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Characterisation of negative-U defects in semiconductors

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Topical Review

Characterisation of negative-U defects insemiconductors

José Coutinhoi3N, Department of Physics, University of Aveiro, Campus Santiago, 3810-193Aveiro, Portugal

E-mail: [email protected]

Vladimir P Markevich and Anthony R PeakerPhoton Science Institute, School of Electrical and Electronic Engineering, TheUniversity of Manchester, Manchester M13 9PL, United Kingdom

Abstract. This review aims at providing a retrospective, as well as adescription of the state-of-the-art and future prospects regarding the theoreticaland experimental characterisation of negative-U defects in semiconductors.This is done by complementing the account with a description of the workthat resulted in some of the most detailed, and yet more complex defectmodels in semiconductors. The essential physics underlying the negative-Ubehaviour is presented, including electronic correlation, electron-phonon coupling,disproportionation, defect transition levels and rates. Techniques for theanalysis of the experimental data and modelling are also introduced, namelydefect statistics, kinetics of carrier capture and emission, defect transformation,configuration coordinate diagrams and other tools. We finally include a showcaseof several works that led to the identification of some of the most impactingnegative-U defects in group-IV and compound semiconductors.

Submitted to: J. Phys.: Condens. MatterKeywords: Semiconductors, Defects, Correlation, Negative-U , Electrical levels

Page 1 of 30 AUTHOR SUBMITTED MANUSCRIPT - JPCM-115996.R1

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Negative-U defects in semiconductors 2

Contents

1 Preface 3

2 Introduction 4

3 Negative-U defects in semiconductors 63.1 Electronic correlation . . . . . . . . . . . 63.2 Disproportionation . . . . . . . . . . . . 83.3 Electronic transition levels of defects . . 93.4 Formation energy diagrams . . . . . . . 103.5 Configuration coordinate diagrams . . . 11

4 Methods of characterisation 134.1 Probing negative-U defects . . . . . . . 13

4.1.1 Elucidation of the nature of adefect by means of analysis of itsoccupancy with charge carriersat equilibrium conditions . . . . 13

4.1.2 Observations of negative-U de-fects in metastable configurations 15

4.1.3 Non-equilibrium occupancy stat-istics for defects with U < 0 . . . 16

4.2 Modelling negative-U defects . . . . . . 184.2.1 Calculation of electronic trans-

ition levels . . . . . . . . . . . . 184.2.2 Calculation of carrier capture

cross-sections in multi-phononassisted transitions . . . . . . . . 18

4.2.3 Calculation of transformationbarriers . . . . . . . . . . . . . . 20

5 Showcase of negative-U defects and theircharacterisation 215.1 BsO2 (LID) complex in silicon . . . . . 215.2 Atomic hydrogen in silicon . . . . . . . . 225.3 DX centres in III-V alloys . . . . . . . . 225.4 The Ga-OAs-Ga defect in GaAs . . . . . 235.5 Electrical levels of selected negative-U

defects in semiconductors . . . . . . . . 25

6 Conclusions 27

References 27

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1. Preface

The presence of defects and contaminants in crystallinesolids, and in particular crystalline semiconductors, isunavoidable. Defects, if not deliberately introduced,necessarily occur due to a fundamental reason: thelowering of free energy by increasing configurationalentropy. Contamination, on the other hand, essentiallydepends on the contact of the sample with alienspecies, and therefore on the specificities of growth andprocessing technologies.

Defects and impurities in semiconductors can beintentionally introduced in order to confer the mater-ial with customised mechanical, electrical, optical ormagnetic functionality. However, even when intention-ally present, they may also turn out to be critical asthe semiconductor can acquire other prominent, butunwanted properties. For instance, while a certainamount of dissolved oxygen is highly beneficial in termsof the mechanical stability of silicon wafers, under cer-tain conditions the oxygen impurities precipitate andform clusters which act as shallow donors and so in-crease the availability of electrons for electrical trans-port at room temperature. This effect alters the elec-trical specifications of the silicon, and in p-type mater-ial type-inversion can occur. Another example is thecreation of silicon vacancies in silicon carbide which canact as single-photon emitters for quantum communica-tions. However, they are also responsible for the intro-duction of several deep electron traps which are highlydetrimental in terms of compensation effects in n-typematerial. Clearly, understanding defect properties, andif possible, their quantum mechanics, is paramount fortechnologies that range from solar power conversion toquantum computing.

Defects can be classified in many ways, butregarding their geometric properties they are usuallydistinguished as point-like and extended. Examplesof point-defects are vacancies, interstitials, antisites(misplaced atoms in compound crystals) and atomic-scale complexes. On the other hand, extendeddefects include dislocations, surfaces and interfaces.Regardless of their size and shape, they change thepotential in the Hamiltonian of the pristine material.Consequently, that often leads to the appearance ofdefect states within a spectral range which would beforbidden to the otherwise perfect crystal.

Unlike crystalline states, defect states are local-ised in real space. They may capture charge carri-ers depending whether the stabilisation energy (for in-stance due to bond formation or exchange interactions)overcomes the repulsive energy between the capturedcharge and other charges already present. In somecases, defects may actually capture more than one car-rier of the same type, but now the stabilisation energyhas to exceed the repulsion from previously captured

charges as well. This extra cost is often referred toas correlation energy (U). We may find an analogy ofthis effect in the sequential ionisation of an atom, forinstance,

He0 + I1 → He+ + e−; (1)He+ + e− + I2 → He++ + 2e−, (2)

where In is the n-th ionization energy of the Heatom. Here I2 > I1, and the difference U = I2 − I1,which essentially accounts for the electron-electronrepulsion and nuclear screening, is obviously positive.Alternatively, U may be defined as

U = E(He0) + E(He++)− 2E(He+), (2)

where E(Heq) is the energy of the He atom in the qcharge state. The fact that U > 0, makes a pair ofHe+ ions more stable than He0 +He++, so that a He+gas will not decompose into a mix of He0 and He++

ensembles.As in the case of isolated atoms, most defects in

semiconductors show positive U -values. However, weknow today, that many effectively show a negative-U correlation between sequential charging events inapparent defiance of electrostatics. It is as if theybecame less ‘eager’ for electrons (holes) immediatelyafter electron (hole) emission!

In the early 1970’s, Anderson introduced the ideaof negative-U to explain the absence of paramagnetismin glassy-semiconductors doped with n- and p-typeimpurities [1]. The material was envisaged as a randomnetwork of three-state bonds whose potential energywas given by V = 1/2 c x2 − λep(n↑ + n↓)x + n↑ n↓ U .Here x represented a bond coordinate, c its respectiveharmonic coefficient, and n↑ and n↓ were spin-up andspin-down bond occupancies (nσ = 0, 1). The lastterm accounted for an electronic Hubbard correlationenergy [2], with impact only when both electrons pairto form the bond (n↑ = n↓ = 1). The strikingingredient of the model was the introduction of anelectron-phonon coupling constant λep connecting thebond displacement with its occupancy. Accordingly,considering atomic relaxations and the definition ofEq. 2, the net effective correlation involving the threestates (n↑+n↓ = 0, 1, 2) is given by Ueff = U−λ2ep/c.It was then noted that for sufficiently strong coupling,Ueff becomes negative, thus providing an explanationfor why in many glasses and polymers, paramagneticn↑ + n↓ = 1 states are unstable against diamagneticn↑ + n↓ = 0, 2 states.

Although the original idea of Anderson was todescribe the formation of bipolarons (where pairingof two electrons and two holes in a localised region isfavoured against single polaron formation), the analogyto the negative-U effect in defects is evident. In fact,


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the concept was extended to defects in semiconductingglasses (like As2Se3) by Street and Mott [3]. Kastnerand co-workers [4] analysed the applicability of themodel to several other compounds, emphasising thatthe dominant contribution to the negative correlationenergy is of chemical origin, or in other words, is drivenby rebonding of atoms.

In 1979, Baraff, Kane and Schülter anticipated nu-merically that neutral, positive and double-positivelycharged vacancies in crystalline silicon (V 0

Si, V+Si and

V ++Si , respectively) formed an ‘Anderson negative-U ’

system [5]. A Jahn-Teller distortion in the neutralcharge state was claimed to be responsible for the ef-fect, being sufficiently stabilising as to render V +

Si un-stable against V 0

Si and V ++Si , irrespective of the posi-

tion of the Fermi energy. A few months later, thesepredictions were confirmed experimentally by Watkinsand Troxell [6]. The following arguments providedthe grounds for the claim: (i) the paramagnetic stateV +Si , as monitored by electron paramagnetic resonance

(EPR), was only observed at cryogenic temperaturesunder photo-excitation. This was an indication thatV +Si is metastable; (ii) Upon turning off the light, the

intensity of the EPR signal bleached at a rate lim-ited by a barrier of 0.05 eV. This figure was inter-preted as the activation barrier for hole emission duringV +Si → V 0

Si + h+; (iii) From deep level transient spec-troscopy (DLTS), a peak with an activation barrier forhole emission of 0.13 eV, with twice the intensity ex-pected for a single hole emission, could be connectedto the following two-hole emission sequence,

V ++Si → V +

Si + h+ → V 0Si + 2h+. (3)

It was then argued that for a negative-U double donor,the hole involved in the first ionisation was boundmore strongly (0.13 eV) than the second one (0.05 eV).Thus, at a temperature where the first hole is emitted,the second hole should follow immediately, naturallyaccounting for the double intensity of the peak.

Baraff, Watkins and others, inspired an entirecommunity towards the characterisation of many othernegative-U defects in semiconductors. This account isabout their work and the observations and conceptsthat followed. It aims at reviewing the literature andpresenting the most recent results regarding experi-mental and theoretical methods involved in the char-acterisation of negative-U defects in semiconductors.

The main text provides a survey regarding severalexperimental and theoretical reports on negative-Udefects (Section 2), an introduction to the physicalconcepts involved in the description of negative-Udefects (Section 3), details regarding experimentaland theoretical methods that are used for theircharacterisation (Section 4), a tabular showcase ofa selection of negative-U defects in semiconductors,accompanied by a detailed description of some of

the most interesting and technologically relevant ones(Section 5), and finally a revision of the main concepts,results and challenges for the future of the topic(Section 6).

2. Introduction

Multi-stability is among the most fascinating prop-erties of point defects in semiconductors [7]. In thepresent context, we refer to a multi-stable defect asone which can be found in several inequivalent atomicconfigurations for a particular charge state. Along thesame lines, a bi-stable defect has two non-degenerateatomic structures in the same charge state. The rel-ative stability as well as formation/annihilation ratesof different configurations of multi-stable defects areoften sensitive to the application of external stimuli(e.g. temperature, electro-magnetic fields or mechan-ical stress). Hence, under favourable conditions, theirpopulations can be shifted from those observed underequilibrium.

Upon changing the atomistic structure of defects,a change of its electronic structure also takes place.It is therefore expected that stimuli-induced defect re-configurations may affect significantly the propertiesof the host material. Exhibition of persistent photo-conductivity, photo-induced capacitance quenching, ortemperature-dependent carrier trapping are symptomsthat are commonly connected to multi-stability. Themagnitude and timescale of these effects depend notonly on the properties of the defects themselves, liketransformation barriers, cross-sections and transitiondipole moments, but also on sample properties like de-fect concentration and distribution, its thermal history,and of course the measurement conditions.

Negative-U defects are intimately tied to multi-stability and metastability. These defects have atleast three charge states, say with electron occupancy|N − 1〉, |N〉, and |N + 1〉. Here N is an arbitraryreference and the |N − m〉 state also includes melectrons at a reservoir with energy EF per electron(Fermi level). The distinction of negative-U defectsis that the intermediate |N〉 state is metastable andnot found under equilibrium conditions. Figure 1shows a schematic phase diagram of such a defect,where transition boundaries are drawn as a functionof the effective correlation energy (Ueff = I2 − I1) andEF. Here Im = E(N+1−m) − E(N+2−m) is the m-thionization energy, with E(N) standing for the energy ofstate |N〉. When Ueff > 0, all three charge states canbe populated under equilibrium conditions, dependingon the location of the Fermi energy with respect to thetransition levels at Ec − I1 and Ec − I2, where Ec isthe conduction band bottom. However, for a negative-U defect, only |N−1〉 or |N+1〉 states are observed, with


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Figure 1. Phase diagram of a three-charge-state defect, withelectron occupancy |N−1〉, |N〉, and |N+1〉. The horizontal axisis the effective correlation energy (Ueff = I2 − I1), whereas thevertical axis is the Fermi energy with respect to the conductionband bottom (Ec −EF). First and second ionisation energies ofthe defect are I1 and I2, respectively. Phase boundaries (bluelines) define the stability borders (Ueff, EF) between differentcharge states. (Adapted from Ref. [8]).

equal populations being found when the Fermi level islocated at Ec − (I1 + I2)/2.

Negative-U defects show strong lattice relaxationeffects upon capture and emission of carriers. Forthat reason, charge state transitions may involveconsiderable barriers and large Franck-Condon shifts,often making optical absorption and luminescencedata hardly comparable to measurements carried outat equilibrium conditions (e.g. Hall effect). Themetastable |N〉 state of negative-U defects has a shortlifetime, being formed only when sample conditionsare away from equilibrium. Although reactions like|N −1〉 → |N〉 + h+ or |N + 1〉 → |N〉 + e− can inprinciple be stimulated with the application of externalperturbations, e.g. injection of current or light pulsing,the population of |N〉 critically depends on a delicatebalance between the time of the measurement, theforward and backward reaction (decay) rates, the latterbeing thermodynamically favoured.

