Defect identification in semiconductors with positron annihilation: [612589]

Defect identification in semiconductors with positron annihilation:
Experiment and theory
Filip Tuomisto *
Department of Applied Physics, Aalto University School of Science, Espoo, Finland
Ilja Makkonen†
COMP Centre of Excellence, Helsinki Institute of Physics and Department of Applied Physics,
Aalto University School of Science, Espoo, Finland
(published 14 November 2013)
Positron annihilation spectroscopy is particularly suitable for studying vacancy-type defects
in semiconductors. Combining state-of-the-art experimental and theoretical methodsallows for detailed identification of the defects and their chemical surroundings. Also charge
states and defect levels in the band gap are accessible. In this review the main experimental and
theoretical analysis techniques are described. The usage of these methods is illustrated through
examples in technologically important elemental and compound semiconductors. Futurechallenges include the analysi s of noncrystalline materials an d of transient defect-related
phenomena.
DOI: 10.1103/RevModPhys.85.1583 PACS numbers: 61.72.J /C0, 78.70.Bj, 71.60.+z, 81.05. /C0t
CONTENTS
I. Introduction 1584
A. Defects in semiconductors 1584
1. Role and formation of defects in
semiconductors 1584
2. Studying defects in semiconductors 1585
B. Positron annihilation spectroscopy 1586
1. Background 15862. Positron annihilation methods 1587
II. Experimental Techniques 1588
A. Positrons in solids 1588
1. Implantation, thermalization, and diffusion 1588
2. Positron states and trapping 1589
3. Trapping model 1590
B. Positron lifetime spectroscopy 1591
1. Experimental details 15912. Data analysis 15933. Information revealed by the positron lifetime 1594
C. Doppler broadening spectroscopy 1595
1. Experimental details 1595
2. Data analysis 1596
3. Chemical information contained in
Doppler spectra 1597
III. Theory and Computational Methods 1598
A. Two-component electron-positron
density-functional theory 1598
B. Modeling localized positrons 1599C. Positron annihilation parameters 1600
1. Annihilation rate and lifetime 16002. Momentum density of annihilating
electron-positron pairs 1601D. Functionals for electron-positron correlation
effects 1602
E. The atomic superposition method 1603
F. Numerical approaches for self-consistent
calculations 1603
IV. Results 1605
A. An overview of results obtained in the past
two decades 1605
1. Elemental semiconductors Si, Ge, and C 16062. Traditional III-V and II-VI semiconductors 16063. Novel semiconductors: III-N, SiC, and ZnO 1607
B. Vacancy-(multi)donor complexes in highly n-type
doped silicon 1608
C. The vacancy-fluorine complex in silicon and
silicon-germanium alloys 1610
D. The EL2 defect in gallium arsenide 1611E. The gallium vacancy–tellurium complex in
gallium arsenide 1613
F. The gallium vacancy and its complexes in
gallium nitride 1614
G. Metal vacancy–nitrogen vacancy complexes in
III-nitrides and their alloys 1615
H. The substitutional lithium-on-zinc-site defect
in zinc oxide 1618
V. Future Challenges 1619
A. Materials with complex crystal structures 1619B. Positron states at interfaces and surfaces 1620C. Positron thermalization and trapping in
nanocrystalline, amorphous, and molecularsystems 1621
D. Pump-probe experiments with positron
annihilation spectroscopy 1622
E. Toward higher slow-positron beam intensity 1623
VI. Summary 1624
Acknowledgments 1624
References 1624
*filip.tuomisto@aalto.fi
†ilja.makkonen@aalto.fiREVIEWS OF MODERN PHYSICS, VOLUME 85, OCTOBER–DECEMBER 2013
1583 0034-6861 =2013=85(4)=1583(49) /C2112013 American Physical Society

I. INTRODUCTION
Positron annihilation spectroscopy has been widely used
for studying defects in semiconductors since the early 1980s,while the first reports dealing with radiation damage in silicon
and germanium had been published already in the 1970s
(Cheng and Yeh, 1973 ;Arifov, Arutyunov, and Ilyasov,
1977 ). The early developments of both experimental and
theoretical approaches applicable to semiconductor studies
were reviewed by Schultz and Lynn (1988) andPuska and
Nieminen (1994) . An introductory book on positron annihi-
lation studies of defects in semiconductors has also been
written by Krause-Rehberg and Leipner (1999) . Our aim in
writing this review is twofold. First, we want to introduce
the basic concepts behind the experimental and theoretical
methods of positron annihilation and review the latest devel-opments that have led to the possibility of identifying defects
in semiconductors with a high level of detail. Second, by
going through a variety of examples in both elemental andcompound semiconductors, we want to illustrate how these
methods can be applied to improve our understanding of the
physics of defects in semiconductors.
The organization of this review is as follows. First, we give
an introduction to defects in semiconductors and the history
and methods of positron annihilation. In the second part, afterbriefly explaining the necessary concepts related to the be-
havior of positrons in solids, we delve into the details of the
experimental methods most used in semiconductor studies:positron lifetime spectroscopy and Doppler broadening spec-
troscopy. Here our aim is to give a frank account of the
strengths and weaknesses of the experimental setups andanalysis methods, hoping to provide useful reference material
for the specialist and at the same time provide the nonpracti-
tioner additional means to assess positron results and inter-pretations. The same approach is applied in the third part
where the theoretical methods are presented. In Secs. IIand
IIIwe go through examples where both experimental and
theoretical positron methods have been applied to study
various semiconductor materials and defects therein. The
focus of these sections is, in addition to showing how thepositron methods work in practice, on the interpretations that
can be made about the defects identified in these technologi-
cally relevant materials. The results are systematically com-pared to the knowledge obtained by other experimental and
theoretical methods in order to give a frame of reference.
Finally we discuss the present challenges and possible futuredirections in semiconductor research with positrons.
It is important to note that we do not attempt to make an
exhaustive review of all positron work on defects in semi-conductors. To cover most of the published works on positron
annihilation in solids, we refer the interested reader to the
reviews by Berko and Hereford (1956) ,Ferrell (1956) ,
Schultz and Lynn (1988) ,Asoka-Kumar, Lynn, and Welch
(1994) ,Puska and Nieminen (1994) ,Krause-Rehberg et al.
(1998) , and Saarinen, Hautoja ¨rvi, and Corbel (1998) .
A number of books have been published on the subject of
positron annihilation in solids, as well as chapters in various
edited volumes. For detailed accounts see the conferenceproceedings of the ICPA (International Conferences on
Positron Annihilation), SLOPOS (International Workshopson Slow Positron Beams), and PSSD/PSD (Positron Studies
on Semiconductors and Defects) and to references therein.
A. Defects in semiconductors
Lattice defects in semiconductors are like spices in your
food: too much is disgusting, too little is worthless, while justthe right kind and amount makes the day. Another commonfeature is that both are typically present in amounts much
smaller than the host. There exists a wide variety of review
articles and books on defects in semiconductors. For a de-tailed picture of the field one is strongly advised to browse theproceedings volumes of the ICDS (International Conferenceson Defects in Semiconductors). A theoretical perspective canbe found in the book by Lannoo and Bourgoin (1981) , while a
recent volume covers many practical issues easily accessible
to the newcomers to the field ( McCluskey and Haller, 2012 ).
1. Role and formation of defects in semiconductors
Defects in crystalline solids are static interruptions to the
periodicity of the crystal. They can be classified by theirspatial extent into point defects that are zero dimensionaland extended defects that can be one dimensional (e.g.,dislocations), two dimensional (e.g., stacking faults), andthree dimensional (e.g., aggregates of impurities). It is not
unusual to have important densities of more than one of these
kinds of defects in a given crystalline material, such as anelemental (e.g., silicon or germanium) or a compound (e.g.,gallium arsenide or zinc oxide) semiconductor. Quite typi-cally they also affect each other’s properties and presence,e.g., the formation of stacking faults in a crystal may inducevacancy defects. In this review, the emphasis is on point
defects in general and on vacancy defects, in particular, as
the positron methods are most sensitive to defects with extraopen volume.
In contrast to metals, in semiconductors very dilute con-
centrations (e.g., less than ppm) of defects may have impor-tant effects on the electrical and optical properties. This is due
to the electronic states created by the defects in the typically
(0.5–5 eV) wide band gap of the semiconductor. Dependingon the position in the gap, electrons can be excited to or fromthese states (from or to the bands or other states in the gap)thermally, electrically, or optically. In practice, the electricaland optical properties of semiconductors are defined bycontrolled introduction of impurities in the host lattice, but
often it is not possible to completely eliminate the formation
of other defects, such as vacancy defects, atoms on interstitiallattice sites, extra impurities, or antisite defects (the latterexist only in compound semiconductors). Either these defectshave a detrimental effect on the targeted property or some-times they can assist in obtaining the desired functionality of
the material. It is also possible for some defects to be neutral
from the point of view of the property to be controlled.
Understanding of the properties of a semiconductor
requires (i) identification of the defects present in the lattice,(ii) their quantification, and (iii) knowledge of the nature ofthe states they introduce in the band gap (i.e., their effectson the properties). Examples of defect properties are sub-
band-gap light absorption and emission, and introduction or
removal of electrons to or from the conduction or the valence1584 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

band. Control of the semiconductor properties requires in
addition that the formation and introduction mechanisms ofthese defects are understood, as well as their other physicalproperties such as how they interact with other defects in thelattice and whether they can be made to move with the hopeof them getting trapped at a neutralizing location or drivenout of the region of interest. It should not be a surprise thatmany different experimental and theoretical methods need to
be employed in order to obtain even a small part of the
required knowledge. Finally, after all this understanding,one needs to be able to manufacture the semiconductormaterial in such a way that desired defects are introducedbut the harmful ones are not. Quite often this is verychallenging.
Usually it is rather straightforward to control the introduc-
tion of the desired impurities in the semiconductor matrix.Dopants can be added to the growth environment in variousways or they can be diffused in or implanted after the growthprocess. The most important limitations are solubility in thecase of in situ or diffusion doping, while implantation is
mostly applicable to close-to-the-surface doping profiles.However, while introduction of dopants is controllable andrequires active measures, other kinds of point defects areformed either thermally, due to kinetic or chemical effects,or as radiation damage in the case of implantation processing.In addition, the growth environment may contain someunwanted impurities that are difficult to control: typicalomnipresent elements are oxygen and hydrogen. Further,for example, in the case of wide-band-gap semiconductors
such as the III-nitrides, where native substrates are not easily
available, the lattice mismatch between the thin film andsubstrate causes initial stresses and strain that are most oftenrelaxed through the generation of dislocations and otherextended defects. There are many ways to try to avoid theformation of the unwanted defects or to try to remove themby postprocessing, such as thermal treatments. Even thoughmany defect-related problems have been identified and solvedover the past 60 years of semiconductor research, the constantquest for faster, cheaper, less power consuming, and newkinds of electronics generates the need for new materialsproperties and hence creates new defect-related challenges.
2. Studying defects in semiconductors
As the existence of defects is what makes semiconductors
such useful materials, defects in semiconductors have been
studied for as long as semiconductors have been known. The
wide variety of methods can be roughly divided into electricalmeasurements, optical spectroscopy, particle beam methods,microscopy, and theoretical calculations. Detailed reviews onthese methods can be found in the literature [see, e.g., Stavola
(1998) ]. In the following we briefly go through the defect
detection, identification, and quantification capabilities of themost used methods in semiconductor defect studies.
Measuring electrical properties from the defect point
of view typically leads to the determination of resistivity(conductivity), free-carrier concentration and mobility, con-centrations of ionized donors and acceptors, and deep carriertraps. By definition, these properties can be considered themost basic properties of a semiconductor. The most popularmethods employed are Hall effect experiments and deep-leveltransient spectroscopy (DLTS) [see, e.g., Svensson, Ryden,
and Lewerentz (1989) ,Dobaczewski et al. (1994) , and Look,
Hemsky, and Sizelove (1999) ]. Optical spectroscopies give
access to another set of basic properties of semiconductors,
namely, the optical absorption and emission that are particu-larly important in optoelectronic device applications such aslight-emitting diodes or laser diodes. Absorption and lumi-
nescence spectroscopies provide detailed information on the
optical transitions between the valence and conduction bandsand on the positions and nature of defect-induced electronicstates in the band gap.
The above techniques provide detailed information on the
electrical and optical properties generated by the defects,but usually they do not allow for direct identification of thedefects in question, and in optical spectroscopy the determi-nation of defect concentrations is challenging ( Reshchikov
and Morkoc ¸, 2005 ). Optical absorption by local vibrational
modes in the infrared (IR) wavelengths can be used toidentify defects through their vibrational frequency finger-prints ( Bergman et al. , 1988 ;Gotz et al. , 1996 ). This method
is particularly useful in the case of hydrogen-related defects
in semiconductors because of the very distinct frequenciesoriginating from the low atomic mass of hydrogen. There is aset of techniques based on photon spectroscopy in the pres-
ence of a magnetic field that are very sensitive to the detailed
atomic structure thanks to the hyperfine interactions. Thesemethods employ the electron spin resonances (ESRs), andrequire the defect to be studied to have a paramagnetic ion-
ized state that can be excited by an external field. Variations
of these experiments include electron paramagnetic reso-nance (EPR), optically detected magnetic resonance, andelectron nuclear double resonance ( Watkins and Corbett,
1964 ). The ESR methods are sensitive to the number of the
active centers (instead of the concentration) and give a verydetailed atomic structure of the defects that are detected.Challenges are encountered with samples with high free-
carrier concentrations due to efficient microwave absorption,
while thin films often have too few active centers in total evenif their concentration is high.
The electrical and optical defect spectroscopy methods are
intrinsically nondestructive, i.e., the semiconductor samples
and their properties are not altered during the measurements.
There is a wide variety of methods based on the use of ionbeams that in turn are destructive, but provide crucial data onthe defect properties. Most common of these are Rutherford
backscattering and nuclear reaction analysis which are very
efficient for detecting and identifying atoms that are not incorrect lattice positions ( Wahl et al. , 1997 ;Yuet al. , 2002 ).
Nondestructive particle beam methods include muon spin
rotation and positron annihilation spectroscopies, of which
the first is particularly useful for modeling behavior ofhydrogen in semiconductors ( Stavola, 1998 ), while the latter
is selectively sensitive to vacancy-type defects. Electron
microscopy methods have already reached (sub-)atomic
resolution; this holds especially for transmission electronmicroscopy (TEM). In addition, the latest advances inthe so-called Zcontrast allow the identification of atomic
species as well ( Pennycook, 2012 ). Hence exact positions of
atoms can be imaged in sample cross sections, providing
direct experimental identification of extended defects andFilip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1585
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

impurities, given the fact that the concentrations (densities)
are high enough, as the typical size of atomic-resolution
images is of the order of 10/C210 nm2. Another challenge
in imaging intrinsic point defects is that they may be created
in the preparation of cross-sectional samples.
Calculations of the electronic structure of semiconductors
and their defects is possible from first principles. By far
the most popular method is the density-functional theory
(supercell calculations) with the electron-electron exchange
and correlation described through the local-density ap-
proximation (LDA) or semilocal generalized-gradient
approximations (GGA). The computing power of modern
supercomputers allows for efficient calculations with rela-tively large supercells (up to 1000 atoms) of the formation
enthalpies and charge transition levels of point defects with
these methods [for reviews, see Van de Walle and Neugebauer
(2004) ,Drabold and Estreicher (2007) ,Janotti and Van de
Walle (2009) , and Van de Walle, Lyons, and Janotti, (2010) ].
However, both the LDA and GGA suffer from predicting
incorrect band gaps and hence the reliability of the predicted
defect levels is often debated. Atomic structures of the defects
seem to be less affected by the different approximations.Rather recently so-called hybrid functionals ( Becke, 1993 ;
Perdew, Ernzerhof, and Burke, 1996 ;Adamo and Barone,
1999 ;Heyd, Scuseria, and Ernzerhof, 2003 ) have been ap-
plied, where part of the exchange and correlation is calculated
within the Hartree-Fock approach to improve the description
of nonlocal effects. This approximation has significantly
improved the predicted band gaps for semiconductors and
allowed for new interpretations for some defect levels. At the
time of writing this review, the computational complexity ofthe hybrid functionals limits the supercell sizes to roughly
100 atoms; hence especially in the case of charged defects
so-called supercell corrections need to be considered ( Makov
and Payne, 1995 ;Schultz, 2000 ;Freysoldt, Neugebauer, and
Van de Walle, 2009 ). With constantly improving computing
power, more accurate approaches, such as the GW quasipar-
ticle approximation and quantum Monte Carlo methods, are
becoming more and more applicable in defect calculations
[see, e.g., Ertekin et al. (2012) andRinke et al. (2012) ].
B. Positron annihilation spectroscopy
Positron annihilation spectroscopy is a characterization
method for probing the local electron density and atomic
structure at the site chosen by the electrostatic interaction
of the positron with its environment. The information on the
structure can be measured in the time and energy spectra of
the positron annihilation radiation. It is thus possible to
investigate experimentally local structures embedded in the
bulk of the material, such as missing atoms (vacancies),clustering of atoms, superlattices and device structures, quan-
tum dots, as well as free volume, and void sizes in polymers
or even biological materials. These imperfections often de-
termine the crucial properties of the materials, such as me-
chanical properties, electrical conductivity, diffusivity, or
light emission. The positron annihilation methods have had
a significant impact on defect spectroscopy in solids by
introducing an experimental technique for the unambiguous
identification of vacancies. Native vacancies have beenobserved at high concentrations in many semiconductors,
and their role in doping and compensation can be quantita-tively discussed.
1. Background
The existence of the antiparticle of the electron, the posi-
tron, was predicted by Dirac (1928) , and its first experimental
observation came in 1932 ( Anderson, 1933 ). Positron-
electron annihilation was eagerly studied throughout the1940s and 1950s, and experimental methods were developed.In the late 1960s it was understood that positrons weresensitive to lattice defects in metals ( MacKenzie et al. ,
1967 ;Bergersen and Stott, 1969 ;Connors and West, 1969 ;
Hodges, 1970 ). The development of variable-energy slow-
positron beams ( Schultz and Lynn, 1988 ) and of the theory
of positrons in semiconductors and defects in the 1980s(Puska and Nieminen, 1994 ) made research on thin films
and coatings accessible to positron spectroscopy and ledto an ongoing growth in interest in these methods for mate-rials research since the early 1990s (Fig. 1). Positron annihi-
lation spectroscopy is nowadays applied in /C24200 research
laboratories worldwide, while there are /C2440operational
slow-positron beams in /C2430research laboratories.
As described above, many techniques are applied to
identify defects in semiconductors on the atomic scale. Theadvantage of the positron annihilation method lies in itsability to selectively detect vacancy-type defects. This isbased on two special properties of the positron: it has apositive charge and it annihilates with electrons. An energeticpositron which has penetrated into a solid rapidly loses itsenergy and then lives for a few hundred picoseconds inthermal equilibrium with the environment. During its thermalmotion the positron interacts with defects, which may lead totrapping into a localized state. Thus the final positron anni-hilation with an electron can happen from various states.Energy and momentum are conserved in the annihilationprocess, where two photons of about 511 keV are emittedinto opposite directions. These photons carry information onthe state of the annihilated positron. The positron lifetime isinversely proportional to the electron density encountered by
1000
800
600
400
200
0Published papers
2010 2000 1990 1980 1970 1960 1950
YearDevelopment of slow positron beams and
theory of positrons in semiconductors and defects
FIG. 1. The number of papers published per year on positrons in
condensed matter physics and materials research. The development
of slow positron beams and of the theory of positrons in semi-
conductors and defects during the 1970s and 1980s predates thestrong increase in research activity. Data from ISI Web of Science.1586 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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the positron. The momentum of the annihilated electron
causes an angular deviation from the 180/C14straight angle
between the two 511 keV photons and creates a Doppler shiftin their energy. Thus the observation of positron annihilationradiation gives experimental information on the electronicand defect structures of solids. For more detailed accountson the positron annihilation in solids in general, see, e.g.,West (1973) andHautoja ¨rvi (1979) .
The sensitivity of positron annihilation spectroscopy to
vacancy-type defects is easy to understand. The free positronin a crystal lattice experiences strong repulsion from thepositive ion cores. An open-volume defect like a vacant
lattice site is therefore an attractive center where the positron
gets trapped. The reduced electron density at the vacant siteincreases the positron lifetime. In addition, the missingvalence and core electrons cause substantial changes in themomentum distribution of the annihilated electrons. Twopositron techniques have been efficiently used in defectstudies in semiconductors, namely, the positron lifetime andthe Doppler broadening of the 511 keV line. There are threemain advantages of positron annihilation spectroscopy whichcan be listed as follows. First, the identification of vacancy-type defects is straightforward. Second, the technique isstrongly supported by theory, since the annihilation character-istics can be calculated from first principles. Finally, positronannihilation can be applied to bulk crystals and thin layers ofany electrical conduction type.
A special feature is that the positron can form a bound state
with an electron in a system with low enough (local) electrondensity ( Mohorovicic, 1934 ;Deutsch, 1951 ;Mogensen,
1995 ;Charlton and Humberston, 2001 ). This hydrogenlike
quasiatom is called positronium (Ps), with a mass of
1:022 MeV =c
2and a diameter of 1:06/C23Ain its ground state
in vacuum. The binding energy of Ps is 6.8 eV in vacuum,i.e., half of the ionization energy of the hydrogen atom.Depending on the spin of the positron relative to the electron,Ps is in either the singlet (antiparallel spins, parapositronium,p-Ps) or triplet (parallel spins, orthopositronium, o-Ps) state.The self-annihilation properties of these two states are verydifferent ( Charlton and Humberston, 2001 ), and the so-called
pick-off annihilation of o-Ps, in which the positron in Psannihilates with an electron from the surroundings, prevailsin matter ( Brandt, Berko, and Walker, 1960 ). Importantly, the
interaction between Ps and matter is predominantly repulsivedue to the electron-electron repulsion. These properties areuseful when porous media or soft condensed matter arestudied with positron annihilation ( Mogensen, 1995 ;Jean,
Mallon, and Schrader, 2003 ).
2. Positron annihilation methods
Positrons can be created in several ways, of which the most
common, in the case of laboratory-scale facilities, is usingradioactive ( /C12
ț) isotopes, such as22Na, which has relatively
low intensity (up to 109positrons =s), but practical half-life of
2.6 years allowing reasonable use of the same source for6–10 years. High-intensity sources (up to 10
12positrons =s)
at large-scale facilities make use of pair production withthe high-energy gamma flux created by a nuclear reactor(Hugenschmidt et al. , 2004 ;Schut et al. ,2 0 0 4 ;Hawari
et al. , 2009 ) or a particle accelerator ( Cassidy et al. ,2 0 0 9 ;Krause-Rehberg et al. , 2011 ). In both cases, the positrons
have a wide and continuous energy spectrum with mean
energies in the hundreds of keV. These fast positrons can be
used directly to probe the bulk (several hundreds of microns)of a material. In order to study thin films and coatings, the
positrons need to be slowed down and if possible monochro-
mated. Many crystal surfaces, such as those of heavy metalelements (e.g., W), have a negative work function for posi-trons, resulting in the fact that thermalized positrons within
the solid can be emitted to the vacuum with an energy of a
few eV if they reach the surface ( Tong, 1972 ). These slow
positrons can be magnetically guided and electrostatically
accelerated to form a variable-energy beam allowing for,
e.g., depth profiling.
The two most used methods in defect studies with positron
annihilation are the positron lifetime spectroscopy and
Doppler broadening (of the positron-electron annihilation
radiation) spectroscopy. These techniques are very efficientin giving important information on vacancy defects in metals
and semiconductors: the vacancy defects can be identified
(sublattice in compounds, size in the case of vacancy clusters,and decoration by impurities), their charge states (in the case
of semiconductors) can be determined, and their concentra-
tions
1can be evaluated in the technologically important range
from 1015–1019cm/C03. Thanks to recent developments in
theoretical calculations, computational studies can be directly
compared with positron experiments, providing possibilitiesfor detailed interpretations of experimental data. The positron
lifetime and Doppler broadening techniques are also widely
used in free-volume studies of molecular materials. Angularcorrelation of annihilation radiation (ACAR) ( Beringer and
Montgomery, 1942 ;Berko, Haghgooie, and Mader, 1977 )i s
used to detect essentially the same phenomenon as Doppler
broadening, namely, the momentum distribution of electron-positron annihilation radiation. The resolution of this tech-
nique is superior to Doppler broadening, but count rates are
correspondingly lower. The better resolution of ACAR allowsfor detailed studies of the electronic structure (e.g., Fermi
surfaces in metals), but does not bring significant improve-
ments in the case of defect studies in semiconductors. Itshould be noted that these three techniques are easily used
in bulk materials employing fast positrons and can be used
with slow-positron beams, although requiring high intensity(except for Doppler broadening).
Positron-induced Auger electron spectroscopy (PAES)
(Weiss et al. , 1989 ;Soininen, Schwab, and Lynn, 1991 )
and reflection high-energy positron diffraction (RHEPD)(Ichimiya, 1992 ;Kawasuso and Okada, 1998 ) are extremely
surface-sensitive positron-using techniques that require the
use of a slow-positron beam, but do not require the measure-ment of positron-electron annihilation radiation. Thesetechniques have important benefits compared to their ‘‘tradi-
tional’’ electron counterparts. In PAES the secondary-electron
background is completely suppressed and the sensitivity isenhanced to only the first atomic layer ( Weiss et al. , 1989 ;
1Here a note on vocabulary is warranted: in the case of defects in
semiconductors, it is typical to speak about concentrations when
densities are meant. Hence concentrations are given in the units ofcm
/C03instead of ppm or ppb.Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1587
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

