Drabold d estreicher s (eds) theory of defects in semiconductors (TAP 104 2006)(ISBN 3540334009)(274s)

Theory has played a critical role in understanding, and thereforecontrolling, the properties of defects.Conversely, the careful experimental studies of defects in Ge, Si, thenmany other

much for the field of defects in semiconductors over the past decades, and convinced so many theorists to calculate beyond

what they thought possible.

Semiconductor materials emerged after World War II and their impact onour lives has grown ever since Semiconductor technology is, to a large ex-tent, the art of defect engineering Today, defect control is often done at theatomic level Theory has played a critical role in understanding, and thereforecontrolling, the properties of defects.

Conversely, the careful experimental studies of defects in Ge, Si, thenmany other semiconductor materials has a huge database of measured quan-tities which allowed theorists to test their methods and approximations.Dramatic improvement in methodology, especially density functional the-ory, along with inexpensive and fast computers, has impedance matched theexperimentalist and theorist in ways unanticipated before the late eighties

As a result, the theory of defects in semiconductors has become quantitative

in many respects Today, more powerful theoretical approaches are still beingdeveloped More importantly perhaps, the tools developed to study defects

in semiconductors are now being adapted to approach many new challengesassociated with nanoscience, a very long list that includes quantum dots,buckyballs and buckytubes, spintronics, interfaces, and many others.Despite the importance of the field, there have been no modern attempts

to treat the computational science of the field in a coherent manner within asingle treatise This is the aim of the present volume

This book brings together several leaders in theoretical research on defects

in semiconductors Although the treatment is tutorial, the level at which thevarious applications are discussed is today’s state-of-the-art in the field.The book begins with a ‘big picture’ view from Manuel Cardona, andcontinues with a brief summary of the historical development of the subject

in Chapter 1 This includes an overview of today’s most commonly usedmethod to describe defects

We have attempted to create a balanced and tutorial treatment of thebasic theory and methodology in Chapters 3-6 They including detailed dis-cussions of the approximations involved, the calculation of electrically-activelevels, and extensions of the theory to finite temperatures Two emergingelectronic structure methodologies of special importance to the field are dis-cussed in Chapters 7 (Quantum Monte-Carlo) and 8 (the GW method) Thencome two chapters on molecular dynamics (MD) In chapter 9, a combina-

tion of high-level and approximate MD is developed, with applications tothe dynamics of extended defect Chapter 10 deals with semiempirical treat-ments of microstructures, including issues such as wafer bonding (Chapters

9 and 10) The book concludes with studies of defects and their role in thephoto-response of topologically disordered (amorphous) systems

The intended audience for the book is graduate students as well as vanced researchers in physics, chemistry, materials science, and engineering

ad-We have sought to provide self-contained descriptions of the work, with tailed references available when needed The book may be used as a text in

de-a prde-acticde-al grde-adude-ate course designed to prepde-are students for resede-arch work

on defects in semiconductors or first-principles theory in materials science

in general The book also serves as a reference for the active theoretical searcher, or as a convenient guide for the experimentalist to keep tabs ontheir theorist colleagues

re-It was a genuine pleasure to edit this volume We are delighted with thecontributions provided in a timely fashion by so many busy and accomplishedpeople We warmly thank all the contributors and hope to have the oppor-tunity to share some nice wine(s) with all of them soon After all,

When Ptolemy, now long ago,

Believed the Earth stood still,

He never would have blundered so

Had he but drunk his fill

He’d then have felt it circulate

And would have learnt to say:

The true way to investigate

Is to drink a bottle a day

(author unknown)

published in Augustus de Morgans A Budget of Paradoxes, (1866).

February 2006

Manuel Cardona 1

1 Early history and contents of the present volume 1

2 Bibliometric studies 7

References 9

1 Defect theory: an armchair history David A Drabold, Stefan K Estreicher 11

1.1 Introduction 11

1.2 The evolution of theory 13

1.3 A sketch of first-principles theory 16

1.3.1 Single particle methods: History 16

1.3.2 Direct approaches to the many-electron problem 18

1.3.3 Hartree and Hartree-Fock approximations 18

1.3.4 Density Functional Theory 19

1.4 The Contributions 22

References 22

2 Supercell methods for defect calculations Risto M Nieminen 27

2.1 Introduction 27

2.2 Density-functional theory 29

2.3 Supercell and other methods 30

2.4 Issues with the supercell method 32

2.5 The exchange-correlation functionals and the semiconducting gap 34 2.6 Core and semicore electrons: pseudopotentials and beyond 38

2.7 Basis sets 39

2.8 Time-dependent and finite-temperature simulations 41

2.9 Jahn-Teller distortions in semiconductor defects 42

2.9.1 Vacancy in silicon 42

2.9.2 Substitutional copper in silicon 44

2.10Vibrational modes 45

2.11Ionisation levels 46

2.12The marker method 48

2.13Brillouin-zone sampling 48

2.14Charged defects and electrostatic corrections 50

2.15Energy-level references and valence-band alignment 53

2.16Examples: the monovacancy and substitutional copper in silicon 53

2.16.1Experiments 55

2.16.2Calculations 56

2.17Summary and conclusions 58

References 59

3 Marker-method calculations for electrical levels using Gaussian-orbital basis-sets J P Goss, M J Shaw, P R Briddon 63

3.1 Introduction 63

3.2 Computational method 65

3.2.1 Gaussian basis-set 65

3.2.2 Choice of exponents 68

3.2.3 Case study: bulk silicon 69

3.2.4 Charge density expansions 73

3.3 Electrical levels 73

3.3.1 Formation energy 74

3.3.2 Calculation of electrical levels using the Marker Method 76

3.4 Application to defects in group-IV materials 77

3.4.1 Chalcogen-hydrogen donors in silicon 77

3.4.2 VO-centers in silicon and germanium 79

3.4.3 Shallow and deep levels in diamond 80

3.5 Summary 82

References 83

4 Dynamical Matrices and Free energies Stefan K Estreicher, Mahdi Sanati 85

4.1 Introduction 85

4.2 Dynamical matrices 87

4.3 Local and pseudolocal modes 88

4.4 Vibrational lifetimes and decay channels 89

4.5 Vibrational free energies and specific heats 92

4.6 Theory of defects at finite temperatures 95

4.7 Discussion 98

References 100

5 The calculation of free energies in semiconductors: defects, transitions and phase diagrams E R Hern´ andez, A Antonelli, L Colombo,, and P Ordej´ on 103

5.1 Introduction 103

5.2 The Calculation of Free energies 104

5.2.1 Thermodynamic integration and adiabatic switching 105

5.2.2 Reversible scaling 108

5.2.3 Phase boundaries and phase diagrams 110

5.3 Applications 113

5.3.1 Thermal properties of defects 113

5.3.2 Melting of Silicon 116

5.3.3 Phase diagrams 119

5.4 Conclusions and outlook 124

References 124

6 Quantum Monte Carlo techniques and defects in semiconductors R.J Needs 127

6.1 Introduction 127

6.2 Quantum Monte Carlo methods 128

6.2.1 The VMC method 128

6.2.2 The DMC method 129

6.2.3 Trial wave functions 131

6.2.4 Optimization of trial wave functions 132

6.2.5 QMC calculations within periodic boundary conditions 133

6.2.6 Using pseudopotentials in QMC calculations 134

6.3 DMC calculations for excited states 134

6.4 Sources of error in DMC calculations 135

6.5 Applications of QMC to the cohesive energies of solids 136

6.6 Applications of QMC to defects in semiconductors 136

6.6.1 Using structures from simpler methods 136

6.6.2 Silicon Self-Interstitial Defects 137

6.6.3 Neutral vacancy in diamond 142

6.6.4 Schottky defects in magnesium oxide 144

6.7 Conclusions 145

References 147

7 Quasiparticle Calculations for Point Defects at Semiconductor Surfaces Arno Schindlmayr, Matthias Scheffler 149

7.1 Introduction 149

7.2 Computational Methods 152

7.2.1 Density-Functional Theory 152

7.2.2 Many-Body Perturbation Theory 155

7.3 Electronic Structure of Defect-Free Surfaces 160

7.4 Defect States 163

7.5 Charge-Transition Levels 168

7.6 Summary 171

References 172

8 Multiscale modelling of defects in semiconductors: a novel molecular dynamics scheme

G´ abor Cs´ anyi, Gianpietro Moras, James R Kermode, Michael C.