Another difficulty for detection of negative-Udefects is related to the fact that the stability of |N−1〉and |N+1〉 states is usually connected to the formationof closed-shell diamagnetic states, and therefore theyare undetectable by EPR [1]. These are amongthe many features which make the characterisationof negative-U defects a challenge, not only in thelaboratory, but also from the perspective of theory andmodelling.

The decade of the 1980’s was particularly prolificregarding the design and conception of experimentscapable of probing the properties of negative-U defects.Watkins and co-workers showed that in a DLTSexperiment, after applying a filling pulse to a Sidiode irradiated with electrons, carriers trapped atsingle vacancies (VSi) and boron interstitials (Bi) in Si,

were subsequently emitted in pairs under reverse bias[6, 9]. This is a particular signature of the negative-U property, which follows from the relatively fast rateof the second emission compared to the slow rate ofthe first one. To large extent, this is determined bythe condition I1 > I2. Hence, a conventional DLTSexperiment can only measure a single transient, with adecay characteristic of the first (slower) emission. Forthe case of Bi, the measured transient correspondedto electron emission from the negatively charged B−i(the deeper |N+1〉 state) [9]. Direct detection of theshallower emission B0

i → B+i + e− (from the |N〉 state)

was not possible. The series of electrical trap-fillingpulses required for DLTS measurements, quickly filledall traps into the |N+1〉 state, no matter how short theapplied pulses were. This was later solved by replacingthe biased injection of electrons by optical injectioninto boron-doped diodes with a mesa structure [10, 11].

A prominent example of a multi-stable defectwhich shows a negative-U ordering of electronic trans-itions is the M-centre in InP. This defect is observedin electron irradiated (1 MeV) undoped, nominally n-type InP, and was found in at least two main forms (Aand B), depending on the bias conditions of the sample[12, 13, 14]. A clear indication that the A-form shows anegative-U ordering of transition levels was found fromthermally stimulated capacitance (TSCAP) measure-ments. While most transitions indicated an ordinaryheating-induced loss of a single charge accumulated ina trap, two of them (labelled A2 and A3), occurredsimultaneously, thus pointing to a very specific featureof negative-U defects: a two-electron emission event.Despite the detailed experimental data already repor-ted for the M-centre, the only atomistic model avail-able in the literature is phenomenological. It consists ofan indium-vacancy-phosphorous-antisite (VInPIn) andphosphorous-vacancy-phosphorous-antisite-pair (VP–2PIn), which were assigned to A and B forms, respect-ively [15].

Another defect showcasing the negative-U effectwas firstly reported by Henry and Lang in order toexplain the observation of persistent photoconductivityin chalcogen-doped AlxGa1−xAs alloys [16, 17]. Therewas no doubt that the defect involved a shallow donorspecies (D). However, because it also had a deepstate at ∼Ec − 0.1 eV, postulated at the time asdue to complexing with an undetermined constituent(X), the defect was labelled ‘DX’ centre. Based onfirst-principles pseudopotential calculations, Chadi andChang came up with a model of a deep DX acceptorstate localised on a broken SAs-Ga (or Al) bond [18].Accordingly, when the Ga (Al) atom connected toneutral SAs (referred to as d0) moves away along the〈111〉 crystallographic axis towards the interstitial site,the total energy increases as expected for a stable


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structure, but the donor state becomes progressivelydeeper. When the level has dropped by more thanthe local repulsive correlation, the total energy is nowEcap ∼ 0.4 eV above the minimum energy of d0. Atthis point a free electron (donated by another SAsdopant in the sample) is captured. The subsequentatomistic relaxation stabilises the negative state byEe = −0.5 eV, withDX− landing at Ecap−Ee = 0.1 eVbelow the d0 state. The authors underlined that thedisproportionation reaction 2d0 → d+ + DX− wasexothermal and therefore indicative of a negative-Udefect. This provided a natural explanation for thelack of EPR involving the DX− state.

A refinement of the model was subsequentlyproposed by Dobaczewski and Kaczor [19], who carriedout a detailed analysis of the photoionisation of DX inTe-doped AlxGa1−xAs. In their work, the intermediatemetastable state could better describe the ionisationkinetics if assumed to be a localised DX0 defect (asopposed to the neutral shallow donor proposed byChadi and Chang [18]). It is however consensualthat the observed persistent photoconductivity can beaccounted for as resulting from the optical ionisationof an equilibrated population of DX− (in the |N+1〉state) into metastable DX0 + e− (|N〉 state), whichquickly converts into d+ + 2e− (|N −1〉 state). Thelatter persists due to a large capture barrier hinderingthe recovery of DX−.

Many other reports of negative-U defects followedthe above examples, including defects with huge impacton the electronic properties of semiconductors anddevices. Prominent examples are the early membersof the thermal double donor family of defects in Si andGe [20, 21], interstitial hydrogen which can be involvedin a multitude of solid-state reactions with defects anddopants in several semiconductors [22, 23], the carbonvacancy in 4H- and 6H-SiC which decisively limitsthe life-time of minority carriers in n-type material[24, 25], or boron-oxygen complexes involved in thelight induced degradation of solar Si [26].

The characterisation of negative-U defects in-volves the experimental monitoring of a two-carrieremission/capture reaction |N + 1〉 |N〉 + e− |N−1〉 + 2e−. This is a multi-step process involvingboth electronic transitions and geometric transforma-tions. The kinetics is usually limited by a slow step,effectively ‘masking’ any subsequent fast steps. Un-der these circumstances, special tricks have been in-troduced in order to access the individual steps ex-perimentally. Examples are the combination of junc-tion spectroscopy with light excitation, or the judiciouscontrol of the amount of free carriers available for cap-ture by using very short and limited injection pulses.From the above, it is also clear that theory, in partic-ular first-principles modelling of the electronic struc-

ture, has played a huge role in unveiling the workingsof negative-U defects and providing guidance for ex-periments. It is in this spirit of collaboration that weintend to introduce the reader to the topic of negative-U defects in semiconductors. Besides presenting a sur-vey regarding what has been achieved so far, we ded-icated much of the content to the concepts and tech-niques (both experimental and theoretical) involved inthe characterisation of this class of defects. We end upwith the identification of several problems, from minorloose ends to completely obscure issues, in the hope ofstirring up those who may feel challenged by the topic.

3. Negative-U defects in semiconductors

3.1. Electronic correlation

Electronic correlation can be thought of as a measureof how much entangled two or more electrons are, oralternatively, how hard electron motion is within thefield of other electrons. It is instructive to look atthe concept from the perspective of the Hartree-Fock(HF) method [27]. Accordingly, the wave functionis represented by a Slater determinant of M spin-orbitals φi(x) = ψi(r)σ(si) with space- and spin-dependence ψi(r) and si = ↑, ↓, respectively, wherei = 1, . . . ,M is a state-index. The manifold x is acomposite of space and spin coordinates. The totalenergy has contributions from single-electron and two-electron components [28],

EHF =∑i

Hi +∑i>j

(Jij −Kij) , (4)

where core integrals,

Hi =

ˆdr3 ψ∗i (r)

(−1

2∇2i −

∑α

Zα|r−Rα|

)ψi(r) (5)

describe a set of independent electrons orbiting inthe field of nuclei located at coordinates Rα. Here,the Born-Oppenheimer approximation holds, so thatRα are fixed (the nuclear kinetic energy vanishes,whereas the nuclear-nuclear repulsion terms add up toan obvious constant which can be included in the resultafter computing the electronic energy). Interactionsbetween electrons, namely their mutual repulsion andexchange contributions are given by the Coulomb andexchange integrals,

Jij =

¨dr3dr′3

|ψi(r)|2 |ψj(r′)|2

|r− r′|(6)

and

Kij =

¨dr3dr′3 ψ∗i (r)ψ∗j (r′)

〈σ(si)|σ(sj)〉|r− r′|

ψi(r′)ψj(r), (7)

respectively, and they obey to the relation Jij ≥Kij ≥ 0. Unlike Jij elements, exchange terms involve


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the interaction between states with parallel spin only.The scalar product 〈σ(si)|σ(sj)〉 = 0 when si 6= sj ,i.e. when the interacting spin-orbitals have oppositespin. This follows from the anti-symmetry of the wavefunction with respect to the exchange of any electroniccoordinate.

Equations 6 and 7 translate that each electroneffectively interacts with an average charge distributiondue to other electrons. This mean-field approximationignores dynamical correlation, i.e., that electroncoordinates are inexorably interdependent, so thatthey instantaneously avoid one another to reduce theirmutual repulsion as much as possible. Nondynamicalcorrelation is also disregarded since the wave functionis approximated by a single Slater determinant. Botheffects were referred to by Löwdin [29] as totalcorrelation energy, Ecorr = E − EHF , where E isthe exact non-relativistic energy of the Schrödingerequation. Ecorr is a very difficult quantity to obtain– a prohibitively large number (millions) of Slaterdeterminants are needed in order to obtain a wavefunction approaching the exact solution, thus makingthe problem intractable, even for today’s largestsupercomputers [30].

If we are solely interested in the ground state,significant progress can be made with the use of densityfunctional theory (DFT) [31, 32, 33], which replaces theHF total energy (a functional of the wave function) bya functional of the electron density ρ [34],

E[ρ(r)] = T0[ρ(r)] +

ˆdr3 vext(r)ρ(r) +

+1

2

¨dr3dr′3

ρ(r)ρ(r′)

|r− r′|+ Exc[ρ(r)]. (8)

Analogous to the core integrals Hi of Eq. 4, theenergy functional accounts for the kinetic energyof an arbitrary number of non-interacting electrons(T0), subject to an external potential (vext), due tonuclear and other non-electronic fields. The electrondensity is obtained from summation over occupiedorbitals, ρ(r) =

∑i=occ |ψi(r)|2 , which satisfy a set

of partial differential equations analogous to single-particle Schrödinger equations. These are solved self-consistently in order to obtain the ground state density(which depends solely on the external potential)[34]. The electron-electron Coulomb repulsion termin Eq. 8) is analogous to the contribution of Jijintegrals in HF, although it now includes an unphysicalself-interaction energy (electrons are repelled bythemselves). In a similar fashion to the exchangeintegrals Kij , the exchange and correlation functionalExc describes the electronic exchange interactions,but also incorporates all remaining effects, includingcorrelation (absent in HF) and the neutralisationof the above-mentioned spurious self-interactions.

Unfortunately, the mathematical form of Exc is notknown (except for an homogeneous electron gas [35,32, 36]). However, several approximations have beenproposed, varying in physical detail and accuracy,and of course in computational load (see for instanceRef. [33] and references therein).

A very simple and intuitive method of treating cor-relation is that proposed by Hubbard for narrow elec-tronic bands [2]. It nicely describes electronic motionwithin systems whose electrons are strongly localisedon atomic orbitals like Mott insulators. Within thesecond-quantisation formalism, the Hubbard Hamilto-nian is given by [37],

HHubb = t∑〈ij〉,σ

c†i,σ ci,σ + U∑i

ni,↑ni,↓ (9)

where 〈ij〉 means that the summation runs overnearest-neighbour atomic sites i and j, while c†i,σ, ci,σand ni,σ stand for creation, annihilation and numberoperators for electrons of spin σ on site i. For stronglylocalised states, electronic motion proceeds via hoppingat a rate proportional to the jump integral t, whichrelates to the band width. This is described by the firstterm of Eq. 9 and involves electron transfer betweenneighbouring sites 〈ij〉. The second term accounts forthe fact that each site is capable of accommodating upto two electrons of opposite spin. However, when asite is fully occupied (ni,↑ = ni,↓ = 1), both electronsinteract with an energy U . Hence, according to theHubbard model, correlation is the energy raise duringan electron transfer event involving two neighbouringsites with identical occupancy (one electron each).

Let us look at the above concept using a pairof oxygen atoms as an example. We know that theenergy of a generic atomic system is a piecewise linearfunction of the electron occupancy (see for instanceRefs. [38]). This is illustrated by the Frost diagram ofFigure 2 (red line), which shows the change in the totalenergy for the sequential reduction of atomic oxygenin contact with an electron reservoir (read the diagramfrom right to left along the horizontal axis). Oxygenis a highly electronegative species with positive firstelectron affinity (A1 = 1.46 eV), and because of that,O− is more stable than the neutral atom. However,capture of an additional electron is not favourable dueto accumulated Coulomb repulsion. This is exhibitedby a negative second affinity, A2 = −7.71 eV. Fromthe diagram, it is also clear that any (even fractional)charge transfer 0 < δ ≤ 1 between a pair of O− anions,2O− → O−1−δ + O−1+δ, raises the total energy bya correlation energy ∆E(N+δ) = (A1 − A2)δ. For thetransfer of a whole electron (δ = 1), we obtain a generalexpression for the total correlation of a reference statewith N electrons.

U (N) = ∆(N+δ) = E(N+1) + E(N−1) − 2E(N), (10)


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Figure 2. Frost diagram showing the first and secondelectron affinities of atomic oxygen (red) [39], along withdisproportionation energetics of hydrogen peroxide (H2O2) inaqueous solution (blue) [40]. Numbers edging solid anddashed line segments represent first- and second-order variations,respectively, of the free-energy per oxygen atom upon oxidation.

where the energies are indexed to the total numberof electrons on each state. For the particular case ofthe O− atomic state, the correlation energy is U =1.46 + 7.71 = 9.17 eV.

