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

This book is dedicated to Manuel Cardona, who has done so much for the field of defects in semiconductors over the past decades, and convinced so many theorists to calculate beyond what they thought possible Preface Semiconductor materials emerged after World War II and their impact on our lives has grown ever since Semiconductor technology is, to a large extent, the art of defect engineering Today, defect control is often done at the atomic level Theory has played a critical role in understanding, and therefore controlling, the properties of defects Conversely, the careful experimental studies of defects in Ge, Si, then many other semiconductor materials has a huge database of measured quantities which allowed theorists to test their methods and approximations Dramatic improvement in methodology, especially density functional theory, along with inexpensive and fast computers, has impedance matched the experimentalist 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 being developed More importantly perhaps, the tools developed to study defects in semiconductors are now being adapted to approach many new challenges associated 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 a single 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 the various applications are discussed is today’s state-of-the-art in the field The book begins with a ‘big picture’ view from Manuel Cardona, and continues with a brief summary of the historical development of the subject in Chapter This includes an overview of today’s most commonly used method to describe defects We have attempted to create a balanced and tutorial treatment of the basic theory and methodology in Chapters 3-6 They including detailed discussions of the approximations involved, the calculation of electrically-active levels, and extensions of the theory to finite temperatures Two emerging electronic structure methodologies of special importance to the field are discussed in Chapters (Quantum Monte-Carlo) and (the GW method) Then come two chapters on molecular dynamics (MD) In chapter 9, a combina- VIII Preface tion of high-level and approximate MD is developed, with applications to the dynamics of extended defect Chapter 10 deals with semiempirical treatments of microstructures, including issues such as wafer bonding (Chapters and 10) The book concludes with studies of defects and their role in the photo-response of topologically disordered (amorphous) systems The intended audience for the book is graduate students as well as advanced researchers in physics, chemistry, materials science, and engineering We have sought to provide self-contained descriptions of the work, with detailed references available when needed The book may be used as a text in a practical graduate course designed to prepare students for research 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 researcher, or as a convenient guide for the experimentalist to keep tabs on their theorist colleagues It was a genuine pleasure to edit this volume We are delighted with the contributions provided in a timely fashion by so many busy and accomplished people We warmly thank all the contributors and hope to have the opportunity 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) Athens, Ohio, Lubbock, Texas, David A Drabold Stefan K Estreicher February 2006 Contents Forewords Manuel Cardona Early history and contents of the present volume Bibliometric studies References 1 Defect theory: an armchair history David A Drabold, Stefan K Estreicher 1.1 Introduction 1.2 The evolution of theory 1.3 A sketch of first-principles theory 1.3.1 Single particle methods: History 1.3.2 Direct approaches to the many-electron problem 1.3.3 Hartree and Hartree-Fock approximations 1.3.4 Density Functional Theory 1.4 The Contributions References 11 11 13 16 16 18 18 19 22 22 Supercell methods for defect calculations Risto M Nieminen 2.1 Introduction 2.2 Density-functional theory 2.3 Supercell and other methods 2.4 Issues with the supercell method 2.5 The exchange-correlation functionals and the semiconducting gap 2.6 Core and semicore electrons: pseudopotentials and beyond 2.7 Basis sets 2.8 Time-dependent and finite-temperature simulations 2.9 Jahn-Teller distortions in semiconductor defects 2.9.1 Vacancy in silicon 2.9.2 Substitutional copper in silicon 2.10Vibrational modes 2.11Ionisation levels 2.12The marker method 2.13Brillouin-zone sampling 27 27 29 30 32 34 38 39 41 42 42 44 45 46 48 48 X Contents 2.14Charged defects and electrostatic corrections 2.15Energy-level references and valence-band alignment 2.16Examples: the monovacancy and substitutional copper in silicon 2.16.1Experiments 2.16.2Calculations 2.17Summary and conclusions References 50 53 53 55 56 58 59 Marker-method calculations for electrical levels using Gaussian-orbital basis-sets J P Goss, M J Shaw, P R Briddon 3.1 Introduction 3.2 Computational method 3.2.1 Gaussian basis-set 3.2.2 Choice of exponents 3.2.3 Case study: bulk silicon 3.2.4 Charge density expansions 3.3 Electrical levels 3.3.1 Formation energy 3.3.2 Calculation of electrical levels using the Marker Method 3.4 Application to defects in group-IV