Payne, Alison Mainwood, Alessandro De Vita 175

8.1 Introduction 175

8.2 A hybrid view 176

8.3 Hybrid simulation 179

8.4 The LOTF scheme 182

8.5 Applications 185

8.6 Summary 190

References 191

9 Empirical molecular dynamics: Possibilities, requirements, and limitations Kurt Scheerschmidt 195

9.1 Introduction: Why empirical molecular dynamics ? 195

9.2 Empirical molecular dynamics: Basic concepts 198

9.2.1 Newtonian equations and numerical integration 198

9.2.2 Particle mechanics and non equilibrium systems 200

9.2.3 Boundary conditions and system control 202

9.2.4 Many body empirical potentials and force fields 203

9.2.5 Determination of properties 205

9.3 Extensions of the empirical molecular dynamics 207

9.3.1 Modified boundary conditions: Elastic embedding 207

9.3.2 Tight-binding based analytic bond-order potentials 209

9.4 Applications 212

9.4.1 Quantum dots: Relaxation, reordering, and stability 212

9.4.2 Bonded interfaces: tailoring electronic or mechanical properties? 215

9.5 Conclusions and outlook 218

References 219

10 Defects in Amorphous Semiconductors: Amorphous Silicon D A Drabold and T A Abtew 225

10.1Introduction 225

10.2Amorphous Semiconductors 225

10.3Defects in Amorphous Semiconductors 228

10.3.1Definition for defect 228

10.3.2Long time dynamics and defect equilibria 230

10.3.3Electronic Aspects of Amorphous Semiconductors 230

10.3.4Electron correlation energy: electron-electron effects 232

10.4Modeling Amorphous Semiconductors 233

10.4.1Forming Structural Models 233

10.4.2Interatomic Potentials 234

10.4.3Lore of approximations in density functional calculations 23510.4.4The electron-lattice interaction 23610.5 Defects in Amorphous Silicon 237References 244

11 Light-Induced Effects in Amorphous and Glassy Solids

S.I Simdyankin, S.R Elliott 247

11.1Photo-induced metastability in Amorphous Solids: an

Experimental Survey 24711.1.1Introduction 24711.1.2Photo-induced effects in chalcogenide glasses 24911.2Theoretical studies of photo-induced excitations in amorphous

materials 25011.2.1Application of the Density-Functional-based Tight-Bindingmethod to the case of amorphous As2S3 251References 261

T A Abtew

Dept of Physics and Astronomy,

Ohio University, Athens, OH 45701,

USA

abtew@helios.phy.ohiou.edu

A Antonelli

Instituto de F´ısica Gleb Wataghin,

Universidade Estadual de Campinas,

Unicamp, 13083-970, Campinas, S˜ao

Paulo, Brazil

aantone@ifi.unicamp.br

P.R Briddon

School of Natural Science, University

of Newcastle, Newcastle upon Tyne,

SLACS (INFM-CNR) and

De-partment of Physics, University of

Cagliari, Cittadella Universitaria,

I-09042 Monserrato (Ca), Italy

colombo@sparc10.dsf.unica.it

Alessandro De Vita

Department of Physics, King’s

College London, Strand, London,

United Kingdom, DEMOCRITOS

National Simulation Center and

CENMAT-UTS, Trieste, Italy

drabold@ohio.edu

S.R Elliott

Department of Chemistry, University

of Cambridge,Lensfield Road, Cam-bridge CB2 1EW, UKsre1@cam.ac.uk

Stefan K Estreicher

Physics Department, Texas TechUniversity, Lubbock TX 79409-1051,USA

stefan.estreicher@ttu.edu

J.P Goss

School of Natural Science, University

of Newcastle, Newcastle upon Tyne,NE1 7RU, UK

J.P.Goss@newcastle.ac.uk

E.R Hern´ andez

Institut de Ci`encia de Materials de

Barcelona (ICMAB–CSIC), Campus

de Bellaterra, 08193 Barcelona,

Spain

ehe@icmab.es

James R Kermode

Cavendish Laboratory, University of

Cambridge, Madingley Road, CB3

0HE, UK

jrk33@cam.ac.uk

Alison Mainwood

Department of Physics, King’s

College London, Strand, London,

UK

alison.mainwood@kcl.ac.uk

Gianpietro Moras

Department of Physics, King’s

College London, Strand, London,

University of Cambridge, Madingley

Road, Cambridge, CB3 0HE, UK

rn11@cam.ac.uk

Risto M Nieminen

COMP/Laboratory of Physics,

Helsinki University of Technology,

POB 1100, FI-02015 HUT, Finland

rni@fyslab.hut.fi

P Ordej´ on

Institut de Ci`encia de Materials de

Barcelona (ICMAB–CSIC), Campus

mcp1@cam.ac.uk

Mahdi Sanati

Physics Department, Texas TechUniversity, Lubbock TX 79409-1051,USA

Matthias Scheffler

Fritz-Haber-Institut der Planck-Gesellschaft, Faraday-weg 4–6, D-14195 Berlin-Dahlem,Germany

M.J Shaw

School of Natural Science, University

of Newcastle, Newcastle upon Tyne,NE1 7RU, UK

M.J.Shaw@newcastle.ac.uk

S.I Simdyankin

Department of Chemistry, University

of Cambridge,Lensfield Road, Cam-bridge CB2 1EW, UKsis24@cam.ac.uk

Manuel Cardona

Max-Planck-Institut fr Festkrperforschung, 70569 Stuttgart, Germany, EuropeanUnion M.Cardona@fkf.mpg.de

Man sollte sich mit Halbleitern nicht beschftigen,

das sind Dreckeffekte –wer wei, ob sie richtig existieren

Wolfgang Pauli, 1931

1 Early history and contents of the present volume

This volume contains a comprehensive description of developments in thefield of Defects in Semiconductors which have taken place during the pasttwo decades Although the field of defects in semiconductors is at least 60years old, it had to wait, in order to reach maturity, for the colossal increase incomputer power that has more recently taken place, following the predictions

of Moores law [1] The ingenuity of computational theorists in developingalgorithms to reduce the intractable many-body problem of defect and host

to one that can be handled with existing and affordable computer power hasalso played a significant role: much of it is described in the present volume Ascomputational power grew, the simplifying assumptions of these algorithms,some of them hard to justify, were reduced The predictive accuracy of thenew calculations then took a great leap forward

In the early days, the real space structure of the defect had to be lated in order to get on with the theory and self-consistency of the electroniccalculations was beyond reach During the past two decades emphasis hasbeen placed in calculating the real space structure of defect plus host andachieving self-consistency in the electronic calculations The results of thesenew calculations have been a great help to experimentalists groping to in-terpret complicated data related to defects I have added up the number ofreferences in the various chapters of the book corresponding to years before

postu-1990 and found that they amount only to 25% of the total number of ences Many of the remaining 75% of references are actually even more recent,having been published after the year 2000 Thus one can say that the contents

refer-of the volume represent the State refer-of the Art in the field Whereas most refer-of thechapters are concerned with defects in crystalline semiconductors, Chapters

10 and 11 deal with defects in amorphous materials, in particular amorphoussilicon, a field about which much less information is available

The three aspects of the defect problem, real space structure, electronicstructure and vibrational properties are discussed in the various chapters

of the book, mainly from the theoretical point of view Defects break thetranslational symmetry of a crystal, a property that already made possiblerather realistic calculations of the host materials half a century ago Smallcrystals and clusters with a relatively small number of atoms (including im-purities and other defects), have become useful to circumvent, in theoreticalcalculations the lack of translational symmetry in the presence of defects

or in amorphous materials The main source of uncertainty in the state ofthe art calculations remains the small number of cluster atoms imposed bythe computational strictures This number is often smaller than that corre-sponding to real world samples, including even nanostructures Clusters with

a number of atoms that can be accommodated by extant computers are thenrepeated periodically so as to obtain a crystal lattice, with a supercell and

a mini-Brillouin zone Although these lattices do not exactly correspond tophysical reality, they enable the use of k-space techniques and are instrumen-tal in keeping computer power to available and affordable levels Anotherwidespread approach is to treat the cluster in real space after passivating thefictitious surface with hydrogen atoms or the like When using these methods

it is a good practice to check convergence with respect to the cluster size byperforming similar calculations for at least two sets of clusters with numbers

of atoms differing, say, by a factor of two

The epigraph above, attributed to Wolfgang Pauli, translates as One should not keep busy with semiconductors, they are dirt effects – Who knows whether they really exist The 24 authors of this book, like many tens of thou-

sands of other physicists and engineers, have fortunately not heeded Pauli’sadvice (given in 1931, 14 years before he received the Nobel Prize) Had theydone it, not only the World would have missed a revolutionary and nowa-days ubiquitous technology, but basic physical science would have lost some

of the most fruitful, beautiful and successful applications of Quantum chanics ‘Dreckeffekte’ is often imprecisely translated as effects of dirt i.e., aseffects of impurities However, effects of structural defects would also fall intothe category of Dreckeffekte In Paulis days applications of semiconductors,including variation of resistivity through doping leading to photocells and rec-tifiers, had been arrived at purely empirically, through some sort of trial anderror alchemy I remember as a child using galena (PbS) detectors in crystalradio sets I had lots of galena from various sources: some of it worked, somenot but nobody seemed to know why Sixty years later, only a few monthsago, I was measuring PbS samples in order to characterize the number ofcarriers (of non-stoichiometric origin involving vacancies) and their type (n

Me-or p) so as to wrap up Me-original research on this canonical material [2] TodayGOOGLE lists 860000 entries under the heading ‘defects in semiconductors’.The Web of Science (WoS) lists 3736 mentions in the title and abstract ofsource articles [3]

The modern science of defects in semiconductors is closely tied to the vention of the transistor at Bell Laboratories in 1948 (by Bardeen, Shockleyand Brattain, [4] Physics Nobel laureates for 1956) Early developments tookplace mainly in the United States, in particular at Bell Laboratories, theLincoln Lab (MIT) and Purdue University Karl Lark-Horovitz, an Austrianimmigrant, started at Purdue a program to investigate the growth and doping(n and p-type) of germanium and all sorts of electrical and optical properties

in-of this element in crystalline form [5] The initial motivation was the ment of germanium detectors for Radar applications During the years 1928till his untimely death in 1958 he built up the Physics Department at Purdueinto the foremost center of academic semiconductor research Work similar tothat at Purdue for germanium was carried out at Bell Labs, also as a spin-off

develop-of the development develop-of silicon rectifiers during World War II At Bell, Scaff

et al [6] discovered that crystalline silicon could be made n- or p-type bydoping with atoms of the fifth (P, As, Sb) or the third (B, Al) column of theperiodic table, respectively n-type dopants were called donors, p-type onesacceptors Pearson and Bardeen performed a rather extensive investigation

of the electrical properties of intrinsic’ and doped silicon [7] These authorsproposed the simplest possible expression for estimating the binding energy

of the so-called hydrogenic energy levels of those impurities: The ionizationenergy of the hydrogen atom (13.6 eV) had to be divided by the square of the

static dielectric constant  ( = 12 for silicon) and multiplied by an effective

mass (typical values m∗ ∼ 0.1) which simulated the presence of a crystalline

potential According to this Ansatz, all donors (acceptors) would have thesame binding energy, a fact which we now know is only approximately true(see Fig 3.5 for diamond)

The simple hydrogenic Ansatz applies to semiconductors with isotropicextrema, so that a unique effective mass can be defined (e.g n-type GaAs)

It does not apply to electrons in either Ge or Si because the conduction bandextrema are strongly anisotropic The hydrogen-like Schr¨odinger equationcan, however, be modified so as to include anisotropic masses, as appliy togermanium and silicon [8] The maximum of the valence bands of most dia-mond and zincblende-like semiconductors occurs at or very close to k=0 It

is four-fould degenerate in the presence of spin-orbit interaction and six-fold

if such interaction is neglected [9] The simple Schr¨odinger equation of thehydrogen atom must be replaced by a set of four coupled equations (withspin-orbit coupling) with effective-mass parameters to be empirically deter-mined [10] Extensive applications of Kohn’s prescriptions were performed byseveral Italian theorists [11]

In the shallow (hydrogenic) level calculations based on effective massHamiltonians the calculated impurity eigenvalues are automatically referred

to the corresponding band edges, thus obviating the need for using a marker,

of the type discussed in Chapter 3 of this book This marker was introduced

in order to avoid errors inherent to the ‘first principles’ calculations, such as

those related to the so called ‘gap problem’ found when using local densityfunctionals to represent many-body exchange and correlation For a way topalliate this problem using the so-called GW approximation see Chapter 7where defects at surfaces are treated.