Strictly speaking, correlation is exclusively anelectronic property, and that is unambiguously definedfor a single atom by Eq. 10. However, for amore complex moiety (e.g. a molecule, atom insolution, defect in a solid, etc.), the atomistic geometry,vibrational modes and electronic structure of ionised|N + 1〉 and |N − 1〉 states may differ substantiallyfrom the those of the reference |N〉-electron state.It is convenient to recall at this point the conceptof effective correlation energy, which besides theelectronic contribution, includes a relaxation energy,

U(N)eff = U (N) −

(∆U

(N+1)rel + ∆U

(N−1)rel

)= U (N) −∆U

(N)rel . (11)

In the above, U (N) is the electronic correlation asdefined at the geometry of the N -electron referencestate. On the other hand, ∆U

(N+1)rel and ∆U

(N−1)rel are

positive relaxation energies, respectively affecting the|N+1〉 and |N−1〉 states just after transitioning from theN-electron state. These quantities reflect ionisation-induced atomic reconfigurations, entropy changes, etc.,and they are analogous to the Franck-Condon energydissipated by a molecule or defect following absorptionor luminescent transitions [41].

3.2. Disproportionation

An electron transfer event involving two identicalmoieties with N electrons can be broken into two steps:(i) a sudden electronic excitation where an electron-hole pair is created – this raises the energy by U (N),and (ii) a subsequent electronic/atomistic relaxationthat lowers the energy by ∆U

(N)rel . According to Eqs. 10

and 11, for a relaxation as large as ∆U(N)rel > U (N),

we have 2E(N) > E(N+1) + E(N−1). In that case, aclose pair of N -electron moieties becomes metastableand disproportionates into N − 1 and N + 1 electronspecies. Disproportionation is a well-known effect inelectrochemistry, either taking place spontaneously orvia thermal activation, eventually with the help of acatalyst [42]. Although disproportionation necessarilyinvolves electron transfer, it is essentially driven byatomic rearrangement. An eloquent example is givenby the reaction

2Cu+ (aq)→ Cu (s) + Cu++ (aq) (12)

where univalent copper (in solution) precipitatesinto solid Cu and cupric ions. Being exothermic,Reaction 12 gives 2E(Cu+) > E(Cu0) + E(Cu++),with E(Cuq) being the free energy per Cu species inthe oxidation state q in its respective phase. In thisreaction, the relaxation energy ∆Urel essentially resultsfrom the formation of Cu bonds in the copper metal.

Another textbook example of a disproportionationreaction is the decomposition of hydrogen peroxideaccelerated at a metallic surface [43],

2H2O−2 (aq)cat−→ 2H2O= (l) + O2 (g). (13)

The superscripts in the componds above stand forthe oxidation state of oxygen. According to thereaction, for every O−-pair in H2O2, one oxygen atomis reduced (to form water), while another becomesoxidised (leading to formation of molecular oxygen).The Frost diagram for Reaction 13 is represented inFigure 2 (blue line). Using Eq. 10, we find that thecorrelation energy per oxygen atom in H2O2 is

Ueff = [E(2H2O) + E(O2)− 2E(H2O2)] /2 (14)= − 1.06 eV (15)

The factor of 1/2 arises due to the fact thateach compound has two oxygen atoms. Thenegative effective correlation energy tells us that O−ions in hydrogen peroxide are metastable againstdisproportionation into O= an O0 in water andmolecular oxygen, respectively. In Figure 2, Ueff isthe free energy difference per oxygen atom, betweenthe H2O2 (aq) state and a mix of H2O (l) and O2 (g)represented by the dashed line. Like in the copperion redox Reaction 12, the effective correlation of O−


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ions in H2O2 is negative due to atomic reconfiguration,more specifically due to proton transfer and formationof oxygen double bonds after electron transfer.

Many other examples are found in the literature,including the solid state disproportionation of tin(II)oxide (SnO) into tin(VI) and tin dioxide (SnO2), againinvolving dramatic structural transformations [44], orthe loss of bond length translational order in severalperovskites due to random disproportionation of themetallic ions in the lattice [45].

3.3. Electronic transition levels of defects

The term electronic transition level of a defect hasbeen given different meanings depending on thecontext. It has been interpreted differently, dependingwhether one refers to single-particle or many-bodycalculations, if it embodies electron-phonon couplingor not, or perhaps if it relates to measurements basedon thermodynamic or kinetic quantities. In the presentcase, an electronic transition level of a defect (or simplydefect level), is denoted as [46],

E(N+1 /N) = E(N+1) − E(N), (16)

and corresponds to the Fermi energy of a material, EF,above which the ground state of N+1 electrons boundto a defect is more stable than any N -electron state(plus one electron at the Fermi reservoir). Analogously,one could say that when EF drops below a E(N+1 /N)level, the (N + 1)-electron state becomes unstableagainst the N -electron state (plus one electron at theFermi reservoir). When the Fermi level coincides withthe defect level, N - and (N + 1)-state populationsare identical. Under such conditions, the followingisothermic conversion between D(N) and D(N+1),

D(N) + e− D(N+1), (17)

shows identical forward and backward rates and occursvia exchange of electrons (e−) with a reservoir withFermi energy EF = E(N+1 /N).

Let us assume that we have a concentration [D]of non-interacting and identical defects in a sampleat temperature T . We also postulate that for theallowed range of EF values, the defects may occur inup to two charge states, with respective concentrations[D(N)] and [D(N+1)], thus possessing a single electronictransition level E(N+1 /N). Under equilibrium, thefraction of defects with N bound electrons is given by[46, 47, 48],

f (N) = Z−1g(N) exp(−E(N)

f /kBT), (18)

where kB is the Boltzmann constant, Z the partitionfunction,

Z =∑m

g(m) exp(−E(m)

f /kBT), (19)

Figure 3. (a) Population fraction of |N〉 and |N ± 1〉 defectstates as a function of the effective correlation U

(N)eff = E(N+

1 /N) − E(N /N−1). The Fermi level is located at half-waybetween E(N+1 /N) and E(N /N−1) transition levels. Otherassumptions are a band gap width of Eg = 1 eV, T = 300 K,and g(N−1) = g(N) = g(N+1) = 1. (b) Transition level diagramfor positive- and negative-U defects (left and right, respectively).Transition levels are shown as horizontal thick lines. Labels inthe shaded areas indicate the most abundant (stable) defect stateunder equilibrium conditions for the different values of EF .

and g(m) is a degeneracy factor [47]. In Eqs. 18 and19, E(m)

f is the formation energy of a stable m-electronstate,

E(m)f = E(m) −mEF (20)

with the second term on the right of Eq. 20 representingthe chemical potential of electrons in a reservoir withFermi energy EF. For our two-state defect this resultsin the well-known distribution function,

f (N+1) =

[1 +

(g(N)

g(N+1)

)exp

(E(N+1 /N)− EF

kBT

)]−1.(21)

Hence, if EF is lowered below E(N +1 /N), thebackward rate of Reaction 17 exceeds the forward rate,and most defects bind N electrons. Conversely, if EFis raised above E(N+1 /N), occupation of the nextavailable bound state becomes energetically favourableand the reaction proceeds to the right. Defined inthis way, an electronic level depends on ground stateenergies E(N) and E(N+1) only. It is a thermodynamicquantity, analogous to a critical chemical potential forwhich a phase transition takes place.

Next we consider the case of a defect with twotransition levels in the band gap. These define theborders between three charge states, say D(N−1), D(N)

and D(N+1), in phase space. Let us inspect thepopulations of the three charge states by varying thelevel positions, but keeping the Fermi level locked at


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midway between E(N /N−1) and E(N+1 /N), i.e.,EF = [E(N+1) − E(N−1)]/2. Using Eq. 18, we readilyarrive at,

f (N) =

[1 +

g(N+1) + g(N−1)

g(N)exp

(−U

(N)effkBT

)]−1(22)

f (N±1) =

[1 +

g(N∓1)

g(N±1)+

g(N)

g(N±1)exp

(U

(N)eff

2kBT

)]−1(23)

with

U(N)eff = E(N+1 /N)− E(N /N−1). (24)

The above fractions are plotted in Figure 3(a),where we assume that g(N−1) = g(N) = g(N+1) = 1.Under these conditions f (N+1) = f (N−1), thus beingrepresented by a common function f (N±1). The abovedegeneracy factors do not influence the location of thetransition levels when extrapolated to T → 0. In thegraph of Figure 3(a) we also assume that the band gapwidth is Eg = 1 eV and T = 300 K.

For a positive-U defect (U (N)eff > 0), we have

f (N) ≈ 1 (see right-hand side of Figure 3(a)) and thepopulation of the other two states is negligible. On theother hand, for a negative-U defect (U (N)

eff < 0), D(N)

is metastable as seen by the negligible probability offinding this state (left-hand side of Figure 3(a)). Underthese conditions, the reaction

2D(N) D(N+1) + D(N−1), (25)

becomes an exothermic disproportionation processwith an energy balance,

∆Er = E(N+1 /N)− E(N /N−1) = U(N)eff . (26)

When the Fermi level is in the middle of the twolevels with negative-U ordering, D(N−1) and D(N+1)

show identical populations f (N±1) = 1/2, implying theexistence of a E(N+1 /N−1) transition level at

E(N+1 /N−1) = [E(N+1 /N) + E(N /N−1)]/2, (27)

which is represented by the thick blue line in theright-hand side of Figure 3(b). The E(N /N−1) andE(N+1 /N) levels involve the metastable D(N) state,so that they cannot correspond to thermodynamictransition levels. Their location is indicated by thearrows in Figure 3(b). Instead, they determinethe position of the thermodynamic E(N+1 / N−1)transition, which involves the exchange of two electronswith the Fermi reservoir. Finally, for the peculiarcase of U (N)

eff = 0, Figure 3(a) indicates that f (N) =f (N±1) = 1/3 so that a triple-point with all statesequally populated is attained. Under these conditions,all states in Reaction 25 are equilibrated underisothermic conditions (∆Er = 0).

3.4. Formation energy diagrams

In the dilute limit, where defect-defect interactions arenegligible, the equilibrium concentration of a defect ona charge state q depends on its formation energy E(q)

fas,

[D(q)] = [D(q)0 ] exp(−E(q)

f /kBT ), (28)

with [D(q)0 ] being the density of degenerate configura-

tions (lattice sites and orientations per unit volume)that the defect has in the sample. In the above, wenow use the charge state q to label the electronic stateof the defect (instead of the number of electrons orholes). The formation energy in Eq. 28 expresses thecost to create an isolated defect by trading electronsand atomic species with electronic and atomic reser-voirs with chemical potentials EF and µi, respectively(see Ref. [49] and references therein),

E(q)f (µi, EF) = E(q) −

∑i

niµi + qEF, (29)

were E(q) is the energy per defect in charge stateq, surrounded by a sufficiently large volume of hostmaterial, enclosing ni atoms of species i (dilute limit).We note that in Eq. 28 we did not consider thecontribution of entropy to the formation energy. Todo so, E

(q)f would have to be replaced by a free

energy of formation, and E(q) in Eq. 29 would becomeH(q) − TS(q), with H(q) and S(q) being the enthalpyand entropy per defect in charge state q, the lateraccounting for vibrational, electronic and magneticentropy terms. While the formation entropy canbe of significant importance, in particular at hightemperatures [50], for defects with deep states inthe gap, electronic excitations depend exponentiallyon large activation energies, so that electronic andmagnetic entropy can often be neglected. More carehas to be taken regarding the vibrational entropy,particularly for negative-U defects where distinctstructures with rather different vibrational spectra canoccur. Although being usually small when comparedto the electronic transitions of deep defects, vibrationalentropy can be easily incorporated in Eq. 29 (see forinstance Estreicher et al. [51]).

Defect formation energies are in principle positivequantities, otherwise the host material would becomeunstable against spontaneous defect creation. Thegraphical representation of the formation energy of adefect as a function of the Fermi level (for a fixed set ofatomic chemical potentials) is a common procedure indefect physics and chemistry. It allows us to comparethe relative binding energy of electrons to differentdefects (with variable charge and stoichiometry) withrespect to a common reference – the Fermi level.

On the left-hand side of Figure 4 we depicta formation energy diagram of the carbon vacancy


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(VC) located on the pseudo-cubic site of 4H-SiC, forEF values in the upper part of the band gap (n-type material). The diagram was constructed solelybased on experimental observations. The origin of thehorizontal axis (right limit of the axis) corresponds tothe conduction band bottom. As we move to the left,EF lowers within the band gap.

The VC defect at the pseudo-cubic site is adouble acceptor that shows negative-U ordering oflevels. These were measured by conventional DLTSand Laplace-DLTS at Ec − E(= /−) = 0.64 eV andEc − E(−/0) = 0.41 eV [24, 53]. From Eq. 27and taking the singly negative state as reference, theacceptor states show an effective negative correlationenergy of Ueff = −0.23 eV, and that implies theexistence of a thermodynamic transition level at,

E(= /0) = [E(= /−) +E(−/0)]/2 = Ec−0.53 eV, (30)

while E(= /−) and E(−/0) are metastable. The E(=/0) level establishes that under equilibrium conditions,only V 0

C or V =C states can be found in n-type 4H-SiC.

This is highlighted in Figure 4 as solid and dashed linesfor the representation of the formation energy of stableand metastable states, respectively. That explainswhy observation of the V −C paramagnetic state requiresoptical excitation [54]. All three relevant transitionsare indicated in the horizontal axis of Figure 4 byvertical lines at the crossing points of the formationenergy segments. The slope of each segment is qEF,clearly distinguishing the charge state of the respectiveV qC defect. The charge neutralisation condition holdsvia trading of (2 − q)e− electrons with the Fermireservoir. The last piece of information needed toconstruct the formation energy diagram of Figure 4is the formation energy of the neutral species (V 0

C).This was measured by means of high temperatureannealing/quenching experiments as E(0)

f = 4.8 eV[55].

