engineering materials and...

Engineering Materials and Processes

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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

Composite Materials D.D.L. Chung

Titanium G. Lütjering and J.C. Williams

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

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

Computational Mechanics of Composite Materials M. Kamiński

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

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

Fuel Cell Technology N. Sammes

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

Computational Quantum Mechanics for Materials Engineers L. Vitos

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

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

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

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

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

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v

Preface

Defect charging can affect numerous aspects of defect properties, including physi-cal structure, rate of diffusion, chemical reactivity, and interactions with the elec-trons 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, ca-talysis for producing chemical products, photocatalysis for environmental reme-diation, 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 re-search, 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 exten-sive reviews where they exist.

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 de-fect engineering.

Urbana, Illinois, USA, July 2008 Edmund G. Seebauer Meredith C. Kratzer

vii

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.

ix

Contents

1 Introduction............................................................................................. 1 References................................................................................................. 3

2 Fundamentals of Defect Ionization and Transport .............................. 5 2.1 Introduction ................................................................................... 5 2.2 Thermodynamics of Defect Charging ........................................... 5

2.2.1 Free Energies, Ionization Levels, and Charged Defect Concentrations................................. 7

2.2.2 Ionization Entropy............................................................ 13 2.2.3 Energetics of Defect Clustering ....................................... 15 2.2.4 Effects of Gas Pressure on Defect Concentration ............ 17

2.3 Thermal Diffusion ......................................................................... 19 2.4 Drift in Electric Fields ................................................................... 24 2.5 Defect Kinetics .............................................................................. 25

2.5.1 Reactions.......................................................................... 25 2.5.2 Charging........................................................................... 29

2.6 Direct Surface-Bulk Coupling ....................................................... 31 2.7 Non-Thermally Stimulated Defect Charging and Formation ........ 32

2.7.1 Photostimulation .............................................................. 32 2.7.2 Ion-Defect Interactions .................................................... 33

References................................................................................................. 34

3 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

x 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...................................................... 63 4.1 Defect Formation........................................................................... 63

4.1.1 Effects of Crystal Structure and Atomic Properties ......... 63 4.1.2 Effects of Stoichiometry .................................................. 66

4.2 Defect Geometry ........................................................................... 68 4.3 Defect Charging............................................................................. 69

4.3.1 Bulk vs. Surface ............................................................... 70 4.3.2 Point Defects vs. Defect Aggregates................................ 71

4.4 Defect Diffusion ............................................................................ 71 References................................................................................................. 72

5 Intrinsic Defects: Structure.................................................................... 73 5.1 Bulk Defects .................................................................................. 73

5.1.1 Silicon .............................................................................. 76 5.1.2 Germanium ...................................................................... 84 5.1.3 Gallium Arsenide ............................................................. 86 5.1.4 Other III–V Semiconductors ............................................ 92 5.1.5 Titanium Dioxide ............................................................. 95 5.1.6 Other Oxide Semiconductors ........................................... 100

5.2 Surface Defects.............................................................................. 105 5.2.1 Silicon .............................................................................. 106 5.2.2 Germanium ...................................................................... 111 5.2.3 Gallium Arsenide ............................................................. 112 5.2.4 Other III–V Semiconductors ............................................ 116 5.2.5 Titanium Dioxide ............................................................. 120 5.2.6 Other Oxide Semiconductors ........................................... 122

References................................................................................................. 123

6 Intrinsic Defects: Ionization Thermodynamics .................................... 131 6.1 Bulk Defects .................................................................................. 131


Contents xi

6.2 Surface Defects.............................................................................. 173 6.2.1 Silicon .............................................................................. 173 6.2.2 Germanium ...................................................................... 176 6.2.3 Gallium Arsenide ............................................................. 178 6.2.4 Other III–V Semiconductors ............................................ 181 6.2.5 Titanium Dioxide ............................................................. 183 6.2.6 Other Oxide Semiconductors ........................................... 185

References................................................................................................. 187

