-doped TiO2 have been considered in multiple charge states. Using total-energy calculations within the local density approximation, Sullivan and Erwin considered the stability of several multiply charged defects in cobalt-doped TiO2 (2003). In “slightly” or 3–6 molar % doped material, CoTi is stable in the (+1), (0), (–1), and (–2) charge states and Coi in the (+2), (+1), and (0) charge states. The complex formed from substitutional cobalt and VO is stable in the (+1) and (0) charge states. The (+1/0), (0/–1), and (–1/–2) ionization levels of CoTi occur at approximately Ev + 0.3 eV, Ev + 1.5 eV, and Ev + 2.2 eV, in that order. The (+2/+1) level of

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Fig. 8.15 Atomic structure of (a) bulk anatase TiO2, (b) the sulfur interstitial, and (c) a close-up view of sulfur interstitial. The large, medium, and small spheres are Ti, S, and O atoms, respectively. The dashed ellipse indicates the split-interstitial S-O pair. The arrows show the relaxation of the neighboring atoms compared to their positions in the bulk. In the close-up view, the bond distances from S to its neighbors are given in angstroms and an additional O, from an adjacent unit cell, appears. Reprinted with permission from Smith MF, Setwong K, Tongpool R et al. (2007) Appl Phys Lett 91: 142107-3. Copyright (2007), American Institute of Physics.

Coi and the (+1/0) level of CoTiVO are located at about Ev + 2.6 eV and Ev + 1.2 to 1.7 eV (where the wide range occurs due to a dependence of the defect formation energy on defect configuration). The formation of neutral CoTi is favorable under O-rich growth conditions, while ionized CoTi and Coi exist in approximately equal concentrations in reduced material. It is worth mentioning that Sullivan and Erwin identify VO+1 as a stable defect in the bulk, albeit one with a negligible concentration (< 10–5 molar %) for all thermodynamically allowed chemical potentials. NaPhattalung et al. later suggested that this state may correspond to the electron filling of the conduction-band edge states, rather than an actual defect state (2006). Instead of inducing a new assortment of charged defects in the bulk, Nd doping drastically narrows the band gap of TiO2. A maximum band gap reduction of 0.55 eV is obtained at a doping concentration 1.5 atomic % (Zhao et al. 2007; Li et al. 2003a). The reduction in band gap width is attributed to the electron states introduced by substitutional Nd3+ ions and has been observed experimentally with near-edge x-ray absorption fine structure measurements and synchrotron radiation photoelectron spectroscopy (Hou et al. 2006), as well as computationally with density functional theory (Wang and Doren 2005). Only one report mentions the existence of non-metal defect ionization levels. Most of the time, the effect of doping on the electronic structure of the semiconductor (in the context of photocatalytic enhancement) is investigated. Low N doping (~1 to 2 atomic %) induces localized N 2p electronic states within the band gap just above the valence band maximum (Batzill et al. 2006; Nakano et al. 2007). Ion implantation of N suppresses the formation of Ti 3d band gap states and, consequently, reduces the amount of trapping centers for photogenerated

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holes. At higher doping levels (~4 atomic %), some band gap narrowing may occur (Yang et al. 2007). Similar behavior is observed in S- and C-doped TiO2 (Tian and Liu 2006; Chen et al. 2008). Smith et al. suggest that the charge state on SO and STi changes from (+2) to (0), and that of Si from (+4) to (0), as the Fermi level rises from the valence band maximum to the conduction band minimum (2007). The exact locations of the ionization levels are not given. 8.1.5.3 Diffusion The mechanisms by which charged extrinsic defects diffuse in rutile and anatase TiO2 are not well understood. There is no information available concerning the effect of the Fermi level in the bulk on defect activation energy; even the migration of neutral extrinsic defects in TiO2 has scarcely been addressed. Sasaki et al. used a radioactive-tracer sectioning technique to examine the diffusion of Co, Ni, Sc, Mn, Cr, and Sc in single crystal rutile (1985). It seems that some generalizations regarding the migration of these species can be made depending on the valence of the diffusing atom. For instance, Co and Ni (both divalent impurities) quickly diffuse as interstitials through the crystal lattice in an anisotropic manner. Trivalent (Cr, Sc) and tetravalent (Zr) impurity atoms also diffuse interstitially via a kick-out mechanism (similar to native titanium atoms). SIMS experiments suggest that the diffusion of Nb is also mediated by interstitials, although the valence of the atom (5) slows the diffusion rate slightly (compared to Ti4+) (Sheppard et al. 2007).