3.5. Configuration coordinate diagrams

While formation energy diagrams capture defect ther-modynamics, they are not so useful when it comesto representing the kinetics of transitions which areactually measured by many techniques, including op-tical spectroscopy and DLTS. In essence, a typicalexperiment involves monitoring the response of asample subjected to the application/removal of an ex-ternal perturbation (e.g. applied field or temperat-ure), leading to a displacement/recovery of the equilib-rium conditions. The measured excitation/relaxationrates of defects in semiconductors may involve absorp-tion/emission of light, the exchange of energy and mo-mentum with phonons and electrons within the sample,or a combination of all these processes. Although be-ing applicable to any type of electronic transitions,

we introduce the concept of a configuration coordin-ate (CC) diagram to illustrate the vibronic nature ofnon-radiative capture and thermal emission of carri-ers, which due to the conservation of energy, involve atrade of phonons with the host lattice [16, 56]. Thisis the most common type of transitions for negative-Udefects, where strong electron-phonon coupling effectsare normally found.

The process of non-radiative phonon assistedtransitions is depicted in the CC diagram of Figure 5,where we assume that the whole system has a singleeffective vibrational degree of freedom. The |N〉ground state finds its minimum at the generalisedatomic coordinate Q(N). The upper parabola, whichis displaced up in energy by the band gap width (Eg),represents an excited state comprising a |N〉 defect plusan uncorrelated electron-hole pair (eventually createdafter illumination of the sample with above-bandgaplight). In the middle of the CC diagram we represent a|N − 1〉 state, which due to electron-phonon coupling,has a minimum-energy coordinate shifted to Q(N−1).To avoid unnecessary complications, we assume thatthe curvature of both |N〉 and |N − 1〉 parabolacorresponds to the same vibrational frequency withquanta ~ω.

Thermal activated conversion of |N〉 into |N − 1〉can be achieved in two ways: either (i) upon capture ofa free-hole followed by multi-phonon emission, or (ii)via emission of a bound electron into the conductionband assisted by multi-phonon capture [57]. In the firstcase the hole capture coefficient is given by

Cp = σp〈vv〉thp, (31)

where σp is the apparent capture cross-section for holestraveling in the valence band top with thermal-averagevelocity 〈vv〉th and density p. In the second case, theemission rate is

en = σn〈vc〉thNcg(N−1)

g(N)exp

(Ec − E(N /N−1)

kBT

), (32)

with Nc being the density of available states in theconduction band and g(N−1)/g(N) the degeneracy ratioof the final to the initial state. The quantities σn and〈vc〉th are electron-analogues of σp and 〈vv〉th.

It is important to note that the above transitionsoccur close to the crossing points of the energy curves.Henry and Lang [16] have shown that the uncertaintyprinciple implies that the adiabatic approximation(upon which the wave function is instantly consistentwith the potential of the oscillating nuclei), breaksdown close to these points, so that transitions mayin fact occur for Q values in a range where |E(N) −E(N−1)| . 60 meV. This feature is incorporated in thetemperature dependence of the effective capture cross-section,

σ = σ∞ exp(−Eσ/kBT ), (33)


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Figure 4. Formation energy diagram (left) along with a configuration coordinate diagram (right) for the carbon vacancy at thepseudo-cubic site in n-type 4H-SiC. Structures B and D are represented as four Si atoms (white circles) forming an approximatetetrahedron viewed along the main crystallographic axis. The vacant site lies below the Si atom at the centre. Atoms connected bysegments are separated by a shorter distance in comparison to equivalent atoms in bulk [52].

Figure 5. Configuration coordinate diagram of a defect witha transition level responsible for an electron trap located atEc − E(N /N−1) and a hole trap at E(N /N−1) − Ev .Each parabola represents a vibronic state that depends on ageneralised coordinate of atoms. Closely spaced horizontalsegments represent effective phonon quanta of energy. At thebottom, we show the |N〉 state, which can emit an electronand become |N − 1〉 + e−, where e− is a free electron at theconduction band bottom. At the top, we show the |N〉 state plusan uncorrelated electron-hole pair. Cn/p are capture coefficientsfor electrons/holes, whereas en are emission rates for electrons.Electron/hole capture barriers are Eσn/p. (Reprinted fromRef. [57], with the permission of AIP Publishing).

with σ∞ being the geometrical (T -independent)capture cross section. Capture rates are thereforethermally activated (see Section 4.2.2 for furtherdetails), and according to Figure 5, the capture barrierfor electrons or holes is

Eσn/p =(Etn/p − EFC)2

EFC, (34)

where the depth of the trap for electrons is

Etn = Ec − E(N /N−1) = Eem,n + EFC, (35)

or for holes,

Etp = E(N /N−1)− Ev = Eem,p + EFC. (36)

In the above, trap energies are divided into twocomponents: a vertical emission and a Franck-Condon relaxation, Eem,n/p and EFC, respectively(see Figure 5). Within a single-mode approximation,the latter relates to the vibrational frequency by theHuang-Rhys factor as EFC = SHR~ω, which forstrongly coupled transitions discloses the dissipationof many phonons (SHR 1) [58].

We can now use our example of the carbonvacancy (double acceptor) in 4H-SiC, and combineits CC and formation energy diagrams to obtain aconsistent and insightful picture of the measurements.The early DLTS experiments by Hemmingsson et al.[24] have found that the conspicuous Z1/2 peakcorresponds to the superposition of two nearly identicalZ1 and Z2 negative-U defects, differing only on the sub-lattice location. These measurements were recentlyrefined by Koizumi et al. [59] in 6H-SiC and byCapan et al. [53] in 4H-SiC, using Laplace-DLTS. Thenegative-U ordering of levels implies that during fillingof the defect, the binding energy of the second electronis higher than that of the first one. On the other hand,


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for the reverse process, the thermal emission of the firstelectron immediately induces a second emission.

The above is better perceived with help ofFigure 4. Accordingly, before pulsing in the majoritycarriers (n-type material), the diode is under reversebias and the Fermi level is lower than the acceptorlevels. The formation energy diagram of Figure 4indicates that the stable state is the neutral one(V 0

C). After applying a filling pulse (zero or forwardbias), the Fermi level edges the conduction bandminimum and the CC diagram on the right-handside of Figure 4 applies. In these conditions thedouble negative state (V =

C ) is more stable and thekinetics of the reaction V 0

C + 2e− → V =C is essentially

limited by the second capture event [24, 53] – thenegative charge accumulated on the defect after thefirst capture makes it less effective for a secondcapture. Interestingly, what makes VC a negative-Udefect are strong pseudo-Jahn-Teller relaxations thatmainly affect the close shell states (neutral and doublenegative) [52], driving V −C to become metastable (seedashed black lines in Figure 4). The relevant structures(labelled B and D), which were found by first-principlescalculations [52], are indicated in the CC diagram,where atoms connected by segments show considerablyshorter distances than those in bulk SiC.

When reverse bias is restored, V =C becomes

unstable, and if the temperature is high enough, twoelectrons are emitted into the conduction band andswept away from the depletion region. We notethat the electron capture barriers for V 0

C and V −C(at the pseudo-cubic site) were found to be as lowas E(0)

σn ∼ 0 eV and E(−)σn ∼ 0.03 eV, respectively

[53]. Considering the relative depth of both acceptorlevels with respect to Ec, electron emission from V =

Chas a higher barrier than that from V −C so thate(=)n < e

(−)n (where e(q)n is the electron emission rate

from V qC). Hence, conventional DLTS is only ableto monitor a transient whose decay reflects the first(slower) emission only. This feature is clearly shownin Figure 4, illustrating a major difficulty regardingthe characterisation of negative-U defects – probing theintermediate metastable state.

In the particular case of VC the metastable stateis the singly negative defect. Access to this statewas achieved by optical excitation [54] and keepingthe sample at low temperatures to avoid carrier re-emission, or providing a small amount of electrons to areverse-biased n-type diode, via short injection [59, 53]or optical pulses [24]. The injection level must be wellbelow saturation limit so that the fraction of defectsin the V =

C state is much smaller than that in the V −Cstate.

Although we can only observe a single emissionpeak by means of junction spectroscopy, the underlying

double emission sequence leads to a variation of thecapacitance twice as large to that observed for atransition involving a single emission. As we will see inthe next Section, this stands as a rather characteristicfeature which is often helpful in the identification ofnegative-U defects.

4. Methods of characterisation

4.1. Probing negative-U defects

For the sake of convenience, below we refer to theeffective correlation energy simply as U ≡ Ueff. It hasalready been mentioned above that under equilibriumconditions, defects with negative-U properties emit orcapture charge carriers by pairs. So, a confirmation ofthe negative-U property of a defect requires evidenceof such paired emission or capture of charge carriers.

The first definitive experimental evidence ofnegative-U properties of a defect in a crystallinesemiconductor, the lattice vacancy in silicon (VSi),was obtained from DLTS measurements [6]. Thistechnique is frequently used for the determinationof the concentration of defects with deep levels.Accordingly, the magnitude of a measured signal(change in sample capacitance due to carrier emissionfrom a defect), is usually directly proportional to theconcentration the defect trap it refers to, ∆C ∼ [D][60]. However, it was found in Ref. [6] that themagnitude of the DLTS signal due to hole emissionfrom V ++

Si was proportional to 2[VSi], so confirmingthat each emission event was in fact a two-hole emissionsequence, so the vacancy in Si is a defect with U < 0.

It should be mentioned that for the conclusionabout ∆C ∼ 2[D] in the recorded DLTS spectra, anindependent method of determination of the vacancyconcentration was used. Reliable independent methodsfor the determination of the concentration of a defectin semiconductor samples for DLTS measurements arerarely possible, so from a conventional analysis ofemission signals in the DLTS spectra, it is usuallynot possible to judge if a carrier emission signal isrelated to a defect with positive or negative effectivecorrelation energy. Simple analysis of temperaturedependencies of carrier emission rates measured withDLTS for a negative-U defect gives only informationabout parameters for the emission (activation energyfor emission and apparent capture cross section) of thefirst, more strongly bound, charge carrier.

4.1.1. Elucidation of the nature of a defect by meansof analysis of its occupancy with charge carriers atequilibrium conditions Solid evidence of negative-Uproperties for a number of defects in semiconductorshas been obtained from studies of their occupancywith charge carriers at equilibrium conditions. The


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general statistics of charge distribution at equilibriumconditions for defects with several trapping levels insemiconductors has been presented in 1957 by Shockleyand Last [46]. It has been shown that the concentrationratio of defects in two charge states differing by oneelectron is

[D(N)]

[D(N−1)]= exp

(−E(N /N−1)− EF

kBT

), (37)

whereN is the number of electrons in the most negativestate and E(N /N− 1) is the energy level of the defect(c.f. Section 3.3).

For a defect with two energy levels, therelationship between them can by represented by

E(N+1 /N) = E(N /N−1) + U. (38)

If U < 0 and |U | kBT , it follows from Eq. 37 thatfor any position of EF, [D(N)] [D(N+1)] + [D(N−1)],i.e. [D(N+1)] + [D(N−1)] ∼= [D], where [D] is the totalconcentration of the defect. Furthermore, the electronoccupancy function of the |N + 1〉 state is

f(N+1)U<0 =

[D(N+1)]

[D]=

=

1 + exp

[2E(N+1 /N−1)− EF

kBT

]−1, (39)

where the quantity E(N+1 /N−1) = [E(N+1 /N) +E(N /N−1)]/2 represents the occupancy level of thedefect with U < 0. When EF > E(N+1 /N−1), thedefect has N + 1 electrons, when EF < E(N+1 /N−1),the defect has N − 1 electrons. It is the differencebetween Equation 39 and the Fermi function, thelater describing the occupancy of single-electron defectlevels, that allows us to determine the nature of a defect(the sign of the effective electron correlation energy)from a study of its occupancy with charge carriers.

The free energy of ionisation with respect toa reference level (for instance the conduction bandbottom), can be presented as

∆E(N+1 /N−1) = Ec − E(N+1 /N−1) (40)= ∆H − T∆S, (41)

where ∆H and ∆S are the changes in enthalpy andentropy due to the (N+1 /N−1) ionisation event, andit is temperature dependent if ∆S(N+1 /N−1) 6= 0.As an illustration, Figure 6 compares dependencies ofcharge occupancy versus Fermi level position in thegap (i) for a defect with U < 0 and Ec − E(N+1 /N−1) = 0.3 eV and (ii) for a defect with U > 0and Ec − E(N+1 /N) = 0.3 eV in n-type silicon,with fully ionised shallow donors providing a densityn = 1× 1015 cm−3 of free electrons. The dependencieshave been calculated assuming that n [D], and

Figure 6. Dependencies of occupancy with electrons (f (N+1))versus the Fermi level position relative to the conduction bandedge for a defect with U < 0 and Ec − E(N+1 /N−1) = 0.3 eVand for a defect with U > 0 and Ec−E(N+1 /N) = 0.3 eV. Thedependencies have been calculated with the use of Eq. 39 (forU < 0) and the Fermi function (for U > 0) upon the assumptionn = 1× 1015 cm−3 [D].

changes in the Fermi level position have been inducedby temperature variations.

The simplest way to probe the occupancy of adefect with charge carriers is to change the temperatureof a semiconductor sample in a certain range andmonitor associated changes either in free carrierconcentration measured by means of Hall effect, or inmagnitude of a signal related to emission or capture ofcharge carriers measured by means of DLTS. Negative-U properties for a number of defects in semiconductorshave been elucidated from analyses of temperaturedependencies of free carrier concentrations, n (p) ∼f(T ). Particularly, evidence of negative-U propertiesof the Si vacancy [61, 62], oxygen-related thermaldouble donors (TDDs) in silicon and germanium [63,20, 21], and complexes consisting of a Si interstitialatom with oxygen dimer (IO2) [64] and interstitialcarbon, oxygen and hydrogen atoms (CiOiH) [65] inSi has been obtained from analyses of temperaturedependencies of electron (hole) concentrations.