Jensen and Weiss, 1990 ;Soininen, Schwab, and Lynn,
1991 ;Hugenschmidt, Mayer, and Schreckenbach, 2010 ). In
RHEPD the positron crystal potential and the positive charge
of the positron give rise to total reflection below a critical
angle, again resulting in enhanced sensitivity to only thetopmost atomic layer at the surface ( Kawasuso and Okada,
1998 ;Kawasuso et al. , 2003 ;Fukaya, Mochizuki, and
Kawasuso, 2012 ). The drawback of these techniques is the
necessity for a high-intensity positron beam as the measure-ment times with laboratory-scale low-intensity beams are fartoo long.
The main technological issues limiting further develop-
ment of laboratory-scale experimental techniques are thepoor efficiency of the moderation process when creating aslow-positron beam (resulting in the necessity for large-scalefacilities with high-intensity sources) and the directional
dispersion of the moderated positrons. The latter is partly
responsible for the small number of scanning positron mi-croprobes (SPMs), comparable to a scanning electron micro-scope (SEM), as the focusing of the beam with reasonable
intensity even at a large-scale facility results in spot sizes of
the order of 5/C22m(Greif et al. , 1997 ;Triftsha ¨user et al. ,
1997 ). Another limitation for the SPM is the lateral straggling
and the positron diffusion length of several hundreds of
nanometers in a perfect crystal that will limit the spot size
even if the focus is improved. The moderator efficiencies arein the 10
/C05–10/C04range for passive crystalline heavy metal
(e.g., W) moderators and in the 10/C03range for solid Ne
moderators ( Mills and Gullikson, 1986 ). As the latter need
to be regenerated weekly, they are somewhat complicated touse. This holds, in particular, when the source and moderatorare floated at a high voltage in order to have the sample
grounded for easy manipulation. Sample manipulation
(temperature, illumination, and bias control) is essential forsophisticated thin-film studies, and hence in many cases theW moderator is better suited.
Further technological limitations arise when the most
powerful technique, positron lifetime spectroscopy, is used
with slow-positron beams. The traditional technique dependson the existence of a start signal (given by the
22Nasource
which emits practically simultaneously with the positron a
high-energy 1.27 MeV photon). The moderation process
strongly limits the usefulness of this start signal. Hence,either a positron-fly-by-detecting sensor must be installedor the beam must be tagged (e.g., by detecting secondary
electrons ejected from the sample surface by positron impact)
or modulated in time in order to retrieve a timing signal.The modulation of the positron beam has been shown to bethe approach of choice, but requires radio-frequency beam
bunching and chopping that have their own complications
(Mills, 1980 ;Scho¨dlbauer et al. , 1987 ;Suzuki et al. , 1992 ;
Tashiro et al. , 2001 ;Reurings and Laakso, 2007 ). The advan-
tage is a time resolution good enough for studying semi-
conductors and metals where the positron lifetimes are an
order of magnitude shorter than in molecular matter. Inprinciple, the beam could be modulated also by trappingthe positrons into a magnetic trap and releasing them at giventime intervals. This approach is, however, better suited for
applications where bunches containing a large amount of
positrons are required, such as in experiments studyingpositron-positron interactions or molecular positronium
(Cassidy and Mills, 2007 ). It should be noted that the
defect-spectroscopic techniques based on the detection ofpositron-electron annihilation radiation rely on the nonexis-
tence of positron-positron interactions. In these measure-
ments, there is only one positron in the sample at any giventime. As positrons in crystalline solids annihilate within atime frame of a few nanoseconds, a maximum intensity ofabout 10
8positrons =sis imposed.
II. EXPERIMENTAL TECHNIQUES
A. Positrons in solids1. Implantation, thermalization, and diffusion
For a full description of the physics of positrons in solids,
seeSchultz and Lynn (1988) andPuska and Nieminen (1994) .
In the following we briefly describe the necessary conceptsand models needed to analyze and interpret the experimental
data.
The stopping profile of energetic positrons emitted by a
radioactive ( /C12
ț) source is exponential ( Brandt and Paulin,
1977 ):
PðxȚ¼/C11expð/C0/C11xȚ;/C11/C2516/C26½g=cm3/C138
E1:4max½MeV/C138cm/C01; (1)
where /C26is the density of the solid and Emaxis the maximum
energy of the continuous /C12țradiation spectrum. The most
common isotope for positron experiments is22Nawith
Emax¼0:54 MeV . Hence for this isotope the characteristic
penetration depth 1=/C11is, e.g., 110/C22min Si and 40/C22min
GaN. This means that positrons implanted directly as emittedfrom the source probe the bulk of a solid. It should be notedthat the average energy of positrons emitted by
22NaisEav¼
0:18 MeV . The (electronic) interactions during the stopping
are the same for positrons as for electrons ( Lennard et al. ,
1995 ), and hence in the case of semiconductors there is
essentially no lattice damage caused by positron implanta-tion. Additionally, a typical total fluence of at most10
12cm/C02is implanted in the samples during an experiment.
This is several orders of magnitude less than typically used inelectron irradiation experiments with the purpose of creatinglattice damage ( Saarinen et al. , 1995 ;Tuomisto, Ranki et al. ,
2007 ;Chen, Betsuyaku, and Kawasuso, 2008 ).
For monoenergetic positrons obtained from a low-energy
positron beam (typical energies are below 50 keV), the stop-ping profile can be described by ( Valkealahti and Nieminen,
1984 )
PðxȚ¼/C0
d
dxexp½/C0ðx=x 0Ț2/C138; (2)
where x0gives the peak position of the profile while the mean
stopping depth is /C22x/C250:886x0due to the asymmetry of the
profile. The mean stopping depth is given as /C22x¼AEn[keV],
where A/C254/C210/C06=/C26½/C22g=cm2/C138andn/C251:6. This de-
scription is very closely matched to the Monte Carlo simula-tions for stopping profiles of low-energy electrons ( Shimizu
and Ze-Jun, 1992 ;Ghosh and Aers, 1995 ;Dapor, 1996 ;
Denison and Farrell, 2004 ;Nyka¨nen et al. , 2012 ). The
mean stopping depth varies from a few nanometers up to a1588 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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few micrometers. Hence low-energy positrons can be used to
study near-surface layers and thin films.
The stopping and the ensuing thermalization of the
positrons are fast processes, taking only 1–3 ps at room
temperature in both metals ( Jensen and Walker, 1990 ) and
semiconductors ( Jorch, Lynn, and McMullen, 1984 ). This is
considerably less than typical positron lifetimes of the orderof 150–300 ps. ACAR experiments have shown that themomentum distribution of annihilating positrons follows thesample temperature down to 10 K ( Hyodo, McMullen, and
Stewart, 1986 ). In a few cases incomplete thermalization may
be important, e.g., positrons implanted at a very low energycan escape the sample nonthermally through the surface [seeGullikson and Mills (1986) ,Nielsen, Lynn, and Chen (1986) ,
Huomo et al. (1987) , and Lynn and Nielsen (1987) and there
are also related problematics in measurements made fornanocrystalline matter; see Sec. V.C]. This needs to be taken
into consideration when interpreting data from near-surface
layers. After implantation and stopping the transport of ther-malized positrons can be quite efficiently described by diffu-sion theory developed for free carriers ( Bergersen et al. ,
1974 ). The positron diffusion coefficient in semiconductors
at room temperature is typically D
ț¼1–2c m2=Vs. The
characteristic diffusion length during the positron lifetime /C28
isLț¼ðDț/C28Ț1=2¼100–200 nm .
2. Positron states and trapping
After implantation and thermalization the positron in a
semiconductor is in a Bloch-like state in a perfect periodiccrystal lattice. The thermalized positron at its ground statecan be described to a good approximation (see Sec. III)b ya
single-particle Schro ¨dinger equation
/C0
1
2m/C3r2cțðrȚțVðrȚcțðrȚ¼EțcțðrȚ; (3)
where the positron potential consists of an electrostatic
Coulomb potential and a term that takes into account theelectron-positron correlation effects. Because of the Coulombrepulsion from positive ion cores, the positron wave function isconcentrated in the interstitial space between the atoms in thelattice. The positron energy band is parabolic and free particle
like ( Boev, Puska, and Nieminen, 1987 ). The effective mass of
the positron is m
/C3/C251:5m0due to phonons and the screening
cloud of electrons ( Bergersen and Pajanne, 1969 ).
The positron lifetime and the Doppler broadening of the
annihilation radiation can also be calculated once the corre-sponding electronic structure of the solid system is known.
The positron annihilation rate /C21, the inverse of the positron
lifetime /C28, can be thought to be proportional to the overlap of
the electron and positron densities:
1
/C28¼/C21¼/C25r2ecZ
jcțðrȚj2n/C0ðrȚdr; (4)
where reis the classical electron radius, cis the velocity of
light, and n/C0ðrȚis the electron density. The momentum
distribution /C26ðpȚof the annihilation radiation is a nonlocal
quantity and requires knowledge of all electron wave
functions cioverlapping with the positron. In the simplest
approximation it can be written in the form/C26ðpȚ¼/C25r2ecX
i/C12/C12/C12/C12/C12/C12/C12/C12Z
dre/C0ip/C1rcțðrȚciðrȚ/C12/C12/C12/C12/C12/C12/C12/C122
; (5)
where the summation goes over occupied electron states.
It should be noted that the momentum distribution /C26ðpȚof
the annihilation radiation is mainly that of the annihilatingelectrons ‘‘seen by the positron,’’ because the momentum ofthe thermalized positron is negligible. The theoretical meth-
ods used to determine the positron’s ground state and the
annihilation parameters are reviewed in Sec. III.
In analogy to free carriers, the positron also has localized
states at lattice imperfections. At vacancy-type defects whereions are missing, the repulsion sensed by the positron islowered and the positron experiences these kinds of defectsas potential wells. As a result, localized positron states at
open-volume defects are formed. The positron ground state at
a vacancy-type defect is generally deep; the binding energyis about 1 eV or more ( Makkonen and Puska, 2007 ). In a
vacancy defect the electron density is locally reduced. This isreflected in the positron lifetime which is longer than in thedefect-free lattice. Hence the positron lifetime measurement
is a probe of vacancy defects in materials. Positron annihila-
tion at a vacancy-type defect also leads to changes in themomentum distribution probed by the Doppler broadeningexperiment. The momentum distribution arising from valenceelectron annihilation becomes slightly narrower due to alower electron density. In addition, the localized positron at
a vacancy has a reduced overlap with ion cores leading to a
considerable decrease in annihilation with high-momentumcore electrons. The localized positron has time to interactwith the host lattice during its lifetime of >150 ps and
enlarge the average open volume of the vacancy by repellingneighboring positive ion cores.
A negatively charged impurity atom or an intrinsic point
defect can bind positrons at shallow states even if these
defects do not contain open volume. Being a positive particle,the positron can be localized at the hydrogenic (Rydberg)state of the Coulomb field around a negatively charged center.The situation is analogous to the binding of an electronto a shallow donor atom. The positron binding energy at the
negative ion can be estimated from simple effective-mass
theory:
E
ion¼13:6e Vm/C3
m0/C15Z2
n2¼10–100 meV ; (6)
where /C15is the dielectric constant, m/C3is the effective mass of
the positron, Zis the charge of the negative ion, and nis the
quantum number. With Z¼1–3andn¼1–4, Eq. ( 6) typi-
cally yields Eion¼10–100 meV , indicating that positrons can
be thermally excited from the Rydberg states at 100–300 K.
The hydrogenic positron state around a negative ion
has a typical extension of 10–100/C23Aand thus positrons
probe the same electron density as in the defect-free lattice.As a consequence, the annihilation characteristics (positronlifetime, momentum density of annihilating pairs) are notdifferent from those in the lattice. Although the negativeions cannot be identified with the experimental parameters,information on their concentration can be obtained in the
positron lifetime and Doppler broadening experiments when
they compete with vacancies in positron trapping.Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1589
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The positron transition from a free Bloch state to a local-
ized state at a defect is called positron trapping. The trappingis analogous to carrier capture. However, it must be fastenough to compete with annihilation. The positron trapping
rate/C20
Dinto a defect Dis proportional to the defect concen-
tration ½D/C138,/C20D¼/C22D½D/C138=Nat, where Natis the atomic den-
sity of the host lattice. The trapping coefficient /C22Ddepends
on the defect and the host lattice. Since the positron bindingenergy at vacancies is typically >1e V , the thermal escape
(detrapping) of positrons from the vacancies can usually be
neglected. Because of the Coulomb repulsion, the trapping
coefficient at positively charged vacancies is so small that thetrapping does not occur during the short positron lifetimeof a few hundred picoseconds ( Puska, Corbel, and Nieminen,
1990 ). Therefore, the positron technique does not detect
vacancies or other defects in their positive charge states.The trapping coefficient at neutral vacancies is typically
/C22
D/C251014–1015s/C01, independent of temperature. The posi-
tron trapping coefficient at negative vacancies is typically/C22
D/C251015–1016s/C01at 300 K. The experimental fingerprint
of a negative vacancy is the increase of /C22Dwith decreas-
ing temperature. The T/C01=2dependence of /C22Dis simply
due to the increase of the amplitude of the free-positron
Coulomb wave in the presence of a negative defect as the
thermal velocity of the positron decreases. The temperaturedependence of /C22
Dallows one to experimentally distinguish
negative vacancy defects from neutral ones.
The positron trapping coefficient /C22ionat the hydrogenic
states around negative ions is of the same order of magnitudeas that at negative vacancies. Furthermore, the trapping co-
efficient exhibits a similar T
/C01=2temperature dependence.
Unlike in the case of vacancy defects, the thermal escape ofpositrons from the negative ions plays a crucial role at usualexperimental temperatures. The principle of detailed balanceyields the following expression for the ratio of detrapping andtrapping rates ( Manninen and Nieminen, 1981 ):
/C14
/C20¼1
cion/C18m/C3kBT
2/C25ℏ2/C193=2
e/C0Eion=kBT; (7)
where cionis the concentration of the negative ions. Typically
the negative ions (shallow traps) influence positron annihila-tion at low temperatures ( T<100 K ), but the ions are not
observed at high temperatures ( T>300 K ), where the escape
rate is high.
It is worth noting that the determination of absolute
defect concentrations based on positron experiments depends
directly on the knowledge of the trapping coefficient, whilethe comparison of defect concentrations (say, in two differ-ently doped semiconductor samples) gives accurate propor-tions even when the trapping coefficient is not known. Inextensively studied semiconductors, such as GaAs, GaN, and
ZnO ( Saarinen et al. , 1995 ;Oila et al. , 2003 ;Tuomisto,
Saarinen, Look, and Farlow, 2005 ), the cross correlation of
optical, electrical, and positron experiments has narroweddown the trapping coefficient of negatively charged cationvacancies to /C22
V/C0/C25ð2–3Ț/C21015s/C01. Theory ( Puska and
Nieminen, 1994 ) predicts that the neutral-vacancy trapping
coefficient should be a factor of 2–3 lower than that
of negatively charged vacancies; hence often the value of
/C22V/C251/C21015s/C01is used for neutral vacancies.3. Trapping model
The practical situation during a measurement, where
only one positron at a time is in the sample, can be described
by a relatively simple kinetic rate model (time-dependent
diffusion equation):
@nðr;tȚ
@t¼Dr2nðr;tȚ/C0/C22țr/C1½nðr;tȚE/C138/C0/C21nðr;tȚțS;
(8)
where nðr;tȚis the probability density of finding a delocal-
ized (free) positron at the position rand time t,Dis the
diffusion coefficient, and /C22țis the positron mobility. In the
sink term /C21is the sum of the ‘‘free’’-positron annihilation rate
/C21Band the trapping rates to the defects in the system /C21¼
/C21BðrȚțP
i/C20iðrȚ. In principle Dand/C22țcan also be functions
of the position r, but only in layered systems where they are
constant throughout the layer (similar to /C21B), while /C20iðrȚis
smooth and follows the defect profile. In the simplest case thesource term SðrȚvanishes, but if positrons are allowed to
escape from the trapped states at the defects, the source termis non-negligible ( S¼P
i/C14inD;i). In practice the three spatial
dimensions can be reduced to just one ( x), as both the lateral
straggling and the spot size of the implanted positrons essen-tially result in the experimental data being spatially averagedin the plane perpendicular to the main implantation direction.
The initial condition nðx;0Țis given by the positron im-
plantation profile PðxȚ(initially the positron is free, so the
probability of finding a positron in a trapped state at a defectis initially /C170), while the boundary conditions can be usu-
ally thought of as being those of a semi-infinite systemassuming thick enough samples so that positrons implantedfrom one side do not reach the other:
nðx;0Ț¼PðxȚ; (9)
lim
x!1nðx; tȚ¼0; (10)
D@n
@x/C12/C12/C12/C12/C12/C12/C12/C12x¼0/C0/C22țEð0Ț¼/C23nð0;tȚ: (11)
Here /C23represents the positron transition rate to states at
the sample surface. The simplest kind of experiment, where
positrons emitted by a radioactive source are directly injected
into a sample, further simplifies the above expressions. This isbecause the data are essentially averaged over a wide spreadof implantation depths, effectively removing the spatial di-mension from Eq. ( 8). The situation can then be described by
a set of rate equations ( Ndifferent defects):
dnB
dt¼/C0/C18
/C21BțX
j/C20j/C19
nBțX
j/C14jnD;j; (12)
dnD;j
dt¼/C20jnB/C0ð/C21D;jț/C14jȚnD;jðj¼1;…;NȚ: (13)
Here the probability of a positron being in the free state is
nBðtȚand the probability of it being in a trapped state at defect
jisnD;jðtȚ. The corresponding annihilation, trapping, and
escape rates are given by /C21B,/C21D;j,/C20j, and/C14j, respectively. In
practice /C14j/C2220only for shallow positron traps (negative-ion-
type defects) at sufficiently high temperatures. The boundary1590 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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and initial conditions in Eq. ( 9) are simplified to nBð0Ț¼1
andnD;jð0Ț¼0. As an example, in the case where positrons
are trapped at one kind of vacancy defect ( V) and one kind of
shallow trap ( st), the above set of equations becomes
dnB
dt¼/C0 ð /C21Bț/C20Vț/C20stȚnBț/C14stnst; (14)
dnV
dt¼/C20VnB/C0/C21VnV; (15)
dnst
dt¼/C20stnB/C0ð/C21stț/C14stȚnst: (16)
Applying the initial condition, the above equations can be
solved and the probability of a positron to be alive at time tis
obtained as
nðtȚ¼nBðtȚțnVðtȚțnstðtȚ¼X
iIiexpð/C21itȚ: (17)
This means that exponential decay should be observed in
experiments. The experimental lifetime spectrum in fact
measures the probability of positron annihilation in the timeinterval t/C1/C1/C1tțdt, and hence the lifetime spectrum
/C0dnðtȚ=dtis in this case composed of a sum of three com-
ponents. The fractions of positron annihilations at variousstates are in this example given by
/C17
B¼1/C0/C17V/C0/C17st; (18)
/C17V¼/C20V
/C21Bț/C20Vțð/C20st=1ț/C14st=/C21stȚ; (19)
/C17st¼/C20st
ð1ț/C14st=/C21stȚ½/C21Bț/C20Vțð/C20st=1ț/C14st=/C21stȚ/C138:(20)
These expressions are useful as they can be compared
with experimentally determ ined time-averaged quantities
such as the average positron lifetime and parametersdescribing the Doppler broadening of annihilation radia-tion, as these parameters ( P) measure the superposition of
the annihilations over all positron states: P¼/C17
BPBțP
j/C17D;jPD;j. Depth-resolved analysis of the latter is possible
when using a conventional slow positron beam ( Schultz and
Lynn, 1988 ). Then one can employ the steady-state version
of Eq. ( 8). Often, especially in the case of thin semicon-
ductor epilayers where the vacancy defect concentrationstend to be high, the diffusion can be neglected altogetherand the positron implantation profile in Eq. ( 2)p r o v i d e sa
sufficient approximation of the depth distribution of the
positron signal. However, in many cases solving the steady-
state version of Eq. ( 8) and fitting it ( van Veen et al. ,1 9 9 1 )
to data measured in layered str uctures provides additional
insight to the experimental observations. For more detaileddiscussions about the trapping model, see, e.g., Saarinen,
Hautoja ¨rvi, and Corbel (1998) and Krause-Rehberg and
Leipner (1999) .
B. Positron lifetime spectroscopy
1. Experimental details
A positron lifetime experiment can be performed in a
relatively simple way by using a radioactive22Napositron
source.22Nadecays through the /C12țprocess, producing apositron and a neutrino, leaving an excited22Nenucleus that
rapidly decays through the emission of a 1.2745 MeV/C13photon. This photon can be used as a start signal for the
positron lifetime measurement, while the stop signal is given
by one of the two 511 keV annihilation /C13photons.
In practice the positron source material is in the form of
NaCl which is typically stored as a water solution. Theexperiment is performed by sandwiching the positron sourcebetween two identical sample pieces. This can be done either
by depositing some of the NaCl directly on one of the
samples, and then placing the other on top of it, or by firstmaking a sealed positron source through packaging some ofthe NaCl in thin foil. Common foil choices are Al, Ni, andsometimes a polymer such as Kapton. The packaging solution
is preferable for reuse of source material, while the metal foils
allow for a wider range of measurement temperatures. Thesource package needs to be made as thin as possible in orderto ensure that a maximal fraction of positrons emitted by thesource enters the samples and that as few positrons as pos-sible annihilate in the source itself: e.g., typical Al-foil
thickness that is used is 1:5/C22m, and at most two layers of
foil are on each side of the deposited NaCl. Typical activityof such a source is 10–30/C22Ci[ð0:3–1Ț/C210
6Bq]. Such a
sample-source-sample sandwich can then be placed on asample holder connected to a temperature control system,
and the experiment can be designed in such a way as to allow
for, e.g., sample illumination.
The lifetime experiment itself is performed by detecting
the/C13photons serving as start and stop signals with two
relatively large scintillating detectors (lateral dimensionsand thicknesses of the order of centimeters) coupled with
photomultiplier tubes [see, e.g., Nissila ¨et al. (2001) ], each of
which is tuned and optimized for one of the two photons.Detector geometry is optimized with respect to overallefficiency (covering as wide a fraction of the solid angleas possible), taking into account restrictions imposed by
different scintillator materials. As an example, plastic
scintillators can be used in a simple collinear geometry wherethe sample-source-sample sandwich is placed on the axisdefined by the two detectors and the detectors are put asclose to each other as possible. On the other hand, special care
needs to be taken when using BaF
2scintillators, which benefit
from a significantly higher detection efficiency and enablebetter resolution, but whose high- ZBa causes strong /C13
scattering ( Becvar et al. , 2000 ).
In a conventional lifetime measurement the two detector
signals are analyzed with analog nuclear instrumentation
electronics: constant fraction discriminators to choose pho-
tons of correct energy, a time-to-amplitude converter, lettingthrough only pulses spaced close enough in time, and amultichannel analyzer (MCA) connected to a measurementcomputer. The result is a histogram of annihilation events as a
function of time differences between the start and stop sig-
nals, i.e., the positron lifetime spectrum. Typical time inter-vals for each MCA channel are of the order of 25 ps. Themodern and considerably cheaper way of doing the same isthrough direct digitization (fast analog-to-digital conversion)of the detector pulses and performing the signal analysis by
software ( Rytso¨la¨et al. , 2002 ;Saito et al. , 2002 ;Nissila ¨et al. ,
2005 ;Becvar, Cizek, and Prochazka, 2008 ). The advantage ofFilip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1591
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the digitized measurement is that it also allows for novel
functionalities such as postmeasurement signal analysis(Rytso ¨la¨et al. , 2002 ;Becvar, Cizek, and Prochazka, 2008 )
and, e.g., efficient pump-probe measurements with high fre-
quencies ( Ma¨ki, Tuomisto et al. , 2011 ;Ma¨kiet al. , 2012 ).
Figure 2shows a typical lifetime spectrum collected with a
positron lifetime spectrometer, containing roughly 2/C210
6
detected emission-annihilation (lifetime) events. The expo-
nential functional form is clearly visible, but there are a fewfeatures that need to be understood prior to analyzing the datain detail. These are (i) the background noise, (ii) the timeresolution, and (iii) annihilations in the source. At first glance
the existence of background noise is striking, as this is a
true coincidence measurement and there should not be anyrandom background noise. However, positrons are notemitted by the source in a deterministic way, and even ifthe source activity is only 10
6Bq, which gives an average
time difference of 1/C22sbetween two decays, some of the
positrons are emitted very rapidly one after the other and
produce false coincidences. Indeed, the background level inthis kind of positron lifetime experiment is completelydetermined by the source activity ( Knoll, 2000 ) and the
peak-to-background ratio can be improved only by reducingthe source activity (and increasing the measurement time).
It is evident from Fig. 2that the transition from the left-
hand-side background to the event peak around t¼0is far
from being sharp, indicating that the time resolution has a
non-negligible width on the scale of the measurement. Thismeans that the experimental data need to be modeled by thesum of exponentials ( 17) convoluted with the timing resolu-
tion function of the measurement setup. Dominant effects onthe timing resolution come from the size of the scintillators
(this is partially an optimization problem) and the settings on
the photomultiplier tubes ( Becvar et al. , 2000 ;Nissila ¨et al. ,
2001 ). The functional form of the timing resolution is im-
portant as well: a Gaussian form (over sufficiently manyorders of magnitude) indicates that the setup produces onlystatistical error in the measurement, does not alter the ex-ponential decay components, and in addition makes data
analysis simpler as fewer parameters need to be fitted ordetermined. In Fig. 2the resolution function of a typical
setup is shown together with the lifetime spectrum. The fullwidth at half maximum (FWHM) defining the Gaussian
resolution function is 250 ps. It should be noted that the
exponential decay component, seen as a slope on thesemilogarithmic plot, corresponds to a lifetime of 160 ps.A Gaussian resolution function with FWHM of 250 ps allows
for reliable determination of exponential decay components
down to roughly 30 ps as long as their intensity is high
enough, while the narrowest (non-Gaussian) resolutionsachieved in experimental setups go down to /C24140 ps
(Becvar, Cizek, and Prochazka, 2008 ). The inset in Fig. 2
points out another important aspect of the resolution function:the data close to t¼0are completely defined by the
functional form of the resolution function, and hence very
rapid decays with low relative intensity (when several decaycomponents are present) cannot be reliably analyzed.
The annihilations in the positron source cause small dis-
tortions in the measured lifetime spectrum and need to betaken into account when analyzing the data. Figures 3and4
show three kinds of spectra: the ‘‘raw’’ data (Fig. 3), two
spectra with subtracted background (upper panel of Fig. 4),
and finally the same two spectra with subtracted sourcecomponents (lower panel of Fig. 4). As an example, the
exponential decay components produced by the source(and by the method of the measurement itself) in the caseof an Al-sealed
22Nasource can be approximated as one
component coming from the heavily dislocated Al foil, the
NaCl, and surface effects at the interfaces between NaCl andAl, and Al and the samples. The contribution of the Al foilcan be estimated from the average Zof the sample and the
thickness of the foil ( Bertolaccini and Zappa, 1967 ), as the
backscattering of positrons from the sample surface strongly
influences the probability of positron stopping in the Al foil.The corresponding lifetime component is 210–215 ps, and fortypical semiconductors, the relative intensity is 1%–3%. TheNaCl and the surface effects produce two kinds of lifetime
components: roughly 400 and 1500 ps. The latter is caused by
positronium formation at the surfaces, and its importance canbe estimated through comparison of background determinedby fitting and averaging. For a carefully made source and
102103104105Counts
6 4 2 0
Time (ns) measured lifetime spectrum
Gaussian resolution function
single exponential function
8FWHM
FIG. 2. Positron lifetime spectrum obtained with a typical spec-
trometer. The dashed and dotted lines show the detector resolution
and the ‘‘ideal’’ spectrum. The inset shows a magnification of thet¼0range.102103104105Counts spectrum with a single component
spectrum with two components
6 4
Time (ns)2 08
FIG. 3. Two ‘‘raw’’ positron lifetime spectra measured in different
samples. The spectra are shown without the left-hand-side back-
ground, as in practice the window for optimal analysis starts aroundt/C25/C00:1n s.1592 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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sample-source-sample sandwich the relative intensity of this
component can be as low as 0.05%, but is still meaningfuldue to the long lifetime. The 400 ps component is often thehardest to evaluate and requires high-quality referencesamples where one can safely assume that no positron trap-ping at vacancy defects should be observed. On the other
hand, the relative intensity of this component is usually in the
range of 2%–6%, making it rather easily fittable. The valuedepends strongly on the material and sample surface quality.
The source corrections, even if typically their total inten-
sity in the measured spectrum is only a few percent, are at theroot of the positron experiments being an inherently com-
parative technique, where changes of some sort are moni-
tored. These changes can be induced, e.g., by change of themeasurement temperature or by difference in the defect con-tent (either identity or density) between samples of the samematerial. This comparative nature is further elucidated whenthe analysis of the lifetime data is considered.
2. Data analysis
Following Eq. ( 17), the experimental positron lifetime
spectrum can be expressed as the convolution of thepreferably Gaussian resolution function RðtȚwith the sum
of exponential functions:
EQ-TARGET ;temp:intralink-;d21;76;155
NðtȚ¼Z1
/C01dsRðsȚ/C18
/C0dn
dtðt/C0sȚ/C19
: (21)
The constant background can simply be neglected in the
following considerations, as its subtraction does not affect theconvolution. The Gaussian form of the resolution functionhas the benefit of not affecting the time expectation value(center of mass) of the spectrum, and of not affecting thevalues of exponential components. In practice, Eq. ( 21) can
be fitted to the experimental data by assuming a number of
exponential components. Components are reliably fittedwhen the annihilation rates (or lifetimes) differ by at least
a factor of 1.3–1.5, meaning that usually at most threecomponents can be extracted. The fitting parameters includethe FWHM of the resolution function and the intensities I
i
and annihilation rates /C21i(slopes on the semilogarithmic plot)
of the lifetime components. The subtraction of source com-ponents is performed through the procedure explained above,prior to final analysis of the data.
In addition to the relatively straightforward fitting of
Eq. (21) to the experimental data, more sophisticated methods
of data analysis can be employed. The two most prominent
approaches are based on the inverse Laplace transformof Eq. ( 21)(van Resandt, Vogel, and Provencher, 1982 ;
Gregory and Zhu, 1990 ) and on the Bayesian-probability-
inspired maximum-entropy method ( Hoffmann et al. , 1993 ;
Shukla, Peter, and Hoffmann, 1993 ). These two methods have
an important common feature, i.e., the number of lifetime
components to be found is not a priori fixed. In addition, both
methods are in principle better suited to cases where insteadof discrete lifetime components, continuous lifetime distri-butions can be expected. The higher level of sophisticationcompared to the straightforward least-squares fitting makesthese methods also somewhat more sensitive to noise in the
experimental data.
Figure 3shows two raw lifetime spectra (each with
2/C210
6annihilation events) measured in two exemplar
samples. Evidently the spectrum plotted with full circleshas an additional component compared to the one plottedwith open circles. After 4 ns, a background is reached in bothspectra, and both the background and the intensity of the
1500 ps component can be determined from the data around
9 ns (denoted by the rectangle in the figure). Figure 4shows
the same lifetime spectra, the upper panel after backgroundsubtraction and the lower panel after subsequent sourcecomponent subtraction (215 ps, 2.8%; 400 ps, 3.8%; and1500 ps, 0.08%). Clearly the data with open circles exhibit
only one exponential component, while two exponential
components can be seen in the data represented by fullcircles.
Fitting the single-component data gives 1=/C21
1¼/C281¼
110/C61p s, which coincides with the ‘‘center of mass’’ of
the spectrum. This measurement is from a natural diamondsample ( Ma¨kiet al. , 2009 ), and the result is typical of that
material. Fitting the two-component data gives 1=/C21
1¼/C281¼
125/C65p s,1=/C212¼/C282¼420/C620 ps , and I2¼1/C0I1¼
ð35/C61Ț%, where the error bars are due to statistical uncer-
tainty in the fitting. The two components mean that positronsannihilate at two distinct states: the simplest assumption isthat one of them is the delocalized state in the lattice and the
other a localized state at a (large) vacancy.
2Hence the set of
Eqs. ( 14) is reduced to two equations:
dnBðtȚ
dt¼/C0/C21BnB/C0/C20VnB; (22)101102103104105Counts
3 2 1 0
Time (ns)4 single component
source effect subtracted
two components
source effect subtracted101102103104105Counts single component
background subtracted
two components
background subtracted
FIG. 4. Single- and two-component positron lifetime spectra after
background subtraction (upper panel) and subsequent source effect
subtraction (lower panel).
2In fact, here this is not strictly the case—the 125 ps lifetime
component is a mixture of the reduced bulk lifetime and the positron
lifetime in dislocations ( Ma¨kiet al. , 2009 ). We omit this detail for
the sake of simplicity of the demonstration of how the reasoningproceeds.Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1593
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dnVðtȚ
dt¼/C0/C21VnVț/C20VnB: (23)
The full solution of this pair of equations [remembering
thatnBð0Ț¼1andnVð0Ț¼0] is of the form
nðtȚ¼I1expð/C0/C211tȚțI2expð/C0/C212tȚ; (24)
where /C211¼/C21Bț/C20Vand/C212¼/C21V. Hence the longer of the
two experimentally determined components is directly the life-time specific to the vacancy defects in question /C28
2¼/C28V, while
the shorter of the components is the ‘‘reduced bulk lifetime,’’
where the apparent positron lifetime in the lattice is shortenedby the trapping process: /C28
1¼/C21/C01
1¼ð/C28/C01
Bț/C20VȚ/C01.
The large number of annihilation events and the stability of
the state-of-the-art positron lifetime spectrometers make the
center of mass of the lifetime spectrum a highly accuratequantity: the standard deviation is as low as 0.2 ps ( Tuomisto,
Saarinen, Look, and Farlow, 2005 ). The importance of this
parameter is accentuated by the fact that it coincides with theaverage positron lifetime defined by the annihilation fractions
[Eq. ( 18)]:
/C28
av¼/C17B/C28Bț/C17V/C28V¼/C28c:m:¼I1/C281țI2/C282; (25)
where /C17Bț/C17V¼I1țI2¼1. This equation relates the
experimental spectrum directly to the kinetic trapping model
and gives the possibility of using the statistically accurateaverage positron lifetime as a key parameter in lifetimeanalysis. Further, the statistical accuracy of /C28
avprovides the
possibility of using it in data interpretation even in caseswhere the fitting of the experimental data represents only abest fit without physical interpretation. In the two-component
data considered here we have /C28
av¼232 ps .
In the simple case considered here the annihilation
fractions are reduced to
/C17B¼/C21B
/C21Bț/C20Vand /C17V¼/C20V
/C21Bț/C20V: (26)
The experimentally determined parameters /C28av,/C28V, and/C28B
give now the possibility to estimate the trapping rate to the
vacancy defect in question. The above considerations give
/C20V¼/C21B/C28av/C0/C28B
/C28V/C0/C28av: (27)
Hence the concentration of the vacancies can be determined
from the experimental data. The form of Eq. ( 27), however,
shows that the concentration determination relies on the
knowledge of /C28B, i.e., on having at hand a reference sample
where no positron trapping at vacancy-type defects is ob-served. The absolute value of /C28
Bis less important, as varia-
tions between spectrometers and source corrections can easilyproduce a 5 ps offset, but being able to measure a baselinewith the setup and source at hand is crucial for the highest
achievable accuracy near the lower sensitivity limit. At more
elevated vacancy concentrations this is not as important. Thesample in the two-component example is a single crystal ZnOsample grown in a way that produces a high concentration ofrelatively large vacancy clusters with /C28
B¼170 ps .
The sensitivity limit is given by the ability to distinguish
the average lifetime from /C28B. The absolute lowest limit is
given by the statistical accuracy of 0.2 ps of /C28av, but is in most
cases limited to roughly 1 ps due to the inherent uncertaintyin source corrections and reference reliability. The sensitivity
is additionally a strong function of both /C28av/C0/C28Band/C28V/C0
/C28avas seen in Fig. 5. It can be seen that the highest sensitivity
to differences in vacancy concentrations is in the midrange ofthe changes in lifetimes, corresponding to roughly 10
17cm/C03
in vacancy concentration. The lower limit for sensitivity is a
result of reliably observing an increased average positron
lifetime of 1 ps above /C28B. The upper sensitivity limit is called
saturation trapping, as it is due to all annihilations comingfrom trapped positrons and corresponds to a vacancy con-centration of roughly 10
19cm/C03. The limit is farther away
from the absolute maximum of /C28av¼/C28Vdue to the higher
uncertainty in the fitting of the lifetime components: itis in practice impossible to distinguish the situation fromsaturation trapping when j/C28
av/C0/C28Vj<10 ps .
The above difficulties related to the source-originated back-
ground level and to the source corrections can be largely avoidedby performing the positron lifetime experiment with a time-
stamped positron beam ( Scho¨dlbauer et al. ,1 9 8 7 ;Suzuki et al. ,
1992 ). The data analysis is, however, faced with other chal-
lenges such as an inherently non-Gaussian resolution function(even if very narrow) in pulsed beams ( Reurings and Laakso,
2007 ). The time-stamped slow-positron beams have the benefit
of enabling lifetime measurements in thin films, layered struc-
tures, and near-surface regions, but for a detailed analysis ofthe lifetime signals the full time-dependent positron diffusionequation [Eq. ( 8)] should be used. It should be noted that, due to
their complexity, only a handful of positron beams capable of
time-resolved experiments have been constructed.
3. Information revealed by the positron lifetime
The lifetime of a positron trapped in a vacancy is longer
than in the ‘‘perfect’’ lattice. This is because the averageelectron density affecting the positron at the vacancy is lowerthan in the bulk, and correspondingly, the annihilation rate is
lower. The lifetime increase for a monovacancy-sized defect is
usually of the order of 20–40 ps ( Saarinen et al. , 1995 ,1999 )
in elemental semiconductors and compound semiconductorswhere the sizes of the constituents are similar (e.g., GaAs).230
220
210
200
190
180
170Average lifetime (ps)
101510161017101810191020
Neutral vacancy concentration (cm -3 )10141015101610171018Negative vacancy concentration (cm -3 ), measurement at 10 KSensitivity (arb. units)
FIG. 5. Vacancy concentration sensitivity of the positron lifetime
measurement in a system where /C28B¼170and/C28V¼230 ps . The
highest sensitivity is midrange, where a small change in vacancy
concentration produces a substantial change in the average positronlifetime.1594 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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In size-mismatched compounds (e.g., GaN) the monovacancy
of the larger atom can produce a lifetime that is 70–80 pslonger than in the bulk ( Saarinen et al. , 1997 ;Tuomisto et al. ,
2003 ), while quite typically the monovacancy of the smaller
atom does not seem to trap positrons. As an example, Table I
shows lifetimes calculated for small unrelaxed vacancy clus-
ters in Si ( Hakala, Puska, and Nieminen, 1998 ); for details on
how calculations are performed, see Sec. III. The lifetime
increases with increasing open volume of the vacancy.
Simultaneously, the core annihilation fraction /C21
c=/C21, i.e., the
fraction of annihilations with core electrons of all annihilationevents, decreases, due to the reduced overlap between thepositron density and the atomic core orbitals. The core anni-hilation fraction is typically quite low and thereby meaning-
less regarding the positron lifetime. However, the effect of the
core electrons is more significant in the high-momentumregion of the momentum density of annihilating electron-positron pairs (see Sec. II.C), where the chemical information
is typically contained. For vacancy clusters larger than 4–5
missing atoms, the positron lifetime is already relatively in-
sensitive to the size. In fact, as the size of the vacancy clusterincreases, the lifetime saturates toward the lifetime of thenegative positronium ion (or the spin-averaged positronium
lifetime), 479 ps ( Mills, 1981 ;Ceeh et al. , 2011 ).
Positron lifetimes of singlet and triplet electron-positron
states at paramagnetic vacancy defects have been consideredtheoretically ( Alatalo, Puska, and Nieminen, 1993 ) but for Si
and GaAs it was concluded that the splitting in the defect
lifetime component is too small to be observed.
In general, with the exception of small vacancy defects
such as monovacancies and divacancies, the positron lifetimecan give only an order-of-magnitude estimate of a vacancydefect’s size. Further, and most importantly, it is very
insensitive to the chemical environments of the detected
vacancy defects. For unambiguous identification of the de-fects, coincidence Doppler broadening measurements (seeSec. Sec. II.C) of the high-momentum part of the momentum
density of annihilating positron-electron pairs are needed.
Also one needs to be able to understand the indirect infor-mation contained in the measured data. Theoretical modelingis especially helpful in this regard.
C. Doppler broadening spectroscopy
1. Experimental details
Doppler broadening spectroscopy does not require time-
resolved experiments and is hence applied regularly in boththe sample-source-sample sandwich kind of experiments de-
scribed in the previous section and with slow-positron beams.The experimental procedure when performing measurementswith slow-positron beams is rather straightforward after such
a beam has been constructed. The positron energy can be
electrostatically tuned, typically in the range 0.1–50 keV. Theannihilation photons are detected with high-purity Ge (HPGe)detectors that possess a high energy resolution. In practice thedetector signals are collected to a computer with a MCA,while digitization of Doppler broadening experiments is a
work in progress. One of the most important aspects in the
instrumentation is effective stabilization of the peak positionthrough online tuning of amplifier gain.
The motion of the annihilating electron-positron pair
causes a Doppler shift in the annihilation radiation /C1E¼
cp
L=2, where pLis the longitudinal momentum component
of the pair in the direction of the 511 keV annihilation photon
emission:
/C26ðpLȚ¼ZZ
dpxdpy/C26ðpȚ: (28)
The electron momentum distribution /C26ðpȚcauses the broad-
ening of the 511 keV annihilation line. The shape of the511 keV peak gives thus the one-dimensional momentumdistribution /C26ðp
LȚof the annihilating electron-positron pairs.
A Doppler shift of 1 keV corresponds to a momentum value
of 0.54 atomic units (a.u.). Another unit used traditionally
is 1 mrad ð¼10/C03m0cȚ, with the correspondence 1 keV ¼
3:91 mrad . This geometric unit has its origin in ACAR
experiments that can be used to detect the same effects, i.e.,the electron momentum distribution. The momentum resolu-tion in these experiments is better than in the energy-resolvedexperiments, but the detection efficiency is much lower. As in
defect spectroscopy it is often necessary to perform large
series of measurements, e.g., as a function of temperature ordetection depth, and Doppler broadening spectroscopy hasbeen the method of choice. ACAR and 2D-ACAR have beenused in defect studies much less ( Saito, Oshiyama, and
Tanigawa, 1991 ;Ambigapathy et al. , 1994 ).
The typical resolution of a HPGe detector is around
1–1.5 keV at 511 keV . This is considerable compared to thetotal width of 2–3 keVof the annihilation peak, meaning that theexperimental line shape is strongly influenced by the detectorresolution (see Fig. 6). In addition, the peak-to-background
ratio is rather low, only about 10
2. Therefore, various shape
parameters, where parts of the annihilation peak are integrated,
are used to characterize the 511 keV line. These parameters are
explained in more detail in Sec. II.C.2 .
The resolution and peak-to-background ratio of the
Doppler broadening measurement can be significantly im-proved by measuring the annihilation radiation with two(collinear) detectors and imposing coincidence conditions
(Alatalo et al. , 1995 ;Asoka-Kumar et al. , 1996 ;Szpala
et al. , 1996 ). The collinearity of the detectors imposes the
first condition: the two annihilation photons need to beemitted in roughly opposite directions. The second conditionis the time coincidence: the two annihilation photons need tooriginate from the same annihilation event. These two con-ditions alone improve the peak-to-background ratio to about
10
4. The finest improvement comes, however, from the third
condition and is obtainable only when two HPGe detectorsTABLE I. Positron lifetimes ( /C28) and relative core electron anni-
hilation rates for the perfect bulk lattice and for the ideal vacancyclusters in Si. /C21
c, and /C21are the core and total annihilation rates,
respectively. Reproduced from Hakala, Puska, and Nieminen
(1998) .
System /C28(ps) /C21c=/C21(%)
Bulk 221 2.19
V 254 1.48
V2 299 0.97
V3 321 0.79
V4 330 0.72
V5 355 0.57Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1595
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