materials 3.4.1 Chalcogen-hydrogen donors in silicon 3.4.2 VO-centers in silicon and germanium 3.4.3 Shallow and deep levels in diamond 3.5 Summary References 63 63 65 65 68 69 73 73 74 76 77 77 79 80 82 83 Dynamical Matrices and Free energies Stefan K Estreicher, Mahdi Sanati 4.1 Introduction 4.2 Dynamical matrices 4.3 Local and pseudolocal modes 4.4 Vibrational lifetimes and decay channels 4.5 Vibrational free energies and specific heats 4.6 Theory of defects at finite temperatures 4.7 Discussion References 85 85 87 88 89 92 95 98 100 The calculation of free energies in semiconductors: defects, transitions and phase diagrams E R Hern´ andez, A Antonelli, L Colombo,, and P Ordej´ on 5.1 Introduction 5.2 The Calculation of Free energies 5.2.1 Thermodynamic integration and adiabatic switching 5.2.2 Reversible scaling 103 103 104 105 108 Contents XI 5.2.3 Phase boundaries and phase diagrams 5.3 Applications 5.3.1 Thermal properties of defects 5.3.2 Melting of Silicon 5.3.3 Phase diagrams 5.4 Conclusions and outlook References 110 113 113 116 119 124 124 Quantum Monte Carlo techniques and defects in semiconductors R.J Needs 6.1 Introduction 6.2 Quantum Monte Carlo methods 6.2.1 The VMC method 6.2.2 The DMC method 6.2.3 Trial wave functions 6.2.4 Optimization of trial wave functions 6.2.5 QMC calculations within periodic boundary conditions 6.2.6 Using pseudopotentials in QMC calculations 6.3 DMC calculations for excited states 6.4 Sources of error in DMC calculations 6.5 Applications of QMC to the cohesive energies of solids 6.6 Applications of QMC to defects in semiconductors 6.6.1 Using structures from simpler methods 6.6.2 Silicon Self-Interstitial Defects 6.6.3 Neutral vacancy in diamond 6.6.4 Schottky defects in magnesium oxide 6.7 Conclusions References 127 127 128 128 129 131 132 133 134 134 135 136 136 136 137 142 144 145 147 Quasiparticle Calculations for Point Defects at Semiconductor Surfaces Arno Schindlmayr, Matthias Scheffler 7.1 Introduction 7.2 Computational Methods 7.2.1 Density-Functional Theory 7.2.2 Many-Body Perturbation Theory 7.3 Electronic Structure of Defect-Free Surfaces 7.4 Defect States 7.5 Charge-Transition Levels 7.6 Summary References 149 149 152 152 155 160 163 168 171 172 XII Contents 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 8.1 Introduction 8.2 A hybrid view 8.3 Hybrid simulation 8.4 The LOTF scheme 8.5 Applications 8.6 Summary References Empirical molecular dynamics: Possibilities, requirements, and limitations Kurt Scheerschmidt 9.1 Introduction: Why empirical molecular dynamics ? 9.2 Empirical molecular dynamics: Basic concepts 9.2.1 Newtonian equations and numerical integration 9.2.2 Particle mechanics and non equilibrium systems 9.2.3 Boundary conditions and system control 9.2.4 Many body empirical potentials and force fields 9.2.5 Determination of properties 9.3 Extensions of the empirical molecular dynamics 9.3.1 Modified boundary conditions: Elastic embedding 9.3.2 Tight-binding based analytic bond-order potentials 9.4 Applications 9.4.1 Quantum dots: Relaxation, reordering, and stability 9.4.2 Bonded interfaces: tailoring electronic or mechanical properties? 9.5 Conclusions and outlook References 10 Defects in Amorphous Semiconductors: Amorphous Silicon D A Drabold and T A Abtew 10.1Introduction 10.2Amorphous Semiconductors 10.3Defects in Amorphous Semiconductors 10.3.1Definition for defect 10.3.2Long time dynamics and defect equilibria 10.3.3Electronic Aspects of Amorphous Semiconductors 10.3.4Electron correlation energy: electron-electron effects 10.4Modeling Amorphous Semiconductors 10.4.1Forming Structural Models 10.4.2Interatomic Potentials 175 175 176 179 182 185 190 191 195 195 198 198 200 202 203 205 207 207 209 212 212 215 218 219 225 225 225 228 228 230 230 232 233 233 234 Contents 10.4.3Lore of approximations in density functional calculations 10.4.4The electron-lattice interaction 10.5 Defects in Amorphous Silicon References 11 Light-Induced Effects in Amorphous and Glassy Solids S.I Simdyankin, S.R Elliott 11.1Photo-induced metastability in Amorphous Solids: an Experimental Survey 11.1.1Introduction 11.1.2Photo-induced effects in chalcogenide glasses 11.2Theoretical studies of photo-induced excitations in amorphous materials 11.2.1Application of the Density-Functional-based Tight-Binding method to the case of amorphous As2 S3 References XIII 235 236 237 244 247 247 247 249 250 251 261 248 S.I Simdyankin and S.R Elliott the light-solid interaction can become non-linear, this optical non-linearity permitting many new phenomena, such as second-harmonic generation, threeand four-wave mixing, to occur Electronic excitation of structural coordination defects, e.g dangling bonds, can be probed by electron spin (paramagnetic) resonance (ES(P)R) The optically-induced phenomena outlined above can be exhibited by all kinds of semiconducting/insulating solid, whether crystalline or amorphous However, certain features unique to the amorphous state in general, and to certain types of amorphous materials in particular, mean that some photo-induced phenomena are special to amorphous semiconducting materials, particularly those which are metastable, i.e which remain after cessation