8.1.6 Other Oxide Semiconductors The investigation of extrinsically-doped ZnO has been prompted by the desire to obtain stable p-type material in a manner compatible with typical microelectronics processing equipment. P-type ZnO is a promising material for blue and ultraviolet light emitted devices due to its wide band gap and low exciton binding energy (Ding et al. 2008). Lithium, for instance, is one of the most attractive dopants for achieving piezoelectricity due to its high diffusivity and low activation energy (Wu et al. 2004). Other group I and group V elements have been considered as viable p-type dopants. Despite this broad range of dopants, researchers have experienced difficulty in preparing high-quality p-type ZnO material in a reproducible manner; p-type ZnO is usually characterized by relatively low-hole concentration, low hole mobility, and instability (Li et al. 2005b). Even nitrogen, one of the most promising elemental dopants, is thought to cause donor states, limited solubility, and metastability (Barnes et al. 2005). Group III atoms including Al, Ga, and In, which introduce donor states into the bulk, have been investigated in order to develop transparent electrode materials for displays and solar cells (Ryoken et al. 2005). As with Si and Ge, impurity hydrogen in the bulk leads to the formation of additional charged defects.

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Interest in doping of UO2 has arisen from two distinct scientific communities. For nuclear applications, it is advantageous to reduce the release of fission gas from irradiated fuel. Gas release is a function of grain size in the bulk, and certain additives are known to affect grain size development, sintering, and thermal expansion (Munir 1981). Metals such as titanium can increase the final bulk density of UO2 and the diffusion coefficients of uranium and xenon in the bulk (Tsuji et al. 1989). Researchers have also considered the behavior of non-metallic solid fission products comprising rare earth (La, Nd, Ce, Y, Sm, Gd, Eu) and alkaline earth (Sr, Ba) elements. The fission product oxides are known to form solid solutions with UO2 (Momin et al. 1991). It is also important to examine how the electrical conductivity of UO2 is affected by the introduction of impurity atoms in the form of solid solutions in order to pursue its use in active components such as solar cells, thermophotovoltaics, diodes, and transistors. Researchers have examined the suitability of elements including B, Al, N, P, Sb, S, Te, and F for microelectronics applications (Meek et al. 2005). CoO has been doped with transition metals and group III elements. These species are known to affect electrical conductivity, Hall effect, chemical diffusion, and Seebeck coefficients. Chromium, an alloying component, has been studied in order to better understand its effect on the oxidation kinetics of CoO (Mrowec and Grzesik 2003). 8.1.6.1 Structure Lithium, sodium, and potassium form substitutional and interstitial defects in ZnO. Hydrogen, a common impurity in ZnO introduced during crystal growth, exists mainly in an interstitial configuration. Lithium substitutes on a zinc lattice site and induces small (~3%) inwards contractions of the surrounding O atoms (Wardle et al. 2005). These authors have considered three types of interstitial sites for Lii: octahedral, tetrahedral, and bond-centered. In wurtzite-ZnO, Lii prefers the caged octahedral configuration and possesses C3v point-symmetry. In zincblende ZnO, the defect resides on a tetrahedral site surrounded by a cage of oxygen atoms. According to Wardle et al., charged defect complexes such as LiiLiZn, VOLii, VZnLiZn, and VOLiZn also arise in the bulk, as well as pairs comprised of LiZn and impurity hydrogen atoms. LiiLiZn can be viewed as a distorted Li2O unit cell encapsulated within wurtzitic ZnO. Nitrogen and phosphorus incorporate into the ZnO crystal lattice much in the same way as the group I atoms. N is the best elemental dopant source for p-type ZnO, as it refrains from forming the NZn–1 antisite, and its AX center (which compensates for acceptors) is only metastable (Park et al. 2002). Donor defects introduced by P and As doping have greater ionization energies and are considerably less stable. The larger atomic radius of phosphorus can also lead to the formation of oxygen vacancies, which inhibit p-type doping. In nitrogen-doped material, interstitial N as well as substitutional N2 on an O site (N2O) may compensate or trap holes. Lee et al. have used DFT and the local density approximation to

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Fig. 8.16 Defect and hole concentrations in ZnO as a function of the total N concentration for a normal N2 gas source. The N concentration is controlled by varying the stoichiometric parameter in the DFT calculations, while the N chemical potential is in the extreme N-rich limit for all concentrations. Reprinted figure with permission from Lee E-C, Kim YS, Jin YG et al. (2001a) Phys Rev B: Condens Matter 64: 085120-4. Copyright (2001) by the American Physical Society.