7 Intrinsic Defects: Diffusion .................................................................... 195 7.1 Bulk Defects .................................................................................. 195

7.1.1 Point Defects.................................................................... 196 7.1.2 Associates and Clusters.................................................... 212

7.2 Surface Defects.............................................................................. 215 7.2.1 Point Defects.................................................................... 215 7.2.2 Associates and Clusters.................................................... 222

7.3 Photostimulated Diffusion............................................................. 222 7.3.1 Photostimulated Diffusion in the Bulk............................. 223 7.3.2 Photostimulated Diffusion on the Surface ....................... 225

References................................................................................................. 226

8 Extrinsic Defects ..................................................................................... 233 8.1 Bulk Defects .................................................................................. 233


8.2 Surface Defects.............................................................................. 277 8.2.1 Silicon .............................................................................. 278 8.2.2 Gallium Arsenide ............................................................. 280 8.2.3 Titanium Dioxide ............................................................. 281

References................................................................................................. 281

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

xiii

List of Abbreviations

AIMPRO Ab initio modeling program APF Atomic packing fraction BOV Bridging oxygen vacancy CBM Conduction band minimum CDB Coincidence Doppler broadening DFT Density functional theory DLTS Deep level transient spectroscopy EELS Electron energy loss spectroscopy ENDOR Electron-nuclear double resonance EPR Electron paramagnetic resonance FC Faulted corner FE Faulted edge FIM Field ion microscopy GGA Generalized gradient approximation HR Hartree–Fock IETS Inelastic electron tunneling spectroscopy KLMC Kinetic lattice Monte Carlo KPFM Kelvin probe force microscopy LDA Local density approximation LDLTS Laplace deep level transient spectroscopy LEED Low energy electron diffraction LMCC Local moment countercharge LSDA Local spin density approximation ML Maximum likelihood ODEPR Optically detected electron paramagnetic resonance OED Oxidation enhanced diffusion PACS Perturbed angular correlation spectroscopy PAS Positron annihilation lifetime spectroscopy PBE Perdew–Burke–Ernzerhof POV In-plane oxygen vacancy

xiv List of Abbreviations

PR Photoreflectance spectroscopy QMC Quantum Monte Carlo RAS Reflectance anisotropy spectroscopy RDS Reflectance difference spectroscopy RHEED Reflection high-energy electron diffraction SDRS Surface differential reflectance spectroscopy SHM Second harmonic microscopy SRDLTS Synchrotron radiation deep level transient spectroscopy SRH Shockley–Read–Hall STM Scanning tunneling microscopy TB Tight-binding TED Transient enhanced diffusion TEM Transmission electron microscopy UFC Unfaulted corner UFE Unfaulted edge VBM Valence band maximum VEPAS Variable-energy positron annihilation spectroscopy XAFS X-ray absorption fine structure

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

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 de-vices (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 perform-ance 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 electri-cal power (Green 1996; Kurtz et al. 1999). To improve material performance, various forms of “defect engineering” have been developed to control defect be-havior 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. Understand-ing surface defects is becoming increasingly important in practical applications – for example, as electronic devices shrink closer to the atomic scale (with the at-tendant increase in surface-to-volume ratios), and as molecular-level control of catalytic reactions becomes increasingly feasible. Of particular importance is de-fect-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 de-fects. (Even less is known about defects at solid–solid interfaces, but some analo-gies 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 di-rectly into the behavior of surface defects (Ditchfield et al. 1998, 2000).

2 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 non-thermally 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 resis-tance in ultrashallow junctions. Such phenomena offer completely new mecha-nisms for defect engineering, as well as new means to study the charging phe-nomenon 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 semicon-ductor 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 mis-matches 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 in-teraction 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 sili-con and III–V defects employs a substantially different notation. To foster a uni-form 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 Materi-

als, 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, Can-berra, 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

4 1 Introduction

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

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

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 de-fects formed either intentionally (via doping or ion implantation, for instance) or accidentally by the introduction of foreign atoms into the semiconductor. In bo-ron-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

6 2 Fundamentals of Defect Ionization and Transport

system. Confusion over this distinction sometimes exists particularly with refer-ence to ionization levels.