of irradiation One general characteristic of amorphous semiconductors of relevance in this connection is the occurrence of disorder-induced spatial localization of electronic states in the band-tail states extending into the gap between the VB and CB [1] The presence of continuous bands of localized states in these tail states, in the energy interval between the “mobility” edges in the VB and CB, marking the transition point between localized and delocalized (extended) electron states [2], as well as localized states deep in the bandgap arising from coordination defects [1], can have a profound influence on the nature of the photon-solid interaction These localized states have an enhanced electron-lattice interaction (see the preceding Chapter by Drabold and Abtew), meaning that optical excitation of such electronic states can have a disproportionate effect on the surrounding atomic structure Secondly, the radiative-recombination lifetime for an optically-created electron-hole pair trapped in localized tail states can be very considerably longer than when the photo-excited carriers are in extended states (as is always the case for crystals), thereby allowing possible non-radiative channels to become significant (e.g involving atomic-structural reconfiguration) Other relevant aspects are more materials specific Optically-induced metastability associated with structural reorganisation is likely to be more prevalent in those materials in which (some) atoms have a low degree of nearest-neighbour connectivity (e.g being two-fold coordinated, rather than four-fold coordinated), thereby imparting a considerable degree of local structural flexibility In addition, structural reorganisation following opticallyinduced electronic excitation is more probable if the (VB) electronic states involved correspond to easily-broken weak bonds One class of materials satisfying the above constraints consists of chalcogenide glasses, namely alloys of the Gp VI chalcogen elements (S, Se, Te) with other (metalloid) elements, e.g B, Ga, P, As, Sb, Si, Ge etc These materials are “lone-pair” semiconductors in which (if the chalcogen content is sufficiently high) the top of the VB comprises chalcogen p-π lone-pair (LP) states [3] Interatomic, “non-bonding” (Van der Waals-like) interactions involving such LP states are appreciably weaker than for normal covalent bonds 11 Light-Induced Effects in Amorphous and Glassy Solids 249 (those states lying deeper in the VB) Since the ground-state electronic configuration of chalcogen atoms is s2 p4 , the occurrence of non-bonding LP electron states means that each chalcogen atom is ideally two-fold coordinated by covalent bonds to its nearest neighbours Thus chalcogenide glasses, having a combination of localized electronic tail states, low atomic coordination and weak bonds associated with optically accessible states at the top of the VB, are ideal candidates for exhibiting photo-induced effects 11.1.2 Photo-induced effects in chalcogenide glasses Amorphous chalcogenide materials exhibit a plethora of photo-induced phenomena Such changes can be variously dynamic (i.e present only whilst a material is illuminated) or metastable (i.e the effects remain after cessation of illumination) Furthermore, the changes may be either scalar or vectoral in nature (respectively independent or dependent on either the polarization state, or the propagation direction, of the inducing light) Finally, metastable changes may be irreversible, or reversible with respect to thermal or optical annealing Examples of dynamic effects are the afore-mentioned phenomena of photoluminescence and photoconductivity [1] and optical non-linearity [4], which are common to all materials (Chalcogenide glasses generally exhibit extremely large optical non-linearities because of the highly electronically polarizable nature of chalcogen atoms present [4].) However, a (scalar) dynamic effect, which is characteristic of chalcogenide glasses, is photo-induced fluidity, wherein the viscosity of a glass (e.g As2 S3 ) decreases on illumination with sub-bandgap light or, in other words, light can cause viscous relaxation in a stressed glass [5, 6] An example of a vectoral dynamic photo-effect in chalcogenide glasses is the opto-mechanical effect [7, 8], wherein linearly-polarized light incident on a chalcogenide-coated clamped microcantilever causes it to displace upwards or downwards, depending on whether the polarization axis is respectively parallel to perpendicular to the cantilever axis, as a result of anisotropic photo-induced strains introduced into the chalcogenide-cantilever bimorph Metastable photo-induced changes are perhaps the most interesting (and applicable) of the effects observed in chalcogenide glasses One such is photodarkening (or bleaching), wherein the optical absorption edge of the material shifts to lower energies (hence the material gets darker at a given wavelength), or to higher energies (bleaching), on illumination with (sub-) bandgap light For reviews, see refs [9, 10] Arsenic-based materials with higher sulphur contents and germanium sulphide materials seem to favour photo-bleaching, for reasons that are not clear at present irreversible