examine the charged point defects in N-doped ZnO (2001a). As shown in Fig. 8.16, at low and high doping concentration NO–1 and (NOZnO)+1 are the primary defects that arise in the bulk. Similar point defects including PO–1, PZn+1, and PZnVZnVZn have been studied in P-doped ZnO (Lee et al. 2006b; Yu et al. 2005). Group III elements also incorporate into the ZnO crystal lattice. In B-doped material, boron impurities form complexes with oxygen interstitials, which stabilizes the native oxygen vacancies in the bulk (Chen et al. 2006). Singly positively charged AlZn defects have been invoked to explain the properties of ZnO films grown via pulsed laser deposition (Ryoken et al. 2005). Interstitial hydrogen can occupy either a bond-centered or anti-bonding site in the ZnO wurtzite crystal lattice; the former is slightly more favorable for all investigated charge states of the defect, however (Van de Walle 2000). The Hi+1 defect prefers to reside where it can form a strong bond to lattice oxygen. Regardless of whether the consequent O-H bond orients parallel or perpendicular to the c axis, the nearest-neighbor Zn atoms are displaced outwards by ~40% of the bond length. In UO2, fission products prefer to occupy cation lattice sites. For instance, it is known that Cs and Rb substitute for uranium atoms in the bulk (Grimes and Catlow 1991). Catlow has obtained theoretical estimates of the energies associated with the substitution of uranium (in different valences), cerium, and plutonium onto UO2 lattice sites (1977). Petit et al. considered several configurations of the Kr atom in UO2 including an interstitial position, substitution on a uranium site, substitution on an oxygen site, substitution on a uranium divacancy, and substitution on a uranium trivacancy (1999). The solution energy is minimized when the krypton atom occupies a neutral uranium trivacancy. The intrinsic conductivity of UO2 can be extended to lower temperatures (~714 K) via the addition of Nb2O5, which leads to the formation of NbU+1 defects (Munir 1981). Trends in the conductivity of La-doped UO2 can be explained in terms of a (2:2:2)+1 cluster that leads to an increased concentration of LaU–1 (Matsui and Naito 1986).

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Titanium added to UO2 to form (U0.993Ti0.007)O2+x adopts an interstitial position in the crystal lattice in the form of Tii+4 (Tsuji et al. 1989). Additionally, larger defect clusters including ((2OiaOib2VO)–4Tii+4)x and ((2OiaOib2VO)–5Tii+4)–1 arise; the former dominates at low partial pressures, while the latter exists in higher concentrations at intermediate and high partial pressures. At least one laboratory has considered the influence of group III to group VII doping on the properties of thin UO2 films (Meek et al. 2005). The authors did not discuss the configurations or charge states of the dopant species within the semiconductor. According to theory, the conductivity of doped UO2 follows a parallel resistor model that predicts a large increase in conductivity upon dopant addition. In practice, it seems that doping with aluminum at 1017 atoms/cm3 increases the electrical conductivity by 75% (in comparison to intrinsic material). When CoO is doped with Cr, Fe, and Ga, additional neutral and charged defects form in the bulk. When Cr, Fe, and Ga ions incorporate into the cation sublattice of CoO, CrCo+1, FeCo+1, GaCo+1 defects are produced. As many of the dopants used in CoO are trivalent, strong associations arise between dopant cations and charge compensating cobalt vacancies (Grimes and Chen 2000). Singly ionized CrCo+1 and native cobalt vacancies (VCo–2) can bind together to form (CrCoVCo)–1. Cr+3 and Ga+3 ions also situate themselves symmetrically around cation vacancies to form (CrCoVCoCrCo)0 and (GaCoVCoGaCo)0. In highly Ga-doped CoO, pairs and triplets of Ga ions and cation vacancies, including (GaCoVCo)0, (GaCoVCo)–1, and (GaCoVCoGaCo)0 occur (Schmackpfeffer and Martin 1993). While Chen et al. (1995) found that the dopants prefer the third-nearest neighbor distance, Grimes and Chen later realized that small and large M3+ cations exhibit a preference for the second and first nearest-neighbor sites, respectively. This makes intuitive sense, as the attractive Coulomb energy of VCo–2 and MCo+1 are minimized if the atoms are in a first neighbor configuration, although conflicting relaxations of nearby oxygen lattice atoms occur. In the second-nearest neighbor configuration, native oxygen atoms can both relax towards M+3 ions and away from neutral cation vacancies. 8.1.6.2 Ionization Levels Few reports exist concerning the ionization levels of charged extrinsic defects in ZnO, UO2, and CoO. Substitional and interstitial defects formed from group I dopants in ZnO have multiple charge states. For Li- and Na-doped ZnO, Lii and Nai are the primary species that prevent p-type doping. The (0/–1) ionization level of LiZn is found 0.25 eV above the valence band maximum (Wardle et al. 2005). The same level has been identified at Ev + 0.50 eV using photoluminescence (Meyer et al. 2004). Wardle et al. also obtain values of Ev + 2.0 eV and Ev + 2.5 eV for the (+1/0) and (–1/0) ionization levels of Lii, respectively. LiiLiZn is only stable in the neutral charge state, while VOLii, VZnLiZn, and VOLiZn support two ((0) and (–1)), three ((0), (–1), and (–2)), and two ((+1) and (–1)) charge states, in that order. On the other hand, interstitial hydrogen in ZnO is thought to be stable in the (+1) charge state for all Fermi energies in both the theoretical and experimental