We have discussed so far the electronic levels of shallow substitutionalimpurities In this volume a number of other defects, such as vacancies, in-terstitial impurities, clusters, etc., will be encountered Energy levels related

to structural defects were first discussed by Lark-Horovitz and coworkers [12]These levels were produced by irradiation with either deuterons, alpha parti-cles or neutrons After irradiation, the material became more p-type It wasthus postulated that the defect levels introduced by the bombardment wereacceptors (vacancies?)

It was also discovered by Lark-Horovitz that neutron bombardment, lowed by annealing in order to reduce structural damage, could be used tocreate electrically active impurities by nuclear transmutation [13] The smallamount of the30Si isotope (∼4%) present in natural Si converts, by neutron

fol-capture, into radioactive 31Si, which decays through β-emission into stable

31P, a donor This technique is still commercially used nowadays for ing very uniform doping concentrations

produc-Since Kohn-Luttinger perturbation theory predicts reasonably well theelectronic levels of shallow impurities (except for the so-called central cellcorrections [14]) this book covers mainly deep impurity levels which not onlyare difficult to calculate for a given real space structure but also require relax-ation of the unperturbed host crystal around the defect Among these deeplevels, native defects such as vacancies and self-interstitials are profusely dis-cussed Most of these levels are related to transition metal atoms, such as

Mn, Cu, and Au (I call Au and Cu transition metals for obvious reasons).The solubility of these transition metal impurities is usually rather low (lessthan 1015cm−3 Exceptions: Cd1−xMnxTe and related alloys) They can go

into the host lattice either as substitutional or as interstitial atoms,1a pointthat can be clarified with EPR and also with ab initio total energy calcu-lations These dopants were used in early applications in order to reducedthe residual conductivity due to shallow levels (because of the fact that tran-sition metal impurities have levels close to the middle of the gap) One caneven nowadays find in the market semi-insulating GaAs obtained by dopingwith chromium I remember having obtained 1957 semi-insulating germa-nium and silicon (doped with either Mn or Au) with carrier concentrationslower than intrinsic (this makes a good exam question!) They were used formeasurements of the low frequency dielectric constants of these materials, in

1 The reader who tries to do a literature search for interstitial gold may be prised by the existence of homonyms: Interstitial gold is important in the treat-ment of prostate cancer It has, of course, nothing to do with our interstitial

sur-gold See Lannon et al., British J Urology 72, 782 (1993).

particular vs temperature and pressure [15] while I was working at Harvard

on my PhD under W Paul

Rough estimates of the positions of deep levels of many impurity elements

in the gap of group IV and II-V semiconductors were obtained by Hjalmarson

et al using Greens function methods [9, 16] In the case of GaAs and relatedmaterials, two kinds of defect complexes, involving structural changes andmetastability have received a lot of attention because of technological impli-cations: the so-called EL2 and DX centers Searching the Web of Science forEL2 one finds 1055 mentions in abstracts and titles of source articles Like-wise 695 mentions are found for the DX centers Chadi and coworkers haveobtained theoretical predictions for the structure of these centers and theirmetastability [17, 18] Although these theoretical models explain a number

of observations related to these centers, there is not yet a general consensusconcerning their structures

An aspect of the defect problem that has not been dealt with explicitly

in this volume is the errors introduced by using non-relativistic Schr¨odingerequations, in particular the neglect of mass-velocity corrections and spin-orbitinteraction (the latter, however, is explicitly included in the Kohn-LuttingerHamiltonian, either in its 4×4 or its 6×6 version) Discrepancies between

calculated and measured gaps are attributed to the ‘gap problem’ inherent

in the local density approximation (LDA) However, already for relativelyheavy atoms (Ge, GaAs) the mass-velocity correction decreases the s-likeconduction levels and, together with the LDA gap problem converts thesemiconductor in a metal in the case of germanium For GaAs it is statedseveral times in this volume that the LDA calculated gap is about half theexperimental one This is for a non-relativistic Hamiltonian Even a scalarrelativistic one reduces the gap even further, to about 0.2eV (experimentalgap: 1.52eV at 4K) [19] This indicates that the gap problem is more seriousthan previously thought on the basis of non-relativistic LDA calculations.Another relativistic effect is the spin-orbit coupling For moderately heavyatoms such as Ge, Ga and As the spin-orbit splitting at the top of the va-lence bands (∼0.3eV) is much larger than the binding energy of hydrogenic

acceptors Hence we can calculate the binding energies of the latter by ing the decoupled 4×4 (J=3/2) and 2×2 (J=1/2) effective mass equations.

solv-This leads to two series of acceptor levels separated by a ‘spin orbit’ splittingbasically equal to that of the band edge states In the case of silicon, how-

ever, the spin-orbit splitting at k=0 (Δ = 0.044eV) is of the order of shallow

impurity binding energies The impurity potential thus couples the J=3/2and J=1/2 bands and the apparent spin-orbit splitting of the correspondingimpurity series becomes smaller than that at the band edges [20, 21] The

difference between band edge spin-orbit splitting (Δ = 0.014eV) and that of

the acceptor levels becomes even larger in diamond Using a simple Greens

functions technique and a Slater-Koster δ-function potential, the impurity

level splittings have been calculated and found to be indeed much smaller

than Δ = 0.014eV This splitting depends strongly on the binding energy of

When an atom of the host lattice of a semiconductor has several stable topes (e.g diamond, Si, Ge, Ga, Zn samples grown with natural material lose,strictly speaking, their translational symmetry In the past 15 years a largenumber of semiconductors have been grown using isotopically pure elements(which have become available in macroscopic and affordable quantities afterthe fall of the Iron Curtain) A different isotope added to an isotopically puresample can thus be considered as an impurity, probably the simplest kind ofdefect possible: Only the atomic mass of such an impurity differs from that

iso-of the host, the electronic properties remain nearly the same.2 The main fect of isotope mass substitution is found in the vibrational frequencies ofhost as well as local vibrational modes: such frequencies are inversely pro-portional to the square root of the vibrating mass (see Chapter 4) Althoughthis effect sounds rather trivial it often induces changes in phonon widths and

ef-in the zero-poef-int anharmonic renormalizations (see Ref 22) which ef-in somecases can be rather drastic and unexpected [23] The structural relaxationaround isotopic impurities is rather small The main such effect corresponds

to a increase of the lattice constant with increasing isotopic mass, about0.015% between 12C and 13C diamond Its origin lies in the change in thezero point renormalization of the lattice constant: ab initio calculations areavailable [24]

The third class of effects of the isotopic impurities refers to electronicstates and energy gaps and their renormalization on account of the electron-phonon interaction The zero point renormalizations also vary like the inversesquare root of the relevant isotopic mass By measuring a gap energy at lowtemperatures for samples with two different isotopic masses, one can extrapo-late to infinite mass and thus determined the unrenormalized value of the gap.Values around 60 meV have been found for Ge and Si For diamond, however,

2 Except for the electron-phonon renormalization of the electronic states and gapswhich is usually rather small See Ref 22

this renormalization seems to be much larger, [25] around 400 meV.3 Thislarge renormalization is a signature of strong electron-phonon interactionwhich seems to be responsible for the superconductivity recently observed

in heavily boron doped (p-type) diamond (Tc higher than 10K) [26, 27] Abinitio calculations of the electronic and vibronic structure of heavily borondoped diamond have been performed and used for estimating the criticaltemperature Tc [28]

2 Bibliometric studies

In the previous section I have already discussed the number of times certain

topics appear in titles, keywords and abstracts in source journals (about 6000

publications chosen by the ISI among ∼100000, as those which contribute

significally to the progress of science) While titles go back to the presentstarting date of the source journal selection (the year 1900), abstracts andkeywords are only collected since 1990 In the Web of Science (WoS) one cancompletely eliminate the latter in order to avoid distortions but, for simplicity,

I kept them in the qualitative survey presented here

In this section, a more detailed bibliometric analysis will be performedusing the WoS which draws on the citation index as the primary data bank