shifts of the optical absorption edge occur in virgin (as-evaporated) thin films of amorphous chalcogenides containing structurally-unstable molecular fragments from the vapour phase [11] that are particularly vulnerable to photo-induced (or thermal) change Reversible photodarkening is observed in bulk glasses and well- 250 S.I Simdyankin and S.R Elliott annealed thin films: optical illumination causes a red-shift of the absorption edge, whilst subsequent thermal annealing to the glass-transition temperature, Tg , erases the effect [9, 10] Photodarkening (bleaching) appears to be associated with photo-induced structural changes, such microscopic changes being manifested as macroscopic volume changes Photo-contraction is particularly prevalent in virgin obliquely-deposited thin films of amorphous chalcogenides having a columnarlike microstructure [12], but photo-expansion is commonly associated with photodarkening in chalcogenide bulk glasses or thin films [9,10] Giant photoexpansion (up to 5% linear expansion) occurs for sub-bandgap illumination [13] Another metastable photo-induced structural change exhibited by chalcogenide materials is photo-induced crystallization and amorphization, as used in rewriteable “phase-change” CDs and DVDs [14, 15] The amorphization process there is believed to result from a photo-induced melting and subsequent very rapid quenching of the material to the glassy state However, illumination of certain crystalline chalcogenide materials (e.g As50 Se50 ) can also cause athermal photo-amorphization [16] Light can also cause “chemical” changes in chalcogenide glasses For example, overlayers of certain metals (notably silver) diffuse into the undoped bulk glass on illumination [17, 18] On the other hand, Ag-containing chalcogenide glasses, rich in Ag, exhibit the opposite effect, namely photo-induced surface deposition, wherein the metal exsolves from the glassy matrix on illumination [19, 20] A particularly interesting metastable vectoral photo-induced effect exhibited by chalcogenide glasses is photo-induced optical anisotropy (POA), first observed in ref [21], and manifested in absorption, reflection (i.e refractive index) and scattering (for a review, see [10, 22]) Illumination of an initially optically isotropic glass by linearly-polarized light causes the material to become dichroic and birefringent In the case of well-annealed thin films and bulk glasses, the effect is completely reversible optically (i.e the induced optical axis is fully rotated when the light-polarization vector is rotated), and the POA can be annealed out thermally (but at a lower temperature than is the case for scalar photodarkening) or optically (using unpolarized or circularly polarized light) 11.2 Theoretical studies of photo-induced excitations in amorphous materials From the above very brief review, it is apparent that amorphous chalcogenide materials exhibit a wide range of interesting photo-induced phenomena Although the experimental phenomenology is, for the most part, well developed, a proper theoretical understanding is still lacking Until very recently, theoretical models have been confined to “hand-waving” models [9, 10], involving simple notions of chemical bonding and the effects of light (e.g bond-breaking and defect formation) However, the predictive, and even descriptive, power of 11 Light-Induced Effects in Amorphous and Glassy Solids 251 such approaches is very limited, and for a microscopic (atomic-level) understanding of photo-induced phenomena in amorphous chalcogenides, proper quantum-mechanical calculations need to be performed Obviously, this is an extremely challenging task, and two essentially opposing methods for making progress in this regard, using different approximations, can be employed The first employs all-electron quantum-chemical calculations on small clusters (a few tens) of atoms, in which the optimized electronic and atomic configurations and energies are found for the ground and electronically-excited states (see, e.g [23–26]) This favours accuracy over dynamics The other method, (discussed in the Chapter by Drabold and Abtew), namely ab initio molecular dynamics (MD) simulations, takes a converse approach: atomic dynamics are followed at the expense of accuracy More detail will be given in the following, but the advantages and disadvantages of these two approaches can be summarized as follows Quantum-chemical methods can only deal with very small clusters (particularly for excited-state calculations) and only initial (ground-state) and final (excited-state) configurations can be studied, not the intermediate dynamics, but the energetics are the most accurate Ab initio MD, on the other hand, provides full information about atomic dynamics, but only over very short time scales (typically a few picoseconds) and for relatively few atoms (typically less than a hundred) In order to make the calculations involved tractable, numerous more-or-less severe approximations need to be invoked (e.g the local density approximation (LDA) in density functional theory (DFT)), which make certain results imprecise (e.g underestimation of the bandgap) Moreover, the Kohn-Sham (KS) orbitals resulting from DFT are