are used (the first two conditions can be achieved by using
one HPGe detector and another, say, scintillating detector
with poorer energy resolution to gate the HPGe detector).This requires that the sum of the energies E
1andE2of the
two coincidentally observed annihilation photons obeys
E1țE2¼2m0c2/C0Eț;/C0/C251:022 MeV ; (29)
i.e., the photons carry the rest mass of the positron and
electron (each 511 keV) reduced by the binding energy ofthe pair E
ț;/C0, which can in fact be neglected as it is of the
order of a few eV. This kind of two-detector measurements
produces a two-dimensional matrix, where on one diagonalone finds the coincidence spectrum with an improved peak-
to-background ratio of about 10
6, while the real resolution
function of the setup can be found on the other diagonal. Theresolution of the measurement is in fact narrowed by a factor
of/C24ffiffiffi
2p
compared to the resolution of the individual detec-
tors. More details about the procedures can be found in, e.g.,
Asoka-Kumar et al. (1996) . Figure 6shows the Doppler-
broadened annihilation peak measured with one HPGedetector, and the same sample measured with a two-HPGedetector coincidence system (data integrated from thediagonal of the 2D matrix). The high background in the
single-HPGe-detector data can be subtracted to some extent
(dashed line in Fig. 6) prior to analysis.
2. Data analysis
The low-momentum shape parameter Sis defined as the
ratio of the counts in the central region of the annihilation
line (see the upper panel of Fig. 7) to the total numberof the counts in the line. In the same way, the high-
momentum parameter Wis the fraction of the counts in the
wing regions of the line. Because of their low momenta (low
degree of localization), mainly valence electrons contribute to
the region of the Sparameter. On the other hand, only core
electrons have momentum values high enough to contributeto the Wparameter. Therefore, SandWare sometimes called
the valence and core annihilation parameters, respectively.
The practical choice of the integration windows for the Sand
Wparameters is straightforward. The Sparameter is defined
in such a way that it retains the statistical accuracy of
collecting a large number of annihilation events (typically
roughly 10
6) to the peak: usually S/C250:5. The lower limit
of the Wparameter integration window is chosen far enough
from the peak center in order to have a minimal contribu-tion from the free-electron distribution shown in Fig. 7.
Prior to parameter integration, the background is subtracted
(shown with a dashed curve in Fig. 6). The functional form is
due to incomplete charge collection in the HPGe detectors,resulting in the background level at a given energy being
proportional to the number of events at higher energies
(Knoll, 2000 ).100101102103104105Counts
-6 -4 -2 0 2 4 6
Electron momentum (a.u.)521 516 511 506 501Gamma energy (keV)
2 x HPGe coincidence single HPGe
single HPGe
background
subtracted
FIG. 6. Doppler broadening spectra obtained with a single HPGe
detector and with two HPGe detectors in coincidence. The highbackground in the single-HPGe-detector measurement can be sub-
tracted; the functional form is dictated by the incomplete charge
collection in the detector (dashed line). The resolution function of atypical HPGe detector is also shown for reference (dotted line).0.0010.010.11Momentum distribution (area normalized)30 20 10 0Electron momentum (10-3 m0c)
Gallium nitride
GaN lattice
Ga vacancy
S W "free electrons"
1.1
1.0
0.9
0.8
0.7
0.6Ratio of momentum distributions
4 3 2 1 0
Electron momentum (a.u.) Experimental Ga vacancy / GaN ratio
Calculated Ga vacancy / GaN ratio
FIG. 7. Folded experimental coincidence Doppler broadeningspectra for the GaN lattice and Ga vacancy. Typical integration
windows for the SandWparameters are shown (upper panel).
Experimental and calculated ‘‘ratio curves’’ of the momentumdensities (lower panel).1596 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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TheSandWparameters are time-averaged (or time-
integrated) quantities and thus behave similarly to the averagepositron lifetime. If saturation trapping at a certain kind ofvacancy defect produces parameters S
V,WVand a reference
sample produces parameters SB,WB, the measurement in a
sample for which /C28aveis given by Eq. ( 25) will give
S¼/C17BSBț/C17VSV;W ¼/C17BWBț/C17VWV: (30)
As a consequence, if SVandSBare known, the vacancy
concentration can also be determined from the ðS; WȚpa-
rameters. Furthermore, the links between these parametersenable the combination of positron lifetime and Doppler
broadening results and various correlations between /C28
av,S,
andWcan be studied. As an example, if a variety of samples
with varying concentrations of the same vacancy defect isstudied, the data will form a line in the ðS; WȚ,ðS; /C28
avȚ, and
ðW;/C28 avȚplanes. It should be noted that the Doppler experi-
ment does not allow the direct extraction of defect-specificparameters in the same way as the positron lifetime experi-
ment, as the data are always a superposition of annihilations
in various states. However, the annihilation fractions that canbe determined in positron lifetime experiments can be used toextract this information from Doppler data in some cases.
It is important to stress that the absolute values of the Sand
Wparameters are meaningless. They depend strongly on the
detector geometry, resolution, calibration, amplifier gain, and
MCA channel width, and in some cases on the direction ofmeasurement. The last point is in fact important: the crystallattice contains natural anisotropy transferred to the electronmomentum distribution, and measurements along differentlattice directions give slightly different results. Hence it iseven more important to perform comparative measurements
and preferably possess a reference sample where no positron
annihilation at vacancy defects is detected. In fact, it is quitetypical to express the SandWparameters as normalized to
theS
BandWBobtained in a reference sample. This reduces
the dependence of the data on the various aspects listed aboveto some extent, but not completely. As an example, even
the normalized parameters depend strongly on the detector
resolution. This means that data measured with one setupcannot be directly compared to those obtained with anothersetup; instead full analysis and interpretation of the results isnecessary prior to comparison.
The coincidence Doppler data shown in Fig. 6are typically
folded along the 0-momentum line (the symmetry of the peak
is evident), and adjacent points are summed together to
improve statistics and reduce noise especially in the wingregions. The result is shown in the upper panel of Fig. 7. The
high-momentum part of the Doppler broadening spectrumarises mainly from annihilations with core electrons andhence contains information on the chemical identity of the
atoms close to the annihilation site of the positron. The
difference between the data obtained in the GaN lattice anda Ga vacancy shown in Fig. 7is evident, but otherwise the
data look rather featureless. A similar normalization proce-dure as in the case of the SandWparameters, where the
whole distribution is normalized to that of the reference(GaN lattice in this case), results in the so-called ‘‘ratio
curve’’ shown in the lower panel of Fig. 7. This procedure
reveals details of the distribution, such as the above-unityvalue of the Ga vacancy data at 0 momentum, the shoulder at
/C241:5a:u:, and the low intensity above 2 a.u. The solid curve
is calculated with methods explained in detail in Sec. III.
3. Chemical information contained in Doppler spectra
Coincidence Doppler broadening measurements made
with a two-detector setup measure the momentum densityof annihilating electron-positron pairs accurately up to a veryhigh momentum of several atomic units or tens of 10
/C03m0c.
This region is dominated by annihilation with core electron
shells. The specificity of the technique to different chemical
elements is seen clearly in bulk measurements made forelemental metals and semiconductors ( Asoka-Kumar et al. ,
1996 ;Myler and Simpson, 1997 ;Ghosh et al. , 2000 ). The
high-momentum region of the spectra contains informationon the chemical elements close to the annihilation sites.
In measurements probing vacancy defects, atoms in their
surroundings can be distinguished from the host lattice atomsthanks to their differing core shells.
As a detailed example of chemical information contained
in spectra measured for vacancy-dopant complexes in semi-conductors, and to show how first-principles modeling canhelp to understand experimental results, we consider a work
on identification of vacancy-antimony complexes in highly
Sb-doped Si ( Rummukainen et al. , 2005 ). In this study,
atomic relaxations of defects were neglected for the mostpart. The inward relaxation of the vacancy defects predictedby the LDA was assumed to be canceled by the repulsiveforce of the positron on the neighboring ions. Figure 8shows
10-410-2100
10-410-2Momentum distribution (1 / a.u.)10-410-2
01234 5
Momentum (a.u.)10-410-2V
V-Sb
V-Sb2
V-P1s2s2p
Sb 4sSb 4pSb 4d
P 2sP 2pSb 4sSb 4pSb 4d(a)
(b)
(c)
(d)
FIG. 8 (color online). Computational Doppler spectra of (a) V,
(b)V-Sb, (c)V-Sb2, and (d) V-Pin Si. The thick (black) solid line
shows the full spectrum. The contributions of individual Si coreshells are shown with thin (black) solid lines, and dopant atoms’
core shell contributions by solid colored lines. Adapted from
Rummukainen et al. , 2005 .Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1597
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