In order to get a feeling for the standing of the various contributors to thisvolume, we could simply perform a citations count (it can be done relatively

easily within the WoS using the cited reference mode However, a more tale

telling index has been recently suggested by J.E Hirsch, [29] the so-calledh-index This index is easily obtained for anyone with access to the WoSgoing back to the first publication of the authors under scrutiny (1974 forNieminen and Shaw) How far back your access to the WoS goes depends onhow much your institution is willing to pay to ISI- Thomson Scientific The

h-index is obtained by using the general search mode of the WoS and ordering

the results of the search for a given individual according to the number ofcitations (there is a function key to order the authors contributions frommost cited to less cited) You then go down the list till the order number of apaper equals its number of citations (you may have to take one more or lesscitation if equality does not exist) The number so obtained is the h-index Itrewards more continued, sustained well cited publications rather than only acouple with a colossal number (such as those that deserve the Nobel Prize).Watch out for possible homonyms although, on the average, they appearseldom They can be purged by hand if the number of terms is not toohigh I had problems with homonyms only for five out of the 24 (excludingmyself) contributors to this volume (Antonelli, Colombo, Hernndez, Sanatiand Shaw) I simply excluded them from the count

3 Theorists: beware (and be aware) of this large renormalization when comparingyour fancy GW calculations of gaps with experimental data

The average h-factor of the remaining 19 authors is h=20 Hirsch tions in Ref 29 that recently elected fellows of the American Physical Soci-ety have typically h∼15-20 Advancement of a physicist to full professor at

men-a reputmen-able US university corresponds to h=18 The high men-avermen-age h men-alremen-adyreveals the high standing of the authors of this book In several cases, theauthors involve a senior partner (h=20, Chapters 3,4,5,7,8,10 and 11) and

a junior colleague I welcome this decision It is a good procedure for ducing junior researchers to the intricacies and ordeals involved in writing

intro-a review intro-article of such extent In this connection, I should mention thintro-atthe h-index is roughly proportional to the scientific age (counted from thefirst publication or the date of the PhD thesis) The values of h given abovefor faculty and NAS membership are appropriate to physicists and chemists.Biomedical scientists often have, everything else being equal, twice as largeh-indexes, whereas engineers and mathematicians (especially the latter) havemuch lower ones

After having discussed the average h-index of our contributors, I wouldlike to mention the range they cover without mentioning specific names.4Theh-indexes of our contributors cover the range 6 ≤ h ≤ 64 Four very junior

authors who have not yet had a chance of being cited have been omitted(one could have set h=0 in their case) Hirsch mentions in his seminal article[29] that election to the National Academy of Sciences of the US is usually

associated with h=45 We therefore must have some potential academicians

among our contributors

Because of the ease in the use of the h-algorithm just described and itsusefulness to evaluate the ‘impact’ of a scientists career, bibliometrists havebeen looking for other applications of the technique Instead of people onecan apply it to journals (provided they are not too large in terms of pub-lished articles), institutions, countries, etc One has to keep in mind that the

resulting h-number always reverts to an analysis of the citations of als which are attached to the investigated items (e.g countries, institutions,

individu-etc.) One can also use the algorithm to survey the importance of keywords ortitle subjects The present volume has 11 chapters and this gives it a certain(albeit small) statistical value to be use in such a survey We thus attach toeach chapter title a couple of keywords and evaluate the corresponding h-index entering these under ‘topic’ in the general search mode of the WoS Inthe table below we list these words, the number of items we find for each set

of them and the corresponding h-index There is considerable arbitrariness inthe procedure to choose the keywords but we must keep in mind that theseapplications are just exploratory and at their very beginning

We display in Table 1 the keywords we have assigned to the eleven ters, the number of terms citing them and the corresponding h-index which

chap-4 Mentioning the h-indexes of the authors, one by one, may be invidious Theinterested reader with access to the WoS can do it by following the prescriptiongiven above

weights them according to the number of times each citing term is cited Onecan draw a number of conclusions from this table Particularly interesting arethe low values of n and h for empirical molecular dynamics, which probablysignals the turn towards ab initio techniques Amorphous semiconductors, in-cluding defect and the metastabilities induced by illumination plus possiblytheir applications to photovoltaics are responsible for the large values of nand h.

Table 1 Keywords assigned (somewhat arbitrarily) to each of the 11 chapters in

the book together with the corresponding number n of source articles citing them

in abstract, keywords or title Also, Hirsch number h which can be assigned to each

of the chapters according to the keywords

Chapter Keyword (topic in WoS) n h

1 defects and semiconductors 3735 76

2 supercell calculations 165 27

5 free energy and defect 494 36

6 Quantum Monte Carlo 2551 71

7 point defect and surface 426 38

8 defect and molecular dynamics 2023 67

9 empirical molecular dynamics 23 7

10 defect and amorphous 4492 77

11 light and amorphous 5747 87

References

1 G.E Moore, Electronics 38, 114 (1965).

2 R Sherwin, R.J.H Clark, R Lauck, and M Cardona, Solid State Commun

134, 265(2005).

3 A source article is one published in a Source Journal as defined by the Thomson Scientific There are about 6000 such journals, including all walks ofscienc.e

ISI-4 J Bardeen and W.H Brattain, Phys Rev 74, 230 (1948).

5 K Lark-Horovitz and V.A Johnson, Phys Rev 69, 258 (1946).

6 J.A Scaff, H.C Theuerer and E.E Schumacher, J Metals: Trans Am Inst

Mining and Metallurgical Engineers 185, 383 (1949).

7 G.L Pearson and J Bardeen, Phys Rev 75, 865 (1949).

8 W Kohn and J.M Luttinger, Phys Rev 97, 1721 (1975).

9 P.Y Yu and M Cardona, Fundamentals of Semiconductors 3rd ed (Springer,Berlin, 2005)

10 W Kohn and D Schechter, Phys Rev 99, 1903 (1955).

11 A Baldereschi and N Lipari, Phys Rev B 9, 1525 (1974).

12 W.E Johnson and K Horovitz, Phys Rev 76, 442 (1949); K Horovitz, E Bleuler, R Davis, and D.Tendam, Phys Rev 73, 1256 (1948).

Lark-13 K Lark-Horovitz, Nucleon-Bombarded Semiconductors, in Semiconducting

Ma-terials (Butterworths, London, 1950) p 47

14 W Kohn, Solid State Physics (Academic, New York, 1957, Vol 5) p 255.

15 M Cardona, W Paul and H Brooks, J Phys Chem Sol 8, 204 (1959).

16 H.P Hjalmarson, P Vogl, D.J Wolford, and J.D Dow, Phys Rev Lett 44,

810 (1980)

17 EL2 center: D.J Chadi and K.J Chang, Phys Rev Lett 60, 2187 (1988).

18 DX center: S.B Chang and D.J Chadi, Phys Rev B 42, 7174 (1990).

19 M Cardona, N.E Christensen and G Fasol, Phys Rev B 38, 1806 (1988).

20 N.O Lipari, Sol St Commun 25, 266 (1978).

21 J Serrano, A Wysmolek, T Ruf and M Cardona Physica B 274, 640 (1999).

22 M Cardona and M.L.V Thewalt, Rev Mod Phys 77, 1173 (2005).

23 J Serrano, F.J Manj´on, A.H Romero, F Widulle, R Lauck, and M Cardona,

Phys Rev Lett 90, 055510 (2003).

24 P Pavone and S Baroni, Sol St Commun 90, 295 (1994).

25 M Cardona, Science and Technology of Advanced Materials, in press See alsoRef 22

26 E.A Ekimov, V.A Sidorov, E.D Bauer, N.N Mel’nik, N.J Curro, J.D

Thomp-son and S.M Stishov, Nature 428, 542 (2004).

27 Y Takano, M Nagao, I Sakaguchi, M Tachiki, T Hatano, K Kobayashi, H

Umezawa, and H Kawarada, Appl Phys Letters 85, 2851 (2004).

28 L Boeri, J Kortus and O.K Andersen Phys Rev Lett 93, 237002 (2004).

29 J.E Hirsch, Proc Nat Acad Sc (USA) 102, 16569 (2005).

David A Drabold and Stefan K Estreicher

1 Dept of Physics and Astronomy, Ohio University, Athens, OH 45701

of a variety of tools, shields and weapons Not long afterwards, the earlymetallurgists realized that sand mixed with a metal is relatively easy to meltand produces glass The Ancient Egyptians discovered that glass beads ofvarious brilliant colors can be obtained by adding trace amounts of specifictransition metals, such as gold for red or cobalt for blue [1]

Defect engineering is not something new However, materials whose chanical, electrical, optical, and magnetic properties are almost entirely con-trolled by defects are relatively new: semiconductors [2, 3] Although, thefirst publication describing the rectifying behavior of a contact dates back to

me-1874 [4], the systematic study of semiconductors begun only during WorldWar II The first task was to grow high-quality Ge (then Si) crystals, that isremoving as many defects as possible The second task was to manipulate theconductivity of the material by adding selected impurities which control thetype and concentration of charge carriers This involved theory to understand

as quantitatively as possible the physics involved Thus, theory has played akey role since the very beginning of this field These early developments havebeen the subject of several excellent reviews [5–8]

For a long time, theory has been trailing the experimental work proximations at all levels were too drastic to allow quantitative predictions.Indeed, modeling a perfect solid is relatively easy since the system is periodic.High-level calculations can be done in the primitive unit cell This periodicity

Ap-is lost when a defect Ap-is present The perturbation to the defect-free material

is often large, in particular when some of the energy eigenvalues of the defectare in the forbidden gap, far from band edges However, in the past decade or

so, theory has become quantitative in many respects Today, theorists oftenpredict geometrical configurations, binding, formation, and various activationenergies, charge and spin densities, vibrational spectra, electrical properties,

and other observable quantities with sufficient accuracy to be useful to perimentalists and sometimes device scientists.