ground-state quantities and hence formally inadmissible for a consideration of excited-state behaviour Nevertheless, use of these theoretical methods has provided very useful atomistic information of help in understanding photo-induced phenomena in chalcogenide glasses Some justification for using both occupied and virtual KS orbitals for qualitative, and sometimes partially quantitative, analysis of electronic structure is provided by comparing the shape and symmetry properties, as well as the energy order, of these orbitals with those obtained by wavefunction-based (e.g at the Hartree-Fock level of theory [27]) and GW [28] calculations There is a very strong similarity between GW and LDA states, and often identical DFT and GW quasiparticle wavefunctions are assumed in practical calculations [29] 11.2.1 Application of the Density-Functional-based Tight-Binding method to the case of amorphous As2 S3 One approximate ab initio-based MD scheme that has proved very useful in understanding the electronic behaviour of chalcogenide glasses is the DensityFunctional-based Tight-Binding (DFTB) method developed by Frauenheim and coworkers Although this is reported in refs [30–32], since this approach is not described elsewhere in this book, we give here a brief description of 252 S.I Simdyankin and S.R Elliott this method and review our recent results on the canonical chalcogenide glass a-As2 S3 obtained by using the DFTB method [33–35] Although the DFTB method is semiempirical, it allows one to improve upon the standard tight-binding description of interatomic interactions by including a DFT-based self-consistent second order in charge fluctuation (SCC) correction to the total energy [31] The flexibility in choosing the desired accuracy while computing the interatomic forces brings about the possibility to perform much faster calculations when high precision is not required, and refine the result if needed As described in ref [36], the SCC-DFTB model is derived from densityfunctional theory (DFT) by a second-order expansion of the DFT total energy functional with respect to the charge-density fluctuations δn = δn(r ) around a given reference density n0 = n0 (r ): occ ˆ |ψi ψi |H E= i + − δ Exc + |r − r | δn δn n0 n0 + Exc [n0 ] − |r − r | δn δn (11.1) n0 Vxc [n0 ]n0 + Eii , ˆ0 = where dr and dr are expressed by and , respectively Here, H ˆ H[n0 ] is the effective Kohn-Sham Hamiltonian evaluated at the reference density and the ψi are Kohn-Sham orbitals Exc and Vxc are the exchangecorrelation energy and potential, respectively and Eii is the core-core repulsion energy To derive the total energy of the SCC-DFTB method, the energy contributions in Eq (11.1) are further subjected to the following approximations: ˆ |ψi are represented in a basis of 1) The Hamiltonian matrix elements ψi |H confined, pseudoatomic orbitals φμ , ciμ φμ ψi = (11.2) μ To determine the basis functions φμ , the atomic DFT problem is solved by adding an additional harmonic potential ( rr0 )2 to confine the basis functions [30] The Hamiltonian matrix elements in this LCAO basis, Hμν , are then calculated as follows The diagonal elements Hμμ are taken to be the atomic eigenvalues and the non-diagonal elements Hμν are calculated in a two-center approximation: = φμ |Tˆ + veff [n0α + n0β ]|φν Hμν μ α, ν β, (11.3) which are tabulated, together with the overlap matrix elements Sμν with respect to the interatomic distance Rαβ veff is the effective Kohn-Sham 11 Light-Induced Effects in Amorphous and Glassy Solids 253 potential and n0α are the densities of the neutral atoms α 2) The charge-density fluctuations δn are written as a superposition of atomic contributions δnα , δn = δnα , (11.4) α which are approximated by the charge fluctuations at the atoms α, Δqα = qα − qα0 qα0 is the number of electrons of the neutral atom α and the qα are determined from a Mulliken-charge analysis The second derivative of the total energy in Eq (11.1) is approximated by a function γαβ , whose functional form for α = β is determined analytically from the Coulomb interaction of two spherical charge distributions, located at Rα and Rβ For α = β, it represents the electron-electron self-interaction on atom α 3) The remaining terms in Eq (11.1), Eii and the energy contributions, which depend on n0 only, are collected in a single energy contribution Erep Erep is then approximated as a sum of short-range repulsive potentials, Erep = U [Rαβ ], (11.5) α=β which depend on the interatomic distances Rαβ With these definitions and approximations, the SCC-DFTB total energy finally reads: ciμ ciν Hμν + Etot = iμν γαβ Δqα Δqβ + Erep (11.6) αβ Applying the variational principle to the energy functional (11.6), one obtains the corresponding Kohn-Sham equations: cνi (Hμν − i Sμν ) = 0, ∀ μ, i (11.7) ν Hμν = φμ |H0 |φν + Sμν (γαζ + γβζ )Δqζ , (11.8) ζ which have to be solved iteratively for the wavefunction expansion coefficients ciμ , since the Hamiltonian matrix elements depend on the ciμ due to the Mulliken charges Analytic first derivatives for the calculation of interatomic forces are readily obtained, and second derivatives of the energy with respect to atomic positions are calculated numerically The repulsive pair potentials U [Rαβ ] are constructed by subtracting the DFT total energy from the SCC-DFTB electronic energy (first two terms on the right-hand side of eq.