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band gap (Van de Walle 2000). In CoO, there is evidence that CrCo has an ionization level at Ev + 0.83 ± 0.08 eV (Gvishi and Tannhauser 1972); it is unclear if it corresponds to the (0/+1) ionization level or the (–1/+1) ionization level. 8.1.6.3 Diffusion Small atoms such as lithium and hydrogen are extremely mobile in ZnO at low temperatures. It is suggested that lithium diffuses in ZnO via an interstitial kick-out mechanism where mobile Li atoms displace substitutional Zn atoms (Garces et al. 2003). This behavior is inferred from the small migration barriers that Lii and Zni are expected to have. Wardle et al. considered two diffusion channels for Lii+1 in w-ZnO: one along the c axis (O-O), and a second within the basal-plane between the octahedral and tetrahedral interstitial sites (O-T-O) (2005). The calculated activation energies for diffusion along the two paths are 0.64 eV and 0.58 eV, in that order. In zincblende ZnO, Lii+1 hops from a tetrahedral site surrounded by oxygen to a tetrahedral site surrounded by Zn; the activation energy is independent of defect charge state and approximately 0.9 eV in magnitude. Experiments and ab initio calculations identify similar energy barriers of migration for positively charged hydrogen interstitials. For instance, Ip et al. obtained a value of 0.17 eV ± 0.12 eV for the activation energy of Hi+1 from SIMS profiles of deuterium-treated material (Ip et al. 2003). Similar deuterium diffusion experiments yield a higher activation energy of 0.17 ± 0.12 eV (Nickel 2006). The activation energy for the diffusion of H in w-ZnO using first-principles methods is 0.4–0.5 eV (Wardle et al. 2006). The migration of fission products in UO2 is of interest to the nuclear energy community. The correspondence between coefficients of self-diffusion and oxygen partial pressure is well understood; the relationship between diffusion coefficients of extrinsic dopants and UO2 stoichiometry is not completely understood. For instance, the dependence of the Xe diffusion coefficient on UO2±x nonstoichiometry is evident in both experimental and theoretical work. At 1,400ºC, the diffusion coefficient of Xe in UO2.12 is about 40 times that in UO2 (Lindner and Matzke 1959). Complication arises due to the fact that the diffusion coefficient depends not only on temperature, but also on irradiation parameters such as burn-up and fission rate (Szuta 1994). The complex relationship between dopant type and concentration, temperature, stoichiometry, and other pertinent parameters is illustrated in Fig. 8.17 (Matzke 1987). An early review paper is an excellent resource for additional information on the diffusion of fission-product rare gases in UO2 (Lawrence 1978). Xenon migration in UO2 is not mediated by uranium or oxygen monovacancies (Matzke 1967). Instead, the migration of xenon in the bulk is mediated by di- and tri- vacancies (Nicoll et al. 1995). The divacancy consists of an anion vacancy bound to a cation vacancy (and has an effective charge of –2) while the trivacancy contains one additional anion vacancy (and has an effective charge of 0) (Ball and Grimes 1990). According to Nicoll et al., the divacancy and trivacancy are relevant to diffusion in hyper- and hypo-stoichiometric material, respectively. After

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Fig. 8.17 Normalized Arrhenius diagram, plotted as Tm/T (where Tm– is the melting point of the semiconductor), of diffusion processes in UO2. Shown are bands to indicate the scatter of different published data, together with their extrapolation to different temperatures, for tracer diffusion of oxygen and uranium, radiation-enhanced metal diffusion, and fission Xe at low and high concentrations. The effect of deviation from stoichiometry on metal diffusion is shown at 1,500ºC. In the upper part of the illustration, typical temperatures ranges of operating fuel are shown. Reprinted figure with permission from Matzke H (1987) J Chem Soc, Faraday Trans II 83: 1136. Reproduced by permission of The Royal Society of Chemistry.