Doppler spectra calculated for the monovacancy ( V)i nS ia s
well as for the vacancy-antimony and vacancy-phosphorus
complexes V-Sb,V-Sb2, and V-P. The neglect of atomic
relaxations enables us to focus solely on the chemical infor-mation contained in the spectra. In this case and in most othersystems containing atoms of more than one kind, the effect of
the chemical surroundings and differing core electron orbitals
is stronger than that of the detailed atomic geometry of thedefects.
Figures 8(a)–8(d) show for V,V-Sb,V-Sb
2, and V-P, re-
spectively, the total spectrum (thick solid lines) as well as the
contributions of the dominant core shells. At high momenta,the dominating shell is the Si 2pin all systems. For the
vacancy-dopant complexes the contributions of the impurity
atoms’ core shells are also shown (thick colored lines). This
calculation, like many others, is made using the model ofAlatalo et al. (1996) [see Eqs. ( 51) and ( 52)] and using a
parametrized positron wave function for annihilating core
electrons. Hence, the contributions of core shells look thesame on the logarithmic scale irrespective of the system.Their magnitudes are determined by the respective partial
annihilation rates /C21
LDA
j. Since the core electron shells of Si
and P, neighboring elements on the periodic table, are from the
positron’s point of view rather similar, the spectra of VandV-P
nearly coincide. The case of Sb is quite different thanks to thepresence of its 4dshell. Its intensity in the Doppler spectrum
increases with an increasing number of Sb nearest neighbors
comparable to or even higher than that of the Si 2pat and below
2 a.u. In a ratio spectrum [see Rummukainen et al. (2005) ] the
‘‘Sb fingerprint‘‘ is seen as a peak at this momentum. The
intensity of the peak correlates with the average number ofSb nearest neighbors around the monovacancy-sized vacancycomplexes in the sample.
The momentum density of annihilating electron-positron
pairs as measured by Doppler broadening or angular correla-
tion measurements is, in general, more anisotropic for thepositron’s delocalized bulk state than for positrons trappedat vacancies. For covalently bonded semiconductors such
as GaAs and Si ( Saito, Oshiyama, and Tanigawa, 1991 ;
Ambigapathy et al. ,1 9 9 4 ;Peng et al. , 1994 ;Hakala, Puska,
and Nieminen, 1998 ), the anisotropic part is significant. As
already discussed, upon positron trapping the (normalized)
spectrum becomes narrower due to increased relative contactwith low-momentum valence electrons at the vacancy. Further,
a vacancy’s spectrum is more isotropic. On the one hand, this is
because a localized positron does not sense much the orderedperiodic lattice with its directed bonds. On the other hand, it isbecause possible asymmetric vacancy defects and defect clus-
ters, which as such would produce an asymmetric momentum
spectrum, are typically oriented randomly in the sample. Themeasurement averages over different orientations and the
anisotropic spectra corresponding to individual orientations
sum up to a more isotropic spectrum. When comparing ex-perimental spectra with computational ones the averaging overorientations has to be performed in the modeling as well
(Hakala, Puska, and Nieminen, 1998 ). The 2D-ACAR tech-
nique and uniaxial stress provided an additional meansto observe the aligning of divacancies in Si in an experiment
(Tang et al. , 1997 ) paralleling the electron paramagnetic
resonance study of Watkins and Corbett (1965) .III. THEORY AND COMPUTATIONAL METHODS
The first models of positrons in solids interacting with
electrons were the electron gas models by Ferrell (1956) ,
Kahana (1960 ,1963) , and Carbotte and Kahana (1965) . The
study proceeded to real inhomogeneous metals involving
high-momentum (umklapp) components and core annihila-tion ( Carbotte, 1966 ;Salvadori and Carbotte, 1969 ;Fujiwara,
Hyodo, and Ohyama, 1972 ;Hede and Carbotte, 1972 ). Since
the early 1970s and the first model for positron trapping(Hodges, 1970 ) defect studies with their trapped positrons
became a separate line of research also on the theory side [for
early work, see Arponen et al. , 1973 ;Manninen et al. , 1975 ;
Gupta and Siegel, 1977 ;Hautoja ¨rviet al. , 1977 ;Gupta and
Siegel, 1980a ,1980b ]. In the theory and modeling aspect of
this review, we focus mostly on theory and methods usefulin theory-assisted defect identification, especially how to
accurately predict positron lifetimes and Doppler spectra
for direct comparisons with experiments. Concerning theoryand models on what happens prior to the annihilation event
(e.g., positron thermalization and trapping mechanisms and
rates), see Puska and Nieminen (1994) .
Important developments for the practical calculation of
positron states and annihilation in solids from the defect studies
point of view included in the 1970s and 1980s, for instance,
the positron pseudopotential theory ( Kubica and Stott, 1974 ;
Nieminen, 1975 ;Stott and Kubica, 1975 ;Stott and West, 1978 ),
two-component electron-positron density-functional models
(Chakraborty, 1981 ;Chakraborty and Siegel, 1983 ;
Nieminen, Boron ´ski, and Lantto, 1985 ;Boron ´ski and
Nieminen, 1986 ), the simple but powerful atomic superposition
method ( Puska and Nieminen, 1983 ), and the applications of
these to timely problems.
Section III.A discusses the formalism of the two-component
electron-positron density-functional theory. Sections III.B and
III.C go deeper into the practical modeling of trapped positrons
and the measurable annihilation parameters such as positron
lifetime and Doppler broadening. Section III.D discusses prac-
tical approximations used in the electron-positron density-functional calculations. Sections III.E andIII.F
discuss different
levels of approximations concerning self-consistency and
numerical implementation, most importantly the atomic super-position method and the application of different band-structure
schemes to the positron problem.
A. Two-component electron-positron density-functional theory
The density-functional theory (DFT) ( Hohenberg and
Kohn, 1964 ) within the Kohn-Sham method ( Kohn and
Sham, 1965 ) is the main workhorse in first-principles materials
modeling [see, e.g., Martin (2004) andDrabold and Estreicher
(2007) ]. Using supercell calculations one can predict bulk
properties and properties of defects in solids, including their
structure, formation enthalpies, diffusion barriers, ionizationlevels, etc. Most of the practical modeling of positrons
annihilating at lattice defects has during the past decades
been made using two-component electron-positron formula-tions of DFT ( Chakraborty, 1981 ;Chakraborty and Siegel,
1983 ;Nieminen, Boron ´ski, and Lantto, 1985 ;Boron ´ski and
Nieminen, 1986 ). Next we discuss the formalism as given by1598 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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Boron ´ski and Nieminen (1986) . The two-component approach
enables modeling of both delocalized and localized positrons,including the effect of the localized positron on the defect’ssurrounding electronic and ionic structure. Furthermore, the
direct connection with the usual one-component DFT and its
approximations, such as those for the exchange-correlationenergy, makes it rather straightforward to implement electron-positron calculations in the codes used by the electronic-structure and materials-modeling communities. The use ofstate-of-the-art electronic-structure packages enables one
also to easily gain, in addition to positron annihilation parame-
ters such as positron lifetime or momentum density of annihi-lating electron-positron pairs, complementary information onthe defects modeled.
In this section, we use the Hartree atomic units ( m¼ℏ¼
e¼4/C25/C15
0¼1).
In the two-component DFT for electron-positron systems,
the fundamental quantities are the electron and positron
densities n/C0ðrȚandnțðrȚ. Practical calculations are based
on a modified Kohn-Sham scheme, in which one models anoninteracting system with the electron and positron densitiesequal to those of the true interacting system using an energyfunctional ansatz of the form
E
tot½n/C0;nț/C138¼F½n/C0/C138țF½nț/C138/C0Z
drdr0n/C0ðrȚnțðr0Ț
jr/C0r0j
țZ
drVextðrȚ½n/C0ðrȚ/C0nțðrȚ/C138
țEe-p
c½n/C0;nț/C138: (31)
In Eq. ( 31) the third term is the classical electrostatic
interaction between electrons and positrons. The fourth
term is the interaction energy with the external potential
(the nuclei are treated within the Born-Oppenheimer approxi-mation). F½n/C138is the one-component functional
F½n/C138¼T
s½n/C138ț1
2Z
drdr0nðrȚnðr0Ț
jr/C0r0jțExc½n/C138; (32)
where Ts½n/C138is the kinetic energy of noninteracting particles
with density nin the two-component system with densities
n/C0ðrȚandnțðrȚ, the second term is the Hartree interaction
energy, and Exc½n/C138is the exchange and correlation energy
between particles of the same kind. Many-body electron-positron interactions are incorporated in the electron-positroncorrelation energy term E
e-p
c½n/C0;nț/C138in Eq. ( 31).
Minimization of the functional equation ( 31) with fixed num-
bers of electrons and positrons leads to the below single-
particle equations for electrons and positrons, respectively:
/C01
2r2ciðrȚț/C20
/C30ðrȚț/C14Exc½n/C0/C138
/C14n/C0ðrȚț/C14Ee-p
c½nț;n/C0/C138
/C14n/C0ðrȚ/C21
ciðrȚ
¼"iciðrȚ; (33)
/C01
2r2cț
iðrȚț/C20
/C0/C30ðrȚț/C14Exc½nț/C138
/C14nțðrȚț/C14Ee-p
c½nț;n/C0/C138
/C14nțðrȚ/C21
/C2cț
iðrȚ¼"ț
icțiðrȚ: (34)
In Eqs. ( 33) and ( 34), the Hartree and external potentials are
incorporated in the term/C30ðrȚ¼Z
dr0n/C0ðr0Ț/C0nțðr0Ț
jr/C0r0jțVextðrȚ: (35)
The densities are obtained by summing over the occupied
orbitals as
n/C0ðrȚ¼X
occjciðrȚj2;n țðrȚ¼X
occjcț
iðrȚj2: (36)
Most often one is interested only in the typical experimen-
tal situation of having only one positron in the sample at any
given time. Then the exchange-correlation energy associated
with the positron corrects only for its Hartree self-interaction(Boron ´ski and Nieminen, 1986 ),
E
xc½nț/C138¼/C01
2Z
drdr0nțðrȚnțðr0Ț
jr/C0r0j: (37)
The corresponding terms in the positron’s potential cancel
accordingly,
/C14Exc½nț/C138
/C14nțðrȚ¼/C0Z
dr0nțðr0Ț
jr/C0r0j: (38)
In the case of a single delocalized positron in an infinite
crystal, it is first assumed that the positron does not affect
the average electron density n/C0ðrȚ. Second, the appropriate
zero-positron-density limit of the electron-positron correla-tion energy is used ( Boron ´ski and Nieminen, 1986 ). After
these assumptions, the two-component calculation can bedone as follows:
(1) The electron density is determined first without the
effect of the positron in a usual one-component DFT
calculation.
(2) The positron’s single-particle wave function is then
solved in the potential
V
țðrȚ¼/C0Z
dr0n/C0ðr0Ț
jr/C0r0j/C0VextðrȚțVcorrðrȚ;(39)
where VcorrðrȚis the zero-positron-density limit of the
electron-positron correlation potential /C14Ee-p
c=/C14n țðrȚ.
Once the electron and positron densities and orbitals are
obtained, one can continue modeling the measurable positron
annihilation characteristics such as the positron lifetime andthe momentum density of annihilating pairs. This will bediscussed in Sec. III.C.
B. Modeling localized positrons
When a positron is localized in a crystal defect,
Eqs. ( 34)–(36) would in principle have to be solved self-
consistently. However, only a very limited number of fullyself-consistent calculations have been made for localizedpositrons in atomistic systems ( Gilgien et al. , 1994 ;Puska,
Seitsonen, and Nieminen, 1995 ;Saito and Oshiyama, 1996 ;
Tang et al. , 1997 ;Makhov and Lewis, 2005 ;Wiktor et al. ,
2013 ). Gilgien et al. used a modification of the original
Boron ´ski and Nieminen (1986) formulation, in which zero-
component limits of the interaction functionals were takeneven for a localized positron. They justified the procedure byexplicit exclusion of self-interaction terms. This is how they
attributed all positron potential terms depending explicitly on
finite positron density. The same approach has also been usedFilip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1599
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by others ( Saito and Oshiyama, 1996 ;Makhov and Lewis,
2005 ). The scheme of Gilgien et al. has been shown to predict
too localized positron densities due to overestimation ofcorrelation effects ( Puska, Seitsonen, and Nieminen, 1995 ).
It may even lead to an unphysical self-trapping phenomenon
in a defect-free crystal ( Gilgien et al. , 1994 ).
The above non-self-consistent method appropriate for de-
localized positrons (the positron does not affect the averageelectron density, and the zero-positron-density limit of theelectron-positron correlation energy used) is routinely ap-
plied also for localized positrons. Because of compensation
and feedback effects [see Puska, Seitsonen, and Nieminen
(1995) ] the two-component approach by Boron ´ski and
Nieminen and the above so-called ‘‘conventional scheme’’give rather similar results regarding the positron lifetime andmomentum density of annihilating electron-positron pairs.Also the energetics, most importantly the relaxations of
vacancy defects in the presence of a localized positron, is
described in a similar way.
During its mean lifetime on the order of 150 ps or more, a
trapped positron has time to influence its surrounding ionicstructure by repelling neighboring ions. When modelingtrapped positrons, the detailed atomic structure of the defect
affects the resulting positron annihilation parameters. As a
first approximation one can use ‘‘ideal’’ unrelaxed geometrieswhere atoms reside on their sites in the pristine lattice.Defect structures relaxed using one-component DFT calcu-lations are not necessarily more accurate since the effect of alocalized positron can change the direction of the displace-ments of the ions neighboring a vacancy from inward to
outward. The force on ion jcan be calculated within the
Born-Oppenheimer approximation from the total derivativeof the total energy with respect to ionic position R
jas
Fj¼/C0 r RjEtot. In the above conventional scheme with no
self-consistency between the electrons and the positrons,
the ground-state total energy can be rewritten in the form
(Laasonen et al. , 1991 )
Etot¼Eț"ț; (40)
where Eis the energy of the electron-ion system from the
one-component DFT calculation, and "țis the positron’s
lowest energy eigenvalue corresponding to the potential ofEq. ( 39). The positron’s contribution to the force on ion jcan
then be calculated with the Hellmann-Feynman theorem as
F
ț
j¼/C0 r Rj"ț¼/C0 r RjhcțjHțjcți
¼/C0 hcțjðrRjHțȚjcți
¼Z
drnțðrȚ½/C0r RjVțðrȚ/C138; (41)
where cțðrȚ,Hț, and VțðrȚare the positron state and the
positron’s single-particle Hamiltonian, and the potential, re-spectively. Here it is assumed that the positron wave functionis normalized and that the basis does not depend explicitly onthe atomic positions.
The Hellmann-Feynman theorem [Eq. ( 41)] can also be
understood in the following manner. Once the electron and
positron densities are calculated quantum mechanically,
forces on ions are determined by classical electrostatics.Therefore, the trapped positron exerts a repulsive force on
the neighboring ions.
When the positron is delocalized in a crystal its energy
band is parabolic with an effective mass close to the free-
electron mass ( Bergersen and Pajanne, 1969 ). For a single
positron in the crystal it is enough to consider only theminimum-energy point /C0ðk¼0Țin the Brillouin zone.
When using the supercell approximation to describe a posi-tron localized at an isolated vacancy defect, the positron’sdiscrete energy state is broadened to a narrow energy band
due to the finite size of the supercell. For a positron at the /C0
point, the derivative of the positron’s wave function vanishesat the cell boundaries between adjacent vacancies, while thewave function itself has a finite amplitude there. As the wavefunction is normalized to unity within the supercell, thepositron density becomes consequently too high at theboundaries and too low at the vacancy center. In other words,
the positron density is too delocalized when the supercell is
not large enough. This has been addressed by Korhonen,
Puska, and Nieminen (1996) . Their remedy is to use two
kpoints in the Brillouin zone, the /C0point and a point from
the edge of the Brillouin zone of the superlattice at the top ofthe energy band. The latter choice implies a boundary con-
dition, which requires the positron wave function to vanish at
the supercell boundary. This enables a much faster conver-gence of the positron density and positron lifetime as afunction of the supercell size. Similar ideas have been pre-sented for minimizing interactions between adjacent cells insupercell calculations of aperiodic systems ( Makov, Shah,
and Payne, 1996 ). For shallow positron traps one should
always check whether the positron density localizes at all
using a large supercell and the /C0point only. Namely, the
scheme by Korhonen et al. can also lead to positron local-
ization for systems where this should actually not happen,namely, when the potential positron trap is shallow and thedispersion of the positron’s energy band becomes similar to
that in a perfect crystal.
C. Positron annihilation parameters
1. Annihilation rate and lifetime
Once the electron and positron densities are solved the
positron annihilation rate /C21and mean lifetime /C28can be
calculated as ( Boron ´ski and Nieminen, 1986 )
/C21¼1
/C28¼/C25r2ecZ
drnțðrȚn/C0ðrȚgðr;r;½nț;n/C0/C138Ț;(42)
where reis the classical radius of the electron and cis
the speed of light.3Here gðr;r;½nț;n/C0/C138Țis the so-called
enhancement factor, the value of the electron-positron pair
correlation function gðr;r0;½nț;n/C0/C138Țat zero distance. The
enhancement factor takes into account the short-range screen-ing of the positron by electrons not accounted for in theaverage one-body densities n
/C0ðrȚandnțðrȚ.
3This is the conventional way to express the prefactor in the
positron literature. It can be related to the fine structure constant /C11
by noting that re¼/C112(in atomic units).1600 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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The electron-positron correlation energy of the two-
component DFT is related to the electron-positron pairdistribution function via ( Jensen and Walker, 1988 )
E
e-p
c½nț;n/C0/C138¼/C0Z1
0d/C21Z
drdr0
/C2n/C0ðrȚnțðr0Țfgðr;r0;½nț;n/C0/C138;/C21Ț/C01g
jr/C0r0j;
(43)
but usually the correlation energy and the enhancement factor
are parametrized separately. The electron-positron correlationenergy is essentially the Coulomb interaction energy of thepositron with its coupling-constant-averaged screening elec-
tron cloud. In Eq. ( 43),/C21is the coupling constant scaling the
electron-positron interaction between noninteracting ( /C21¼0)
and interacting ( /C21¼1) limits, but the densities correspond to
the full interaction case.
Typically, also in the case of localized positrons when
using the conventional scheme, the enhancement factor isapproximated within the LDA and evaluated at the zero-
positron-density limit appropriate for delocalized positrons.
Then the annihilation rate reads ( Chakraborty, 1981 ;
Boron ´ski and Nieminen, 1986 ;Jensen, 1989 )
/C21¼/C25r
2ecZ
drnțðrȚn/C0ðrȚ/C13ðn/C0ðrȚȚ; (44)
where /C13ðn/C0ðrȚȚis the zero-positron-density limit of the LDA
enhancement factor.
2. Momentum density of annihilating electron-positron pairs
The quantity measured in Doppler broadening and ACAR
experiments is the momentum density of annihilatingelectron-positron pairs /C26ðpȚ. Its integral gives the total anni-
hilation rate,
/C21¼1
ð2/C25Ț3Z
dp/C26ðpȚ: (45)
The experiments measure only certain projections of
the momentum density. Doppler broadening and one-dimensional angular correlation experiments measure one-dimensional projections of the momentum density. In aDoppler broadening experiment, the detected energy shift/C1Eis related to the longitudinal momentum component p
L
of the annihilating pair as /C1E¼pLc=2. Angular correlation
experiments measure the deviation of the photons from 180/C14
by the small angle /C18¼pT=m0c. Here m0is the electron mass
andpTis the measured transverse momentum component.
The one-dimensional projection of the momentum densitydetected by these techniques can be written as
/C26ðp
zȚ¼ZZ
dpxdpy/C26ðpȚ; (46)
where pzis the measured momentum component ( pLor
pTdepending on the technique), and pyandpyare the
corresponding transverse components. On the other hand,
two-dimensional angular correlation measurements measure
a two-dimensional projection,
/C26ðpx;pyȚ¼Z
dpz/C26ðpȚ: (47)When comparing measured spectra (Doppler, 1D-ACAR,
2D-ACAR) with corresponding computational ones, one firsthas to either convolute the theoretical ones with the experi-mental resolution function or correspondingly deconvolute
the experimental ones.
Electronic-structure calculations done within DFT provide
direct access only to the one-body densities, not to the truemany-body wave function of the interacting system. Strictlyspeaking, this or at least the relevant reduced two-bodyelectron-positron density matrix would be required to calcu-
late the momentum density of annihilating electron-positron
pairs. A formal but unfortunately impractical relation existsfor expressing general operator ground-state expectation val-ues including momentum densities of the interacting systemwithin the Kohn-Sham density-functional formalism ( Bauer,
1983 ;Barbiellini et al. , 1999 ). This theorem is a general-
ization of the Lam-Platzman correlation correction routinely
used in x-ray Compton scattering ( Lam and Platzman, 1974 ).
For a noninteracting system the momentum density reads
/C26ðpȚ¼/C25r
2ecX
j/C12/C12/C12/C12/C12/C12/C12/C12Z
dre/C0ip/C1rcțðrȚcjðrȚ/C12/C12/C12/C12/C12/C12/C12/C122
: (48)
Here and in the models discussed below, the summation is
over occupied single-particle orbitals. This is the so-calledindependent-particle model (IPM) neglecting effects of short-
range electron-positron correlations, i.e., the screening of the
positron by electrons. There exist a number of improvedapproximations for calculating the momentum density ofannihilating electron-positron pairs using the single-particleorbitals of the noninteracting Kohn-Sham system. A generalexpression for periodic systems typically applied within theLDA can be written as ( Sˇob, Sormann, and Kuriplach, 2003 )
/C26ðpȚ¼X
jk/C14ðp/C0k/C0GȚ
/C2/C12/C12/C12/C12/C12/C12/C12/C12Z
dre/C0ip/C1rcțðrȚcjkðrȚffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gepðr;jkȚq /C12/C12/C12/C12/C12/C12/C12/C122
;(49)
where kis the crystal momentum of the electron’s Bloch
state,Gis a reciprocal lattice vector, and gepðr;jkȚis a two-
particle enhancement function depending on the position r,k,
and band index j. In defect studies with coincidence Doppler
broadening spectroscopy, in which the most important signalis often contained at high momenta and is due to core electronannihilation and umklapp components ( G/C2220) of valence
electrons, the correct description of the kdependence of the
enhancement is less important than in Fermi surface studies
with 2D-ACAR [for reviews on the topic and discussion
see, for e.g., Mijnarends and Bansil (1993) ,West (1993) ,
Kontrym-Sznajd (2009) ,Laverock et al. (2010) , and
Kontrym-Sznajd, Sormann, and Boron ´ski (2012) ]. More im-
portant is how well the core electron annihilation and the G
dependence of the enhancement are described. Most modelsused do not involve any explicit kdependence at all.
Sˇob (1978 ,1979) , and Mijnarends and Singru (1979)
parametrized a phenomenological momentum-dependent en-hancement factor based on the electron gas results of Kahana
(1960 ,1963) . They write the enhancement in terms of energy
instead of the momentum by equating p=p
F¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ejk=EFq
,
where Ejkis the one-particle energy of an electron in theFilip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1601
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jth band at Bloch vector kmeasured from the bottom of the
conduction band, and pFandEFare the Fermi momentum
and Fermi energy, respectively. Below, in the context ofmodels without explicit kdependence, we drop this index
for simplicity.
Daniuk et al. (1987) and Jarlborg and Singh (1987)
parametrized position-dependent LDA enhancement factorsincorporating effects of short-range electron-positroncorrelations,
/C26ðpȚ¼/C25r
2ecX
j/C12/C12/C12/C12/C12/C12/C12/C12Z
dre/C0ip/C1rcțðrȚcjðrȚffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
/C13ðn/C0ðrȚȚq /C12/C12/C12/C12/C12/C12/C12/C122
:
(50)
A similar approach has also been used for core states
(Daniuk et al. , 1989 ;Daniuk, S ˇob, and Rubaszek, 1991 ;
Alatalo et al. , 1995 ).
A state-dependent enhancement factor was parametrized
and used to describe core electron annihilation by Alatalo
et al. (1996) and later applied also for valence electrons
(Barbiellini et al. , 1997 )
/C26ðpȚ¼/C25r2ecX
j/C13j/C12/C12/C12/C12/C12/C12/C12/C12Z
dre/C0ip/C1rcțðrȚcjðrȚ/C12/C12/C12/C12/C12/C12/C12/C122
: (51)
Here /C13j¼/C21LDA
j=/C21IPMj, where
/C21LDA
j¼/C25r2ecZ
drnțðrȚjcjðrȚj2/C13ðn/C0ðrȚȚ; (52)
and/C21IPM
jis calculated similarly with /C13/C171. The scheme can
be applied instead of the LDA equally well within the GGA
for elecron-positron systems (see Sec. III.D ). Recently, the
empirical model of Laverock et al. (2010) combined the
above model of Alatalo et al. with an energy-dependent
enhancement factor.
Tang et al. (2005) also applied together with Eq. ( 50)
within the LDA of Puska, Seitsonen, and Nieminen (1995)
theGW self-energy correction to correct electron occupation
numbers.
D. Functionals for electron-positron correlation effects
The true many-body interactions responsible for the
short-range screening of the positron by the electrons are
taken into account in the two-component DFT only in an
approximate manner. One takes input from many-bodycalculations made for electron-positron systems and parame-trizes the unknown functionals, the electron-positron corre-lation energy E
e-p
c½n/C0;nț/C138, and the enhancement factor
gðr;r;½nț;n/C0/C138Ț, which incorporate the effects of the screen-
ing into the energy functional and on the positron annihilation
rate. The starting point is usually the LDA, in which thequantities are approximated using functions of the localelectron and positron densities. The electron-positron corre-lation energy within the LDA is then written with the help ofthe correlation energy density per unit volume F
e-p
cðn/C0;nțȚ,
in a homogenous electron-positron gas with densities n/C0and
nț(Boron ´ski and Nieminen, 1986 ;Puska, Seitsonen, and
Nieminen, 1995 )
Ee-p
c½n/C0;nț/C138¼Z
drFe-p
cðn/C0ðrȚ;nțðrȚȚ; (53)i.e., assuming that at each point in space the local correlation
energy density is equal to that in a homogenous system withdensities equal to the local ones. Similarly, the finite-positron-density LDA enhancement factor /C13ðn
/C0ðrȚ;nțðrȚȚis taken
from results for homogenous systems. In the zero-positron-
density limit, the LDA correlation energy can be expressedusing the correlation energy per particle as in the case ofone-component DFT ( Boron ´ski and Nieminen, 1986 ),
E
e-p
c½n/C0;nț!0/C138¼Z
drnțðrȚ"e-p
cðn/C0ðrȚ;nț!0Ț;
(54)
where "e-p
cðn/C0;nț!0Țis the correlation energy per
positron in a homogenous electron-positron gas with electrondensity n
/C0and a single delocalized positron. In this limit, the
LDA enhancement factor becomes simply a function of thelocal electron density /C13ðnðrȚȚ; see Eq. ( 44).
For the case of finite densities, there exist LDA parametri-
zations by Boron ´ski and Nieminen (1986) and Puska,
Seitsonen, and Nieminen (1995) . Both are based on multicom-
ponent Fermi hypernetted-chain calculations by Lantto (1987) ,
and the zero-positron-density limit calculations of Arponen and
Pajanne (1979a ,1979b) .H o w e v e r ,B o r o n ´ski and Nieminen had
only the data for n
ț!0,nț¼n/C0=2,a n d nț¼n/C0avail-
able. The zero-positron-density limit has been parametrized by
Boron ´ski and Nieminen (1986) using correlation energy from
Arponen and Pajanne (1979a ,1979b) and the contact density
ofLantto (1987) . The calculations of Arponen and Pajanne are
based on correcting the results of the random-phase approxi-mation in a boson formalism. Barbiellini et al. (1995 ,1996)
reparametrized the LDA enhancement factor to be consistentwith the data by Arponen and Pajanne and used this parame-
trization as a basis for their gradient corrections. Recently,
Drummond et al. (2011) parametrized a LDA for correlation
energy and enhancement factor using their own quantumMonte Carlo results. This work followed an earlier one bythe same group ( Drummond et al., 2010 ) in which the screening
of the positron was modeled by applying one-component DFT
in the positron’s reference frame.
Before the time of ab initio positron lifetime calculations,
enhancement models were empirical, based on dividing thetotal electron density nðrȚinto core n
cðrȚand valence nvðrȚ
components, and, when applicable, to a delectron component
ndðrȚand using a different enhancement factor for each of
these [see, e.g., Puska and Nieminen (1983) ]. Puska and
Nieminen used a constant enhancement factor for core
electrons, and the delectron enhancement factors were
determined by fitting to measured bulk lifetime data. Jensen
(1989) was the first to evaluate the positron annihilation rate
using the LDA of Eq. ( 44), using the total density and without
adjustable parameters.
Most models describing electron-positron correlation
energy and enhancement effects derive information frommany-body modeling made for metallic systems, i.e., thehomogenous electron-positron gas. The screening of thepositrons in semiconductors and insulators with less efficientscreening is less understood. Puska et al. (1989) introduced
semiempirical models for semiconductors and insulators. The
parameter in the semiempirical semiconductor model, based
on the earlier work of Brandt and Reinheimer (1970) , is the1602 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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high-frequency dielectric constant, whereas in the insulator
model the enhancement is connected to the atomic polar-izability. Previously a model describing the screening ofpositive point charges in semiconductors ( Brandt and
Reinheimer, 1971 ) had been adapted to describe electron-
positron correlations ( Puska et al. , 1986 ). In this case the
free parameter in the model was fitted to reproduce theexperimental bulk lifetime.
Barbiellini et al. (1995 ,1996) approached the correlation
problem by adding gradient corrections to the LDA correla-
tion energy and enhancement factor. Their approach involves
one semiempirical parameter chosen by requiring goodagreement between calculated and measured positron life-times for a large number of different solids. For predictingabsolute values of positron lifetimes this approach is pres-ently the most accurate one [see, e.g., Campillo Robles,
Ogando, and Plazaola (2007) ], although it has been shown
that it might need further improvements in the very low-
density regime ( Mitroy and Barbiellini, 2002 ). Analysis of
the behavior of this GGA model helped to further explainwhy the annihilation rates for core electrons can be welldescribed by the independent-particle limit [see, e.g.,Alatalo et al. (1996) ].
A completely nonlocal model based on the original work
byGunnarsson, Jonson, and Lundqvist (1979) is the
weighted-density approximation (WDA) formulated forthe electron-positron case by Jensen and Walker (1988) and
Rubaszek (1991) and applied for positron surface states.
More recent applications on bulk solids include Rubaszek,
Szotek, and Temmerman (1998 ,2000 ,2001 ,2002) . The
WDA has not, however, been applied for positrons trapped
at vacancy defects.
E. The atomic superposition method
The so-called atomic superposition (ATSUP) method
(Puska and Nieminen, 1983 ) is a simple but already very
applicable method for modeling positron states and annihila-tion in solids. In essence, the method is based on a fullynon-self-consistent approach to the two-component DFTpresented above. The electron density is approximated as a
superposition of densities of neutral free atoms,
n
/C0ðrȚ/C25X
Rnat/C0ðjr/C0RjȚ; (55)
and the positron’s potential is a sum of Coulomb potentials
due to the superimposed free atoms and a LDA correlationpotential similar to that in Eq. ( 39),
V
țðrȚ/C25X
RVat
Coulðjr/C0RjȚ țVcorrðn/C0ðrȚȚ: (56)
Positron lifetimes calculated with the atomic superposition
method agree remarkably well with those calculated withmore self-consistent methods [see, e.g., Puska et al. , 1986 ;
Puska, 1991 ;Campillo Robles, Ogando, and Plazaola, 2007 ],
even within a few picoseconds, as long as lattice parametersand ionic positions are the same and the same functionals areused for the correlation energy and enhancement factor. This
is due to a simple compensating feedback effect ( Puska and
Nieminen, 1983 ,1994 ). The positron density will follow anychanges in the electron density, for example, due to an
improved description, keeping their mutual overlap, the an-
nihilation rate [see Eq. ( 42)], constant.
First calculations of coincidence Doppler broadening spec-
tra were done within the ATSUP method ( Alatalo et al. , 1995 ,
1996 ;Asoka-Kumar et al. , 1996 ). The results of the method are
rather accurate at high momenta dominated by the core shells,
which are rigid and do not depend much on the chemical
environment of the atoms. When modeling Doppler broad-ening one typically assumes in addition to the spherical sym-
metry of the core electron shells the positron wave function to
be spherically symmetric around nuclei. Its decay toward the
nucleus is parametrized using all-electron calculations done,
for example, with the linear-muffin-tin-orbital method(Alatalo et al. , 1995 ,1996 ). The same approach can be used
when including the core electron contribution in calcula-
tions employing the frozen-core approximation, such as
the plane-wave pseudopotential method or the projector-
augmented-wave method (see Sec. III.F).
The atomic superposition method is computationally cheap
and it still remains an applicable method also in modeling
Doppler broadening as long as results are compared withexperiments only at high momenta [ >ð2–2:5Ța:u:] domi-
nated by core electron annihilation. For instance, the
MIKA
Doppler software package ( Torsti et al. , 2003 ,2006 ) imple-
ments calculations with the atomic superposition method and
provides also the possibility of coupling the code with DFT
codes in order to perform more self-consistent calculations
with accurate electron densities.
F. Numerical approaches for self-consistent calculations
Practical modeling of positron states and annihilation in
solids going beyond the atomic superposition method ( Puska
and Nieminen, 1983 , Sec. III.E), positron pseudopotential
(Kubica and Stott, 1974 ;Stott and Kubica, 1975 ), or jellium
models [see, e.g., Manninen et al. (1975) andBoron ´ski and
Nieminen (1986) ] is typically based on some standard band-
structure method for performing the underlying electronic-
structure calculation. Depending on the level of sophistication
and self-consistency, the positron is either treated within thesame code or added in the system in a postprocessing manner
(see the end of Sec. III.A ). The numerical representation of the
positron wave function is in a sense less complicated than those
of electrons, since in the typical case of only one positron in the
lattice there are no orthogonality requirements, and the wavefunction is of stype, soft and nodeless. On the other hand, the
basis used for the positron has to work well in regions in which
the positron wave function has the largest amplitude, such as
interstitial regions and vacancies. In non-all-electron methods
where the frozen-core approximation is used, such as in plane-
wave pseudopotential calculations, the annihilation with core
electrons typically is described using frozen-core densities andorbitals, and when modeling momentum densities of annihi-
lating electron-positron pairs, a spherically symmetric ap-
proximation for the positron wave function is used. Periodic
boundary conditions and the supercell approximation are used
in most numerical implementations also when one is modelingpositrons trapped at isolated vacancy-type defects. Figure 9
shows an example of a model for a Ga vacancy in a 96-atomFilip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1603
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