When thermally generated or artificial point defects are introduced into a per-fect 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

[ ]0 0 0

0 0

0

exp exp expf f fX X X

X X

X G S HS kT k kT

θ θ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− −⎣ ⎦ = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (2.1)

where [S] is the concentration of available lattice sites in the crystal, 0Xθ is the number of degrees of internal freedom of the defect on a lattice site, and 0

fXG , 0

fXH ,

and 0fXS 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 degener-acy or equivalent geometric configurations at the same site (Pichler 2004). Typi-cally only the spin degeneracy is of direct interest for defect charging. For simplic-ity, 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

[ ] [ ][ ][ ]

222 exp

bX

XEX X

XS kT

θ⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

(2.2)

where 2

bXE denotes the binding energy of the X2 defect, and the degeneracy factor

2Xθ 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 dis-cussed 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 pres-sure 2OP . However, it becomes necessary to incorporate several additional vari-ables 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

[ ] 1 1exp exp[ ]

MM

M

V S HPM k kT

α− Δ Δ⎡ ⎤ ⎡ ⎤= −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ (2.3)

and that of vacancies in the oxygen sublattice by

[ ]2

2 1 2 1exp exp[ ]

OO

O

V S S H HPO k kT

α− Δ − Δ Δ − Δ⎡ ⎤ ⎡ ⎤= −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ (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 tempera-ture dependent, the empirical values of n are typically determined from tempera-ture-dependent electrical conductivity measurements. ΔS1 and ΔS2 may contain contributions from the vibrational entropy of the crystal resulting from the addi-tion of VM, VO, and extra oxygen atoms, as well as the standard Gibbs free entropy of the oxygen molecule in the gas phase 2OSΔ . 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 de-pend upon the value of the chemical potential (or more colloquially, “Fermi en-ergy” 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., dan-gling 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 elec-tron 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 rich-ness 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 con-duction 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 wavefunc-tions, 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 dielec-tric 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.


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

( ) ( ) /

11 FE E kT

f Ee −

=+

(2.5)

where k is again Boltzmann’s constant, f (E) the probability that an available en-ergy 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 elec-tron 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 terminol-ogy 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

ln2 2

C V VF

C

E E kT NEN

+= + (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 approxi-mately in the middle of the band gap.

The product of the two charge-carrier concentrations is independent of the Fermi level and obeys

2 2i in p 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 elec-tron and hole concentrations can also be approximated by:

exp F CC

E En NkT−⎛ ⎞= ⎜ ⎟

⎝ ⎠ (2.8)

and

exp V FV

E Ep NkT−⎛ ⎞= ⎜ ⎟

⎝ ⎠ (2.9)


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:

0 1 1X X h− +↔ + . (2.10)

Equation 2.10 is equally valid for point defects such as vacancies and self-interstitials as it is for divacancies and substitutional extrinsic defects. The law of mass action implies that

( ) 1

1

1

[ ]

1 exp FX

XXE Eg X

kT−

−

−⎡ ⎤ =⎣ ⎦ −⎡ ⎤+ ⎢ ⎥⎣ ⎦

, (2.11)

where [X] is the concentration of the defect in all charge states, g is an overall degeneracy factor, and 1XE − is the ionization level for the singly ionized acceptor. This expression can be simplified when │ 1XE − – 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:

1 1

0

1

0exp FX X

X

X E EkTX

θθ

− −

−⎡ ⎤ −⎡ ⎤⎣ ⎦ = ⎢ ⎥⎡ ⎤ ⎣ ⎦⎣ ⎦, (2.12)

where 1Xθ − and 0Xθ 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

0 1 1X X e+ −↔ + , (2.13)

where the concentration of X+1 can be determined from Eq. 2.14:

( ) 1

1

1

[ ]