(11.6)) with respect to the bond distance Rαβ for a small set of suitable reference systems To summarize, in order to determine the appropriate parameters for a new element, the following steps have to be taken First, DFT calculations have to 254 S.I Simdyankin and S.R Elliott be performed for the neutral atom to determine the LCAO basis functions φμ and the reference densities n0α Here the confinement radius can in principle be chosen to be different for the density (r0n ) and each type of atomic orbital (r0s,p,d) The value of r0 is usually taken to be the same for s- and p-functions In a minimal basis, this yields a total number of two adjustable parameters for elements in the first and second rows, while there are three if d-functions are included After this, the different matrix elements can be calculated and the pair potentials U [Rαβ ] are obtained as stated above for every combination of the new element with the ones already parameterized The single-particle KS occupied, ψi , and unoccupied, ψj , orbitals and the corresponding KS excitation energies ωij = j − i are sometimes used for qualitative analysis of electronic excitations Although the KS excitation energies can be considered as an approximation to true excitation energies ωI [37] (see also end of the introduction to Sec 11.2), the correction terms to this approximation can be very significant in many cases Modelling electronic excitations by varying the occupation of the KS orbitals results in mixed quantum states with undefined contributions from singlet and triplet states We have performed modelling of electronic excitations at the level of KS orbitals and energies and have attempted to validate the results by using more exact methods One such method is based on time-dependent densityfunctional response theory (TD-DFRT) [38, 39], where the true excitation energies are found by solving the following eigenvalue problem: √ √ I I [ωij δik δjl δστ + ωij Kijσ,klτ ωkl ]Fijσ = ωI2 Fklτ (11.9) ijσ Here σ and τ are spin indices The indices i, k correspond to occupied, and j, l to unoccupied, KS orbitals, respectively The coupling matrix K is defined as (see ref [40] for a more detailed description of the method): Kijσ,klτ = ψi (r)ψj (r) δ Exc + |r − r | δnσ δnτ ψk (r )ψl (r ), (11.10) where we use a notation consistent with that in Eq (11.2) Our structural models of a-As2 S3 were obtained by a “melt-and-quench” procedure [35] similar to the one described in the preceding chapter by Drabold and Abtew The structural quality of models can be assessed by examining the radial distribution function (RDF) and the structure factor (see fig 11.1) Apart from general good agreement with experimental data, an interesting feature revealed by fig 11.1(a) is the shape of the first peak of the RDF The shoulders on both sides of this peak indicate the presence of homopolar (As-As or S-S) bonds in the material Such bonds (chemical defects), as well as coordination (or topological) defects, are easily detectable in computer models An example of a configuration featuring an As-As bond is shown in fig 11.2 Special significance can be attributed to the presence of 11 Light-Induced Effects in Amorphous and Glassy Solids 255 simulation experiment g(r) 0 (a) r [Å] simulation experiment −1 Q(S(Q)−1) [Å ] −1 −2 (b) 10 12 14 16 18 20 22 24 −1 Q [Å ] Fig 11.1 (a) Pair-correlation functions for a 200-atom model of a-As2 S3 and the neutron-diffraction experiment (ref [41]) (b) Reduced structure factors (interference functions) for the same model and experiment five-membered rings in models with an appreciable concentration of homopolar bonds (models with all-heteropolar bonds contain only an even number of atoms in all rings) When such rings share some of the bonds, the resulting local structure is close to that of cage-like molecules (e.g As4 S4 ), as found in the vapor phase and in some chalcogenide molecular crystals Fig 11.2(a) shows two such bond-sharing rings Upon breaking the two bonds connecting the rings to the rest of the network, the distance between the two freed arsenic atoms (connected by a dashed line in fig 11.2(a)) could be reduced, thus producing another As-As homopolar bond and this group of atoms would then form an As4 S4 molecule (shown in fig 11.2(b)) Evidence of the presence of such molecules in bulk Asx S1−x glasses from Raman-scattering experiments has recently been reported in ref [42] Our result shows that the As4 S4 frag- 256 (a) S.I Simdyankin and S.R Elliott (b) Fig 11.2 (a) Fragment of a 200-atom model of a-As2 S3 : two bond-sharing fivemembered rings and the two AsS3 groups connected to this structure The dangling bonds show where the displayed configuration connects to the rest of the amorphous network The dashed line connects two As atoms, which, if brought nearer together, would close up to form an As4 S4 molecule shown in panel (b) The shading of the As atoms (all with three neighbors) is darker than that of the S atoms (all with two neighbors) ments may not only form discrete cage-like molecules but also be embedded