ex-Furthermore, the theoretical tools developed to study defects in ductors can be easily extended to other areas of materials theory, includ-ing many fields of nanoscience It is the need to understand the properties

semicon-of defects in semiconductors, in particular silicon, that has allowed theory

to develop as much as it did One key reason for this was the ity of microscopic experimental data, ranging from electron paramagneticresonance (EPR) to vibrational spectroscopy, photoluminescence (PL), orelectrical data, all of which provided critical tests for theory at every step.The word ‘defect’ means a native defect (vacancy, self-interstitial, an-tisite, ), an impurity (atom of a different kind than the host atoms), orany combination of those isolated defects: small clusters, aggregates, or evenlarger defect structures such as precipitates, interfaces, grain boundaries, sur-faces, etc However, nanometer-size defects play many important roles andare the building blocks of larger defect structures Therefore, understandingthe properties of defects begins at the atomic scale

availabil-There are many examples of the beneficial or detrimental roles of fects Oxygen and nitrogen pin dislocations in Si and allow wafers to undergo

de-a rde-ange of processing steps without brede-aking [10] Smde-all oxygen precipitde-atesprovide internal gettering sites for transition metals, but some oxygen clustersare unwanted donors which must be annealed out [11] Shallow dopants areoften implanted They contribute electrons to the conduction band or holes tothe valence band Native defects, such as vacancies or self-interstitials, pro-mote or prevent the diffusion of selected impurities, in particular dopants.Self-interstitial precipitates may release self-interstitials which in turn pro-mote the transient enhanced diffusion of dopants [12] Transition metal impu-rities are often associated with electron-hole recombination centers Hydro-gen, almost always present at various stage of device processing, passivatesthe electrical activity of dopants and of many deep-level defects, or formsextended defect structures known as platelets [13] Mg-doped GaN must beannealed at rather high temperatures to break up the {Mg, H} complexes which prevent p-type doping [14] Magnetic impurities such as Mn can ren-

der a semiconductor ferromagnetic The list goes on

Much of the microscopic information about defects comes from cal, optical, and/or magnetic experimental probes The electrical data areoften obtained from capacitance techniques such as deep-level transient spec-troscopy (DLTS) The sensitivity of DLTS is very high and the presence

electri-of defects in concentrations as low as 1011cm−3 can be detected However,

even in conjunction with uniaxial stress experiments, these data provide little

or no elemental and structural information and, by themselves, are cient to identify the defect responsible for electrical activity local vibrationalmode (LVM) spectroscopy, Raman, and Fourier-transform infrared absorp-tion (FTIR), often give sharp lines characteristic of the Raman- or IR-active

insuffi-LVMs of impurities lighter than the host atoms When uniaxial stress, ing, and isotope substitution studies are performed, the experimental dataprovide a wealth of critical information about a defect This information can

anneal-be correlated e.g with DLTS annealing data However, Raman and FTIRare not as sensitive as DLTS In the case of Raman, over 1017cm−1 defects

centers must be present in the surface layer exposed to the laser In the case

of FTIR, some 1016cm−3 defect centers are needed, although much higher

sensitivities have been obtained from multiple-internal reflection FTIR [15]Photoluminescence is much more sensitive, sometimes down to 1011cm−1,

but the spectra can be more complicated to interpret [6] Finally, magneticprobes such as EPR are wonderfully detailed and a lot of defect-specific datacan be extracted: identification of the element(s) involved in the defect andits immediate surrounding, symmetry, spin density maps, etc However, thesensitivity of EPR is rather low, of the order of 1016cm−3 Further, localized

gap levels in semiconductors often prefer to be empty or doubly occupied asmost defect centers in semiconductors are unstable in a spin 1

2 state Thesample must be illuminated in order to create an EPR-active version of thedefect under study [17, 18]

This introductory chapter contains brief reviews of the evolution of theory[20, 31] since its early days and of the key ingredients of today’s state-of-the-art theory It concludes with an overview of the content of this book

1.2 The evolution of theory

The first device-related problem that required understanding was the creation

of electrons or holes by dopants These (mostly substitutional) impurities are

a small perturbation to the perfect crystal and are well described by EffectiveMass Theory (EMT) [21] The Schr¨odinger equation for the nearly-free chargecarrier, trapped very close to a parabolic band edge, is written in hydrogenicform with an effective mass determined by the curvature of the band Thecalculated binding energy of the charge carrier is that of a hydrogen atom butreduced by the square of the dielectric constant As a result, the associatedwavefunction is substantially delocalized, with an effective Bohr radius some

100 times larger than that of the free hydrogen atom

EMT has been refined in a variety of ways [22] and provided a basicunderstanding of doping However, it cannot be extended to defects thathave energy eigenvalues far from band edges These so-called ‘deep-level’defects are not weak perturbations to the crystal and often involve substantialrelaxations and distortions The first such defects to be studied were thebyproducts of radiation damage, a hot issue in the early days of the cold war.EPR data became available for the vacancy [17, 23] and the divacancy, [24]

in silicon (the Si self-interstitial has never been detected) Transition metal(TM) impurities, which are common impurities and active recombinationcenters, have also been studied by EPR [25]

In most charge states, the undistorted vacancy (Td symmetry) or vacancy (D3d symmetry) is an orbital triplet or doublet, respectively, andtherefore should undergo Jahn-Teller distortions The EPR studies showedthat this is indeed the case Although interstitial oxygen, the most commonimpurity in Czochralski-grown Si, was known to be at a puckered bond-centered site [26, 27], it was not realized how much energy is involved inrelaxations and distortions It was believed that the chemistry of defects insemiconductors is well described in first order by assuming high symmetry,undistorted, lattice sites Relaxations and distortions were believed to be asecond-order correction The important issue then was to correctly predicttrends in the spin densities and electrical activities of specific defects centers

di-in order to expladi-in the EPR and electrical data (see e.g Refs [28, 29]) Thecritical importance of carefully optimizing the geometry around defects andthe magnitudes of the relaxation energies were not fully realized until the1980s [30] The host atom displacements can be of several tenths of an ˚A,and the chemical rebonding can lead to energy changes as large as severaleVs (undistorted vs relaxed structures)

The first theoretical tool used to describe localized defects in

semiconduc-tors involved Green’s functions [2, 31–33] These calculations begin with

the Hamiltonian H0 of the perfect crystal Its eigenvalues give the crystalsband structure and the eigenfunctions are Bloch or Wannier functions Inprinciple, the defect-free host crystal is perfectly described The localized de-fect is represented by a Hamiltonian H which includes the defect potential

V The Green’s function is G(E) = 1/(E − H ) Therefore, the perturbed

en-ergies E coincide with its poles The new eigenvalues include the gap levels ofthe defect and the corresponding eigenfunctions are the defect wavefunctions

In principle, Green’s functions provide an ideal description of the defect inits crystalline environment In practice, there are many difficulties associatedwith the Hamiltonian, the construction of perfect-crystal eigenfunctions thatcan be used as a basis set for the defect calculation, [34] and the construction

of the defect potential itself This is especially true for those defects whichinduce large lattice relaxations and/or distortions

The first successful Green’s functions calculations for semiconductors dateback to the late 1970s [35–37] They were used to study charged defects,[38,38] calculate forces, [40–42] total energies, [43,44] and LVMs [45,46] Thesecalculations also provided important clues about the role of native defects inimpurity diffusion [47] However, while Green’s functions do provide a near-ideal description of the defect in a crystal, their implementation is difficultand not very intuitive Clusters or supercells are much easier to use andprovide a physically and chemically appealing description of the defect andits immediate surroundings Green’s functions have mostly been abandonedsince the mid 1980s, but a rebirth within the GW formalism [48] is now takingplace (see the chapter by Schindlmayr and Scheffler)

In order to describe the distortions around a vacancy, Friedel et al [50]

completely ignored the host crystal and limited their description to rigidlinear combinations of atomic orbitals (LCAO) Messmer and Watkins [51]expanded this approach to linear combinations of dangling-bond states Thesesimple quantum-chemical descriptions provided a much-needed insight and

a correct, albeit qualitative, explanation of the EPR data Here, the defectwas assumed to be so localized that the entire crystal could be ignored in 0th

order

The natural extension of this work was to include a few host atoms around

the defect, thus defining a cluster These types of calculations were

per-formed in real space with basis sets consisting of localized functions such asGaussians or LCAOs The dangling bonds on the surface atoms must be tied

up in some way, most often with H atoms However, without the underlyingcrystal and its periodicity, the band structure is missing and the defect’s en-ergy eigenvalues cannot be placed within a gap Further, the finite size of thecluster artificially confines the wavefunctions This affects charged defects themost, as the charge tends to distribute itself on the surface of the cluster.However, the local covalent interactions are well described

The Schr¨odinger equation for a cluster containing a defect can be solvedusing almost any electronic structure methods The early work was empiri-cal or semiempirical, with heavily approximated quantum chemical methods

At first, extended H¨uckel theory [52, 53] then self-consistent semiempiricalHartree-Fock: CNDO, [54] MNDO, [55] MINDO [56] Geometries could beoptimized, albeit often with symmetry assumptions The methods sufferedfrom a variety of problems such as cluster size and surface effects, basis setlimitations, lack of electron correlation, and the use of adjustable parameters.Their values are normally fitted to atomic or molecular data, and transfer-ability is a big issue

In order to bypass the surface problem, cyclic clusters have been designed,mostly in conjunction with semiempirical Hartree-Fock Cyclic clusters can beviewed as clusters to which Born-von Karman periodic boundary conditionsare applied [57, 58] These boundary conditions can be difficult to handle, inparticular when 3- and 4-center interactions are included [59]

DeLeo and co-workers extensively used the scattering-Xα method in

clus-ters [60,61] to study trends for inclus-terstitial TM impurities and hydrogen-alkalimetal complexes The results provided qualitative insight into these issues.Ultimately, the method proved difficult to bring to self-consistency and therather arbitrarily defined muffin-tin spheres rendered it poorly suited to thecalculation of total energies vs atomic positions

The method of Partial Retention of Diatomic Differential Overlap [62, 63](PRDDO) was first used for defects in diamond and silicon in the mid-1980’s