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all, the ambient partial pressure affects the concentration of oxygen vacancies in the bulk; these defects react with divacancies to form trivacancies. The behavior of Cr, Fe, and Ga diffusion in CoO is similar to that of cation selfdiffusion. That is to say that oxygen partial pressure also affects the diffusion coefficient of impurity atoms. For pressures greater than 10–3 atm, Fe diffuses at a similar rate to Co, with a corresponding activation energy of 0.59 eV (Hoshino and Peterson 1985). At low oxygen partial pressures (10–7 to 10–3 atm), the diffusion coefficient of Fe is about two times larger than that of Co. Schmackpfeffer and Martin, in their study of the defect structure of Ga-doped CoO, also looked at tracer diffusion in (Co1–xGax)O for 0 ≤ x ≤ 0.03 (Schmackpfeffer and Martin 1993). They associated the non-linear increase in Co and Ga diffusion coefficients with dopant concentration with the transition from complex-dominated to cation vacancydominated defect chemistry. The activation energy for Ga diffusion (1.6 eV) is larger than that of Co diffusion (1.3 eV) at low oxygen partial pressures; the activation energies of diffusion decrease strongly with dopant fraction. For instance, VCo–2 has a diffusion coefficient that is approximately 50% larger than that of VCo–1.

8.2 Surface Defects Extrinsic defects occur on semiconductor surfaces in forms analogous to the bulk. In some sense, any adsorbed atom or molecule can be viewed as an extrinsic defect. We will not adopt such an expansive view here, but rather will limit the discussion to elements that find extensive use as intentional bulk dopants or for which there is significant literature on ionization of the adsorbate. In comparison to the bulk analogs, the literature base for surface extrinsic defects is much smaller. Still less is understood about the charging of these defects. Yet, novel surface modification techniques, such as those that restrict dopant incorporation to the uppermost atomic layers of the substrate (Weir et al. 1995), are undoubtedly affected by the presence of charged surface defects. The dependence of thermal surface diffusion on charged surface species has already been discussed in Chap. 7. Also, the chemical modification of semiconductor surfaces can alter the stability of certain intrinsic defects; the adsorption and thermal decomposition of C24H12 and C60 on Si(001)-(2×1) induces Si dimer vacancy defects (Yates et al. 2006). Due to the small body of literature on the subject, only the behavior of charged extrinsic defects on Si, GaAs, and TiO2 surfaces will be examined here. Especially with doped material, researchers tend to define the term “surface” loosely. Consequently we will define the “surface” as a region several atomic layers deep where material and electronic properties are significantly perturbed by truncation of the bulk. This designation is necessary, as reports of surface defect charging often distinguish between the behavior of defects in the first, second, and third atomic layers. As many semiconductor surfaces are so open that second and third layer atoms still participate in reconstructions and are “visible” from the surface, it makes sense to consider the effects of defect substitution in these layers, where appropriate.

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8.2.1 Silicon For the most part, Group III dopant atoms preferentially adsorb in the trenches between surface dimer rows on Si(100). At critical doping levels (typically greater than 0.5 monolayer), impurity ad-dimers reorder to form a (2×2) reconstructions (Northrup et al. 1991). This behavior has been attributed to a whole host of factors, including valence, atomic size, and electronegativity. The first two ideas can be ruled out, as impurities with a wide range of atomic sizes, as well as those of difference valence (Sn, for example), behave similarly under certain circumstances. Instead, Ramamoorthy et al. suggest that miscibility and electronegativity can be invoked to explain the unfavorable incorporation of Al, Ga, and In into the Si(100) surface (1998). From binary phase diagrams, it can be seen that P, As, Sb, and B react exothermically with bulk Si to form stoichiometric compounds, while Al, Ga, In, and Sn have extremely small miscibility with Si at all temperatures. As the electronegativities of Al, Ga, In, and Sn are approximately equal to that of Si (refer to Table 4.3), there is little charge transfer between the two species and only weak covalent bonds form. As boron behaves differently from the other group III adatoms on the silicon surface, which has many consequences for device fabrication, its behavior will be discussed here in extra detail. Boron-induced reconstructions have been observed using LEED (Sardela et al. 1991) and STM (Wang and Hamers 1995) and investigated via ab initio total-energy calculations (Chang and Stott 1996; Fritsch et al. 1998). According to Wang and Hamers, the deposition and thermal decomposition of boron on Si(100) results in ordered reconstructions with (4×4) symmetry. The p-type character of the surface scales with B coverage, which indicates that at least some of the boron atoms ionize. The isolated substitutional boron atom can be located in an assortment of sites within the first, second, or third layer of the surface. According to DFT calculations performed by Ramamoorthy et al., the lowest energy configuration involves boron substituting for silicon at a second layer site (1999). The small atomic size of the dopant boron atom leads to large relaxations of the nearest-neighbor silicon atoms. Another structural model, the “rotated-dimer” model proposed by Fritsch et al. (Fig. 8.18), where boron atoms substitute for two silicon atoms in the third atomic layer and two silicon atoms in the first, significantly lowers the surface energy per unit cell and agrees with the periodicity, location, and orientation of the features seen in the STM experiments. Group V atoms readily incorporate into the Si(111) and Si(100) surfaces, yet have no effect on their surface reconstructions. Phosphorus, the most common n-type dopant for bulk silicon, can form bonds with surface silicon adatoms and cause both point and line defects. Individual P donors are observed on the n-type Si(111)-(2×1) surface at room temperature (Trappmann et al. 1997; Trappmann et al. 1999). The defects marked by arrows in Fig. 8.19 scale in number with the P doping concentration. Si of various doping concentrations was cleaved to expose either these charged dopant atoms (for highly doped material) or no P-induced defects (for slightly doped material). Also, the features appear as protrusions at