into the amorphous network We verified that the vibrational signatures of the As4 S4 fragment from models and are similar to those from an isolated As4 S4 molecule, apart from a few very symmetric modes of the latter The observed tendency for formation of quasi-molecular structural groups suggests that amorphous chalcogenides can be viewed as nanostructured materials Localization of the electronic states near the optical band-gap edges is of great interest for studies of photoinduced phenomena The inverse participation ratios (IPR, defined in the chapter by Drabold and Abtew) for a model of a-As2 S3 are shown in Fig 11.3 The general picture is that, at the top of the valence band, the eigenstates are predominantly localized at what can be called sulphur-rich regions, where several sulphur atoms are closer than about 3.45 ˚ A, i.e their interatomic distances are on the low-r side of the second peak in g(r) shown in Fig 11.1(a) or some of these atoms form homopolar S-S bonds For instance, most of the HOMO (highest occupied molecular orbital) level in this model is localized at two sulphur atoms separated by 3.42 ˚ A and which are part of the molecule-like fragment depicted in Fig 11.2(a) By inspecting the projected (local) IPRs in Fig 11.3(b) at the optical gap edges, it is seen that the IPRs are greatest for the S atoms It appears that the localization at the top of the valence band is facilitated by the proximity of the lone-pair p orbitals in the sulphur-rich regions This observation is consistent with a result for a-GeS2 [44] 0.8 As, s As, p As, d 0.6 Total 0.4 0.8 0.2 LEDOS [eV-1] 0.8 257 simulation experiment S, s S, p S, d EDOS [eV-1] LEDOS [eV-1] 11 Light-Induced Effects in Amorphous and Glassy Solids 0.6 0.4 0.6 0.2 0.4 0.2 0 -15 -10 -5 -15 -10 -5 -15 -10 -5 (a) E [eV] E [eV] -15 -10 E [eV] -5 5 E [eV] 0.2 As, s As, p As, d IPR 0.15 0.2 0.1 Total 0.05 0.15 IPR 0.2 S, s S, p S, d 0.1 IPR 0.15 0.1 0.05 0.05 0 -15 -10 -5 -15 -10 -5 -15 -10 -5 (b) E [eV] E [eV] E [eV] -15 -10 -5 E [eV] Fig 11.3 Local and total electronic density of states (a) and inverse participation ratios (b) for a 200-atom model of a-As2 S3 The Fermi energy is at the energy origin The experimental data in (a) are obtained from ref [43] At the bottom of the conduction band, the states tend to localize at various anomalous local configurations, such as four-membered rings, S-S homopolar bonds (some of these bonds are in five-membered rings) and valencealternation pairs (coordination defects) For example, the LUMO (lowest unoccupied molecular orbital) state for the structure depicted in fig 11.5(a,b) is localized on an overcoordinated S atom (with three bonded neighbours) We found that a basis set of s, p and d Slater-type orbitals for all atoms is an essential prerequisite for the observation of overcoordinated defects [36,45] It is possible to analyse individual contributions of orbitals of each type to the total EDOS and IPR, and these contributions are shown in Fig 11.3 As mentioned above, in the context of photoinduced metastability, a great deal of significance is attributed to the presence of topological and/or chemical defects [9] It is therefore imperative to create models both with and 258 S.I Simdyankin and S.R Elliott C G F E B A H I D (a) C B F A E G I D H (b) (c) Fig 11.4 Planar trigonal, [S3 ] (marked by the letter ‘A’), and “seesaw”, [As4 ]− (marked by letter ‘E’), configurations in (a) a fragment of a 60-atom model of aAs2 S3 (the dangling bonds show where the displayed configuration connects to the rest (not shown) of the network) and, (b) and (c), charged isolated clusters(the dangling bonds are terminated with hydrogen atoms) The shading of the atoms is the same as in fig 11.2 + without such defects in a theoretical investigation that attempts to be conclusive Defect-free models can be produced by “surgical” manipulations For example, atoms with undesired coordination can be removed from the model, and, in order to eliminate chemical defects, one can iteratively apply the following algorithm First, a sulphur atom is inserted in the middle of each As-As homopolar bond Second, each S-S bond is replaced by a single sulphur atom located at its mid-point so that each local As-S-S-As configuration is turned into As-S-As Third, the distance between each newly introduced S atom and its two nearest arsenic atoms in the newly created As-S-As units is reduced in order to increase the bonding character of the As-S bonds stretched by the above manipulation Fourth, the modified configuration is relaxed in an MD run We have found that only a few iterations can be sufficient in order to obtain models with all-heteropolar (As-S) bonds Elimination of the defects in our models removes some electronic states in the optical band gap As a result, the band gap broadens, which can be viewed as an artificial bleaching of the material The states at the band edges, however, are still localized due to disorder It is possible that exposure to light may lead to bond breaking and defect creation even in such “all-heteropolar” 11 Light-Induced Effects in Amorphous and Glassy Solids 259 E E C C B B A A D D (a) (c) (b) (d) Fig 11.5 An As4 S10 H8 cluster containing both