1 exp F X

XXE Eg X

kT+

+

+⎡ ⎤ =⎣ ⎦ −⎡ ⎤+ ⎢ ⎥⎣ ⎦

(2.14)

or Eq. 2.15, when │EF – 1XE + │>> kT:

1 1

0

1

0exp FX X

X

X E EkTX

θθ

+ +

+⎡ ⎤ −⎡ ⎤⎣ ⎦ = ⎢ ⎥⎡ ⎤ ⎣ ⎦⎣ ⎦. (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


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 degen-eracy factor). For example, if 1Xθ − = 0Xθ in Eq. 2.12, then [X–1] = [X0] when EF = 1XE − .

The degeneracy factors in Eqs. 2.11 and 2.14 are usually concerned with differ-ences 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 degen-eracy, 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 con-structed by induction from the single charge states. As an example, the concentra-tion of the multiply charged acceptor X−2 with ionization level 2XE − is:

2 2 1

0

2

0

2exp FX X X

X

X E E EkTX

θθ

− − −−⎡ ⎤ + −⎡ ⎤⎣ ⎦ = −⎢ ⎥⎡ ⎤ ⎣ ⎦⎣ ⎦

, (2.16)

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

2 2 1

0

2

0

2exp FX X X

X

X E E EkTX

θθ

+ + ++⎡ ⎤ − −⎡ ⎤⎣ ⎦ = −⎢ ⎥⎡ ⎤ ⎣ ⎦⎣ ⎦

. (2.17)

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

11 0f f

XX XG G E −− = + (2.18)

where 0 0 0f f fX X XG H TS= − (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

q q qf f

FX e XG G qE−+ = − , (2.19)


where q is the charge state of defect X and EF is the Fermi energy. When consider-ing 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

11 1 0f f f

FXX X XG G G E E−− −Δ = − = − . (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

2 12 2 0 2f f fFX XX X XG G G E E E− −− −Δ = − = + − (2.21)

and

11 1 0f f f

F XX X XG G G E E ++ +Δ = − = − . (2.22)

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.


These free energies of defect ionization can be decomposed into corresponding enthalpies and entropies of ionization, q

fXHΔ and q

fXSΔ :

q q qf f fX X XG H T SΔ = Δ − Δ . (2.23)

Note that qfXGΔ , q

fXHΔ , and q

fXSΔ 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 q

fXHΔ 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 q

fXHΔ 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):

( ) ( ) ( )2

0g gTE T E

Tα

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 nega-tive standard entropy of that reaction (Van Vechten 1980),

gcv

ES

T∂

≡ −Δ∂

. (2.25)

Then the definition ΔG = ΔH – TΔS implies

( ) ( ) ( )gcv g

E TH T E T T

T∂

Δ = −∂

(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:

( ) ( )( )

2

20cv gTH T E

Tαβ

βΔ = +

+. (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 ob-tained from Eq. 2.25, when combined with q

fXHΔ at T = 0 K as deduced from ex-

periment 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

( ) ( ) ( )0q qf f

cvX XH T H H TΔ = Δ + Δ . (2.28)

The enthalpy of ionization at 0 K, ( )0qfXHΔ , 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.


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 con-figurational degeneracy, lattice mode softening due to bond cleavage, and ioniza-tion (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 q

fXSΔ can be very large for certain kinds of native defects such as vacan-

cies. The main contribution to qfXSΔ 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 frequen-cies (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 cor-responds to the standard chemical potential for creating a delocalized hole at the valence band maximum and a delocalized electron at the conduction band minimum.