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Fig. 8.18 Atomic arrangement of the β-c(4×4) (a) and α-c(4×4) reconstruction (b) of Si(001):B in the rotated-dimer model illustrated in a top view of the first three atomic layers. Reprinted figure with permission from Fritsch J, Page JB, Schmidt KE et al. (1998) Phys Rev B: Condens Matter 57: 9750. Copyright (1998) by the American Physical Society.

Fig. 8.19 Simultaneously acquired STM images (150 Å × 150 Å) at +1.1V (a) and –1.1 V (b) for a sample with a P doping concentration of 6 × 1019 cm–3. P-induced defects are marked by arrows, and scale with doping concentration. Reprinted with permission from Trappmann T, Surgers C, v. Lohneysen H, “Investigation of the (111) surface of P-doped Si by scanning tunneling microscopy,” (1999) Appl Phys A A68: 169, Fig. 5. Copyright (1999) by Springer-Verlag.

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positive voltage and indentations at negative voltage, which is indicative of their electronic, rather than topological, nature. On Si(100), in contrast to Si(111), neutral P-P dimers form at low dopant concentrations, while line defects (perpendicular to the P dimer rows) arise at P coverages greater than one monolayer. According to Wang et al., the formation of the P-Si dimer has an energy of 0.1 eV/dimer on the Si(100)-(2×1) surface (1994). Consequently, the P-Si bond is 0.05 eV more stable than the average energy of a Si-Si dimer and a P-P dimer. Similar behavior has been predicted for As on Si(100) (Ramamoorthy et al. 1998).

8.2.2 Gallium Arsenide Experimental evidence exists for charged extrinsic defects on GaAs(110). Highresolution STM images reveal isolated defects, SiAs acceptors, SiGa donors, and complexes consisting of Si dopant atoms and native Ga vacancies form on Si-doped GaAs (Gebauer et al. 1997). The SiAs and SiGa defects adopt charge states of (–1) (Domke et al. 1996) and (+1) (Zheng et al. 1994) on the surface, respectively. The concentration of defects complexes increases with increasing Ga vacancy and Si concentration. As no local band bending is observed, it is suggested that the defect complex is uncharged on the surface. Consequently, it must be formed from a negatively charged Ga vacancy and a positively charged SiGa defect; in the bulk, this complex is negatively charged (Northrup and Zhang 1993). STM images of all of these defects are depicted in Fig. 8.20. A similar structure comprised of ZnGa and an anion vacancy has been observed on Zn-doped GaAs(110) (Ebert et al. 1996).

Fig. 8.20 Images of occupied (upper frames) and empty (lower frames) density of states of the major defects on Si-doped GaAs(110) including (a) the gallium vacancy (b) the SiGa donor (c) the SiAs acceptor and (d) the SiGaVGa complex. Reprinted figure with permission from Domke C, Ebert P, Heinrich M et al. (1996) Phys Rev B: Condens Matter 54: 10289. Copyright (1996) by the American Physical Society.

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8.2.3 Titanium Dioxide Dopants and impurities in TiO2 can occupy sites on the (110) surface. Elements including V, Nb, and Ta may substitute for Ti and form positively charged surface defects (Batzill et al. 2002). The substitution of V is more favorable than that of Nb and Ta due to its larger atomic size; smaller atoms cause less elastic distortion of the crystal lattice and are not as likely to segregate to the semiconductor surface. Nitrogen doping has an effect on the local surface structure of the rutile TiO2(110) surface. Domains of white stripes, which are visible in STM images and characteristic of the (1×2) reconstruction, scale in density with N doping level (Batzill et al. 2006). The existence of this reconstruction, which is also observed for heavily reduced material, suggests that N facilitates the formation of oxygen vacancies on the TiO2 surface. Chlorine preferentially adsorbs on rutile TiO2(110) by filling oxygen vacancy sites along bridging oxygen rows (Batzill et al. 2003). The change in XPS spectra following Cl adsorption (Hebenstreit et al. 2002) and DFT calculations (Vogtenhuber et al. 2002) indicate that the adatom is negatively charged and results in upward local band bending. Chlorine adsorption is suppressed on regions of the surface where electronegative Cl atoms compete for electrons with positively charged subsurface defects.