It is self-consistent, contains no semiempirical parameters, and allows etry optimizations to be performed without symmetry assumptions Conver-gence is very efficient and relatively large clusters (44 host atoms) could be

geom-used However, PRDDO is a minimal basis set technique and ignores electroncorrelation Its earliest success was to demonstrate [30] the stability of bond-centered hydrogen in diamond and silicon It was not expected at all that animpurity as light as H could indeed force a Si-Si bond to stretch by a over1˚A Substantial progress in the theory of defects in semiconductors occurred

in the mid 1980’s with the combination of periodic supercells to representthe host crystal, ab-initio type pseudopotentials [78–80] for the core regions,

DF theory for the valence regions, and ab-initio molecular dynamics (MD)simulations [81, 82] for nuclear motion This combination is now referred to

as ‘first-principles’ in opposition to ‘semiempirical’ There are parameters inthe theory They include the size of the supercell, k-point sampling, type andsize of the basis set, chosen by the user, as well as the parameters associatedwith the basis sets and pseudopotentials However, these parameters and userinputs are not fitted to an experimental database Instead, some are deter-mined self-consistently, other are calculated from first principles or obtainedfrom high-level atomic calculations Note that the first supercell calculationswere done in the 1970’s in conjunction with approximate electronic structuremethods [83–85] As much as 1.5 to 1.6˚A PRDDO was used to study clustersize and surface effects [64] and many defects [65] It provided good inputgeometries for single point ab-initio Hartree-Fock calculations [66] However,

it suffered from the problems associated with all Hartree-Fock techniques,such as unreasonably large gaps and inaccurate LVMs A number of researchgroups have used Hartree-Fock and post-Hartree-Fock techniques [67–69] tostudy defects in clusters, but these efforts have now been mostly abandoned.Density-functional (DF) theory [70, 71] with local basis sets [72] in largeclusters allowed more quantitative predictions The DF-based AIMPRO code[73, 74] uses Gaussian basis sets, has been applied to many defect problems(this code handles periodic supercells as well) In addition to geometries andenergetics, rather accurate LVMs for light impurities can be predicted [75,76]Large clusters have been used [77] to study the distortions around a vacancy

or divacancy in Si However, all clusters suffer from the surface problem andlack of periodicity

1.3 A sketch of first-principles theory

The theoretical approach known as ‘first-principles’ has proven to be a olutionary tool to predict quantitatively some key properties of defects Anelementary exposition of the theory follows

rev-1.3.1 Single particle methods: History

After the Born-Oppenheimer approximation is made, so that electronic andionic degrees of freedom are separated, we face the time-independent many-electron problem [71]:

of science, P A M Dirac is said to have implied that chemistry was just anapplication of Equation 1.1, though he also acknowledged that the equationwas intractable It is true that the quantum mechanics of the many-electronproblem is beautifully and succinctly represented in Equation 1.1.

Kohn gives an interesting argument [71] stating that even in principle,Equation 1.1 is hopeless as a practical tool for calculation if the number ofelectrons exceeds of order 103 His argument is a development of a paper ofVan Vleck [86, 87] and points out that it appears to be fundamentally im-

possible to obtain an approximate many-electron function Ψ with significant

overlap with the “true” many-body wavefunction for large systems He ther points out that the sheer dimensionality of the problem rapidly makes

fur-it unrepresentable on any conceivable computer Thus, a credible case can

be made that for large systems it does not even make sense to estimate Ψ directly Kohn has named this the “exponential wall” To some extent this is disconcerting, because of the simplicity of the form of Equation 1.1, but it

points to the need for new concepts if we are to make sense of solids – to saynothing of defects!

Empirical experience with solids also suggests that the unfathomable plexity of the many-body wavefunction is unnecessary If all of the informa-

com-tion contained in Ψ was really required for estimating the properties of solids

that we care about, e.g experimental observables, molecular physics wouldreach exhaustion with tiny molecules, and solid state physics would neverget off the ground at all The fact is that many characteristics of materialsare independent of system size For example, in a macroscopic sample, theelectronic density of states has the same form for a system with N atoms

and an identically prepared one with 2N atoms Yet Ψ is immeasurably more

complex for the second system than the first Thus, the additional complexity

of the 2N system ψ must be completely irrelevant to our observable, in this

case the density of states

The saturation of complexity of the preceding paragraph is connected toKohn’s “principle of nearsightedness” [88], which states that in fact quantum

mechanics in the solid state is intrinsically local (how local depends sensitively

on the system [89, 90]) The natural gauge of this locality is the decay of

the density operator in the position representation: ρ(x, x ) = x|ˆρ|x  : a

function of|x − x  |, decaying as a power law in metals and exponentially in

systems with a gap For systems with a gap, the exponential fall off enablesaccurate calculations of all local properties by undertaking a calculation in a

finite volume determined by the rate of decay The decay in semiconductorsand insulators can be exploited to produce efficient order-N methods forcomputing total energies and forces, with computational cost scaling linearlywith the number of atoms or electrons [68].

1.3.2 Direct approaches to the many-electron problem

While this book emphasizes single-particle methods, there is one importantexception Richard Needs shows in Chapter 6 that remarkable progress can

be made for the computation of expectation values of observables using

Quan-tum Monte Carlo methods, with no essential approximations to Equation 1.1.This is a promising class of methods that offers the most accurate calcula-tions available today for complex systems Several groups are advancing thesemethods, and even the stochastic calculation of forces is becoming possible.One can be certain that Quantum Monte Carlo will play an important role insystems needing the most accurate calculations available, and certainly this

is can be the case for the theory of defects

1.3.3 Hartree and Hartree-Fock approximations

In 1928, Hartree [92] started with Equation 1.1 with a view to extractinguseful single-particle equations from it He used the variational principlefor the ground state wavefunction adopting a simple product trial function:

Ψ (x1, x2, , xn) = ψ1(x 1)ψ2(x 2) ψ n(x n) The product ansatz did not

en-force Fermion antisymmetry requirements; this was built in with a Slater terminant ansatz as proposed by Fock in 1930 [93]: the Hartree-Fock method.These methods map the many-body ground state problem onto a set ofchallenging single-particle equations The structure of the Hartree equations

de-is a Schr¨odinger equation with a potential depending upon all the orbitals:

the sum is over occupied states This appealing equation prescribes an

effec-tive Coulomb field for electron l arising from all of the other electrons Since

computing the potential requires knowledge of the other wavefunctions, theequation must be solved self-consistently with a scheme of iteration While

the equation is intuitive, it is highly approximate Curiously, it will turn out

that the equations of density functional theory have the mathematical

struc-ture of Equation 1.2, but with quite a different (and far more predictive) V ef f

(see Equation 2.7 below), derived from a very different point of view.The Hartree-Fock approximation includes that part of the exchange en-ergy implied by the exclusion principle and has well known analytic problems

at the Fermi surface in metals [68] and a tendency to exaggerate charge tuations at atomic sites in molecules or solids [94] In molecular calculations,

fluc-the correlation energy (roughly speaking, what is missing from Hartree-Fock)can be estimated in perturbation theory This is computationally expensive:

the most popular fourth order perturbation theory (MP4) scales like N7,

where N eis the number of electrons Such methods are important for ular systems, but challenging to apply to condensed systems

molec-1.3.4 Density Functional Theory

Thomas-Fermi Model

Not long after the dawn of quantum mechanics, Thomas and Fermi [96] gested a key role for the electron density distribution as the determiner ofthe total energy of an inhomogeneous electron gas This was the first serious

sug-attempt to express the energy as a functional of the electron density ρ: E[ρ] =

is sufficiently small Variation of Equation 1.3 with respect to ρ with fixed

electron number leads to the coupled self-consistent Thomas Fermi equations(see e.g., the treatment by Fulde [94]) The coarse treatment of the kineticenergy greatly limits the predictive power of the method – for example, theshell structure of atoms is not predicted [97] However, the Thomas-Fermimodel is conceptually a density functional theory

Modern Density Functional Theory

For atomistic force calculations on solids, the overwhelming method of choice

is density functional theory, due to Kohn, Hohenberg and Sham [98] Thischapter only sketches the concepts in broad outline, as a detailed treatmentfocused on defect calculations is available in this book (Niemenen chapter).For additional discussion we strongly recommend the books of Martin [68]and Fulde [94]

The following statements embody the foundation of zero-temperaturedensity functional theory:

(1) The ground state energy of a many electron system is a functional of

the electron density ρ(x):

continuing work to determine it The practical utility of this result derivesfrom:

(2) The functional E[ρ] is minimized by the true ground state electron

density

It remains to estimate the functional F [ρ], which in conjunction with the variational principle (2), enables real calculations To estimate F [ρ], the usual

procedure is to note that we already know some of the major contributions

to F [ρ], and decompose the functional in the form:

F [ρ] = e2/2



d3x d3x  ρ(x) ρ(x  )/ |x − x  | + T ni (ρ) + E xc (ρ) (1.5)Here, the integral is just the electrostatic (Hartree) interaction of the

electrons, T ni is the kinetic energy of a noninteracting electron gas of density ρ, and E xc (ρ) is yet another unknown functional, the exchange-

correlation functional, which includes non classical effects of the

interact-ing electrons Eq 2.5 is difficult to evaluate directly in terms of ρ, because

of the term T ni Thus, one introduces single electron orbitals|χ i , for which

T ni=

i occ χ i |−¯h2

/2m ∇2|χ i , and ρ =i occ |χ i(x)|2is the charge density

of the physically relevant interacting system The value of this tion is that E xc (ρ) is smooth and reasonably slowly varying, and therefore a

decomposi-functional that we can successfully approximate: we have included the most

difficult and rapidly varying parts of F in T niand the Hartree integral, as can

be seen from essentially exact many-body calculations on the homogeneouselectron gas [99] The Hartree and non-interacting kinetic energy terms areeasy to compute and if one makes the local density approximation, that is as-

suming that the electron density islocally uniform) With information about

the homogeneous electron gas, the functional (Eq 2.5) is fully specified.With noninteracting orbitals|χ i , (with ρ(x) = 2i occ |x|χ i |2), thenthe minimum principle plus the constraint that χ i |χ j  = δ ij can be trans-lated into an eigenvalue problem for the|χ i :

{−¯h2∇2/2m + Veff[ρ(x)] }|χ i  =  i |χ i , (1.6)where the effective density-dependent (in practical calculations, orbital-

dependent) potential Veff is:

system), the ground state electronic charge density ρ(x), and related ground

state properties like the forces In particular, it is tempting to interpret the

|χ i  and  ias genuine electronic eigenstates and energies, and indeed this canoften be useful Such identifications are not rigorous [68] It is instructive tonote that the starting point of density functional theory was to depart fromthe use of orbitals and formulate the electronic structure problem rigorously

in terms of the electron density ρ; yet a practical implementation (which

enables an accurate estimate of the electronic kinetic energy) led us diately back to orbitals! This illustrates why it would be very worthwhile

imme-to know F (ρ), or at least the kinetic energy functional since we would then

have a theory with a structure close to Thomas-Fermi form and would

there-fore be able to seek one function ρ rather than the cumbersome collection of

orthonormal|χ i .