GaN supercell used by Hautakangas et al. (2006) . Care has to
always be taken so that the results are converged with respectto the size of the supercell. A ground state with a localizedpositron can be confirmed only by visualization of the real-space positron density n
țðrȚas in Fig. 9. A lifetime increase
compared to the bulk value is not a safe indicator alone.
Here we list some band-structure methods which have been
applied to model positrons states and annihilation in solids.We mostly focus on the so-called projector-augmented-wavemethod ( Blo¨chl, 1994 ), which currently is the most useful one
when modeling coincidence Doppler broadening spectra forpositrons localized at defects in solids.
Berko and Plaskett (1958) used the Wigner-Seitz band-
structure method for describing positron annihilation in
metals.
The augmented plane-wave (APW) method by Slater
(1937) was used by several groups ( Loucks, 1966 ;Wakoh,
Kubo, and Yamashita, 1975 ;Kubo, Wakoh, and Yamashita,
1976 ;Gupta and Siegel, 1977 ,1980a ,1980b ) in a non-self-
consistent fashion. The supercell approximation was used for
localized positrons first by Gupta and Siegel (1977) . Also the
linearized (LAPW) ( Andersen, 1975 )(Daniuk, 1983 ;Daniuk
et al. , 1987 ) and the full-potential version (FP-LAPW)
(Baruah, Zope, and Kshirsagar, 1999 ;Tang, Hasegawa,
Nagai, and Saito, 2002 ;Tang, Hasegawa, Nagai, Saito, and
Kawazoe, 2002 ) have been applied. The FP-LAPW method is
highly accurate and treats also the core electrons and their
annihilation fully self-consistently using the true symmetryof electron and positron states. However, we are not awareof any calculations made for localized positrons with theFP-LAPW method.
The linear-muffin-tin orbital (LMTO) method ( Andersen,
1975 ) is another all-electron method which has been widely
used in the positron community, in both defect identificationand Fermiology applications [for bulk studies see, e.g., Singhand Jarlborg (1985) ,Jarlborg and Singh (1987) ,Sterne and
Kaiser (1991) , and Barbiellini, Dugdale, and Jarlborg
(2003) ]. Positrons localized at vacancies have been studied
(Puska et al. , 1989 ,1994 ;Puska, 1991 ;Alatalo, Puska, and
Nieminen, 1993 ;Plazaola, Seitsonen, and Puska, 1994 ;
Korhonen, Puska, and Nieminen, 1996 ;Barbiellini et al. ,
1997 ), also using the LMTO Green’s function method
(Puska et al. , 1986
). In most of the works the so-called
atomic-sphere approximation ( Skriver, 1984 ;Andersen,
Jepsen, and Glo ¨tzel, 1985 ) has been made but also the
full-potential variant has been used ( Korhonen, Puska, and
Nieminen, 1996 ).
The Korringa-Kohn-Rostoker (KKR) Green’s function
method ( Korringa, 1947 ;Kohn and Rostoker, 1954 ) has also
been applied in bulk studies ( Hanssen and Mijnarends, 1986 ;
Mijnarends and Rabou, 1986 ;Bansil et al. , 1988 ;Mijnarends
and Bansil, 1990 ;Bansil, Mijnarends, and Smedskjaer, 1991 ;
Mijnarends et al. , 1998 ;Eijt et al. , 2006 ), and within the
coherent potential approximation (CPA) ( Soven, 1967 ,1969 ;
Velicky ´, Kirkpatrick, and Ehrenreich, 1968 ) for alloys
(Mijnarends et al. , 1987 ;Smedskjaer et al. , 1987 ). Further,
some of the more approximate methods used earlier ( Hubbard,
1969 ;Hubbard and Mijnarends, 1972 ), for example, by
Mijnarends (1973) ,Singru and Mijnarends (1974) , and
Mijnarends and Singru (1979) were based on the KKR method.
Ishibashi et al. (1997) used localized orbitals in all-
electron calculations. The positron state was solved on agrid using finite differences.
Finite-difference and finite-element methods are very use-
ful in positron calculations, since the region where accuracyis needed is different between electrons and the positron.
Whereas electronic orbitals oscillate rapidly close to nuclei,
from the positron’s point of view this region is unimportantbecause of the vanishingly small positron density there. Moreimportant are the less repulsive regions in the interstitialspace and open-volume defects.
Puska and Nieminen (1983) used in the first atomic super-
position calculations the finite-difference method of Kimball
and Shortley (1934) . The finite-difference method has also
been used with the conjugate-gradient solver ( Seitsonen,
Puska, and Nieminen, 1995 ) and the Rayleigh quotient multi-
grid method ( Heiskanen et al. , 2001 ); see Torsti et al. (2003 ,
2006) .
Pask et al. (1999 ,2001) andSterne, Pask, and Klein
(1999) used finite-element modeling for electron-positron
systems.
Especially when modeling positrons trapped in vacancy
defects it becomes necessary to use rather large supercellswith even hundreds of atoms and relax the defect structuresmodeled. For these purposes, the method of choice has in totalenergy calculations traditionally been the plane-wave pseudo-potential method with either norm-conserving [see, e.g.,Hamann, Schlu ¨ter, and Chiang (1979) ,Kerker (1980) ,
Bachelet, Hamann, and Schlu ¨ter (1982) ,Vanderbilt (1985) ,
andTroullier and Martins (1991) ] or ultrasoft pseudopotentials
(Blo¨chl, 1990 ;Vanderbilt, 1990 ), or lately especially the
closely related all-electron technique, the projector-aug-
mented-wave (PAW) method ( Blo¨chl, 1994 ), discussed in
more detail in the next paragraph. From the point of view ofpositron defect calculations these methods also bring flexibil-ity. Also, they do not involve any other shape approximations
FIG. 9 (color online). An isosurface of the positron density for a
positron localized at a Ga vacancy in GaN in a 96-atom supercell
model. The Ga atoms are shown by light shaded and the N atoms by
dark shaded spheres. Based on the calculations of Hautakangas
et al. (2006) . From Torsti et al. , 2006 .1604 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