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289

Index

A Ab initio modeling program, 144 acceptor defect, 7, 9 deep acceptor, 233 shallow acceptor, 233 activation energy, 24, 198, 216 adatom, 5, 105 adatom diffusion, 216 adsorbate, 184 adsorption, 32, 221 AIMPRO, See Ab initio modeling program alkaline earth elements, 272 amphoteric dopant, 256 antisite, 74, 89, 93, 114, 153 APF, See atomic packing fraction Arrhenius form, 20 atomic packing fraction, 2, 64 atomic radius, 66, 243 B ballistic model for diffusion, 202 band bending, 31, 116, 176, 178, 219, 280, 281 band gap energy, 13, 131, 144, 156, 161, 166 narrowing of, 268, 270 underestimation of, 49, 55 bias error, 56 binding energy, 6, 15, 28 Boltzmann statistics, 8

boundary conditions, 49 Brouwer diagram, 19 C capture cross section, 30, 42 carrier concentration, 8, 41 CBM, See conduction band minimum charge additivity, 2, 71 chemical potential, 7 cluster diffusion, 212, 222 cluster dissociation energies, 28 coincidence Doppler broadening, 41 conduction band, 8, 33 conduction band minimum, 13, 30, 233 conductivity, 43 confidence interval, 53 continuum approach, 22 correction schemes, 49 Coulomb energy, 16, 49 covalent radius, 65, 66 covalent semiconductor, 68 crystal structure, 64 anatase, 65, 95 brookite, 95 diamond, 65, 76, 84 fluorite, 65, 100, 122 hexagonal, 64, 92 rocksalt, 65, 100 rutile, 65, 95 wurtzite, 65, 100, 116 zincblende, 64, 86, 92, 100, 116 cuboctahedral cluster, 104

291

292 D dangling bond, 7, 32 deep level transient spectroscopy, 41 high-resolution deep level transient spectroscopy, 80 Laplace deep level transient spectroscopy, 42 synchrotron radiation deep level transient spectroscopy, 42 defect aggregate, 71 defect associate, 75, 95 defect cluster, 5, 15, 75, 120, 274 defect complex, 75, 236 defect engineering, 1 defect pair, 234 degeneracy, 9 degeneracy factor, 10 degree of localization, 14 degrees of freedom, 6 demarcation level, 30 density functional theory, 47, 131 DFT, See density functional theory diffusion, 71, 195 diffusion coefficient, 20, 21 diffusion measurements, 44, 196 diffusion pathway, 21 di-interstitial, 82, 91 di-interstitial diffusion, 213 dimer vacancy, See divacancy divacancy, 79, 86, 90, 99, 108 mixed divacancy, 90, 102 split-off dimer defect, 109, 175 split-off dimer vacancy, 112 divacancy diffusion, 213, 214 DLTS, See deep level transient spectroscopy donor defect, 7, 9 deep donor, 233 shallow donor, 233, 261 doping, 8, 233 dynamic random access memory, 235 E EELS, See electron energy loss spectroscopy eigenstates, 7, 16 EL2 defect, 89, 115 electric fields, 7, 24 electron energy loss spectroscopy, 45

Index electron paramagnetic resonance, 39, 80, 82, 147 optically detected electron paramagnetic resonance, 160 electronegativity, 68, 218, 243, 278 electron-hole pair, 12, 29 electron-lattice coupling, 2, 68 electron-nuclear double resonance, 40 ENDOR, See electron-nuclear double resonance enthalpy of formation, 12, 15 enthalpy of ionization, 12 enthalpy of migration, 20 entropy of formation, 218 entropy of ionization, 13, 174 entropy of migration, 20 extended defect, 75 F Fermi energy, 7, 8, 9, 10 Fermi-Dirac statistics, 8, 9 ferromagnetism, 256, 257, 269 field ion microscopy, 23 field-assisted diffusion, 25 FIM, See field ion microscopy fission products, 273, 275 Frank-Turnbull mechanism, 197, 260 Frenkel defect formation, 162, 170 Frenkel pair, 83, 90, 95, 144 G generalized gradient approximation, 48, 135, 169 GGA, See generalized gradient approximation Gibbs free energy, 6, 11, 151 Green’s function calculations, 50, 133, 134 H Hall-effect measurements, 133, 141 Hartree-Fock, 50 HF, See Hartree-Fock hopping diffusivity, 23 hopping frequency, 20 hydrogen impurities, 235, 249 I illumination, 223, 225 inelastic electron tunneling spectroscopy, 45

Index

293

interstitial, 73, 78, 85, 87, 94, 97, 101 surface interstitial, See adatom interstitial diffusion, 198, 205, 207, 209, 211, 243, 253 intrinsic material, 8 introduction, 1 ion bombardment, 33 ion implantation, 1, 28, 259 ionic radius, 66, 172 ionic semiconductor, 68 ionization level, 9, 10, 15, 54 ionization level error, 55