The initiation of modern first-principles theory and its development into

a standard method with widespread application was due to Car and rinello [81] They developed a powerful method for simultaneously solvingthe electronic problem and evolving the positions of the ions The method isusually applied to plane wave basis sets though the key ideas are indepen-dent of the choice of representation One of the early application to defects insilicon was the diffusion of bond-centered hydrogen [101] An alternative ab-initio approach to MD simulations, based on a tight-binding perspective, wasproposed by Sankey and co-workers [82] Their basis sets consist of pseudo-

Par-atomic orbitals with s, p, d, symmetry The wavefunctions are truncated

beyond some radius and renormalized The early version of this code wasnot self-consistent and was restricted to minimum basis sets A more recentversion [102] is self-consistent and can accommodate expanded and polarizedbasis sets This is also the case for the highly flexible SIESTA code [10, 11](Spanish Initiative for the Electronic Structure with Thousands of Atoms).Although basis sets of local orbitals (typically, LCAOs) are highly intuitiveand allow population analysis and other chemical information to be calcu-lated, they are less complete than plane wave basis sets The latter can easily

be checked for convergence On the other hand, when an atom such as Si is

given two sets of 3s and 3p plus a set of 3d orbitals, the basis set is sufficient

to describe quite well virtually all the chemical interactions of this element,

as the contribution of the n = 4 shell of Si is exceedingly small, except

un-der extreme conditions that ground-state theory is not capable of handlinganyway Many details of the implementation of density functional methods

is given in the contribution of Nieminen in Chapter 2

The power of these methods is that they yield parameter free estimatesfor the structure of defects and even topologically disordered systems, provideaccurate estimates of total energies, formation energies, vibrational states, de-

fect dynamics, and with suitable caveats information about electronic

struc-ture, defect levels, localization, etc There are many subtle aspects to their

applications to defect physics, partly because high accuracy is often required

in such studies

1.4 The Contributions

In Chapter 2, Nieminen carefully discusses the use of periodic boundary ditions in supercells calculations, detailing both strengths and weaknesses,and other basic features of these calculations In Chapter 3, Goss and cowork-ers discuss the marker method to extract electronic energy levels, and includeseveral important applications In Chapter 4, Estreicher and Sanati describethe calculation and remarkable utility of vibrational modes in systems con-taining defects, including novel analysis of finite-temperature properties ofdefects Then, in Chapter 5, Hernandez and co-workers work out a propertheory of free energies and phase diagrams for semiconductors Chapter 6describes the most rigorous attack of the book on the quantum many-bodyproblem, as Needs explores Quantum Monte Carlo methods and their promisefor defects in semiconductors Schindlmayr and Scheffler describe the theoryand application of self-energy corrected density functional theory, the ‘GW’approximation in Chapter 7 Like the work of Needs, this technique has pre-dictive power beyond DFT Csanyi and coworkers present a multiscale mod-eling approach in Chapter 8 with applications Scheerschmidt offers a com-prehensive view of molecular dynamics in Chapter 9, focusing on empiricalmethods, though much of his chapter is applicable to first principles simula-tion as well In Chapter 10, Drabold and Abtew discuss defects in amorphoussemiconductors with a special emphasis on hydrogenated amorphous silicon.Last but not least, in Chapter 11, Simdyankin and Elliott write on the theory

con-of light-induced effects in amorphous materials, an area con-of great basic andpractical interest, which nevertheless depends very much upon the defectspresent in the material

References

1 J.L Mass, M.T Wypyski, and R.E Stone, Ma Res Soc Bull 26, 38 (2001).

2 A.M Stoneham, Theory of Defects in Solids (Clarendon, Oxford, 1985).

3 H.J Queisser and E.E Haller, Science 281, 945 (1998).

4 F Braun, Ann Phys Chem (Leipzig) 153, 556 (1874).

5 A.B Fowler, Phys Today, October 1993, p 59

6 F Seitz, Phys Today, January 1995, p 22

7 M Riordan and I Hoddeson, Phys Today, December 1997, p 42

8 I.M Ross, Phys Today, December 1997, p 34

9 J.R Chelikowsky, Interface 14, 24 (2005).

10 Oxygen in Silicon, ed F Shimura, Semic and Semimetals 42 (Academic,

Boston, 1994)

11 Early Stages of Oxygen Precipitation in Silicon, ed R Jones (Kluwer,

Dor-drecht, 1996)

12 R Scholtz, U G¨osele, J.Y.Y.Huh, and T.Y Tan, Appl Phys Lett 72, 200

(1998)

13 S.K Estreicher, Mat Sci Engr R 14, 319 (1995)

14 S.J Pearton, in GaN and Related Materials, ed S.J Pearton (Gordon and

Breach, Amsterdam, 1997), p 333

15 F Jiang, M Stavola, A Rohatgi, D Kim, J Holt, H Atwater, and J Kalejs,

Appl Phys Lett 83, 931 (2003)

16 G Davies, Phys Rep 176, 83 (1989)

17 G.D Watkins, in Deep Centers in Semiconductors, ed S.T Pantelides

(Gor-don and Breach, New York, 1986), p 147

18 Y.V Gorelkinskii and N.N Nevinnyi, Physica B 170, 155 (1991) and Mat Sci Engr B 36, 133 (1996)

19 S.T Pantelides, in Deep Centers in Semiconductors, ed S.T Pantelides

(Gor-don and Breach, New York, 1986)

20 S.K Estreicher, Materials Today 6 (6), 26 (2003)

21 W Kohn, Sol St Phys 5, 257 (1957)

22 S.T Pantelides, Rev Mod Phys 50, 797 (1978)

23 G.D Watkins, in Radiation Damage in Semiconductors (Dunod, Paris, 1964),

p 97

24 G.D Watkins and J.W Corbett, Phys Rev 138, A543 (1965)

25 G.W Ludwig and H.H Woodbury, Sol St Phys 13, 223 (1962)

26 W Kaiser, P.H Keck, and C.F Lange, Phys Rev 101, 1264 (1956)

27 J.W Corbett, R.S McDonald, and G.D Watkins, J Phys Chem Sol 25,

873 (1964)

28 G.G DeLeo, G.D Watkins, and W Beal Fowler, Phys Rev B 25, 4972 (1982)

29 A Zunger and U Lindefelt, Phys Rev B 26, 5989 (1982)

30 T.L Estle, S.K Estreicher, and D.S Marynick, Hyp Inter 32, 637 (1986) and Phys Rev Lett 58, 1547 (1987)

31 S.T Pantelides in Deep Centers in Semiconductors, ed S.T Pantelides

(Gor-don and Breach, New York, 1986)

32 G.F Koster and J.C Slater, Phys Rev 95, 1167 (1954) and 96, 1208 (1954)

33 J Callaway, J Math Phys 5, 783 (1964)

34 J Callaway and A.J Hughes, Phys Rev 156, 860 (1967) and 164, 1043

(1967)

35 J Bernholc and S.T Pantelides, Phys Rev B 18, 1780 (1978)

36 J Bernholc, N.O Lipari, and S.T Pantelides, Phys Rev Lett 41, 895 (1978)

37 G.A Baraff and N Schl¨uter, Phys Rev Lett 41, 892 (1978)

38 G.A Baraff, E.O Kane, and M Schl¨uter, Phys Rev Lett 43, 956 (1979)

39 G.A Baraff, and M Schl¨uter, Phys Rev B 30, 1853 (1984)

40 M Scheffler, J.P Vigneron, and G.B Bachelet, Phys Rev Lett 49, 1756

(1982)

41 U Lindefelt, Phys Rev B 28, 4510 (1983)

42 U Lindefelt and A Zunger, Phys Rev B 30, 1102 (1984)

43 G.A Baraff and M Schl¨uter, Phys Rev B 28, 2296 (1983)

44 R Car, P.J Kelly, A Oshiyama, and S.T Pantelides, Phys Rev Lett 52,

1814 (1984)

45 D.N Talwar, M Vandevyver, and M Zigone, J Phys C 13, 3775 (1980)

46 R.M Feenstra, R.J Hauenstein, and T.C McGill, Phys Rev B 28, 5793

(1983)

47 R Car, P.J Kelly, A Oshiyama, and S.T Pantelides, Phys Rev Lett 54 360

(1985)

48 F Aryasetiawan and O Gunnarsson, Rep Prog Phys 61, 237 (1998)

49 M Scheffler and A Schindlmayr, in Theory of Defects in Semiconductors, ed.

D A Drabold and S.K Estreicher (Springer, Berlin, 2006)