than the one of spherically symmetric ion cores inherent in
the frozen-core approximation. Plane-wave pseudopotentialmethods have been used extensively both within the frame-work of Car and Parrinello (1985) [see, e.g., Laasonen et al.
(1991) ,Alatalo, Puska, and Nieminen (1993) ,Gilgien et al.
(1994) , and Puska, Seitsonen, and Nieminen (1995) ] and/or
in other Born-Oppenheimer calculations ( Hakala, Puska, and
Nieminen, 1998 ;Ishibashi et al. , 1999 ;Saarinen et al. , 1999 ;
Ishibashi and Kohyama, 2003 ). The downside related to
momentum-density calculations is the inaccuracy of the
high-momentum components of valence wave functions.
The increasing sophistication of the plane-wave pseudo-
potential method has led to better computational accuracywith smaller computational cost and smaller plane-wavebases. However, in order to accurately describe the wavefunction’s high-momentum components, which are essentialin defect identification, a small plane-wave basis is not
enough. To exemplify, the coincidence Doppler broadening
measurements can measure the Doppler spectra accurately upto a momentum of 6–8 a.u. Since the Doppler spectrum is aone-dimensional projection of the three-dimensional momen-tum density of annihilating electron-positron pairs, an evenhigher momentum-space cutoff on the order of 10 a.u. is
needed. This corresponds to a plane-wave kinetic energy
cutoff of /C241400 eV and implies high requirements for the
numerical accuracy of the high-momentum plane-wave com-ponents. Calculations with such high cutoffs (cf. typicalvalues of 250–400 eV) are not affordable. Further, since thepseudo wave functions are ‘‘softened,’’ they do not representthe accurate all-electron wave functions even in the limit of a
complete basis set. Techniques exist to reconstruct the all-
electron wave functions from norm-conserving or ultrasoftwave functions ( Meyer et al. , 1995 ;Delaney, Kra ´lik, and
Louie, 1998 ;Hete´nyiet al. , 2001 ;Ishibashi, 2004 )i na
postprocessing fashion. Furthermore, the PAW method isespecially useful since it involves a well-defined relation
between soft pseudo wave functions j~/C9irepresented in a
plane-wave basis and the accurate all-electron wave functionsj/C9i(Blo¨chl, 1994 ),
j/C9i¼j ~/C9ițX
iðj/C30ii/C0j ~/C30iiȚh~pij~/C9i: (57)
Here j/C30ii,j~/C30ii, and j~piiare atom-centered all-electron and
pseudo partial waves and projector functions, respectively.The index iis a shorthand index referring to the atomic site,
angular momentum quantum numbers, and reference energy.Equation ( 57) is applied as a postcorrection after a self-
consistent calculation made using effectively the soft pseudo
wave functions j~/C9i, either on a dense real-space grid or in
momentum space using an expanded plane-wave basis(Ishibashi, 2004 ;Makkonen, Hakala, and Puska, 2005 ,
2006 ). In the case of positrons localized at vacancies the
transformation needs to be applied only within the first fewcoordination shells around the vacancy. Since the positronwave function is smooth, it can be represented using only aplane-wave expansion or a real-space point grid. In this ap-proach, an accurate enough numerical potential for the posi-tron is obtained from, instead of the all-electron density never
calculated explicitly within the method, a sum of the core
density, pseudovalence charge density, and atom-centeredcompensation charges of the PAW method ( Makkonen,
Hakala, and Puska, 2006 ).
Positron lifetimes or momentum densities of annihilating
electron-positron pairs have been calculated using PAW
implementations based on such modern DFT packages as
the quantum materials simulator (
QMAS )(Ishibashi, 2004 ;
Ishibashi et al. , 2007 ), Vienna ab initio simulation package
(VASP )(Kresse and Hafner, 1993 ;Kresse and Furthmu ¨ller,
1996a ,1996b ;Kresse and Joubert, 1999 ) [see, e.g.,
Makkonen, Hakala, and Puska, 2005 ,2006 ], and ABINIT
(Gonze et al. , 2009 ;Wiktor et al. , 2013 ).
IV. RESULTS
In this section, we present a series of examples showing
how the combination of positron lifetime experiments, regu-
lar and coincidence Doppler broadening experiments, sample
state manipulation, and theoretical calculations has been usedto identify and elucidate properties of defects in semiconduc-tors. As an introduction, we review the types of studiesperformed and results obtained with positron annihilationmethods on defects in various semiconductor material fami-
lies over the past 20 years of active research in this field. The
core of this section then consists of a detailed account ofseven cases: (i) vacancy-(multi-)donor complexes in silicon,(ii) the vacancy-fluorine complex in silicon and silicon-germanium alloys, (iii) the ‘‘EL2’’ defect in gallium arsenide,(iv) the gallium vacancy–tellurium complex in gallium arsen-ide, (v) the gallium vacancy and its complexes in gallium
nitride, (vi) metal vacancy–nitrogen vacancy complexes in
III-nitrides and their alloys, and (vii) the substitutionallithium-on-zinc-site defect in zinc oxide. These cases havebeen chosen in order to illustrate different approaches andfeatures observed in the experimental spectra. We alsodiscuss the physical properties and importance of each of
the above defects, relating the positron results to other
experimental and theoretical findings. Future challenges arediscussed in the next section.
A. An overview of results obtained in the past two decades
The examples given in the next sections concentrate on
cases where a detailed identification of the vacancy-type de-fects has provided a basis for further work. In the majority ofsituations, however, detailed identification of the atomic struc-
ture of the defects observed with positrons is not possible. The
two most typical cases are (i) the presence of multiple kinds of(vacancy) defects with high concentrations, and (ii) the inabil-ity to define a proper reference (i.e., ‘‘defect-free’’ sample).These situations often coexist, as they tend to be related to thechallenges in material growth, e.g., typical of early stages of a
development of a new semiconductor. Another issue is that
many semiconductors can be grown only as thin films, as is thecase for the majority of alloyed systems such as Si
1/C0xGexor
Al1/C0xGaxAs. Also some compounds, such as InN, cannot be
found as bulk crystals (substrates). Hence, lifetime measure-ments for finding a suitable reference sample are complicated.In addition, epitaxial growth has to be performed on non-native
substrates, increasing the probability of lattice mismatch and
ensuing high extended defect densities. Luckily, insightfulFilip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1605
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studies on the formation of defects and on their properties can
be performed even without detailed identification.
Typical studies, where it is enough to know the vacancy
character (sublattice in compounds) and size, concentrate onthe effects of growth parameters on the vacancy formation.Such parameters can be the growth temperature, stoichiome-try, the addition of dopants or surfactants, or the choice andprocessing of the substrate or buffer layer. Thermal formationof vacancies can be studied through either analyzing growthdata or postgrowth thermal annealings. The latter can be usedfor studying the migration kinetics of in-grown defects, butthese kinds of studies are more convenient in irradiated
or ion-implanted samples, where defects are often better
defined. Defects introduced in ion implantation processingare by themselves an important field of study, as they dra-matically affect the doping efficiency achieved by implanta-tion. High-energy electron irradiation is of particular usewhen controlled defect introduction is needed, and it oftenprovides a baseline for defect identification.
1. Elemental semiconductors Si, Ge, and C
The formation enthalpy of a monovacancy in Si is pre-
dicted to be rather high, above 3 eV ( Puska et al. , 1998 ;
Probert and Payne, 2003 ) for all charge states and Fermi level
positions within the band gap. This prediction is in goodagreement with experimental findings from both diffusion(Bracht et al. , 2003 ) and positron annihilation ( Ranki and
Saarinen, 2004 ;Kuitunen, Saarinen, and Tuomisto, 2007 )
studies where thermal formation of vacancies requires rather
high temperatures, even when the vacancies are formed right
next to donors in highly n-type material. Hence, the vast
majority of positron annihilation studies on vacancies insilicon have been performed on irradiated or implanted ma-terial, with possibly the only exception being highly n-type
doped Si (see Sec. IV.B). Irradiation and implantation studies
have enabled the identification of the monovacancy (at lowtemperature) and determination of its migration barrier ingood agreement with EPR studies ( Ma¨kinen, Rajainma ¨ki,
and Linderoth, 1990 ;Watkins, 2000 ). The divacancy can be
detected and identified after room-temperature irradiation,and also the optical ionization levels and dissociation energyhave been determined ( Kauppinen et al. , 1998 ). Larger va-
cancy clusters have been observed in ion-implanted ( Avalos
and Dannefaer, 1996 ) and neutron-irradiated ( Meng et al. ,
1994 ) material. Complexes of vacancies with impurities, in
particular, donors such as As, Sb, or P, have been studied a
great deal, as well as the V-O(Kauppinen et al. , 1998 ) and
V-Fsystems. Vacancy-donor and vacancy-fluorine complexes
are discussed in more detail in Secs. IV.B andIV.C. Vacancies
and vacancy clusters in ion implantation processing havebeen studied mostly from the point of view of affecting thediffusion of either interstitials or boron.
Germanium has been studied less than Si also with positron
annihilation spectroscopies, although lately some revival canbe seen. Monovacancies and divacancies have been identified(Corbel, Moser, and Stucky, 1985 ;Polity and Rudolf, 1999 ;
Kuitunen et al. , 2008 ;Slotte et al. , 2008 ,2011 ) and their
stability at and below room temperature investigated.Vacancies have also been found to pair with donors in Gewith positron annihilation ( Arutyunov and Emtsev, 2007 ),while thorough investigations are still to be performed.
Even though germanium exhibits interesting properties inion implantation, i.e., it is easily amorphized and the recrys-tallization temperature is low ( Hickey et al. ,2 0 0 7 ), surpris-
ingly few positron studies seem to have been performed(Krause-Rehberg et al.
, 1993 ;Slotte et al. , 2008 ). The situ-
ation is completely different for the Si-Ge alloys which havebeen studied to a large extent. Vacancy-donor complexes
introduced by irradiation of n-type material ( Sihto et al. ,
2003 ;Rummukainen et al. , 2006 ;Kuitunen, Tuomisto, and
Slotte, 2007 ), vacancy-fluorine complexes produced by
F implantation ( Edwardson et al. , 2012 ), and the particular-
ities related to the random alloy nature of Si-Ge alloys(Shoukri et al. , 2005 ;Ferragut et al. , 2010 ;Kilpela ¨inen
et al. , 2010 ,2011 ) have been studied in more detail.
Point defects in diamond have been studied quite actively,
but not so much from the semiconducting properties pointof view. Studies have concentrated on elucidating the funda-mental properties of the crystal such as the mobility ofisolated vacancy defects introduced by irradiation ( Uedono
et al. , 1999 ;Puet al. , 2000 ;Iakoubovskii, Dannefaer, and
Stesmans, 2005 ;Dannefaer and Iakoubovskii, 2008 ), or the
vacancy-related coloration of natural and synthetic diamond(Nilen et al. , 1997 ;Dannefaer, Pu, and Kerr, 2001 ;Ma¨ki
et al. , 2009 ;Ma¨ki, Tuomisto et al. , 2011 ). Nitrogen-vacancy
centers, which have high potential in quantum computing(Maurer et al. , 2012 ) and, in particular, in hypersensitive
high-dynamic range magnetometry ( Waldherr et al. ,2 0 1 2 ),
have received some attention ( Dannefaer, 2009 ;Botsoa et al. ,
2011 ). The formation of vacancy-donor complexes, which are
potentially important in limiting the n-type dopability of
diamond, seems not to have been addressed with positronmethods.
2. Traditional III-V and II-VI semiconductors
Compound semiconductors have provided a particularly
fruitful ground for studying point defects and their effectson the electrical and optical properties of these materials.The formation enthalpies of, e.g., the vacancy defects onboth sublattices tend to be clearly lower than in Si, and inaddition changes in growth stoichiometry can further increasethe probability of defect formation. Further, these defects tendto be stable well above room temperature and they produce amultitude of localized states in the band gap, hence controllingmany of the crucial material properties. By traditional III-V
and II-VI semiconductors we mean the III-phosphides,
III-arsenides, and III-antimonides, and II-selenides andII-tellurides. The body of work with any defect spectroscopyon these materials is large. An important property of thetraditional III-V and II-VI compounds is that neither of theconstituent atoms is ‘‘small,’’ i.e., positrons can be at least inprinciple trapped by vacancy defects on either sublattice. Thisis in contrast to the ‘‘novel’’ compound semiconductors, wherethe anion is typically small (N, O, or C); see Sec. IV.A.3 .
The studies of vacancy defects in GaAs range from in-
grown and irradiation-induced defects in n-type and semi-
insulating crystals to the defects formed in low-temperaturemolecular beam epitaxy (LT-MBE) of thin films. The studieshave shown that negatively charged Ga vacancies and theircomplexes with donors are important compensating centers in1606 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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n-type doped GaAs ( Dannefaer, Hogg, and Kerr, 1984 ;
Corbel et al. , 1988 ;Laine et al. , 1996 ;Gebauer et al. ,
1999 ). Arsenic vacancies are observed in unintentionally
n-type as-grown GaAs crystals ( Saarinen et al. , 1991 ), while
they coexist with Ga vacancies and EL2 defects in semi-insulating GaAs ( Kuisma et al. , 1996 ). The metastable EL2
and DX centers in arsenides have both been shown to have
vacancy character ( Krause, Saarinen et al. , 1990 ;Ma¨kinen
et al. , 1993 ). Thermal annealing experiments have been
performed to elucidate vacancy formation mechanisms(Bondarenko et al. , 2005 ) and the role of copper in determin-
ing the vacancy distribution in GaAs crystals ( Elsayed et al. ,
2008 ,2011 ). LT-MBE GaAs is typically grown in conditions
that favor the formation of Ga sublattice defects: Ga vacan-cies and As antisites ( Bliss et al. , 1992 ;Keeble et al. , 1993 ;
Sto¨rmer et al. , 1996 ;Fleischer et al. , 1997 ;Gebauer et al. ,
1997 ;Laine et al. , 1999 ). The Ga vacancy-related defects
follow the growth stoichiometry and compensate the n-type
doping. Similar effects have been found in the magnetically
doped dilute magnetic semiconductor Ga
1/C0xMnxAs
(Tuomisto et al. , 2004 ) that is also grown by LT-MBE.
In III-phosphides (such as GaP or InP) the vacancies on
both sublattices have been identified, and these have animportant effect on the electrical properties in undoped,
n-type, and p-type material ( Mahony, Mascher, and Puff,
1996 ;Bretagnon, Dannefaer, and Kerr, 1997 ;Dekker et al. ,
2002 ). As an example, the V
P-Znpairs have been shown to
form through vacancy migration from the crystal surface in
Zn-doped InP ( Slotte et al. , 2003 ), while the In vacancies
formed through thermal annealings render undoped InP semi-insulating through compensation of residual donors ( Deng
et al. , 2003 ). Alloys of III-V semiconductors, such as GaInP
(Dekker et al. , 2002 ), AlGaAs ( Ma¨kinen et al. , 1993 ), or
InGaAsP ( Pinkney et al. , 1998 ) have also been studied to
some extent, as well as the so-called diluted nitrides where atmost a few percent of nitrogen replaces the group-V compo-
nent ( Toivonen et al. , 2003 ). Cation vacancies are typically
found to be strongly correlated with the optoelectronic prop-erties in these studies. Much of the positron research in III-Vsemiconductors is covered by Saarinen, Hautoja ¨rvi, and
Corbel (1998) andKrause-Rehberg and Leipner (1999) .
The most important early positron results in II-VI com-
pound semiconductors have been reviewed by Krause-
Rehberg et al. (1998) . The interest in studying defects in
these materials stemmed from the doping asymmetry: bulk
crystals of ZnSe, CdSe, ZnS, and CdS tend to always be
ntype, irrespective of impurity content, while ZnTe is ptype.
The II-VI compounds appeared promising for various deviceapplications ranging from particle detectors to optoelectronic
devices; their technological breakthroughs have been limited
due to this issue. Only CdTe is easy to dope either way. Thesituation in thin films is clearly better: e.g., p-type ZnSe can
be grown by MBE and hence p-njunctions can be fabricated
(Park et al. , 1990 ). Vacancy defects on both sublattices have
been shown to exist depending on doping in ZnS
xSe1/C0x
(Saarinen et al. , 1996 ;Oila et al. , 1999 ;Desgardin et al. ,
2000 ;Gebauer et al. , 2002 ). Cation vacancies complexed
with impurities in CdTe and Hg1/C0xCdxTehave been identi-
fied as important defects controlling the conductivity in bulk
crystals ( Krause, Klimakow et al. , 1990 ;Kauppinen et al. ,1997 ), while both cation vacancies and divacancies have been
found in thin films ( Liszkay et al. , 1994 ;Keeble et al. , 2011 ).
3. Novel semiconductors: III-N, SiC, and ZnO
The novel, often wide-band-gap, compound semiconduc-
tors are characterized by large size mismatch between thecation and anion. III-nitrides, SiC, and ZnO are goodexamples of such semiconductors. While the traditional
III-V and II-VI compounds tend to crystallize in a cubic
(zinc blende) structure, these novel compounds prefer hex-agonal symmetry (e.g., the wurtzite structure) and a smallerlattice constant. Another common aspect in these novelcompounds is that anion (N, O, and C) vacancies have beenblamed for various properties of the materials, while theirdirect observation is difficult or even impossible. Complexoxides suffer from the same syndrome. Unfortunately, due tothe small size and natural positive charge of the anion vacan-cies, they are mostly elusive to positron annihilation spectros-copies as well. Cation vacancies, on the other hand, arereadily observable, and in fact they have been shown to beresponsible for many optoelectronic properties of these com-pounds. Positron annihilation spectroscopies have beenwidely applied to study III-nitrides, SiC, and ZnO-relatedmaterials.
GaN is by far the most studied of the III-nitrides. Early
results of positron studies have been reviewed by Saarinen
(2000) and bulk GaN crystal studies by Tuomisto (2010) .
Section IV.F discusses the identification and studies of Ga
vacancies and their complexes in GaN, while Sec. IV.G
covers the cation-anion vacancy clusters found in InN and
InGaN alloys. A wide variety of studies of Ga-vacancydefects generated during epitaxial growth by MBE andmetal-organic vapor phase deposition (MOCVD) have shownthat similar effects related to growth temperature and stoichi-ometry can be found as in the traditional compounds, but inGaN oxygen impurities play a decisive role ( Rummukainen
et al. , 2004 ;Hautakangas et al. , 2006 ;Tuomisto, Paskova
et al. , 2007 ), and possibly also hydrogen ( Hautakangas et al. ,
2006 ;Nyka¨nenet al. , 2012 ).
The hexagonal symmetry of the wurtzite lattice brings an
additional property to the crystal compared to cubic lattices,namely, spontaneous polarization. Indeed the polarity of thegrowth surface strongly affects the Ga-vacancy defect for-mation and impurity incorporation in GaN ( Rummukainen
et al. , 2004 ;Tuomisto, Saarinen, Lucznik et al. , 2005 ;
Tuomisto, Paskova et al. , 2007 ). Some evidence of N
vacancies with positrons has been found in irradiated(Tuomisto, Ranki et al. , 2007 ) and Mg-doped GaN samples
(Hautakangas et al. , 2003 ), but further studies are clearly
required. Thanks to the abundance of defects generated dur-
ing growth of GaN or other III-nitrides, irradiation and ionimplantation studies are rather scarce ( Tuomisto, Pelli et al. ,
2007 ;Tuomisto, Ranki et al. , 2007 ;Uedono et al. , 2007 ;
Moutanabbir et al. , 2010 ;Ma¨ki, Makkonen et al. , 2011 ).
Technologically important alloys such as InGaN andAlGaN have also been studied ( Slotte et al. , 2007 ;Chichibu
et al. , 2011 ;Uedono et al.
, 2012 ). The cation vacancies and
their complexes found in these studies act as compensatingcenters in n-type material, as nonradiative recombination
centers, and have sometimes been correlated with parasiticFilip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1607
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

deep-level yellow luminescence in GaN ( Saarinen et al. ,
1997 ).
ZnO is the II-VI counterpart of GaN and has been the
object of a multitude of positron studies. Zn-vacancy defects
have been identified in detail in both irradiated andas-grown material ( Tuomisto et al. , 2003 ;Tuomisto,
Saarinen, Look, and Farlow, 2005 ), while some indirect
evidence of O vacancies has also been found ( Tuomisto,
Saarinen, Look, and Farlow, 2005 ;Selim et al. , 2007 ). The
high resistance of ZnO’s electrical and optical properties todeterioration under particle irradiation, the so-called radia-
tion hardness, has generated significant interest in under-
standing the defect structure and behavior under thermaltreatments. Indeed electron and ion irradiation experimentswith subsequent thermal annealings ( Tuomisto, Saarinen,
Look, and Farlow, 2005 ;Chen et al. , 2007 ;Chen,
Betsuyaku, and Kawasuso, 2008 ;Zubiaga et al. , 2008 ;
Knutsen et al. , 2012 ) have shown that the radiation hardness
originates from the high mobility of Zn sublattice damagealready at room temperature in ZnO. The quest for p-type
ZnO has led to many doping-by-implantation studies ( Chen
et al. , 2004 ,2005a ,2005b ;Chen, Maekawa et al. , 2005 ;
Børseth et al. , 2006 ,2008 ;Neuvonen et al. ,2 0 0 9 ,2011 )
where the Zn sublattice damage (Zn vacancies) has been
shown to strongly interact with the implanted impurities.
Section IV.H discusses the role of Li in positron studies of
ZnO—this abundant impurity in bulk crystals grown by thehydrothermal method has led to some scatter in positron
data published over the years ( Johansen, Zubiaga,
Makkonen et al. , 2011 ).
Silicon carbide (SiC) has the interesting property of exist-
ing in hundreds of crystalline forms, called polytypes, formedthrough different stackings of hexagonal planes. The simplest
structures correspond to the zinc blende and wurtzite struc-
tures typical of III-Vand II-VI compounds, called 3Cand2H,
respectively. The 3Cstructure is the only cubic structure of
SiC. The 6
HSiC polytype is the most studied ( 6Hhas triple
periodicity in the stacking sequence of the hexagonal planes
compared to 2H), with 4Hand3Cgathering more and more
interest. Silicon vacancies, silicon-carbon divacancies, andvacancy clusters have been studied in as-grown, irradiated,
and implanted SiC ( Brauer et al. , 1996 ;Kawasuso et al. ,
1996 ;Polity, Huth, and Lausmann, 1999 ;Ling, Beling, and
Fung, 2000 ;Henry et al. , 2003 ;Aavikko et al. , 2007 ). The
optical ionization levels of vacancy defects have also been
studied ( Arpiainen et al. , 2002 ) using positron spectroscop-
ies. The detailed identification of the various vacancy defectshas generated some discussion and even controversy ( Rempel
et al. , 2002 ;Kuriplach et al. , 2003 ). A probable reason for the
relatively strong disagreement between some of the experi-
mental and theoretical results is the crystal structure of SiC isthat most of the experimental studies have been performed in6HSiC (some in 4HSiC), which has three (two) nonequi-
valent lattice sites on both sublattices. Hence the detailed
balance between the defects on these lattice sites can affectthe results quite strongly already for monovacancy defects(Wiktor et al. , 2013 ), not to mention divacancies. For larger
vacancy clusters the situation is the same as in other semi-
conductors, as the number of possible atomic configurations
is already large for simpler structures.B. Vacancy-(multi)donor complexes in highly n-type
doped silicon
The decrease of the size of Si field-effect transistors requires
extremely high doping densities in the drain and source regions.At donor concentrations above 10
20cm/C03, however, the free-
electron concentration stops increasing with doping. This elec-trical deactivation was naturally attributed to the formation ofcompensating defects, while their identification has been muchdebated ( Fahey, Griffin, and Plummer, 1989 ;Nylandsted Larsen
et al.,1 9 9 3 ;Packan, 1999 ). Many computational first-principles
studies have addressed the electrical properties of these defectsas well as their energetics including diffusion barriers [see,e.g., Pandey et al. (1988) ,Ramamoorthy and Pantelides
(1996) ,Xie and Chen (1999) ,Christoph Mueller, Alonso, and
Fichtner (2003) ,a n d V ollenweider, Sahli, and Fichtner (2010) ].
Vacancy-impurity complexes were observed quite early in posi-tron annihilation experiments ( Lawther et al. ,1 9 9 5 ) but their
exact structure remained unknown. Coincidence Dopplerbroadening experiments combined with theoretical calculationsprovided the optimal method for identification of these defects.In fact, in the original study where V-As
ncomplexes with
n¼1–3were identified in Czochralski-grown highly As-doped
Si (Saarinen et al. ,1 9 9 9 ), the coincidence experiments were
performed with only one HPGe detector, with the gating signal
provided by a NaI detector. This kind of coincidence experiment
does not reduce the background as much as the coincidenceexperiment with two HPGe detectors ( Alatalo et al. ,1 9 9 6 ;
Asoka-Kumar et al. ,1 9 9 6 ;Szpala et al. ,1 9 9 6 ), nor does
it narrow the energy resolution, but it provided informationaccurate enough in this case.
Figure 10shows positron lifetime spectra in a float-zone-
(FZ-)grown undoped Si sample and in two 2-MeV electron-irradiated samples: one undoped FZ grown and one As doped
FIG. 10. Positron lifetime spectra in as-grown and in 2-MeV
electron-irradiated Si samples. Positrons annihilate in the as-grown
sample with a single lifetime of 220 ps corresponding to delocalizedpositrons in the lattice. In the irradiated samples the experiments
reveal vacancies with positron lifetimes of 250 ps ( V-Aspair in Cz
Si:As sample doped with ½As/C138¼1020 cm
/C03) and 300 ps (diva-
cancy in undoped FZ Si sample). From Saarinen and Ranki, 2003 .1608 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

(½As/C138¼1020cm/C03) grown by the Czhochralski (Cz) method.
Positrons annihilate in the as-grown sample with a singlelifetime of 220 ps corresponding to delocalized positrons in
the lattice. In the irradiated samples, the experiments reveal
vacancies with positron lifetimes of 250 ps (Cz-grown Si:As)and 300 ps (undoped FZ-grown Si), corresponding tomonovacancy-sized defects and divacancies, respectively
(Kauppinen et al. , 1998 ;Saarinen et al. , 1999 ). Isolated
monovacancies are very mobile in Si already at room tem-perature ( Watkins, 1986 ), and hence in undoped FZ-grown Si
the vacancy defects that survive after irradiation are divacan-
cies formed through the migration process, while in As-doped
Cz-grown Si the mobile monovacancies find As atoms andform stable vacancy-donor complexes ( Lawther et al. , 1995 ;
Saarinen et al. , 1999 ). Positron lifetime spectra in Cz-grown
Si doped with As ( 10
19cm/C03)o rP( 1020cm/C03) also have a
single component of about 220 ps corresponding the bulklifetime /C28
Bin Si. Hence, these materials are free of vacancies
trapping positrons. The average positron lifetime is clearly
higher in as-grown Si doped with As to a higher level
(½As/C138¼1020cm/C03), giving /C28av¼232 ps at room tempera-
ture. Furthermore, the lifetime spectrum has two components,the longer of which is /C28
2¼250/C63p s. Both /C28avand/C282
are almost constant as a function of temperature. The two-
componential lifetime spectrum and the increase of /C28ave
above the bulk lifetime /C28Bare clear signs that native vacan-
cies exist in Sið½As/C138¼1020cm/C03Ț. The second lifetime
component /C282¼250/C63p s is characteristic of the positron
annihilations at a monovacancy defect.
In 2-MeV electron-irradiated samples the average positron
lifetime is longer than in as-grown samples, indicating thatirradiation-induced vacancies are observed (Fig. 10). In
Sið½P/C138¼10
20cm/C03Țirradiated to the fluence 5/C21017cm/C02
the lifetime spectrum can be decomposed and the vacancy
component /C282¼250/C63p s is obtained ( Saarinen et al. ,
1999 ). Irradiated Sið½As/C138¼1020cm/C03) exhibits only a single
positron lifetime of about 247/C62p s, almost independently of
the irradiation fluence. This behavior can be explained by a totalpositron trapping at irradiation-induced vacancy defects. Whenthe vacancy concentration exceeds 10
18cm/C03, all positrons
annihilate at the irradiation-induced vacancy defects with the
lifetime 247/C62p s and no annihilations take place at the
delocalized bulk states or at the native vacancies detected beforeirradiation. The vacancy concentration of /C2110
18cm/C03is con-
sistent with the expected introduction rate in electron-irradiated
heavily n-type Si. The same positron lifetime characteristic
of a monovacancy, /C28V¼248/C63p s, is in fact observed for
three different types of samples: (i) as-grown Sið½As/C138¼
1020cm/C03Ț, (ii) electron-irradiated Sið½As/C138¼1020cm/C03Ț,
and (iii) electron-irradiated Sið½P/C138¼1020cm/C03Ț(Saarinen
et al. ,1 9 9 9 ).
Doppler broadening experiments using the two-detector
coincidence technique have been used to identify these mono-
vacancy defects in more detail. In the samples containing
vacancy defects, the Doppler broadening represents the super-imposed distribution /C26ðpȚ¼ð1/C0/C17Ț/C26
BðpȚț/C17V/C26VðpȚ, where
/C26BðpȚand/C26VðpȚare the momentum distributions in the
lattice and at the vacancy, respectively. The lifetime results
(Fig. 10) can be used to determine the fraction of positrons
annihilating at vacancies /C17V¼ð/C28ave/C0/C28BȚ=ð/C28V/C0/C28BȚ[see Eqs. ( 26) and ( 27)]. Since the momentum distribution in
the lattice /C26BðpȚcan be measured in the reference sample, the
distributions /C26VðpȚat vacancies can be decomposed from
the measured spectrum /C26ðpȚ. They are shown in Fig. 11for
the monovacancies observed in as-grown Sið½As/C138¼
1020cm/C03Țas well as in irradiated Sið½As/C138¼1020cm/C03Ț
andSið½P/C138¼1020cm/C03Ț.
The momentum distributions at vacancies /C26VðpȚindicate
large differences in the higher momenta ( p>2a:u:), where
the annihilation with core electrons is the most important
contribution (Fig. 11). Since the core electron momentum
distribution is a specific characteristic of a given atom, thedifferences between the spectra in Fig. 11indicate different
atomic environments of the vacancy in each of the three
cases. Because in both Si ( Z¼14) and P ( Z¼15) the 2p
electrons constitute the outermost core electron shell, the core
electron momentum distributions of these elements are very
similar. The crucial difference in the core electron structuresof Si, P, and As is the presence of ten 3delectrons in As. The
overlap of positrons with the As 3delectrons is much stronger
than with the more localized Si or P 2pelectrons. The large
intensity of the core electron momentum distribution is thus aclear sign of As atoms surrounding the vacancy. The 2-MeV
electron irradiation creates vacancies and interstitials as pri-
mary defects, both of which are mobile at room temperature(Watkins, 1986 ).
In heavily n-type Si the donor atom may capture the
vacancy and form a vacancy-impurity pair ( Watkins, 1986 ).
The monovacancy detected in heavily P-doped Si is thus the
V-Ppair. Similarly, it is natural to associate the electron-
irradiation-induced vacancy in Sið½As/C138¼10
20cm/C03Țwith a
V-Aspair. An even stronger signal from As is seen in the as-
grown Sið½As/C138¼1020cm/C03Ț. A linear extrapolation of the
intensity of the distribution suggests that the native complexisV-As
3, i.e., the vacancy is surrounded by three As atoms.
These identifications are confirmed by theoretical results that
are in very good agreement at both low and high momenta(Fig. 11). The theoretical calculations strongly support the
FIG. 11. Experimental (markers) and calculated (solid curves)
coincidence Doppler broadening spectra in vacancy-donor com-plexes in Si, showing the perfect match between theory and
experiment. Adapted from Saarinen and Ranki, 2003 .Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1609
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