defect centers [As4 ]− (marked by letter ‘A’) and [S3 ]+ (‘B’) The shading of the atoms is the same as in Fig 11.2 The black solid lines signify elongated bonds (a) Optimized ground-state geometry Bond lengths are (˚ A): AC=3.00, AD=2.32, and BE=2.4 (b) Isosurfaces corresponding to the value of 0.025 of electron density in the HOMO (darker red surface) and LUMO (lighter cyan surface) states for the structure shown in (a) (c) Optimized geometry in the first singlet excited state Bond lengths are (˚ A): AC = 2.43, AD = 2.44, and BE = 2.82 (d) Same as (b), but for the structure shown in (c) models Indeed, As-S bond elongation/breaking has been observed in allheteropolar clusters [23,24] and cage-like molecules (unpublished) Such bond breaking under irradiation can lead to the creation of self-trapped excitons (STEs), i.e oppositely charged defect pairs, as shown in ref [29] for silicon dioxide where, following Si-O bond breaking, the hole is localized at the oxygen defect centre and the electron at the silicon defect centre In analogy with the case of SiO2 , it is possible that positively charged chalcogen defects and negatively charged non-chalcogen defects are introduced in amorphous chalcogenides by a similar mechanism In the As-S system, such STEs can possibly lead to creation of the [As4 ]− -[S3 ]+ defect pairs described below These defects, as well as those which are always present in real materials and “as-prepared” models, may mediate local structural rearrangements upon photon absorption Creation of additional states at optical band-gap edges (photodarkening) of a-As2 S3 exposed to light has long been attributed to the generation by illumination of defects in excess of the thermal equi- 260 S.I Simdyankin and S.R Elliott librium concentration [9], but the exact mechanism of defect creation and the nature of the defects are still enigmatic Possible candidate defect types, notably valence-alternation pairs, have been proposed over the years [9] Normally, such defect pairs contain singly-coordinated chalcogen atoms having distinct spectroscopic signatures [46] Experimentally, the concentration of these defects is estimated to be rather small [47], i.e 1017 cm−3 , compared with the atomic density of about × 1025 cm−3 , in order quantitatively to account for the observed magnitude of the photo-induced effects In our simulations, in addition to the charged coordination defects previously proposed to exist in chalcogenide glasses, a novel defect pair, [As4 ]− [S3 ]+ (see fig 11.4), consisting of a four-fold coordinated arsenic site in a “seesaw” configuration and a three-fold coordinated sulfur site in a nearplanar trigonal configuration, was found in several models [33] Such defect pairs are unusual in two ways First, there is an excess of negative charge in the vicinity of the normally electropositive pnictogen (As) atoms and, second, there are no under-coordinated atoms with dangling bonds in these local configurations The latter peculiarity may be the reason why such defect pairs have not yet been identified experimentally These defect pairs, however, are consistent with the STEs described above Although electronic excitations, where one electron is promoted from HOMO to LUMO Kohn-Sham states [48], are not especially realistic, we simulate such excitations in order qualitatively to assess defect stability with respect to (optically-induced) electronic excitations In some models, [S3 ]+ − + S− defect pairs are converted into [As4 ] -[S3 ] pairs as a result of the electronic excitation The presence of these defect pairs introduces additional localized states at the optical band-gap edges with energies quantitatively consistent with the phenomenon of photodarkening [34] − A possible mechanism of conversion of [S3 ]+ -S− defect pairs into [As4 ] + [S3 ] pairs is illustrated in fig 11.5, which shows an As4 S10 H8 cluster containing an [S3 ]+ -S− pair in the ground state Geometry optimization in the first singlet excited state within the linear-response approximation to timedependent (TD) density-functional theory (which gives a much better description of excited states compared with HOMO-to-LUMO electron excitations [40]) leads to a redistribution of electron density, so that the singly coordinated S atoms become attached to a normally coordinated As atom thus forming an [As4 ]− defect At the same time, the [S3 ]+ defect breaks up (we observed that bond breaking/elongation in all our models generally occurs at the groups of atoms where the LUMO is localized, indicating the expected antibonding character of LUMO states.) 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The Web of Science (WoS) lists 3736 mentions in the title and abstract of source articles [3] Forewords The modern science of defects in semiconductors is closely... tools developed to study defects in semiconductors can be easily extended to other areas of materials theory, including many fields of nanoscience It is the need to understand the properties of defects. .. realized how much energy is involved in relaxations and distortions It was believed that the chemistry of defects in semiconductors is well described in first order by assuming high symmetry, undistorted,