The experimental n (p) v.s. T dependenciescan be analysed either by methods based on solvingcharge carrier neutrality equations [66] or by thedifferential method proposed by Hoffmann [67, 68].According to the differential method, the concentrationof a defect and position of its energy level in thegap can be determined from a dependency of X =kBT (dn/dEF) versus EF in semiconductors of n-type[X = kBT (dp/dEF) in semiconductors of p-type]. TheX(EF) dependencies look like spectra with spectral


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Figure 7. Temperature dependencies of free electronconcentrations, n, (left part) and its derivative X =kBT (dn/dEF) (right part) versus the Fermi level with respectto the conduction band edge F = Ec − EF , for an oxygen-richGe crystal subjected to heat-treatments at 623 K for (1) 5 min,(2) 10 min, (3) 25 min, (4) 60 min, (5) 120 min, (6) 240 min,(7) 480 min, (8) 1020 min, and (9) 1860 min. (Reproduced withpermission from Litvinov et al. [21], © 1988, WILEY-VCHVerlag GmbH & Co.).

bands induced by ionisation of defects. The magnitudeof a peak (Xm) is proportional to the concentrationof a defect, while the peak position (Em) correspondsto the position of an energy level in the gap. It hasbeen shown in Refs. [67] and [68] that for defects withU > 0, Xm = 0.25[D] and the half-width of a band,δEF, is about 3.5kBTm, where Tm is the temperaturewhich corresponds to Em. For defects with U < 0 onthe other hand, Xm = [D] and δEF ≈ 1.8kBTm. So,an analysis of the half-width of bands in the X(EF)dependencies can be considered as a quick test on thesign of effective correlation energy.

As an example, Fig. 7 shows n(103/T ) and X(F )dependences with F = Ec − EF, for an oxygen-richGe crystal, which was subjected to heat-treatments ofdifferent durations at 623 K [21]. Such heat-treatmentsare known to result in the introduction of oxygen-related thermal double donors (TDDs), which consistof a family of subsequently formed (at least nine) defectspecies in Ge:O crystals [69]. An analysis of the X(F )spectra in Fig. 7 showed that the half-width of thethree appearing bands is about 1.8kBTm, so giving solidevidence that the first three species of the TDD family

of defects in Ge are centres with negative-U [21].From investigations of the occupancy of negative-

U defects with charge carriers at equilibrium condi-tions, only a value of free energy for the two-electroncharge state change, ∆E(N+1 /N−1), can be determ-ined. Further information about the E(N+1 /N) andE(N /N− 1) values, charges states, structural config-urations and energy barriers between them can be ob-tained from investigations of non-equilibrium processesof carrier emission and capture.

4.1.2. Observations of negative-U defects in metastableconfigurations For the majority of defects with U <0, rather large energy barriers exist between theiratomic configurations. Because of these barriers,under certain conditions, it is possible to ‘freeze’ anegative-U defect in a metastable atomic configurationwith a shallower level, and to obtain informationabout the electronic properties of the defect in thisconfiguration. Such ‘freezing’ or, in other words,deep-shallow excitation can be induced by differentmethods, e.g., by (i) injection of minority carriers incertain temperature ranges either by above-bandgap-energy light pulses or forward bias pulses in n-pdiodes, (ii) a quick change in temperature (quenching),usually from higher to lower T , (iii) cooling down ofa semiconductor sample under external illuminationwith photons having above-bandgap energies, (iv)cooling down an n-p diode with an applied reversebias voltage. The significant changes in n (p) v.s.T dependencies, in DLTS, EPR and various optical(infrared absorption, photoluminescence, etc.) spectrainduced by the ‘freezing’/excitation methods listedabove, usually indicate the presence of metastable(negative-U) defect states in a semiconductor material.From analysis of the appearing signals in the excitedspectra, electronic and structural characteristics of anegative-U defect in the shallow level configuration canbe determined.

In EPR studies, a number of defects, e.g.,positively charged vacancy and interstitial boron insilicon [6, 9, 10], Se-related DX centre in AlSb [70] orcarbon vacancy in 4H-SiC [54], could only be observedin the spectra after external illumination with above-band-gap light. It has even been argued in Ref. [70]that the absence of an EPR signal in the samples cooledin the dark and its appearance after illumination canbe considered as a direct evidence of the negative-Uproperties. It should be noted, however, that sucheffects can be related to a defect with a single deeplevel in the gap and large energy barrier for capture ofa charge carrier.

In Si and Ge crystals containing oxygen-relatedthermal double donors, significant changes in n(T )dependencies and IR absorption spectra have been


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observed after cooling the investigated samples downunder external illumination with energy hν ≥ Eg[63, 20, 71, 21, 72, 73, 69]. In the ‘illuminated’IR absorption spectra of Si crystals, three sets ofabsorption bands, which have not been seen in thespectra recorded after cooling down in the dark, weredetected [20, 71, 72, 73]. From the analysis of positionsof the electronic-transition related absorption linesin the ‘illuminated’ spectra, the exact locations ofenergy levels of three bistable TDD species (negative-U defects) in the shallow donor configuration could bedetermined in both Si and Ge [20, 71, 72, 73, 69].

Changes in DLTS spectra that resulted fromdifferent cooling down conditions of investigateddiodes have been reported for semiconductor samplescontaining both defects with negative-U properties andmetastable defects with U > 0 [74, 75, 76, 73, 64]. Thespectra are usually recorded upon heating the diodesup after their cooling down either with or without areverse bias applied. Cooling down with the appliedreverse bias, when there are no free electrons/holes inthe probed ‘depletion’ region, results in ‘freezing’ of anegative-U defect in a configuration with a shallowerlevel, which is the minimum energy configuration athigh temperatures. So, typically, in the DLTS spectrarecorded after cooling down the diodes with the appliedreverse bias, a signal due to charge emission from theshallower level appears, and a signal due to emissionfrom the deeper level either disappears or decreases.As an example, Figure 8 compares the DLTS spectrafor a Czochralski silicon (Cz-Si) sample containing theIO2 complex (with I standing for a Si self-interstitial).This complex is a negative-U defect with its first andsecond donor levels being at E(0/+) = Ev + 0.12 eVand E(+/ + +) = Ev + 0.36 eV [64, 77]. Thespectra were recorded after cooling down with eitherbias-on or bias-off. Cooling down with the appliedreverse bias resulted in the disappearance of the DLTSsignal due to hole emission from the double positivelycharged state (the minimum energy configuration atlow temperatures) of the IO2 defect and appearanceof the signal due to hole emission from the metastablesingle positively charged state.

Further, back transformations from a metastableto stable configuration can be induced by e.g. anincrease in temperature if majority charge carriers areavailable. From studies of kinetics of these backwardprocesses at different temperatures in semiconductorcrystals with different free carrier concentrations, acomprehensive set of information about the electronicstructure and concentration of defects with negative-Ucan be obtained.

4.1.3. Non-equilibrium occupancy statistics for defectswith U < 0 The non-equilibrium occupancy statistics

Figure 8. DLTS spectra for a boron-doped Cz-Si sample withlow carbon content ([C] ≤ 1015 cm−3) after its irradiation with4 MeV electrons. The dose of irradiation was 4 × 1014 cm−2.The spectra were recorded upon heating the sample from 35 Kto 300 K with the measurement settings shown in the graph.Spectrum 1 was recorded after cooling the sample down withoutreverse bias applied to the Schottky diode, spectrum 2 wasrecorded after cooling down with the reverse bias on.

for defects with U < 0 have been developed in Refs. [78,79, 80]. For an elucidation of the details of thesestatistics let us consider a total concentration Nd of anamphoteric defect (having acceptor and donor levels)with negative-U level ordering, which exchange chargecarriers (electrons) with the conduction band. Figure 9shows a general configuration-coordinate diagram forsuch a defect. In the absence of minority carriers,transitions between the stable acceptor (A−) and donor(D+) states occur through the metastable X0 and D0

states according to the following sequence of reactions:

A− A0 + e− X0 + e− D+ + 2e− (42)

The changes in the density of defect states and in thefree electron concentration can be described by thefollowing set of differential equations,

d[A−]

dt= + cX0 [X0]− eA− [A−],

d[X0]

dt= − cX0 [X0] + eA− [A−]− ωXD[X0] + ωDX[D0],

d[D0]

dt= + cD+ [D+] + eD0 [D0] + ωXD[X0]− ωDX[D0],

d[D+]

dt= − cD+ [D+] + eD0 [D0],

dndt

= − cX0 [X0] + eA− [A−]− cD+ [D+] + eD0 [D0],

where emission and capture rates, eQ and cQ, are


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Figure 9. Configuration coordinate diagram for an amphotericdefect (having an acceptor and a donor levels) with U < 0.

defined as

eQ = cQ′(Nc/n) exp[−∆E(Q/Q′)/kBT ] (43)

and

cQ = σ∞Qn〈vc〉th exp(−EσQ/kBT ), (44)

respectively, with Q and Q′ referring to neighbouringconfigurations with different charge. Structuraltransformations X D without a charge state change,occur at a rate ωQQ′ = ω∞,QQ′ exp(−EQQ′/kBT ),where now Q and Q′ refer to neutral X or D states.Defect concentrations [Qq] of configuration Q in chargestate q, are subject to

∑q[Q

q] = Nd. For thetransformation process, ω∞,QQ′ and EQQ′ are the high-temperature attempt frequency and transformationbarrier, respectively.

The full solution of the above set of the differentialequations is not an easy task. It should be noted,however, that some of the reaction rates are usuallymuch faster than others, so some of the states are inquasi-equilibrium. Furthermore, upon the assumptionthat Nd n ≈ ‘const’, the above set of equationscan be transformed to a set of first order differentialequations, which can be solved analytically. Belowwe will consider in more detail the changes in thedensity of the A− state, which is usually monitoredin DLTS measurements. The changes in the density ofthe A− state at a constant temperature after an initialdeviation from equilibrium, ∆[A−]0, can be expressedby the following equation,

∆[A−](t) = ∆[A−]0 exp(−t/τ), (45)

where τ is the characteristic time of the decay process.The characteristic time depends on state transitionrates as [80]:

τ−1 =ωXDeA− + cX0ωDXfD

ωDX + cX0

, (46)

where

fD =[D0]

[D0] + [D+]=

[1 + exp

(−EF−E(0/+)

kBT

)]−1,(47)

=

[1 +

Nc

nexp

(−∆E(0/+)

kBT

)]−1, (48)

is the occupancy function for the defect in the donorconfiguration. Equation 46 can be expressed in acommon way as

τ−1 = eeff + ceff, (49)

with

eeff =eA−

1 + cX0/ωXD, (50)

and

ceff =cX0ωDXω

−1XDfD

1 + cX0/ωXD. (51)

An analysis of Eqs. 46, 49, 50 and 51 indicates thatseveral terms, with their specific activation energiesand power dependence on the free carrier concentra-tion, can dominate the temperature dependence of τ−1.Particularly influential factors are the Fermi level posi-tion with respect to E(−/0), E(0/+) and E(−/+), andthe cX0/ωXD ratio. The effective emission rate, eeff, isthe dominant term in Eq. 49 when the Fermi level is be-low the E(−/+) = Ec−[∆E(−/0)+∆E(0/+)]/2 occu-pancy level of a negative-U defect. At these conditions,τ−1 = eeff. When cX0 ωXD, τ−1 = eA− , so the tem-perature dependence of τ−1 can be described by a com-monly used equation for single electron emission. Forthe cases of cX0 ωXD, we have τ−1 = ωXDeA−/cX0 ,so the transformation rate (actually the rate of occu-pancy of the A− state) is inversely proportional to thefree electron concentration. Such unusual dependenceof τ−1 versus n has been clearly observed for a complexconsisting of a substitutional boron atom and oxygendimer in silicon [26, 81].

The capture term in Eq. 49 is dominant whenEF > E(−/+). In this case, τ−1 = ceff and up to fourspecific terms with different values of activation energyand power dependence on n can occur, depending onthe cX0/ωXD ratio and the Fermi level position withrespect to E(0/+). For a number of defects, e.g.bistable TDDs in Si and Ge crystals, under certainconditions, τ−1 ∼ n2 dependencies have been observed


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[78, 21, 79, 82]. Such dependencies clearly show a two-fold change in the charge state of a defect upon captureof carriers, so indicating its negative-U properties.

From the analysis of temperature dependenciesof transition rates between configurations of a defectwith U < 0 monitored by measurements of changes ineither free carrier concentration or density of a specificdefect state in semiconductor crystals with differentn (p), it is possible to elucidate the electronic structureof the defect and determine practically all its energydifferences and transformation barriers. This has beendone for a number of defects in Ge and Si crystals[78, 21, 79, 80, 64, 82, 26, 81].

4.2. Modelling negative-U defects

4.2.1. Calculation of electronic transition levels Thecalculation of electronic transition levels is essentially aproblem of finding the energy of an electronic reservoir(Fermi level), for which the exchange of electrons witha defective sample becomes energetically favourable(see Section 3.3). This is a topic which has been widelyrevised in the past (see for instance Ref. [49] and [83]),so we leave here the essential features. We start byrecalling Eq. 29, which determines the energy neededto create a defect in a crystal (defect formation energy)

E(q)f (R, µi, EF) = E(q)(R)−

∑i

niµi + qEF. (52)

Here, E(q)(R) is the energy of a large portion of asample enclosing a single defect, including its electronicwave functions. The defective sample volume is madeof ni atoms of species i with chemical potential µi andcollective atomic coordinate R = Rα (α being anatomic index). It can be easily shown that a transitionenergy level between charges states q and q′ (with qbeing more negative) is located at

E(q/q′) = −E(q)(R)− E(q′)(R′)

q − q′. (53)

Equation 53 accounts for the fact that charge states qand q′ may correspond to radically different atomisticgeometries R and R′.