experimental defect identifications that (i) vacancies com-
plexed with single donor impurities are detected in electron-irradiated P and As-doped Si, and (ii) the native defect in
Sið½As/C138¼10
20cm/C03Țis a vacancy surrounded by three As
atoms.
The existence of V-As3complexes in heavily As-doped
Cz-grown Si is consistent with the defect formationand diffusion mechanisms described theoretically
(Ramamoorthy and Pantelides, 1996 ). The calculated forma-
tion energies of V-As
n(n>2) complexes are negative, sug-
gesting that total deactivation of As takes place at any dopinglevel ( Pandey et al. , 1988 ;Ramamoorthy and Pantelides,
1996 ). The n-type conductivity of Si(As) is possible only
because the creation of defect complexes is limited by kinetic
processes such as the migration of defects. At high tempera-ture the diffusion of As starts by the formation of V-Aspairs,
which can migrate to form V-As
2complexes ( Mathiot and
Pfister, 1983 ). The calculations ( Ramamoorthy and
Pantelides, 1996 ) predict that these complexes can diffuse
until they stop at the substitutional As forming the V-As3
complex. In addition, no V-As3are found at the lower doping
level of ½As/C138¼1019cm/C03(Saarinen et al. , 1999 ), most likely
because the average distance between the donor atoms is too
large and the migrating V-AsandV-As2may dissociate
before creating larger complexes. In fact, similar observationsand conclusions can be made in heavily P-doped and
Sb-doped Si as well ( Ranki, Pelli, and Saarinen, 2004 ;
Rummukainen et al. , 2005 ). The formation mechanism of
the vacancy-donor complexes ( V-D) has been studied in
annealing experiments of the electron-irradiated samples.
A general scheme for vacancy-donor complex formation
can be deduced from positron experiments ( Ranki, Pelli, and
Saarinen, 2004 ). The electron irradiation creates vacancy-
donor pairs ( V-D
1) that start to migrate at 400–450 K. The
migrating and dissociating V-D1defects form more stable
V-D2defects and some divacancies. The divacancies start to
anneal away soon after they are formed, around 450–500 K,
and at 600 K there are only V-D2defects left. In the case of
As and P doping, the V-D2defects start to migrate at 650–
700 K forming V-D3defects. The formation of V-D2and
V-D3depends heavily on the doping concentration. The
vacancy-impurity complex formation can be explained by
the ring diffusion mechanism, where the opposite chargesof the vacancy and the donor atom bind the migrating com-plex together ( Ramamoorthy and Pantelides, 1996 ;Pankratov
et al. , 1997 ;Xie and Chen, 1999 ;Ranki, Nissila ¨, and
Saarinen, 2002 ). These observations are consistent with re-
sults obtained by EPR and deep-level transient spectroscopy(Nylandsted Larsen, Christensen, and Petersen, 1999 ). The
estimated activation energies are also in very good agreement
with theoretical calculations of migration barriers ( Xie and
Chen, 1999 ;Vollenweider, Sahli, and Fichtner, 2010 ).
C. The vacancy-fluorine complex in silicon and
silicon-germanium alloys
The effect of fluorine on the behavior of vacancies ( V) and
interstitials ( I) in Si has been of great interest ( El Mubarek
and Ashburn, 2003 ). It has been incorporated in Si in several
device processes through ion implantation ( Ma, 1992 ;Williams and Ashburn, 1992 ;Downey, Osburn, and
Marcus, 1997 ). Fluorine in silicon exhibits the ability to
retard the diffusion of boron, either when coimplanted asBF
2, or as separate implants. This allows a strategy for
ultrashallow junction technologies. The key is the formationof fluorine-vacancy ( F-V) and fluorine-interstitial ( F-I) com-
plexes. The incomplete picture of the basic behavior of F in Sihas stood in the way of the realization of its full potential.
Positron annihilation spectroscopy has been used to study the
effects of F on the vacancies in implantation processing of Siand Si-Ge alloys. These studies provide an illustrativeexample of the study of ion implantation damage and sub-sequent identification of the defects created by processing.
Figure 12shows the Sparameter as a function of positron
implantation energy for a selection of F-implanted ( E
ion¼
0:5 MeV ) and subsequently annealed Cz-grown Si samples
(Pi, Burrows, and Coleman, 2003 ). The trapping of positrons
at vacancy defects leads to a higher Sparameter compared
with that for the defect-free lattice. The Svalues presented in
Fig.12have been normalized with respect to that for a bulk
virgin Si sample for which Sis thus 1. The 0.5 MeV implan-
tation energy of F ions corresponds to a projected range Rp¼
0:9/C22m. The vacancy distribution extends to around 2:3/C22m.
This abnormally deep distribution of vacancies is probablydue to F atoms strongly reacting with interstitials during ionimplantation and allowing the survival of most of the vacan-cies. The dip that appears around 11 keV (Fig. 12) in the S
curve for the as-implanted sample and deepens strongly in theannealings corresponds to the region close to R
p. It is known
from earlier studies that F and O reduce Swhen they are
associated with vacancies because of the large momenta of
(a)
Incident Positron Energy (keV)0 5 10 15 20 25 30SParameter
0.880.900.920.940.960.981.001.021.041.061.08
(b)as-implanted
30 s
8.5 min
30 min
20 h
43 h
67 h
0.920.940.960.981.001.021.041.06Mean Depth ( µµµµm)
0.0 0.2 0.5 1.0 1.5 2.0 3.0 4.0
125 has-implanted
8.5min
1 h
4 h
43 h
90 h 30 s
FIG. 12. Sparameter depth profiles for F-implanted Si samples
annealed at (a) 400/C14Cfor up to 67 h and (b) 700/C14Cfor up to 125 h.
From Pi, Burrows, and Coleman, 2003 .1610 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

their outermost-shell electrons ( Coleman, Chilton, and Baker,
1990 ;Fujinami, 1996 ;Uedono et al. , 1997 ). Hence based on
these depth profiles it seems evident that F decorates somevacancies around R
palready right after implantation, and this
effect becomes stronger after the annealings.
Figure 13shows coincidence Dopper broadening data
(ratio curves) from similarly implanted FZ-grown Si samples.
The ratio curve for the as-implanted sample is typical of
undecorated vacancy defects (or in the case of silicon, thedivacancy) with the low-momentum region having intensityabove 1 and the high-momentum region below 1. The dataobtained in the F-implanted samples after annealing show a
very strong effect at high momenta (ratios up to 2.5 at around
1.5 a.u.), while the Sparameter region at low momenta goes
below 1. This behavior is typical of impurities with outer-shell electrons with large momenta (such as O or F) or withlarge number of outer-shell electrons (e.g., As, see previous
section). The concentrations of 3dimpurities are very low in
FZ-grown Si, as is the concentration of O, and hence it isnatural to assign the strong peak at 1.5 a.u. as the F finger-print. Figure 14shows similar data for F-implanted Si-Gealloys. The figure shows that the effects of Ge, O, and F in the
silicon lattice, although qualitatively quite similar, can be
separated. The differences in the positions of the signatures
(511 keV corresponds to 0 a.u., and 522 keV to 6 a.u.)
between Figs. 13and14originate from different detector
resolutions: the data in Fig. 13have been measured with a
two-HPGe-detector coincidence setup that narrows the reso-
lution by a factor of /C241:4compared to the single-detector
measurement that was used for acquiring the data in Fig. 14.
The fast convergence to a ratio of /C241above /C24516 keV
(2.7 a.u.) in Fig. 14, not observed in Fig. 13, is due to the
dominance of the high background in the single-detector
measurement compared to the two-HPGe-detector coinci-
dence measurement.
The fluorine-vacancy complexes have indeed been pre-
dicted as the cause for the reduction of diffusion of boron
(Diebel and Dunham, 2004 ;Lopez et al. , 2005 ;Lopez and
Fiorentini, 2006 ;Vollenweider, Sahli, and Fichtner, 2009 ),
and they have been experimentally found also by EPR
(Umeda et al. , 2010 ). TEM has been used to follow the
generation of larger F precipitates in solid-phase epitaxy ofSi (Boninelli et al. , 2006 ,2008 ), consistent with the predicted
vacancy-fluorine complex dynamics. The boron-diffusion-
reducing properties of fluorine and the strong vacancy-
fluorine interactions extend from Si also to Si-Ge alloys
(El Mubarek et al. , 2005 ) and Ge ( Jung et al. , 2012 ). The
positron annihilation experiments have given a significant
contribution to the understanding of the phenomena related
to fluorine implantation and its effects on vacancy passivation
and suppression of boron transient enhanced diffusion.
D. The EL2 defect in gallium arsenide
One of the most prominent examples of the application of
positron annihilation spectroscopy in identification of tech-
nologically important defects is the investigation of the so-
called EL2 midgap donor defect in GaAs. The importance of
this defect stemmed from its central role in the growth ofundoped semi-insulating (SI) GaAs ( Martin and Makram-
Ebeid, 1986 ;Kaminska and Weber, 1993 ). A key property
of EL2 is its optically induced metastability: it can be perma-
nently converted to the neutral metastable state EL2* under
0.8–1.5 eV illumination at temperatures below 100 K. The
photoquenching occurs without generation of any new elec-
trical or optical signals that could be associated with themetastable state EL2*. Identification of the atomic structure
of this defect was the focus of a considerable effort in the late
1980s and early 1990s. Another metastable defect whose
identification proved to be a challenging task (around the
same time) was the so-called DX center ( Mooney, 1990 ;
Ma¨kinen et al. , 1993 )i nAl
1/C0xGaxAsthat produces persistent
photoconductivity.
Figure 15shows results of positron experiments where SI
GaAs crystals with two different EL2 concentrations were
cooled in darkness and illuminated in situ with 1.2 eV light
(Krause, Saarinen et al. , 1990 ). To check that the EL2 defects
were photoquenched by the illumination, infrared absorption
was also measured. It is clearly seen that after the photo-quenching of the EL2 centers at low temperatures (below
100 K) there is a clear increase of the positron annihilation
FIG. 13. Coincidence Doppler broadening spectra, normalized to
the spectrum for silicon, showing relative intensity vs electron
momentum. Peaks in the data from annealed samples match the‘‘fingerprint’’ of fluorine. From Simpson et al. , 2004 .
FIG. 14 (color online). Ratios of the annealed samples of relaxed
10% and 30% Ge and multilayer 30% Ge at /C242 keV . Best fits are
shown on top of data. Ratios of implanted SiO 2=Si,F=Si,Ge=Si,
andV2inSi=Siare shown for reference. All spectra are divided by a
Si spectrum. From Edwardson et al. , 2012 .Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1611
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

parameters /C28avandS. This property is reproducible and has
been since observed in various SI GaAs crystals, indicatingthat vacancy defects are generated by the photoquenching ofEL2. In addition, the concentration of these vacancy defects iscorrelated with the total EL2 concentrations ( Le Berre et al. ,
1994 ;Saarinen et al. , 1994 ). Further, both the time constant
and photon energy dependence of the effect show that thevacancy generation is due to the EL2 conversion to themetastable state ( Saarinen et al. , 1994 ). As an example,
Fig. 16shows the optical cross section for the creation of
the metastable state of the EL2 defect and the metastable
vacancy. A similar approach can be used to study the photo-
ionization levels of vacancy defects ( Kuisma et al. , 1996 ).
The identification of the metastable vacancy defect was
possible by comparison of the second lifetime componentextracted from the lifetime spectra after illumination, whichin SI GaAs was found to be /C28
2¼247/C63p s. In this particu-
lar state of the sample, two kinds of vacancies may trap
positrons: the Ga vacancies often found as native defects inSI GaAs and the metastable vacancies observed after illumi-nation. It should be noted that in Fig. 15the average positron
lifetime in dark is a bit longer than the bulk lifetime /C28
B¼
230 ps of GaAs. Hence the second lifetime /C282should be a
superposition of the lifetimes at Ga vacancies /C28V¼260 ps
and at the metastable vacancies with the positron lifetime /C28V/C3.Taking into account the positron trapping at native Ga vacan-
cies, one could estimate that the positron lifetime at themetastable vacancy is /C28
V/C3/C24245 ps (Saarinen et al. ,1 9 9 4 ).
This lifetime is clearly shorter than the values at the Ga
vacancies and As vacancies (260–300 ps). Hence the meta-stable structure of the EL2 defect has an associated openvolume that is smaller than that of a monovacancy in GaAs.These observations gave strong support to the model where
the EL2 defect consists of an isolated As antisite defect that
relaxes toward the interstitial position in the metastable state.The reaction As
Ga!VGațAsicreates the VGa-Asipair that
is the metastable vacancy V/C3with smaller open volume than
that of the isolated Ga vacancy ( Chadi and Chang, 1988 ;
Dabrowski and Scheffler, 1988 ). There is no evidence of
positron trapping at the stable state of EL2, consistent withthe idea that open volume is not present then. Theoreticalcalculations of the positron states ( Laasonen et al. , 1991 ) also
show that the proposed atomic configuration of the meta-
stable EL2 indeed localizes a positron, with a specific lifetimein between the bulk and Ga-vacancy lifetimes.
The work on the EL2 defect also demonstrates the capa-
bility of positron annihilation methods to identify transition
levels in the band gap of a semiconductor. In the above case,the optically induced transition involves a strong latticerelaxation that makes the defect appear in positron annihila-tion experiments. Another possibility is the change in the
charge state of the vacancy defect, affecting the sensitivity
of positrons (i.e., the positron trapping coefficient and its
FIG. 15. The average positron lifetime /C28avand annihilation line-
shape parameter Sin semi-insulating GaAs as functions of isochro-
nal annealing temperature after 1.2-eV illumination at 25 K. Theillumination transforms EL2 into the metastable state and corre-
sponding changes in positron parameters are indicated by arrows.
The open triangles with dashed lines below 100 K represent thereference levels where EL2 is in the stable state. The normalized
infrared absorption coefficient is shown in the top panel. All
measurements have been made in darkness at 25 K. From Krause,
Saarinen et al. , 1990 .
FIG. 16. Optical cross section for the creation of the metastable
vacancy as a function of the photon energy in a SI GaAs sample.The data in the upper part are obtained from IR absorption
measurements and in the lower part from positron lifetime mea-
surements. Measurement temperature was 25 K. From Saarinen
et al. , 1994 .1612 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

temperature behavior) to the defect in question. The effects of
vacancy charge states are discussed in more detail in the nextsections. Good examples of observation of photoionization ofdefect levels include the work on vacancies in SiC ( Arpiainen
et al. , 2002 ) and vacancy clusters in diamond ( Dannefaer, Pu,
and Kerr, 2001 ;Ma¨kiet al. , 2009 ).
E. The gallium vacancy–tellurium complex in gallium arsenide
The studies of Te-doped GaAs provide an instructive
example of how temperature-dependent positron lifetimemeasurements show the competition of negatively charged
vacancies and negative ions in positron trapping at low
temperatures, and how the changes in their relative concen-trations affect the data. In addition, coincidence Dopplerbroadening studies of n-type GaAs show how to distinguish
between the two possible sublattices of the observed vacan-cies, as positron lifetime experiments alone are not enoughfor this task: the same lifetime can be produced by a Ga
vacancy, an As vacancy, and a vacancy-dopant complex when
all atoms are on lattice sites. Te is incorporated in the Assublattice only ( Hurle, 1979 ).
Figure 17from Gebauer et al. (1999) shows positron
lifetime data measured in Te-doped Cz-grown GaAssamples with varying carrier concentration ( n¼5/C2
10
16–1018cm/C03). The Zn-doped GaAs reference does not
show any trapping of positrons at vacancies. The slightincrease of the average positron lifetime with measurementtemperature comes from the thermal expansion of the lattice(Le Berre et al. , 1995 ). The average positron lifetime /C28
avin
the Te-doped samples is above /C28B¼229–230 ps found in
GaAs:Zn, indicating positron trapping at vacancies. The
temperature dependence of /C28avin GaAs:Te is typical whenpositrons are trapped at negatively charged vacancies and
negative ions. The negative ions with a lifetime close to /C28B
trap positrons in their shallow potential only at low tempera-
ture. Negative ions can be attributed to intrinsic defects, such
asGaAs, or extrinsic impurities, but positron annihilation
alone does not allow their detailed identification. Withincreasing temperature positrons escape from the ions anda larger fraction annihilates at vacancies, causing the increaseof/C28
avbetween 100 and 200 K. The decrease of /C28avat
T>200 K in the medium-doped samples indicates positron
trapping at negative vacancies.
The data are fitted with a model taking into account
positron trapping and detrapping at the shallow Rydbergstates around negative ions and vacancies as well as the
T
/C01=2dependence for positron trapping at negatively charged
defects. The temperature dependence of positron trapping isdiscussed in Sec. II, while a detailed discussion can be found
in, e.g., Krause-Rehberg and Leipner (1999) . The parameters
describing the temperature dependence of positron trappingare similar in all samples, the binding energy of positrons to
the Rydberg states was E
ion¼65/C620 meV , and only the
concentrations of the ions and vacancies relative to each otherchange ( Gebauer et al. , 1999 ). Positron trapping at vacancies
is practically saturated in the most highly doped sample andhence /C28
avereflects the slight decrease of the defect-related
lifetime /C28Dwith temperature. This might be attributed to
lattice expansion too, although the effect is larger than inthe reference. It is important that /C28
D¼254/C63p s at 300 K
is the same in all samples and exhibits the same temperaturedependence, suggesting that the vacancies are similar in allsamples. This suggestion is confirmed by the observation that
the Doppler broadening parameters ( SandW) change line-
arly with the average positron lifetime in these samples(Gebauer et al. , 1999 ).
Figure 18from Gebauer et al. (1999) shows the high-
momentum part of the annihilation momentum distribution
ratio curve for the vacancies in GaAs:Te compared to those
found in GaAs:Si. The core annihilation is more intense inGaAs:Te than in GaAs:Si in the momentum range p
L¼
ð10–20Ț/C210/C03m0c(1.3–2.6 a.u.). The observations of the
momentum distribution can be explained as follows. In bulk
GaAs, the dominating contribution to the core annihilation
comes from Ga 3delectrons ( Z¼31)(Alatalo et al. , 1996 ).
The As 3delectrons ( Z¼33) are more tightly bound and
hence the momentum distribution is broader and the intensityof the core annihilation is reduced. Positron annihilation attheSi
Ga-VGacomplexes in GaAs:Si occurs mainly with 3d
electrons from As. Thus, the momentum distribution should
be broader compared to the bulk. This is, in fact, observed. Incontrast, at As vacancies the momentum distribution shouldbe narrower and more intense because annihilation occursmainly with the Ga 3delectrons.
In tellurium the main contribution to the core annihilation
comes from 4delectrons that are less strongly bound than
the As 3delectrons in GaAs. They contribute therefore to the
core annihilation more at lower momenta and have a steepermomentum distribution. The theoretical calculations shownin Fig. 18(b) demonstrate this effect very well. A similar
difference has been noted by comparing results from bulk
InP, GaSb, and GaAs ( Alatalo et al. ,1 9 9 5 ) and for
FIG. 17. (a) Average positron lifetime /C28aveand (b) defect-related
lifetime /C28defvs measurement temperature in Te-doped GaAs
compared to a GaAs:Zn reference. Spectral decomposition was
not reliable for T>350 K for the 5/C21016cm/C03doped sample
(/C28avis close to /C28b). Lines are fits according to the temperature-
dependent trapping model. From Gebauer et al. , 1999 .Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1613
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