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 for-mation of the electron-hole pair obeys (Thurmond 1975):

( ) ( )( )2

2cvcv

T TES TT T

α ββ+∂ΔΔ = − =

∂ +. (2.29)

Ionization of a defect represents another mechanism for creating two new carri-ers 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 ioniza-tion 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 esti-mate, Van Vechten and Thurmond examined experimental data for the entropies of optical transitions in Si, Ge, GaAs and GaP between various points in the Bril-louin zone. These data were derived from the temperature dependence of the vari-ous gaps as determined by optical reflectance. For Si the reported entropies suf-fered 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,

( ) ( )qf

cvXS T S TΔ ≈ Δ (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 suscep-tibilities. (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 ( )qfXS TΔ regard-

less 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 ( )qfXS TΔ is that, as T

increases and Eg decreases, free energies referenced to the valence band maximum


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, di-interstitial, vacancy-dopant pair, etc. Numerous methods and approximations for calculating the formation enthalpy of a defect pair exist in the literature; this sec-tion 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 en-ergy.” For simplicity, the following discussion will distinguish between two com-ponents of the binding energy of (XX)2q. The Coulombic interaction and the short-range “chemical” interaction between defects Xq and Xq sum to yield the binding energy of the pair:

( ) ( )2 2, q q

q qb b Coulombic X XH XX H XXΔ = Δ +Φ . (2.32)

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 va-cancy 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.


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 approxi-mation 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

( )2 2

2,

04q

b Coulombicr

q eH XXr

απε ε

Δ = , (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 dis-tance, 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 con-stant of the semiconductor, εr, which is a measure of the polarizability of the lat-tice. 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 consid-ered, the short-range “chemical” interaction between Xq and Xq, q qX XΦ . This com-ponent is especially important for semiconductors having primarily covalent bond-ing 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 approxi-mation is clearly not applicable. The non-Coulombic interactions between the defects can be summed into the term q qX XΦ , 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). Ac-cording to this model

( ) 6expq q q q

q q q q

q q q q

X X X XX X X X

X X X X

r Cr A

rρ⎛ ⎞

Φ = − −⎜ ⎟⎝ ⎠

, (2.34)

where q qX Xr is the nearest neighbor distance between Xq and Xq, and q qX XA , q qX Xρ , and q qX XC are adjustable parameters. The parameters were selected to reproduce


the unit cell volumes of various relates oxides. Also, the pair interaction in a cova-lent material as a function of radial separation can be expressed as

( ) ( )0 exp / 1q qX Xr r rβ⎡ ⎤Φ = Φ − −⎣ ⎦, (2.35)

where q qX Xr 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 experi-mental 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 be-hind 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 anni-hilate oxygen vacancies. Analogous phenomena occur in other compound semi-conductors 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

( )0 0 gM OMO M O MO≡ + ↔ (2.36)

( ) ( )0 02

12

ggM OM O M O+ ↔ + . (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 re-mains vacant, creating a metal vacancy:

( ) 0 02

12

gO MO O V↔ + . (2.38)


If the metal vacancy were to subsequently ionize to VM−1, the concentration of

VM−1

could be described as a function of oxygen partial pressure according to

2

1/ 211 O

MO

K PV

p 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 de-nominator, whereas for the neutral state p would not appear at all.

Oxides can also exchange metal atoms with the gas phase, although most ex-perimental 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 approxi-mations to equilibrium conditions. When PM is high, metal atoms fill metal vacan-cies in the bulk and create vacancies in the oxygen sublattice:

( ) 0 0gM OM M V↔ + . (2.40)

Once oxygen vacancies ionize into the (+1) charge state, their concentration is given by

210M

OM

K PVn 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 elec-trical charge, even though charged defects exist in the bulk:

1 1M On V p V− +⎡ ⎤ ⎡ ⎤+ = +⎣ ⎦ ⎣ ⎦ . (2.42)

One final equilibrium expression,

in p K∗ = (2.43)

arises from the equilibration of electrons and holes in the crystal. The ionized de-fects in MO are now described by a series of algebraic equations containing seven variables: n, p, 1

MV −⎡ ⎤⎣ ⎦, 1

OV +⎡ ⎤⎣ ⎦, PM, 2OP , and T, where T is the absolute temperature of the system. Normally T and 2OP 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 con-stants; no separate contributions to the free energy, entropy, or enthalpy of ioniza-tion 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.

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