The large chunk of material with a defectreferred above is normally approximated to a defectivesupercell. The inherent periodic boundary conditionsimply the cell to be neutral, even when we add(remove) electrons to (from) a defect state located inthe gap. Technically, a uniform background counter-charge of density −q/Ω (with Ω being the supercellvolume) is superimposed to the electronic density. Thisimplies that the aperiodic energy E(q) that entersin Equations 52 and 53, is related to the periodicenergy E(q) obtained from the supercell calculationas E(q) = E(q) + E

(q)pcc, where E

(q)pcc is a correction

often referred to as periodic charge correction. Severalschemes have been proposed for the calculation of E(q)

pcc(see for instance Ref. [84] and references therein).

Of course, in a semiconductor, transitions areonly observable if they are located in the range Ev <E(q/q′) < Ec. We have therefore to calculate Ev orEc in order to cast the levels as measurable quantities,i.e., E(q/q′)−Ev or Ec−E(q/q′). One possibility is toassume that Ev,c = εv,c,bulk, which are the highestoccupied (εv,bulk) and lowest unoccupied (εc,bulk) statesof a single-particle Hamiltonian (like the Kohn-Shamequations). Another option is to follow the ∆SCF(delta self-consistent field) method to obtain Ev =

E(0)bulk− E

(+)bulk or Ec = E

(−)bulk− E

(0)bulk from total energies

of bulk supercells. These must be identical in sizeand shape to those used to obtain E(q)(R) values. Inany case, we should be aware that the accuracy ofthe calculated defect levels, including the energies ofthe band gap edges, strongly depends on the qualityof the Hamiltonian. Particularly important is thelevel of detail put in the description of the exchangeand correlation interactions between electrons (see forinstance Ref. [85]).

4.2.2. Calculation of carrier capture cross-sections inmulti-phonon assisted transitions Inelastic scatteringof free carriers by defects essentially consists of thetransfer of energy and momentum of a propagatingelectron or hole, to another carrier bound to a defect(trap-Auger process), its conversion into radiation(luminescence process), into nuclear motion (e.g.capture enhanced defect migration) or heat dissipation(multi-phonon emission process) [58]. Non-radiativecapture of carriers via multi-phonon emission (MPE)is therefore a special case of inelastic scattering –the energy drop of the traveling carrier is fullydissipated into atomic vibrations. This is the mostcommon capture mechanism at deep traps involvingnegative-U defects. Other important processes,like recombination, generation or emission, can bedescribed either as a sequence of consecutive captureevents, or by reversing the operation using detailedbalance.

The relevant quantity to be evaluated is thecarrier capture rate, cn/p, which relates to theapparent capture cross-section of Eq. 33 as cn =σn〈vc〉th n for the case of electrons (with an obviousanalogous expression for holes). While first-principlescalculations of electronic transition levels have beenroutinely reported in the literature, the calculation ofcapture rates is clearly lagging behind. For a historicalaccount regarding the development MPE theory, wedivert the reader to Refs. [58, 86, 87]. In general, thecalculation of the non-radiative transition rate betweenfree and bound states starts with the description of


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a many-body Hamiltonian H and the choice of apractical basis to describe the total initial and finalwave functions involved, Ψi and Ψf, respectively. Theinitial state represents the defect plus uncorrelatedfree carrier, whereas the final state stands for thetrapped state. The static approximation is a ratherconvenient starting point in the context of popularelectronic structure methods like Hartree-Fock anddensity functional theory. Here, the electronic wavefunctions ψs;R0(r) (s being an electronic state indexand r all electronic degrees of freedom) are foundfrom a static Hamiltonian HR0

, constructed for a fixedatomic geometry R0. The nuclear wave functionsare treated separately. Assuming the harmonicapproximation, we write the initial state as Ψim =ψi;R0(r, Qik)χim(Qik) and likewise, the final stateas Ψfm = ψf;R0(r, Qfk)χfn(Qfk). Here, m and nare quantum numbers for the vibrational states χi andχf, respectively, while k indexes are used to identifythe 3N individual vibrational modes Qi,fk, N beingthe number of atoms in the sample.

The transition probability between states Ψim andΨfn relates to the off-diagonal matrix elements of theHamiltonian when perturbed by a change in geometry(away fromR0) induced by the promoting modes Qikof the initial electronic state. Such quantities maybe obtained by first-order expansion of H in a Taylorseries of Qik (linear coupling approximation),

∆Him,fn =∑k

〈ψi,R0 |∂H

∂Qik|ψf,R0

〉〈χim|∆Qik |χfm〉 , (54)

where ∆Qik = Qik − Q0k with Q0k representinga generalised coordinate, equivalent to the collectivenuclear coordinate R0 in the Qik basis. Thetransition (capture) rate is then cast in the form ofFermi’s golden rule,

c =2π

~∑m,n

wm(T )|∆Him,fn|2δ(∆Eim,fn), (55)

which sums up all possible vibronic transitions,weighted by the occupancy fraction wm of thepromoting vibronic states. Under thermal equilibriumwm is simply a normalised Boltzmann distribution.The allowed transitions in Equation 55 are alsorestricted to those which conserve the energy, hence theuse of a Dirac delta δ(∆Eim,fn) = δ(Ei−Ef +m~ωm−n~ωn). In practice, the δ function is replaced by anormalised Gaussian function with full width ∼ kBTto account for thermal broadening.

Up to this point, several bold approximations werealready made. Still, the formulation of ∆Him,fn asits stands in Eq. 54, makes its calculation a rathercumbersome exercise. The problem lies essentially onthe identification of the vibrational modes which leadto non-vanishing 〈χi|∆Qik|χf〉 terms. To give an idea,

using a triply hydrogenated vacancy (VH3) defect insilicon as a benchmark, Barmparis and co-workers [87]used a Monte Carlo sampling scheme, being able tofind convergence for ∆Him,fn when about 12 distinctphonon modes were considered.

Motivated by the fact that a judicious choice of asingle effective phonon mode resulted in a promisingagreement between the observed and calculatedluminescence intensity of radiative transitions (see forinstance Refs. [88] and [89]), several authors evaluatedthe transition matrix elements within the single-phonon approximation. Accordingly, the effectivemode is the one that maximises the coupling to thedistortion during the capture process,

Q2 =∑α

mαλ2|∆Rα|2. (56)

Equation 56 is a linear interpolation betweeninitial and final atomic coordinates ∆Rα = Rf,α−Ri,αwith α running over all N atoms with mass mα. Thedisplacement amplitude along the effective mode isgoverned by a unitless factor 0 ≤ λ ≤ 1. The unitsof Q are amu1/2 Å (amu being the atomic mass unit).From the effective mode we arrive at the correspondingeffective vibrational frequency ωi,f of the initial orfinal states by extracting ωi,f = ∂Ei,f/∂Q from totalenergy calculations. With this in mind, Equation 55simplifies into a factorisation of purely electronic andvibrational terms,

c =2π

~W 2

if Gif(T ), (57)

where the transition matrix element is now Wif =〈ψi|∂H/∂Q|ψf〉 and Gif is the temperature-dependentline shape factor

Gif(T ) =∑m,n

wm(T )|〈χim|χfn〉|2δ(∆Eim,fn). (58)

Equations 57 and 58 translate the Franck-Condonapproximation, which separates an instantaneous elec-tronic transition from the sluggish phonon dissipationprocess that follows. The calculation of Gif boils downto the Frank-Condon integrals 〈χim|χfn〉 for all vibra-tional promoting and accepting states. For that, re-cursive techniques have been proposed [90, 91, 92, 93]and applied on different contexts [94, 88, 95]. Samplingmethods have also been applied [87], although inclusionof a few phonons implied the evaluation of millions of〈χim|χfn〉 configurations that matched the electronicenergy off-set.

Regarding the evaluation of Wif, different ap-proaches have also been used. Alkauskas et al. [86]replaced the many-body Hamiltonian and wave func-tions by single-particle counterparts h and φi,f, whichwere taken from the Kohn-Sham equations of density-functional theory. Thus, Wif becomes

Wif = 〈φi|∂h/∂Q|φf〉, (59)


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which can be solved by first-order perturbation theory[86]. Alternatively, a phenomenological approach waspursued by Krasikov et al. [95] who used Eq. 88 in thepaper of Henry and Lang [16],

|Wif|2 =2π

Ω

(~2

2m∗

)3/2

ε1/2, (60)

where Ω is the crystal volume, m∗ is the effectivemass of the free carrier and ε is the energy separationbetween the initial and final states near the intersectionpoint (see discussion in Section 3.5).

4.2.3. Calculation of transformation barriers Defectreorientation and transformation are important pro-cesses that occur during the experimental characterisa-tion of negative-U defects. Understanding the under-lying physics of such processes is fundamental in orderto explain and ultimately to control defect behaviour.The problem is usually addressed in the spirit of trans-ition state theory [96], and usually involves finding thelowest free energy barrier that separates specific react-ants from products, or in the case of defects, initialfrom final structures.

Although being a rather challenging problem,particularly when we have no clue about themechanisms involved, the search for a saddle-pointalong a potential energy landscape can be investigatedin a number of ways. In general, defect transformationmechanisms are investigated via constrained-relaxationof the atomistic structure, ideally using a highlyaccurate quantum-mechanical method to evaluate theforces. Among the existing algorithms we highlight:

The dimer method [97], which requires knowledgeof a single minimum energy configuration, performsa search for a nearby saddle point. This method isparticularly useful to look for saddle points along pathswith unknown final configurations. The dimer methodworks on two structures (the dimer) separated by asmall distance in configurational space. The directionalong the lowest curvature of the potential energy isobtained by rotation of the dimer, and by finding theforces for each structure. This result is used to movethe dimer uphill, from a stable configuration to a nearsaddle point.

The Lanczos method was originally referred toas ART algorithm by Malek and Mousseau [98] andis a close relative to the dimer method. It can beused to search for a saddle point close to an arbitraryinitial configuration. Unlike the dimer method, theLanczos scheme estimates the Hessian eigenvector thatcorresponds to the lowest eigenvalue by expansion ofthe potential energy surface in a Krylov subspace.After this stage, the method proceeds in the same wayas the dimer.

Figure 10. Atomic structure of stable and metastabletrivacancy defects in silicon. The transformation mechanismis depicted in four stages (a), (b), (c) and (d) by the arrowsindicating the motion of three Si atoms at the core (shown inblack for a better perception). Perfect lattice sites are drawnwith a dashed line. Broken bonds are shown as solid red sticks.(Reprinted with permission from Ref. [101]. © 2012 by theAmerican Physical Society).

The nudged elastic band (NEB) method [99, 100]is a robust algorithm that works when the initial andfinal states of the mechanism are known a priori. Itbasically involves the relaxation of a structure sequence(referred to as intermediate images) along the multi-dimensional path that separates the end-structures.Usually a starting guess for the intermediate images isobtained by linear interpolation of the end-structures.The latter are kept fixed during the relaxationprocess. During the relaxation of each image, equalspacing between neighbouring images is maintained byinclusion of artificial spring forces.

An example of a defect-related mechanism studiedusing the NEB method is depicted in Figure 10,showing the transformation of the bistable trivacancy(V3) complex in Si [101]. This is one of themost important radiation products in heavy particleirradiated Si [102]. Right after irradiation the observeddefect has the structure depicted in Figure 10(a). TheSi broken bonds on this structure are responsible forseveral deep donor and acceptor levels observed byDLTS [102]. However, storage for a few weeks of theirradiated Si samples at room temperature result inthe transformation to the structure of Figure 10(d).This is accompanied by the disappearance of the deeplevels and the introduction of shallow electron traps at75 meV bellow the conduction band bottom.

The energy barrier for the above transformationwas estimated using the NEB method as 1.15 eV


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and 1.14 eV for neutral and negatively charged V3,respectively [101]. These figures match very well theirrespective experimental counterparts of 1.16 eV and1.15 eV [102].

5. Showcase of negative-U defects and theircharacterisation

In this section we provide some examples of negative-U centres in widely used semiconductors. In theselection we try to illustrate a range of behaviour, theconvergence of theory and experiment, and cases whichhave provided particular challenges in understanding.Among this selection are examples of negative-Ucentres which are of importance in semiconductordevices. The section is divided in two parts, firstlymini reviews of published work on four defects, 1)BsO2 (LID) complex in silicon, 2) atomic hydrogen insilicon, 3) DX centres in III-V alloys and 4) The Ga-OAs-Ga defect in GaAs. This is followed by a tablewhich covers a wider range of important negative-Ucentres providing basic parameters and key references.The table includes the four defects listed above andnegative-U defects referred to previously in the text toillustrate particular aspects of negative-U behaviour,and theoretical and experimental challenges in theirstudy.

5.1. BsO2 (LID) complex in silicon

A complex consisting of a substitutional boron atomand an oxygen dimer (BsO2) in Si was suggested tobe responsible for light induced degradation (LID) ofsolar cells produced from silicon crystals, which containboron and oxygen impurity atoms [103, 104, 105]. Thesuggestion was based on the experimental observationsof linear and quadratic dependencies of concentrationof recombination active defects responsible for LID onsubstitutional boron and interstitial oxygen concentra-tions, respectively [103, 104]. However, for almost twodecades, there was no clear understanding of the elec-tronic structure of the complex or the mechanisms of itsformation and transformations [105]. Answers to theabove uncertainties have been found recently in a com-plex study consisting of a range of experimental tech-niques (various junction spectroscopy methods, photo-luminescence, microwave detected photo-conductancedecays) and ab-initio modelling [26, 81], where it wasconcluded that BsO2 has negative-U properties.