N

J

P

Jahn-Teller distortion, 74, 77, 80, 86, 91

lattice constant, 76, 84, 86, 95, 100, 147 lattice-mode softening, 13 LDA, See local density approximation LDLTS, See deep level transient spectroscopy LEED, See low energy electron diffraction linear muffin-tin orbital calculations, 50 local density approximation, 48, 135, 142 local spin density approximation, 48, 144 low energy electron diffraction, 45

PACS, See perturbed angular correlation spectroscopy pair diffusion, 21, 26, 243 parameter sensitivity analysis, 51 partial pressure, 17, 19, 66, 163, 167 PAS, See positron annihilation spectroscopy Perdew-Burke-Ernzerhof, 142 perturbed angular correlation spectroscopy, 144 photocatalysis, 1, 266 photoreflectance spectroscopy, 45 photostimulated diffusion, 222, 225 photostimulation, 7, 32 point-group symmetry, 75, 113 positron annihilation spectroscopy, 40, 80, 149 positron lifetime, 41 PR, See photoreflectance spectroscopy pre-exponential factor, 20, 24, 71, 200, 202, 218 pseudopotential, 49 p-type doping, 234, 249, 261, 271

M

R

maximum a posteriori estimation, 51 maximum likelihood, 27, 51, 131, 136, 141, 142, 145, 146, 148, 151, 152, 155, 171, 238, 243 mesoscale diffusion, 215, 216 mesoscale diffusivity, 23 mesoscopic point-like defect, 76 mixed divacancy, 214 ML, See maximum likelihood Monte Carlo calculations, 51 kinetic lattice Monte Carlo, 51, 75 quantum Monte Carlo, 51

radiative charge exchange, 29 rare earth elements, 272 RAS, See reflectance anisotropy spectroscopy reaction kinetics, 25 dissociation kinetics, 27 rebonding, 68, 69, 109, 175 recombination center, 30, 234, 235 reflectance anisotropy spectroscopy, 45 reflection high-energy electron diffraction, 45, 107

K Kelvin probe force microscopy, 46 kick-out mechanism, 26, 197, 204, 243, 264, 271 k-point sampling, 54, 55 Kröger-Vink notation, 2 L

negative-U behavior, 132, 133, 135, 138, 146, 149, 151, 153, 157, 173 non-Fickian diffusion, 22 non-thermal diffusion, 173 n-type doping, 234, 239, 250, 261 O oxidation enhanced diffusion, 82, 213 oxygen impurities, 235, 249

294 relaxation, 2, 68, 74 RHEED, See reflection high-energy electron diffraction S scanning Kelvin probe microscopy, 75 scanning tunneling microscopy, 23, 45, 56, 107, 109, 118, 179, 220 Schottky defect formation, 162, 170 second harmonic microscopy, 47, 107 SHM, See second harmonic microscopy Shockley-Read-Hall (SRH) model, 30 site-to-site hopping, 198, 216 spin degeneracy, 6 SRDLTS, See deep level transient spectroscopy STM, See scanning tunneling microscopy stoichiometry, 6, 67, 156, 157, 162, 180, 209 substitutional defect, 73, 233, 235 supercell, 48, 54, 158 surface differential reflectance spectroscopy, 45 surface diffusion, 22, 46, 71, 215 surface reconstructions, 69, 105, 113, 117 surface-bulk coupling, 31 T TB, See tight-binding TED, See transient enhanced diffusion TEM, See transmission electron microscopy temperature, 12, 14, 50, 132, 151, 163, 165 tetra-interstitial, 83 thermal charge exchange, 29 thermal diffusion, 19

Index tight-binding, 50, 160 tracer diffusion coefficient, 44 transient enhanced diffusion, 45, 52, 82, 213, 242 transition metals, 235, 237, 239, 247, 250, 256, 261, 262, 264, 266, 272 transmission electron microscopy, 45 tri-interstitial, 82 trivacancy, 81 U unfaulted edge vacancy, 174 V vacancy, 73, 76, 84, 87, 93, 96, 100, 108, 112, 113, 117, 123 bridging oxygen vacancy, 121 faulted corner vacancy, 108 faulted edge vacancy, 108 in-plane oxygen vacancy, 121 split vacancy, 73, 76 unfaulted corner vacancy, 108 unfaulted edge vacancy, 108 vacancy cluster, 105 vacancy diffusion, 199, 201, 202, 205, 207, 209, 211, 219, 220, 221, 243, 253, 259 vacancy island, 105 valence band, 8, 33 valence band maximum, 13, 14 variable-energy positron annihilation spectroscopy, 41 variance, 52 VBM, See valence band maximum W weighting factor, 52 Willis cluster, 104