50 J Friedel M Lannoo, and G Leman, Phys Rev 164, 1056 (1967)

51 R.P Messmer and G.D Watkins, Phys Rev Lett 25, 656 (1970) and Phys Rev B 7, 2568 (1973)

52 F.P.Larkins, J Phys C: Sol St Phys 4, 3065 (1971)

53 R.P Messmer and G.D Watkins, Proc Reading Conf on Radiation age and Defects in Semiconductors, e.d J.E.Whitehouse (Insitute of Physics,1973), p 255

Dam-54 A Mainwood, J Phys C: Sol St Phys 11, 2703 (1978)

55 J.W Corbett, S.N Sahu, T.S Shi, and L.C Snyder, Phys Lett 93A, 303

(1983)

56 P.De´ak and L.C Snyder, Radiation Effects and Defects in Solids 111-112, 77

(1989)

57 P De´ak and L.C Snyder, Phys Rev B 36, 9619 (1987)

58 P De´ak, L.C Snyder, and J.W Corbett, Phys Rev B 45, 11612 (1992)

59 J Mir´o, P De´ak, C.P Ewels, and R Jones, J Phys.: Cond Matt 9, 9555

(1997)

60 G.G DeLeo, G.D Watkins, and W.B Fowler, Phys Rev B 23, 1819 (1981)

61 G.G DeLeo, W.B Fowler, and G.D Watkins, Phys Rev B 29, 1851 (1984)

62 T.A Halgren and W.N Lipscomb, J Chem Phys 58, 1569 (1973)

63 D.S Marynick and W.N Lipscomb, Proc Nat Acad Sci (USA) 79, 1341

(1982)

64 S.K Estreicher, A.K Ray, J.L Fry, and D.S Marynick, Phys Rev Lett 55,

1976 (1985)

65 See e.g S.K Estreicher, Phys Rev B 41, 9886 (1990) and 41, 5447 (1990)

66 See e.g S.K Estreicher, Phys Rev B 60, 5375 (1999)

67 A.A Bonapasta, A Lapiccirella, N Tomassini, and M Capizzi, Phys Rev B

36, 6228 (1987)

68 E Artacho and F Yndur´ain, Sol St Comm 72, 393 (1989)

69 R Luchsinger, P.F Meier, N Paschedag, H.U Suter, and Y Zhou, Phil

Trans R Soc Lond A 350, 203 (1995)

70 P Hohenberg and W Kohn, Phys Rev 136B, 468 (1964); W Kohn and L.J Sham, Phys Rev 140A, 1133 (1965); L.J Sham and W Kohn, Phys Rev.

145, 561 (1966)

71 W Kohn, Rev Mod Phys 71, 1253 (1999)

72 See, e.g., M Saito and A Oshiyama, Phys Rev B 38, 10711 (1988)

73 R Jones and A Sayyash, J Phys C 19, L653 (1986)

74 R Jones and P.R Briddon, in Identification of Defects in Semiconductors, ed.

M Stavola, Semiconductors and Semimetals 51 (Academic, Boston, 1998)

75 J.D Holbech, B Bech Nielsen, R Jones, P Sitch, and S ¨Oberg, Phys Rev

Lett 71, 875 (1993)

76 R Jones, B.J Coomer, and P.R Briddon, J Phys : Condens Matter 16,

S2643 (2004)

77 S Og¨ut and J.R Chelikowsky, Phys Rev Lett 83, 3852 (1999) and Phys Rev B 64, 245206 (2001)

78 D.R Hamann, M Schl¨uter, and C Chiang, Phys Rev Lett 43, 1494 (1979)

79 G.B Bachelet, D.R Hamann, and M Schl¨uter, Phys Rev B 26, 4199 (1982)

80 L Kleinman and D.M Bylander, Phys Rev Lett 48, 1425 (1982)

81 R Car and M Parrinello, Phys Rev Lett 55, 2471 (1985)

82 O Sankey and D.J Niklewski, Phys Rev B 40, 3979 (1989)

83 A Zunger and A Katzir, Phys Rev B 11, 2378 (1975)

84 S.G Louie, M Schl¨uter, J.R Chelikowsky, and M.L Cohen, Phys Rev B 13,

1654 (1976)

85 W Pickett, M.L Cohen, and C Kittel, Phys Rev B 20, 5050 (1979)

86 J H Van Vleck, Phys Rev 49 232 (1936).

87 In this prescient paper on Heisenberg’s theory of ferromagnetism, Van Vleckintroduces what later was called the ‘Van Vleck Catastrophe’, and emphasizesthe fundamentally non-local character of quantum mechanics as expressed

in Equation 1.1, the associated factorial growth in complexity, and its direimplications to attempts to compute many-particle states for large systems

88 W Kohn, Phys Rev Lett, 76 3168 (1996).

89 X P Li, R W Nunes and D Vanderbilt, Phys Rev B 47 10891 (1993).

90 S N Taraskin, D A Drabold and S R Elliott, Phys Rev Lett 88 196405

(2002)

91 See R M Martin, Electronic Structure, Basic Theory and Practical

Applica-tions, Cambridge University Press, Cambridge (2004).

92 D R Hartree, Proc Cambridge Philos Soc 24 89 (1928).

98 P Hohenberg and W Kohn, Phys Rev 136 B864 (1964); W Kohn and L.

J Sham, Phys Rev 140 A1133 (1965).

99 D M Ceperley and G J Alder, Phys Rev Lett 45, 566 (1980).

100 P Ordej´on, E Artacho and J M Soler, Phys Rev B 53, R10441 (1996);

101 F Buda, G.L Chiarotti, R Car, and M Parrinello, Phys Rev Lett 63, 4294

Risto M Nieminen

COMP/Laboratory of Physics, Helsinki University of Technology, POB 1100,FI-02015 HUT, Finland rni@fyslab.hut.fi

2.1 Introduction

An important role for theory and computation in studies of defects in

semi-conductors is to provide means for reliable, robust interpretation of defect fingerprints observed by many different experimental techniques, such as

deep-level transient spectroscopy, various methods based on positron lation (PA), local-vibrational-mode (LVM) spectroscopy, and spin-resonancetechniques [1] High-resolution studies can provide a dizzying zoo of defect-related features, the interpretation and assignment of which requires accuratecalculations of both the defect electronic and atomic properties as well asquantitative theory and computation also for probe itself

annihi-Materials processing can also draw significant advantages of the predictivecomputational studies The kinetics of defect diffusion and reactions duringthermal treatment depend crucially on the atomic-scale energetics (for-mation energies, migration barriers, etc., derived from the bond-breaking,bond-making and other chemical interactions between the atoms in question.Accurately calculated estimates for defect and impurity energetics can consid-erably facilitate the design of strategies for doping and thermal processing toachieve the desired materials properties Parameter values calculated atom-

istically, from the electronic degrees of freedom, can be fed into multiscale modeling methods for defect evolution, such as kinetic Monte Carlo, cellular

automaton, or phase-field simulation tools [2]

A defect in otherwise perfect material breaks the crystalline symmetryand introduces the possibility of localized electronic states in the funda-mental semiconducting or insulating gap Depending on the position of thehost-material Fermi level of the host semiconductor, these states are eitherunoccupied or occupied by one or more electrons, depending on their degen-eracy and the spin assignment The defects thus appear in different chargestates, with varying degrees of charge localization around the defect center

To achieve a new ground state, the neighboring atoms relax around the defect

to new equilibrium positions For a point defect a new point symmetry groupcan be defined

The energy levels in the gap, also known as ‘ionization levels’, tion energy levels’ or ‘transition levels’, correspond to the Fermi level posi-tions where the ground state of the system changes from one charge state

‘occupa-to another Their values can be computationally estimated by the ‘ΔSCF’ approach, i.e from total-energy differences between different charge states.

These gap levels can be probed experimentally by temperature-dependentHall conductivity measurements, deep-level-transient spectroscopy (DLTS),and photoluminescence (PL) The nature of the defect-related gap-state wavefunctions can be examined with electron paramagnetic resonance (EPR) andelectron-nuclear double-resonance (ENDOR) measurements

Experimentally, the defect energy levels can often be located with the cision of the order of 0.01 eV with respect to host material band edges Theproper interpretation of the levels and the physical features associated withthem pose demanding challenges for theory and computation At present,

pre-the computational accuracy of pre-the level position is typically a few tenths of

an eV It follows that sometimes different calculations for the same physicalsystem lead to very different conclusions and interpretations of the experi-ments Defect calculations may also fail to predict correctly the symmetry-breaking distortions around defects undisputedly revealed by experiments.While theoretical modeling has had considerable success and a major impact

in semiconductor physics, there are still severe limitations to its capabilities.The purpose of this chapter is to point out and discuss these

Density-functional theory (DFT) [3] is the workhorse of atomic-scale putational materials science and is also widely used to study defects in semi-conductors, predict their structures and energetics, vibrational and diffusionaldynamics, and elucidate their electronic and optical properties, as observed

com-by the various experimental probes The central quantity in DFT is the tron density The ground-state total energy is a functional of the electrondensity and can be obtained via variational minimization of the functional.The wave-mechanical kinetic energy part of the total-energy functional isobtained not from the density directly but through a mapping to a non-interacting Kohn-Sham system The mean-field electrostatic Hartree energy

elec-is obtained from the density, as are in principle the remaining exchange andcorrelation terms, coded into the exchange-correlation functional

The purpose of this chapter is to examine critically the methodologicalstatus of calculations of defects in semiconductors, especially those based

on the so-called supercell methods As will be discussed in more detail low, there are three main sources of error in such calculations The first isthe proper quantum-mechanical treatment of electron-electron interactions,which at the DFT level is dependent on the choice of the exchange-correlationfunctional The popular local- or semi-local density approximations lead tounderestimation, sometimes serious, of the semiconducting gap The under-lying reasons for this are the neglect of self-interactions and the unphysicalcontinuity of the exchange-correlation energy functional as a function of leveloccupation number This has naturally serious consequences to the mapping

be-of the calculated defect electronic levels onto the experimental energy gap.The second source of errors is related to the geometrical description of the