Figure 11 shows the configuration-coordinatediagram and atomic configurations of the BsO2

complex in different charge states. The negative-U properties of the complex are related to thetransformations of the oxygen dimer in the vicinityof a substitutional boron atom between the so called‘squared’ configuration (D and X in Figure 11,

Figure 11. Configuration coordinate diagram and atomicconfigurations of the BsO2 complex in Si on different chargestates. (Reprinted from Ref. [26], with the permission of AIPPublishing).

with two three-fold coordinated oxygen atoms) and‘staggered’ configurations (configurations A, A′, andA′′ in Figure 11, with two-fold coordinated oxygenatoms). In p-type Si crystals when EF < Ev + 0.3 eV,the BsO2 centre is in the positive charge state (stateD+ in Figure 11). In this state, the defect caneither capture an electron (created by either light orforward bias pulses) or emit a hole (thermally whenthe temperature is high enough) and transform to theshallow acceptor A− state according to the followingsequence of transitions: D+ + h+ + e− X0 + h+ A0 + h+ A− + 2h+. Kinetics of the forward andback transitions between the D+ and A− have beenmonitored at different temperatures in the p-type Sicrystals with different hole concentrations and analysedwith the use of Equations 49, 50 and 51 [26, 81].From this analysis, the values of energy differencesand barriers between different configurations of thecomplex have been derived.

Further, it has been found that upon relativelylong (tens of hours at room temperature) minority-


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Figure 12. Illustration of the minimum-energy atomicconfigurations for hydrogen interstitials in silicon, as derivedfrom first-principles calculations (a) the positive charge andneutral charge states, and (b) the negative charge state.

carrier-injection treatments the BsO2 defect transformsto a metastable configuration with a shallow acceptorlevel and stronger recombination activity (than thatfor Bs and the A state of the BsO2 complex). Apossible mechanism of this transformation has beenconsidered in Ref. [26]. Ab-initio modelling resultshave shown the existence of two metastable shallowacceptor configurations (A′ and A′′ configurations inFig. 11) of the BsO2 complex [26]. So, it appears thatthe BsO2 complex is a multi-stable defect with at leastthree experimentally confirmed atomic configurationsfor the neutral charge state (X0, A0, and A′0/A′′0).Because of the existence of a rather high energybarrier between the A′0/A′′0 and A0 configurations themetastable A′0/A′′0 configurations are long-living atroom temperature. Only at temperatures above 150 Cthe A′/A′′ → A/(D) reactions occur efficiently.

5.2. Atomic hydrogen in silicon

Hydrogen is a very common impurity in silicon. Itis quite mobile at room temperature and extremelyreactive in its atomic form. As a result, it readilycombines with other impurities and itself, leaving onlya small fraction of isolated hydrogen atoms. Theseexist in three charge states and exhibit negative-Ubehaviour with the H-related (0/+) donor level lyingabove the (−/0) acceptor level, where neutral H0

is a metastable species [106]. Ab-initio calculationspredict that both H0 and H+ have minimum energypositions at the Si-Si bond centre (BC), while H−is located at an interstitial tetrahedral (T) latticesite [107]. Representative structures are shown inFigure 12. The resultant negative-U characteristic ofisolated hydrogen is technologically very significant,impacting on hydrogen’s ability to passivate defectsand its diffusivity. Several comprehensive reviews havebeen published which include the negative-U aspectsof hydrogen in silicon [22, 108, 109, 110].

The experimental study of isolated hydrogen insilicon has proved to be very difficult and it is only bythe application of multiple techniques and calculationsthat a good understanding has been achieved. The

fundamental problem is that hydrogen is both highlyreactive and very mobile. The end result has beenthat some experiments claiming to characterise isolatedhydrogen actually derive from hydrogen complexes,commonly H-O and H-C. A methodology to avoid this,which has met with success, is to implant protonsat low temperature and undertake measurements atlow temperature. The technique was first used byStein [111] who showed unambiguously that the lineat 1998 cm−1 seen in LVM absorption studies wasrelated to hydrogen at the bond centre position. Lowtemperature implantation at T = 45 K was used byHolm [112] to study n-type Si using DLTS, and anEc−E(0/+) = 0.16 eV donor level (E3′) was observed,being later ascribed to bond-centred H(BC) . This wasfollowed by detailed studies by Bonde-Nielsen et al.[113] on hydrogen near the T site, which quantified theimpact of oxygen on nearby sites and determined theenergy of the hydrogen acceptor to be Ec −E(−/0) =0.68 eV.

5.3. DX centres in III-V alloys

DX centres are a special case of negative-U systemsin which the addition of dopants, expected toform shallow donors in III-V alloys, gives rise topersistent photoconductivity and deep acceptors. Thephenomenon was first reported in detail by Crafordet al. in 1968 [114] describing Hall effect measurementsof GaAs1−xPx with x ≈ 0.3 doped with Te orS. The phenomenon was erroneously believed to bethe result of a donor ‘D’ reacting with an unknowndefect ‘X’ to produce a deep state (hence ‘DX’).This was a result of studies of AlxGa1−xAs withx > 0.22 doped with Si, Ge, Sn, S, Se or Teusing Hall, photo-capacitance and DLTS (see forexample Refs. [16, 17]). The story of the DX centresis a classic research history spanning well over 25years starting with a technological driver of achievingadequately high doping in III-V heterojunction lasersand transistors and involving some of the world’sleading semiconductor groups combining theory andexperiment. After many plausible explanations wereexplored it is now evident that the DX centre is aclassic example of negative-U behaviour. The saga upto 1990 has been reviewed by Mooney [115] and somekey issues are covered in the introductory section ofthe current paper.

The most extensively studied DX centre is thatof Si in AlxGa1−xAs. DX behaviour is observedfor the case where x > 0.22 or at lower aluminiumconcentrations under hydrostatic pressure, for exampleDX behaviour of Si in GaAs is observed at pressuresP > 2 GPa. In the DX configuration the siliconaccepts an electron and is negatively charged. Thecharge state has been determined experimentally by


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Figure 13. Configuration coordinate diagram of the Si DXcentre in AlxGa1−xAs. In the DX configuration (left), theSi atom is displaced from its substitutional site. In theshallow-donor configuration (right), the Si atom occupies thesubstitutional site.

Dobaczewski et al. [19] from direct measurement ofthe capture cross section. As shown in Figure 13, thischange of charge state results in the relaxation of theSi from its substitutional position on the group III site,to an interstitial site neighbouring three As atoms.In this siting, the Si does not act as a conventionalthermally ionised donor and does not contribute to theconductivity.

If the system is exposed to light of a sufficientlyhigh energy, electrons are excited from the Si to theconduction band and the DX centre reverts to thesubstitutional site where it is a metastable neutraldonor. At temperatures T < 180 K the electrondoes not have enough energy to surmount the barrier(Ea ∼ 0.2 eV) shown by the arrow in Fig. 13, and sothe additional conductivity resulting from the shallowdonor state induced by the light remains for hoursor at lower temperatures for days or even weeks.This bizarre behaviour is referred to as persistentphotoconductivity and is a signature of DX behaviour.

The identification of the substitutional-interstitialmotion evolved from a sequence of calculationsand experiments discussed in Mooney’s review [115].However, a series of revealing experiments publishedsubsequently provided unambiguous detail of thisbehaviour [116, 117]. In this work, high resolutionLaplace DLTS was analysed to provide detail of theinfluence of the local environment on the electronthermal emission process fromDX centres in a range ofcompositions of AlxGa1−xAs. The direct comparisonof this process for centres related to Si which can

replace gallium or aluminium, with that observed fora group-VI donor (tellurium), which resides in thearsenic sublattice, enabled the configuration of atomsto be determined when the centre is in the groundstate. This shows that the DX(Si) defect ionisationprocess is associated with interstitial-substitutionalmotion of the silicon atom. However, in the caseof the DX(Te) centre, the spectra shows peaksfrom two groups which are attributed to interstitialsubstitutional motion of the neighbouring aluminiumor gallium atoms, respectively.

DX behaviour has been observed in many III-V and II-VI semiconductors, and recently, persistentphotoconductivity has been reported in oxides (e.g.ZnO and SrTiO3), which has been interpreted as DXbehaviour associated with negative-U characteristics[118, 119].

5.4. The Ga-OAs-Ga defect in GaAs

The Ga-OAs-Ga defect in GaAs is often referred to assubstitutional oxygen (on the arsenic site) or arsenic-vacancy-oxygen (VAs-O) complex [120]. It shows upas a group of three local vibrational mode (LVM)absorption bands with identical fine structure. Thebands, labelled A, B′ and B, appear at 731 cm−1,714 cm−1 and 715 cm−1. Band B′ emerges duringthe intermediate stages of optical-induced conversionfrom A to B using light with hν > 0.8 eV (belowband-gap light). This was interpreted as an electron-transfer process from EL2 to the conduction band,and subsequent capture by the Ga-OAs-Ga complex.EL2 is an As antisite-related defect, responsible for apinning E(0/+) level, effectively working as a reservoirof electrons at Ec − 0.75 eV [121]. Hence, A, B′ and Bbands were ascribed to LVMs of the same complex incharge states q, q − 1 and q − 2, respectively [122].

The A and B bands were first reported in semi-insulating material and assigned to EL2 by Song et al.[123]. However, this connection could not be correctand was quickly supplanted by a model involving oneO atom with two equivalent Ga-O bonds [124]. Thatway, the triple-peak fine structure with intensity ratioof about 2:3:1 on each band (from high to low frequencypeaks), could be explained by a combination of naturaloccurring Ga isotopes. Considering that 69Ga and71Ga are found in a proportion of about 3:2, a naturalpopulation of 69Ga-X-69Ga, 69Ga-X-71Ga, and 71Ga-X-71Ga units corresponds to an intensity ratio of 9 :(2 × 6) : 4 = 2.25 : 3 : 1, respectively [124, 125].Based on sample history arguments and semi-empiricalcalculations, the X element was suggested to be oxygen[124]. This was confirmed shortly after with theobservation of a 18O-isotope shift of the bands bySchneider and co-workers [125].

Besides possessing a Ga-O-Ga unit with two


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Figure 14. Configuration coordinate diagram for the Ga-O-Ga defect in GaAs based on experimental observations. Arrowheads indicate electron capture and emission processes describedin the text.

equivalent Ga-O bonds, we know from stress-splittingFTIR absorption that the A band involves a 〈110〉-oriented vibration dipole [126] and the defect shouldhave C2v symmetry. These experiments also concludedthat Ga-OAs-Ga does not reorient like VO in Si.

The negative-U nature of the Ga-OAs-Ga defectwas found by Alt, who performed a series of enlight-ening experiments, combining Fourier-transform infra-red (FTIR) absorption, photo-excitation using mono-chromatic light with energy spanning the band gapand annealing [122, 127]. The configuration coordinatediagram presented in Figure 14 was constructed fromthe experimental data. Upon cooling the sample fromroom temperature to 10 K in dark, only band A was ob-served, indicating that this should be the ground statein semi-insulating GaAs (for EF = Ec−0.75 eV). StateA is here assumed to have a reference charge state q,thus being referred to as Aq.

After applying a short pulse of light with 0.8 eV <hν < 1.5 eV, band A was partially quenched andthat corresponded to a proportional growth of bandB′. As referred to above, this was assigned to thephotoionisation of EL2 followed by electron capture byGa-OAs-Ga. Importantly, band B was not visible atthis point. This process is indicated in Figure 14 bythe arrow heads with labels (1) and (2). The B′ bandwas therefore assigned to the q−1 charge state (Bq−1).

Applying a 95 K dark annealing treatment tosamples where only A and B′ bands were present,led to the quenching of B′ and to the concurrentgrowth of bands A and B. The argument was thatby warming the sample at 95 K, electrons bound toB′q−1 were thermally promoted into the conduction

band, thus forming Aq again (process with arrow head(5) in Figure 14). The relatively small capture crosssection of ionized EL2, also allowed free electrons tobe captured by other B′q−1 defects, thus leading to thegrowth of band B. This was ascribed to the formationof a q − 2 charge state, namely Bq−2 (process number(3) in Figure 14). The activation energy for thisconversion was measured as Ea = 0.15 eV, and wasassigned to the thermal emission barrier of B′q−1 →Aq + e− [122, 128]. As pointed out by Alt [122],process (3) and (5) consisted on a disproportionationof metastable B′q−1 with negative-U ordering of levels.It was also noted that full A → B band conversioncould be achieved by subjecting the samples for longillumination times with 0.8 eV < hν < 1.5 eV atT < 70 K. At these temperatures, the thermal emissionchannel (5) in Figure 14 was suppressed and all defectsended up in state Bq−2.

At higher temperatures (T ≥ 180 K), band Bstarts to decrease in intensity at a rate identical tothe recovery of band A. Full recovery of band Ais achieved when B vanishes. This was suggested toresult from a two-electron emission sequence Bq−2 →B′q−1 + e− → Aq + 2e−, where the limiting stepis the first one (with larger barrier). The aboveemission sequence is represented in Figure 14 as stepswith arrow heads (4) and (5), respectively. At thesetemperatures, the thermally emitted electrons wereshown to be captured and trapped at the deeeper EL2state (step (6) of Figure 14). The activation energy forthe B→ A conversion was measured as Ea = 0.60 eV,thus suggesting that the binding energy of the secondelectron captured by the Ga-OAs-Ga defect, exceededthe binding energy first one (0.15 eV).

We note that a more cautious analysis of theabove picture would include the capture barriers forsteps (2) and (3). DLTS measurements give activationenergies of 0.58 eV and 0.14 eV for electron emissionfrom Bq−2 B′q−1, respectively [129]. As far as we areaware, capture barriers were not measured. However,judging from Hall effect measurements, which find adisproportionating (q−2/q) transition at Ec−0.43 eV,and considering the above emission barriers, the sum offirst and second capture barriers should be in the meVrange [130, 127]. Also interestingly, based on its DLTSfingerprint, electron emission from Bq−2 was assignedto EL3, a previously known deep trap with unknowncomposition [129].

The first atomistic model claiming to account forthe above data was a 〈001〉-displaced substitutionaloxygen atom at the arsenic site (OAs) [125], resemblingthe vacancy-oxygen complex in silicon. OAs possessesa Ga-O-Ga unit aligned along the 〈110〉 direction,thus being compliant with the conclusions from thepolarised absorption experiments [126]. Jones and


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Öberg [131] inspected the OAs defect in H-terminatedGaAs spherical clusters using local density functionaltheory. The paramagnetic O=

As state was found to bemetastable and was ascribed to band B′. Bands A andB were therefore connected to diamagnetic states withO−As and O−3As , respectively. While the calculations ofRef. [131] accounted for the experimentally observedred-shift of the oxygen vibrational frequency of B withrespect to A, this result could not be reproduced bysubsequent calculations [132, 133, 134]. Additionally,and perhaps more problematic for the OAs model, isthat it cannot explain the lack of reorientation of thedefect under the effect of uniaxial stress (although thiswas speculated to be due to a nearly isotropic stresstensor [126]).

Alternatively, Pesola et al. [134] proposed a defectmodel in which the two Ga radicals (not connectedto oxygen) in the off-site OAs defect are replaced byAs. This complex was referred to as (AsGa)2-OAs

and incorporates a Ga-O-Ga unit like in OAs. Thefirst-principles calculations of Ref. [134] indicated that(AsGa)2-OAs shows a negative-U ordering of E(0/+)and E(−/0) transitions, and the calculated O-LVMfrequencies of charge states +, 0 and − closely followthe observed frequency variations of bands A, B′ andB, respectively. The model naturally accounts for thelack of reorientation under uniaxial stress. However, itclashes with optically detected electron-nuclear doubleresonance (OD-ENDOR), which reported that as muchas 60% of spin-density of the B′0 state (with spin-1/2) is localized on the first shell of Ga atoms[135] – Figure 2 of Ref. [134] clearly shows that the(AsGa)2-OAs model has most of the density on As shellsand virtually none on Ga shells.

The origin for the negative-U ordering of levelsin the Ga-OAs-Ga complex is not clear. Thesimpler OAs substitution, although incompatiblewith the vibrational absorption data, could explainthe effect based on strong charge-dependent Ga-Gareconstructions. This feature was found theoreticallyfor the As vacancy (VAs) in GaAs [136], and used as ajustification for V 0

As and V=As showing a negative-U . On

the other hand, the (AsGa)2-OAs model has a mid-gapstate localised on two As radicals, which were suggestedto induce strong forces on neighbouring atoms as afunction of occupancy [134]. This effect does not finda parallel in the (positive-U) Ga vacancy (VAs), whoseAs radicals induce weak deformations [136].

Recently, the OAs defect was investigated usingmodern hybrid density functional theory [137]. O+

Asand O0

As were found to be most stable at the perfecttetrahedral site. The highest occupied state of O0

As,which is paramagnetic, is an s-like orbital on theoxygen atom, and therefore not compliant with theOD-ENDOR data. O−As had an off-site Ga-OAs-Ga

configuration with a localised acceptor state on areconstructed Ga-Ga pair. The E(0/+) and E(−/0)levels were calculated at Ec−0.71 eV and Ec−0.54 eV,thus showing a positive-U ordering. Clearly the Ga-OAs-Ga defect should be revisited by both theory andexperiments.

5.5. Electrical levels of selected negative-U defects insemiconductors

Table 1 lists the electronic characteristics of a selectionof negative-U defects in semiconductors. It should benoted that the table is not comprehensive. We onlyconsidered defects for which the negative-U behaviouris confirmed by solid experimental results, where theirelectronic properties are relatively well understood andthe atomic structures in different charge states havebeen predicted by experiments and ab-initio modelling.Negative-U properties have been suggested for anumber of other defects in semiconductor materials,e.g., for interstitial boron-interstitial oxygen complexand hydrogen-related shallow donors in silicon [138,76]. Besides silicon, atomic hydrogen has also beenclaimed to show negative-U behaviour in many othersemiconductors [139, 140]. Further examples could beenumerated. However, no solid experimental evidenceconfirming these suggestions has been presented yet.Perhaps it is worth mentioning the interesting case ofthe Si trivacancy, which has been briefly discussed inSection 4.2.3. It has been argued in Refs. [102, 101]that V3 in Si can be considered as a defect withnegative-U properties as it has the E(= /−) acceptorlevel at Ec − 0.21 eV below the E(−/0) level atEc − 0.07 eV. However, the levels mentioned aboveare energy levels of V3 in different configurations (partof the hexagonal ring and four-fold coordinated shownin Figures 10(a) and 10(d), respectively), which areseparated by rather high energy barriers. Becauseof these barriers the paired emission or capture ofelectrons (a signature of negative-U defects) has notbeen observed for the V3 defect, and therefore, it hasnot been included into the table.

The transition levels of Table 1 are cast asE(q / q+1) and E(q−1 / q), where q is a referencecharge state. For negative-U defects E(q / q+1) >E(q−1 / q), and the q charge state corresponds to themetastable state which, under equilibrium conditions,disproportionates into stable q + 1 and q − 1 chargestates.


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Tab

le1.

Electricallevels

ofselected

negative-U

defectsin

semicon

ductorsan

drespective

keyreferences.Allenergies

ineV

.Starred(*)defectsarediscussedin

detaileither

below

orelsewhere

inthis

pape

r.

Host

Defect

q/q

+1

E(q/q

+1)

q−

1/q

E(q−

1/q)

Com

ment

Key

references

SiIsolated

vacancy(*)

+/

++

Ev

+0.

130/

+Ev

+0.

05DLT

S,EPR,p

-typ

e,low

tempe

rature

irradiation

[5,6

]

SiInterstitial

B(*)

0/+

Ec−

0.13

−/0

Ec

+0.

37DLT

S,ODLT

S,EPR,p

-an

dn-type

,n+

p&

p+ndiod

es,low

tempe

rature

irradiation

[9,1

0,11

]

SiHyd

rogen-Oxy

gen

0/+

Ec−

0.16

−/0

Ec−

0.79

DLT

S,n-type

Cz,

low

tempe

rature

proton

implan

tation

[113

]

SiHyd

rogen(*)

0/+

Ec−

0.16

−/0

Ec−

0.68

Ctran

sients,P

Hdissociation

,EPR,

n-type

,low

tempe

rature

proton

implan

tation

[106

]

SiBsO

2(*)

0/+

Ev

+0.

56−/0

Ev

+0.

04DLT

S,LD

LTS,

admittance

spectroscopy,p

type

CZ,

heat-treatments

at40

0-50

0 C

[26,

81]

SiC

iOiH

0/+

Ec−

0.04

3−/0

Ec−

0.11

Halle

ffect,D

LTS,

FT-IR,n

-typ

eCz,

heat-treatments

350 C

electron

irradiated

hydrog

enated

samples

[141

,80,

142]

SiSelf-interstitial

(I)

+/

++

Ec−

0.39

0/+

?DLT

S,EPR,p

-typ

e,low

tempe

rature

implan

tation

withalph

apa

rticles

[143

]

SiIO

2(*)

+/

++

Ev

+0.

360/

+Ev

+0.

12Halle

ffect,D

LTS,

FT-IR,c

arbo

nlean

p-type

Cz,

room

tempe

rature

electron

irradiation

[144

,64,

77]

SiO-related

TDDs

+/

++

Ec−

0.16

0/+

TDD-0,E

c−

0.75

Halle

ffect,p

hotocond

uctivity,F

T-IR,

n-type

Cz,

heat

treatm

ents

400-50

0 C

[63,

79,2

0]0/

+TDD-1,E

c−

0.48

0/+

TDD-2,E

c−

0.29

Ge

O-related

TDDs

+/

++

Ec−

0.04

0/+

TDD-0,E

c−

0.48

Halle

ffect,p

hotocond

uctivity,F

T-IR,

ntype

Cz,

heat

treatm

ents

300-40

0 C

[21,

145]

0/+

TDD-1,E

c−

0.34

0/+

TDD-2,E

c−

0.26

4H-SiC

Cvacancy(*)

−/0

Ec−

0.42

=/−

Ec−

0.64

DLT

S,La

place-DLT

S,EPR,n

-typ

e,traces

inalla

s-grow

nsamples

[24,

54,5

3]

6H-SiC

Cvacancy(*)

−/0

Ec−

0.20

=/−

Ec−

0.43

DLT

S,La

place-DLT

S,n-type

,traces

inalla

s-grow

nsamples

[25,

59]

InP

M-center

+/

++

Ec−

0.18

0/+

Ec−

0.42

TSC

AP,

DLT

S,op

ticalion

isation,

irradiated

samples

[12,

13,1

4,15

]

GaA

sGa-OAs-Ga(*)

0/+

Ec−

0.14

−/0

Ec−

0.58

DLT

S,op

ticalion

isation

[122

,129

,134

]GaA

sPDX

centre

Seetext,p

rope

rtiesdepe

ndon

alloycompo

sition

Halle

ffect,o

ptical

absorption

[114

]AlG

aAs

DX

centre

Seetext,p

rope

rtiesdepe

ndon

alloycompo

sition

DLT

S,ph

oto-capa

citance,

Halle

ffect

[146

,17,

18,1

9,14

7]Zn

OOxy

genvacancy

+/

++

Ec−

0.4

0/+

Ec−

1.6

DLT

S,theory

[148

,149

]GaS

bDX(S)

0/+

Ec−

0.22

−/0

Ec−

0.25

DLT

S[19]


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6. Conclusions

The present review begins with an historical perspect-ive of the topic of ‘negative-U defects in semiconduct-ors’. We start from the early 1980’s, where the firstreports of this class of defects emerged in the literat-ure. The account includes several difficulties identifiedat the time, mostly due to meta-stability, or in somecases multi-stability, of one of the three charge statesinvolved. We provide the reader with an introductionto the essential physics of the workings of negative-Udefects (electronic correlation, electron-phonon coup-ling, disproportionation and defect transition levels).We present the state-of-the-art regarding the experi-mental and theoretical methods that are currently usedfor the characterisation of negative-U defects. Thesesubjects are presented in detail and they are accompan-ied with several successful studies from the literature.This includes experimental methods based either ondefect thermodynamics or defect kinetics (monitoredby several techniques, like Hall effect, DLTS or op-tical absorption), as well as numerical methods thatcan help us in the construction of detailed configura-tion coordinate diagrams (with the calculation of trans-ition levels, transition rates and transformation barri-ers). To wrap up, we showcase a few experimentaland theoretical analyses of some of the most impactingnegative-U defects in group-IV and compound semi-conductors, leaving a summary of the characteristics ofmany others, along with the most relevant referencesin a tabular format.

A negative-U defect has a metastable state withN electrons that disproportionate into N − 1 andN + 1 species under equilibrium conditions. Thisis only possible in many-atom systems, and thanksto the conversion of electron correlation energy intobond formation energy, either during ionisation orcapture of carriers. In these defects, the stable N − 1and N + 1 states are normally diamagnetic close-shellsystems, making them unnoticeable by paramagneticresonance techniques, unless some sort of excitationis provided to the sample. The metastability ofthe N -electron state works like a barrier separatingthe N − 1 and N + 1 species. When one of thelatter is a shallow donor or a shallow acceptor, theiroccupation via deep-shallow excitation induced byillumination, usually results in a concentration increaseof free carriers. This effect is often measurable asa persistent photoconductivity, which depending onthe temperature, may last for several hours, until thedefects surmounts a barrier associated with a chargecapture event followed by transformation and returnto the ground state configuration.

The above features make junction spectroscopymethods a primary choice for the characterisationof negative-U defects in semiconductors. However,

the complexity of their behaviour often requires theusage of complementary techniques, as well as specialtricks, like illumination or thermal quenching, whichallow researchers to play with thermodynamics andkinetics, and by doing so, to isolate the defect in oneof its three states, or to investigate the conversionsbetween them. Defect modelling has also beenpivotal. Since the pioneering work on the siliconvacancy, theories, numerical algorithms and hardwarehave evolved dramatically. Density functional relatedmethods are perhaps the most successful cases. Thesehave been able to provide us with a detailed quantummechanical description of the interactions betweentrapped carriers and the many-body cloud of electronscomprising the defective semiconductor.

Finally, we would like to emphasise the followingconcluding remark – we have achieved a rather deepunderstanding of the electronic and atomistic structureof many negative-U defects in semiconductors. Forsure, that was achieved thanks to concurrent andcombined experimental and theoretical work.

Acknowledgments

JC thanks the support of the i3N project, Refs.UIDB/50025/2020 and UIDP/50025/2020, financed bythe FCT/MEC in Portugal. VM and ARP have beensupported by the UK EPSRC for this work underprojects EP/M024911/1 and EP/P015581/1

ORCID iDs

J Coutinho https://orcid.org/0000-0003-0280-366XV P Markevich https://orcid.org/0000-0002-2503-6144A R Peaker https://orcid.org/0000-0001-7667-4624

All authors contributed equally to this work.

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characterisation of negative-u defects in semiconductors · 2020. 4. 22. · topical review...

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