Zn-impurity–P-vacancy complexes in InP ( Alatalo et al. ,
1996 ). The shape of the momentum distribution measured
in GaAs:Te therefore indicates that the vacancies areneighbored by Te atoms. Because Te resides on the Assublattice, the vacancy must be on the Ga sublattice. Hencethe vacancies in GaAs:Te can be identified as Ga-vacancy–Te
Ascomplexes. This assignment of the reduced positron
lifetime at the Ga vacancies in GaAs:Te (254 ps) compared
to the Ga vacancies in GaAs:Si (262 ps) is natural, as the largeTe atom can be expected to decrease the open volume of theneighboring Ga vacancy.
F. The gallium vacancy and its complexes in gallium nitride
Positron studies of GaN and the III-nitrides started in the
second half of the 1990s, with five papers published bydifferent groups in 1997 ( Cho et al. , 1997 ;Dannefaer, Puff,
and Kerr, 1997 ;Jorgensen et al. , 1997 ;Saarinen et al. , 1997 ;
Suzuki et al. , 1997 ). These reports give an instructive
example of how a new material is studied and how the results
can be interpreted just by using knowledge acquired inprevious studies of another material. GaAs was a naturalmaterial of comparison, as it was the compound semiconduc-tor that had been studied extensively with positron annihila-tion spectroscopy. The identification of Ga vacancies bySaarinen et al. (1997) was possible thanks to the growth of
high-nitrogen-pressure bulk GaN single crystals of sufficient
size for experimentation.Figure 19shows the positron lifetime data obtained as a
function of temperature in such crystals. The average positron
lifetime in the GaN bulk crystal is constant /C28
av¼167 ps (/C28av
in the figure) at temperatures T¼10–150 K but increases up
to/C28av¼191 ps at 500 K. The lifetime spectra recorded at
200–500 K can be decomposed into two components. The
longer-lifetime component is constant, /C282¼235/C65p s
(Fig. 19), as a function of temperature. By also fixing this
lifetime, the spectra measured at 10–200 K could be decom-
posed. The lifetime component /C281is a constant, /C281¼164/C6
1p s, at 10–150 K and then decreases to about /C281¼140 ps at
500 K. The two-component lifetime spectrum implies that
positrons in the GaN bulk crystals annihilate either from adelocalized state in the lattice or as localized at vacancy
defects. The positrons trapped at vacancies annihilate with
the longer lifetime /C28
V¼/C282¼235/C65p s. The decrease of
the average lifetime at low temperatures indicates that the
fraction /C17Vof positron annihilations at vacancies decreases.
When /C17V!0, the component /C281approaches the lifetime
value /C28Bof delocalized positrons in the lattice. At 10 K, one
can see that /C281¼164/C61p s and/C28av¼167 ps . The positron
lifetime in the GaN lattice can be interpreted to lie between
these values, i.e., /C28B¼166/C61p s.
By comparing to atomic superposition calculations that
showed that N vacancies could not explain the increase in
the lifetime of the trapped positrons, it was concluded that
Ga-vacancy-related defects are responsible for the /C282¼
235 ps lifetime component. Doppler broadening measure-
ments were also performed on these crystals, and early valuesfor the SandWparameters specific to the Ga-vacancy-
related defects could be proposed. It should, of course, be
FIG. 18. (a) High-momentum part of the positron annihilation
momentum distribution (normalized by taking the ratio to aGaAs:Zn reference) for the vacancies in GaAs:Te and GaAs:Si.
The spectra (total area 3:5/C210
7counts) were brought to unity and
scaled to full trapping at the vacancies before normalization. Linesresult from smoothing and serve to guide the eye only. (b) Ratio ofthe momentum density to bulk GaAs for different vacancies in
GaAs from theoretical calculations. The curves for V
Ga-TeAsand
VAs-SiGacomplexes are highlighted to emphasize the good agree-
ment with the respective experimental data in GaAs:Te and GaAs:
Si. The theoretical curves are not accurate for pL<15/C210/C03m0c
and hence are omitted. From Gebauer et al. , 1999 .
FIG. 19. The average positron lifetime /C28avand the lifetime compo-
nent/C282vs measurement temperature for GaN bulk crystal. The
lifetime component /C282could be decomposed only at T>200 K .
The solid lines are drawn to guide the eye. From Saarinen et al. , 1997 .1614 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

noted that these values depend strongly on the experimental
geometry and detector details. In that work the Ga-vacancycomplexes found in the bulk GaN crystals and MOCVD
GaN thin epilayers on sapphire were shown to be correlated
with the presence of yellow luminescence often observed inn-type GaN.
The vacancy defects in GaN and III-nitrides have since
their identification been studied with positron annihilationspectroscopy by many research groups, with a total paper
count amounting to about 200 in the past 15 years. The
Ga-vacancy-related defects are now known to be thedominant intrinsic acceptor-type (compensating) defects inboth unintentionally and intentionally n-type doped GaN,
irrespective of the method of growth ( Saarinen, 2000 ;
Tuomisto, 2010 ). When Mg is a contaminant, it also com-
pensates for the n-type conductivity (and at high intentional
concentrations produces highly resistive and/or p-type GaN).
The compensating Ga-vacancy-related defects exhibitnegative charge states, with transitions deep in the widebandgap of GaN, and they have been shown to contribute to
both nonradiative recombination processes and luminescent
processes such as the parasitic yellow luminescence. A widebody of experimental and theoretical research exists on thesedefects ( Neugebauer and Van de Walle, 1994 ,1996 ;
Kaufmann et al. , 1999 ;Armitage et al. , 2003 ;Chow et al. ,
2004 ;Limpijumnong and Van de Walle, 2004 ;Van de Walle
and Neugebauer, 2004 ;Reshchikov and Morkoc ¸, 2005 ).
A more detailed identification of the Ga-vacancy-related
defects was obtained relatively recently, thanks to the combi-nation of coincidence Doppler broadening experiments andstate-of-the-art theoretical calculations ( Hautakangas et al. ,
2006 ). Electron irradiation experiments ( Tuomisto, Ranki
et al. , 2007 ) were used to produce isolated Ga vacancies,
and the data could be compared to GaN samples grown bydifferent methods. Figure 7shows the electron momentum
distribution ratio curve of the Ga vacancy in 2-MeV electron-
irradiated GaN. The signal of a clean Ga vacancy was ob-
tained by decomposing the original Doppler broadeningspectrum by determining the fraction of annihilations ofdelocalized positrons /C17
V¼40% from the positron lifetime
measurement. The calculated curve for the isolated Ga va-
cancy correlates well with the experimental one through the
whole momentum range, supporting the identification of theisolated Ga vacancy. The main contribution in the rangebetween 2 and 4 a.u. arises from annihilations with Ga 3d
electrons. The decrease in intensity at this momentum regionis due to the reduced intensity of Ga 3delectrons in a Ga
vacancy. The good agreement at both valence and core
electron regions manifests the accuracy and predictive powerof the theoretical calculations.
Figure 20shows experimental ratio curves also for other
vacancy-related defects in representative GaN samples grown
by different methods. The curves are not similar, indicating
that different complexes can be distinguished. In the momen-tum range between 2 and 4 a.u. the data have clear order. Theintensity of irradiated GaN is the lowest while the oxygen-doped GaN has higher intensity. This effect can be attributedto oxygen surrounding the Ga vacancy in the defect complex:
the O atom is smaller than N and thus contributes more at
high electron momentum. The same behavior can be seen inthe calculated momentum ratio curves. The difference be-
tween V
Ga-ONand isolated VGaarises from the valence elec-
tron states derived from the atomic 2porbitals. The data in
MOCVD GaN at a high-momentum region where the intensityis the highest of all samples are best explained by additionalhydrogen in the V
GaandVGa-ONcomplexes. The contribution
of hydrogen cannot be completely ruled out in the case of
O-doped hydride vapor phase epitaxy GaN either, but in thosesamples it was also shown that the Ga-vacancy concentrationis clearly correlated with the O concentration ( Hautakangas
et al. , 2006 ). Lately the presence of vacancy-hydrogen com-
plexes in MOCVD GaN has been observed to be directlyrelated to nonradiative recombination ( Nyka¨nenet al. , 2012 ).
G. Metal vacancy–nitrogen vacancy complexes in III-nitrides
and their alloys
Metal vacancies and their complexes have been studied
rather extensively in the III-nitrides and their alloys; see,e.g., Stampfl et al. (2000) ,Limpijumnong and Van de Walle
(2004) ,Hautakangas et al. (2006) ,Duan and Stampfl
(2008 ,2009a ,2009b) ,Son et al. (2009) ,Van de Walle,
Lyons, and Janotti (2010) ,Ma¨ki, Makkonen et al. (2011) ,
Rauch, Makkonen, and Tuomisto (2011) , and Janotti, Lyons,
and Van de Walle (2012) . The vast majority of the studies are
devoted to the binary compounds GaN, InN, and AlN, butsome also make an attempt to understand the defect structurein ternary alloys. In fact, this is not a trivial task as the randomalloys in principle exhibit a wide variety of local defectsurroundings.
A systematic theoretical study of various defect spectra
(ratio curves) in the case of InN is shown in Fig. 21(Rauch,
Makkonen, and Tuomisto, 2011 ). The ratio curve for the V
In
exhibits a distinct line shape with a maximum of /C241:08at the
peak center region (0 a.u.). For momenta above 0.6 a.u. the
spectrum drops below 1 and a clear shoulder is visible at1.2 a.u., which has been determined by atomic superpositioncalculations to stem from annihilations with N 2pelectrons.
At/C243:3a:u:a second broad peak appears with an intensity of
/C240:8relative to the InN lattice. Positron annihilation char-
acteristics of the 2V
Indefect are very similar to those of the1.1
1.0
0.9
0.8
0.7
0.6
0.5Experimental intensity ratio
3 2 1 0
Momentum (a.u.)GaN
O doped HVPE n-GaN
MOCVD n-GaN
Irradiated GaN
4
FIG. 20. Experimental coincidence Doppler ratio curves for
irradiation-induced and two kinds of in-grown Ga vacancies.
Adapted from Hautakangas et al. , 2006 .Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1615
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

isolated VIn. For the 3VIncomplex the ratio curve changes
significantly with an increased peak maximum and a morepronounced drop at 2.1 a.u. Nevertheless, further analysisshows that the relative line shapes of the 3V
InandVInare
very similar and the spectrum of the VIncan be essentially
reproduced from the 3VInspectrum by assuming a positron
annihilation fraction of /C17/C250:8.T h e 3VInandVInare there-
fore practically indistinguishable. Isolated VNand pure VN
clusters do not localize positrons.
The calculated positron annihilation characteristics for the
relaxed lattice structures of a variety of mixed vacancy com-plexes in InN, namely, V
In-VN,VIn-2VN,2VIn-VN, andVIn-3VN
are shown in Fig. 21(b) . A systematic trend compared to the
isolated VInis visible in the spectra when adding an increasing
amount of VNaround a single VIn. A strong increase of the zero
momentum maximum to over 1.15 for the VIn-3VNis visible,
which is related to the increase in open volume. At the sametime, the intensity of the shoulder at 1.2 a.u. decreases with anincreasing number of V
Nuntil it entirely disappears forVIn-3VN. The ratio curve of 2VIn-VNis close to that of
VIn-VNfor lower momentum values but starts to deviate at
around 1.4 a.u. with lower intensities at higher momenta, dueto reduced annihilation with In 4delectrons. Figure 21(c)
shows data for relaxed defect structures for the V
In-ON,
VIn-3ON, and VIn-SiIncomplexes. The peak maximum de-
creases with increasing number of O ions, while the intensityin the spectral range above 0.9 a.u. increases, including theshoulder at 1.2 a.u. and the peak at 3.4 a.u. The form of theV
In-ONratio curve resembles the case of VIntrapping with a
reduced annihilation fraction of /C17/C250:8. The spectrum of the
VIn-SiIncomplex is very close to that of VInand hence hardly
distinguishable in experiments. The case is different for theratio curve of V
In-3ONwhich possesses distinct features with
shoulders at 1.2 and 3.6 a.u., respectively, which should bemeasurable in coincidence Doppler measurements.
Figure 22shows experimental ratio curves measured
in selected InN samples. Sample I is a MBE-grown InN
FIG. 21 (color online). Ratio curves of the calculated momentum
densities of annihilating e-ppairs in selected vacancy complexes
in InN. All spectra are convoluted with a Gaussian of 0.53 a.u.
FWHM (except V0:66 a:u:
In ,FWHM ¼0:66 a:u:) and divided by the
momentum-density spectrum of the InN lattice. From Rauch,
Makkonen, and Tuomisto, 2011 .
FIG. 22 (color online). Experimental coincidence Doppler spectra
of the investigated samples in the layer (a), (b) and interface
(c) region. The data have been divided by a suitable reference
spectrum for the InN lattice. Computational ratio curves are shownfor comparison. From Rauch, Makkonen, and Tuomisto, 2011 .1616 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

irradiated with 2-MeV He ions to a fluence of 9/C21015cm/C02;
samples II and III are as-grown Si-doped and undoped InNlayers deposited by MBE and MOCVD, respectively.
Samples I and II show a strong change in the Doppler broad-
ening signal when approaching the interface, a commonfeature in many InN samples. Therefore, the interface regionis investigated separately in these two samples. The extrapo-lated ratio curve of sample I is shown in Fig. 22(a) . This is in
agreement with the spectrum of the isolated V
Infor most of
the spectral range. In the central region of the peak slightly
higher intensities are found in the calculated spectrum com-pared to the experimental one. This region is mostly sensitiveto the size of the open volume of the positron trap, with higherintensities for larger volumes.
The extrapolated ratio curves of both sample II and the as-
measured spectrum of sample III show a very similar lineshape in Fig. 22(b) . The bigger scatter in the former is due to
a smaller annihilation fraction. Compared to sample I, the as-grown samples II and III show several differences in theirratio curves. First, the intensity in the peak center region is
clearly increased. The intensity difference from the InN
lattice is thereby magnified by /C2435% compared to the spec-
trum of sample I. Second, a significant decrease of theshoulder at 1.2 a.u. is visible, also with high statistical accu-racy. Third, the drop at 2 a.u. is less pronounced, followed by
slightly higher intensities in the high-momentum region of the
spectrum. A comparison with the calculated defect spectra inFig.21(b) reveals that these changes coincide with the effects
of the decoration of a V
InbyVN. In particular, the character-
istic decrease of the shoulder at 1.2 a.u. in ratio curves of theexperimental spectra cannot be correlated with any other
calculated vacancy defect complex.
At higher implantation energies strong changes in the
Doppler broadening signal are observed for samples I andII, and the extrapolated ratio curves shown in Fig. 22(c) .I n
both samples a strong increase in the peak center intensity to
/C241:12is visible, which is over twofold compared to that
observed in the irradiated layer. Additionally, the signal dropsstraight to the minimum at 2 a.u. without showing any longerthe shoulder which is visible in the layer region of bothsamples. The observed trends are qualitatively very similar
to those of samples II and III, but are intensified. The induced
changes can be associated with an increase in the decorationofV
InwithVN. When comparing to the calculated momentum
distributions, the best agreement is found for the spectrum oftheV
In-3VNcomplex.
The changes in the Doppler broadening spectra can natu-
rally also be seen in the ðS; WȚparameters. In particular, the
decoration of the cation vacancies by the N vacancies isobserved as a shift toward the right in the ðS; WȚplane.
This behavior has been observed in GaN ( Hautakangas
et al. , 2006 ), InN ( Rauch, Makkonen, and Tuomisto, 2011 ),
and also the ternary alloys, and can be used to detect V
III-VN
complexes. Figure 23shows both experimental and theoreti-
cally predicted ðS; WȚdata in InGaN samples ( Uedono et al. ,
2012 ). These data, together with the data on InN, have one
remarkable feature that highlights the importance of perform-ing both accurate experiments and state-of-the-art theoretical
calculations. In elemental semiconductors and compound
semiconductors with components of roughly the same size(such as GaAs), the positron lifetime is quite monotonically
correlated with SandWparameters: an increase in lifetime
means an increase in Sand a decrease in W. However, it
seems that in strongly cation-anion-mismatched compounds,as exemplified by the results in InN, cation-anion vacancy
complexes can have very different Sparameters while
producing similar positron lifetimes.
As an example, the predicted relative Sparameter specific
to the V
In-2VNcomplex is S¼1:082(S¼1:055forVIn), a
value often associated with very large vacancy complexes,while both are observed to exhibit the positron lifetime of a
monovacancy-sized defect (the calculated lifetime for the two
defects is the same within a couple of picoseconds). Hencecomparisons with traditional knowledge acquired in Si
and GaAs do not necessarily provide the best reference forFIG. 23 (color online). (a) The ðS; WȚvalues corresponding to the
annihilation of positrons in the delocalized state (defect free, DF)and that of positrons trapped by cation vacancies ( V
InorVGa)
calculated using ordered InxGa1xN(x¼0;…;1with steps of
0.1). The ðS; WȚvalues for VInVNandVGaVNinIn0:5Ga0:5Nare
also shown. Arrows show the effect of VNcoupled with cation
vacancies. (b) The ðS; WȚvalue for the cation vacancies in
SQS-In0:5Ga0:5N[special-quasirandom structure SQS)] and experi-
mentally obtained SWrelationship for InxGa1xN. The xvalues for
the sample are shown. The ðS; WȚvalues for MBE-grown GaN and
HVPE-grown GaN are also shown. The dotted lines connecting the
ðS; WȚvalues for DF and cation vacancies show the effect of the
trapping of positrons by the cation vacancies in ordered In0:5Ga0:5N.
From Uedono et al. , 2012 .Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1617
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

interpretations in these kinds of compounds. Further, even if
the different VIn-nVNcomplexes give different points in the
ðS; WȚplot, the most dramatic difference is seen in the range
between the SandWparameter windows. This highlights the
importance of performing advanced theoretical calculations
for detailed identification of defects.
H. The substitutional lithium-on-zinc-site defect in zinc oxide
As in the case of nitrogen vacancies in III-nitrides, it is
often observed in calculations that small vacancy defects donot trap positrons. However, some cases bring surprises, suchas, for example, the Li
Zndefect. Figure 24shows positron
lifetime spectra recorded in two high-quality ZnO bulkcrystals grown by the vapor phase (VP) and hydrothermalmethods (HT) ( Tuomisto et al. , 2003 ;Johansen, Zubiaga,
Makkonen et al. , 2011 ). In both as-grown samples the posi-
tron lifetime spectrum has a single component: 170/C61p s
for VP and 184/C61p s for HT. The value in the VP samples
corresponds to positron annihilations in the defect-free lat-tice. The electron-irradiated VP sample has two lifetimecomponents, the longer of which ( /C28
2¼230 ps ) is due to
positrons annihilating as trapped at in-grown Zn vacancies(Tuomisto et al. , 2003 ;Tuomisto, Saarinen, Look, and
Farlow, 2005 ).
The experimental results are often presented in terms of the
average positron lifetime /C28
avdefined as the time expectation
value of the experimental spectrum [see Eq. ( 25)], and it
coincides with the center of mass of the spectrum. The latter
property makes the average lifetime a statistically accurate
parameter. Hence it can be correctly calculated from theintensity and lifetime values even if the decomposition rep-resented only a good fit to the experimental data without anyphysical meaning. On the other hand, the decomposition isimportant, as, for example, when comparing the data in theas-grown HT sample and the electron-irradiated VP sample:the average positron lifetime is the same, but the spectra areclearly different (Fig. 24). In the HT sample the longer life-
time is caused by Li
Znwhose defect-specific lifetime is very
close to the average positron lifetime in the sample ( Johansen,
Zubiaga, Makkonen et al. , 2011 ), while in the electron-
irradiated VP sample Zn vacancies are the cause of the
increased lifetime.
The lifetime result for the HT-grown ZnO sample shown in
Fig.24is the source of a wide scatter in reported ZnO lattice
lifetimes. The reason for this is that in most HT-grown ZnOsamples a single lifetime component is observed, but the
values tend to be 10–15 ps higher than, for example, in
melt-grown (MG) ZnO or VP-grown ZnO ( Puff et al. ,
1995 ;Brauer et al. , 2007 ;Chen et al. , 2007 ;Tuomisto and
Look, 2007 ). It was recently shown ( Johansen, Zubiaga,
Makkonen et al. , 2011 ) that this lifetime component, which
is rather close to the ZnO lattice lifetime, is in fact related toLi impurities present in high concentrations in typical HT-
grown ZnO. Li on the Zn site ( Li
Zn) is theoretically predicted
to be the stable form of Li in n-type ZnO ( Wardle, Goss, and
Briddon, 2005 ;Carvalho et al. , 2009 ), and indeed state-of-
the-art calculations show that positrons can be trapped at LiZn
(i.e., the positron density is strongly localized at the defect),
producing a lifetime 6–8 ps longer than in the ZnO lattice
(Johansen, Zubiaga, Makkonen et al. , 2011 ). This result is in
fact slightly surprising, as it is generally thought that positronlocalization in a deep state requires at least a monovacancy-sized open volume. This result suggests that the observedtrapping could be possible also in other cases where the Zof
the substitutional atom is much smaller than that of the hostatom. In other words, from the positron point of view Li
Znis
essentially VZndecorated by Li.
The case for the LiZnmodel becomes much stronger when
one considers the coincidence Doppler broadening data ob-tained in various samples. Figure 25shows the ratio curves
measured in the HT-grown and Li in-diffused MG samples
101102103104105Counts (normalized)
2.0 1.5 1.0 0.5 0.0
Time (ns) VP-ZnO, 1-component spectrum
τave = τCM = τ1 = 170 ps irradiated VP-ZnO
2-component spectrum
τ1 = 155 ps, τ2 = 230 ps
τave = τCM = 184 ps
HT-ZnO
1-component spectrum
τave = τCM = τ1 = 184 ps
FIG. 24. Positron lifetime spectra measured for an electron-
irradiated VP-grown ZnO sample ( Zubiaga et al. , 2008 ), a typical
spectrum for HT-grown ZnO (as well as for Li-enriched MG-grown
ZnO samples) and the bulk lifetime spectrum as measured in the
VP-grown ZnO reference sample. From Johansen, Zubiaga,
Makkonen et al. , 2011 .1.1
1.0
0.9
0.8
0.7Intensity ratio to ZnO lattice
4 3 2 1 0
Electron momentum (a.u.) HT ZnO
MG ZnO
MG ZnO + in-diffused Li VZn (theory)
VZn (experiment)
LiZn (theory)
Li-H-Li (theory)
FIG. 25. Coincidence Doppler broadening measurements for
as-grown HT ZnO, as-grown MG ZnO, and Li-doped MG ZnO as
compared to the theoretical values obtained for VZn,LiZn, and the
LiZn-H-LiZncomplex. The experimental data obtained for VZnare
shown for comparison. From Johansen, Zubiaga, Makkonen
et al. , 2011 .1618 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

(normalized to the spectrum measured for the VP reference),
together with the corresponding ratio curves obtained theoreti-
cally for VZn, substitutional LiZn, and the LiZn-H-LiZncomplex
proposed by Sann et al. (2006) . Also the OH-LiZnandLiZn-Lii
complexes have been calculated, using the ground-state con-
figurations predicted by Wardle, Goss, and Briddon (2005) ,b u t
they are not shown in Fig. 25as the LiZn-Liipair is not found to
be active as a positron trap while the curve for OH-LiZnis
indistinguishable from the one for LiZn.
For comparison, Fig. 25also includes data obtained for an
irradiated sample with a high concentration of VZn, illustrat-
ing that the features of the experimental and theoretical ratio
curves (such as the shoulder at 1–1.2 a.u.) agree very well.
From this, it is evident that the experimental ratio curvesobtained in the Li-rich materials cannot be explained by
assuming nonsaturated trapping by V
Zn. On the other hand,
the theoretical curves for the LiZn,LiZn-H-LiZn, and OH-LiZn
complexes all show excellent agreement with the data
obtained from the Li-indiffused MG sample (the as-grownMG sample has some near-surface Zn vacancy defects). The
differences between the V
ZnandLiZnratio curves are largely
explained by the smaller open volume seen by the positron in
the latter case. Li repels the positron toward neighboring ion
cores, thereby increasing the high-momentum intensity rela-tive to the V
Znspectrum. The direct contribution of the Li
orbitals to the LiZnspectrum can be quantified by considering
the system as a superposition of free atoms and decomposing
the total annihilation rate to contributions due to different
atomic orbitals. The Li contribution turns out to be only 5%of the total annihilation rate. Furthermore, the direct Li
contribution to the Doppler spectrum is rather featureless.
In conclusion, these calculations indicate that there is no clear
‘‘Li fingerprint,’’ which would provide the possibility to
unambiguously identify Li-related defects in a more generalcase. However, the flat region with the ratio slightly above 1.0
extending from 0 to 1.3 a.u. is unique for Li
Zn. It is evident
from the data that LiZnoccurs as the dominant trap also in
HT-grown ZnO, but with a detectable contribution from VZn.
Importantly, all Li atoms present in n-type HT ZnO reside
on the Zn site and the resulting open volume is thus respon-
sible for the increase in the single-component positron life-
time observed in as-grown HT ZnO as compared to samplesproduced by other growth techniques yielding material with
low Li concentrations. This also explains the discrepancy in
the reported values for the bulk positron lifetime in ZnO. It
should be noted that the Li-related signal has been observed
to disappear when the samples are hydrogenated ( Johansen,
Zubiaga, Tuomisto et al. , 2011 ) suggesting that Li
Zntraps
hydrogen, turns neutral, and becomes less attractive to posi-trons. At the same time the remaining V
Zndefects have been
shown to become efficient hydrogen traps. Further work is
necessary in order to fully elucidate the role of residual
hydrogen impurities and their interaction with intrinsic
open-volume defects in ZnO. Be in GaN may behavesimilarly ( Leeet al. , 2006 ;Lany and Zunger, 2010 )
V. FUTURE CHALLENGES
Even if the methods presented in this review allow for
advanced and detailed identification of certain types ofvacancy defects in semiconductor materials (crystalline
solids in general), there is a wide variety of important devel-
opments to be realized in order to unleash the full potential of
positron annihilation spectroscopy in materials research. Wehave chosen five ‘‘challenges,’’ in both the development of
theoretical and experimental methods, which we find to be
the most promising regarding our own interests. By no means
is this meant as an exhaustive listing of all possible (or
probable) developments in the field.
The first challenge is in studying electronic materials with
complex crystal structures, such as complex oxides
(Sec. V.A). In these materials the main issue is the large
number of different lattice sites for monovacancy defects,
making the defect identification complicated. The two further
challenges concern the development of theoretical methodsthat would be crucial for detailed interpretation of experi-
mental results in systems where the 3D periodicity is broken,
namely, surfaces and interfaces between crystalline solids
(Sec. V.B), and nanocrystalline, amorphous, and molecular
systems (Sec. V.C). Reports on positron experiments in these
areas are becoming more and more numerous, while the
theoretical descriptions of positron states, thermalization,
and trapping are not well established. The two last challengesconcern developments in experimental methods: setups al-
lowing for advanced sample state manipulation eventually
allowing pump-probe experiments with light, bias, magnetic
field, temperature, and pressure as the pump (Sec. V.D); and
the issue of slow-positron beam intensity that can be solvedby large-scale facilities providing intense sources or by im-
proving positron moderation efficiency (retaining the quality)
in laboratory-scale facilities (Sec. V.E).
A. Materials with complex crystal structures
Already a relatively moderate additional degree of com-
plexity in the crystalline structure of a semiconductor mate-
rial creates challenges in defect identification. This is true in
general as well, but holds particularly for positron annihila-tion spectroscopy. In the following we discuss two kinds of
complex crystal structures: multielement compounds and
semiconductor alloys (or mixed crystals). The first are strictly
periodic, but characterized by relatively large unit cells. The
second are characterized by nonperiodic (with various de-grees of randomness) distribution of atoms on the lattice sites.
By multielement compounds we mean all compound semi-
conductors that are more complex than just AB. Examples
include materials with chalcogenide structure (such as
ZnGeAs
2) and complex oxides (such as SrTiO 3). Some two-
component compounds also fall into this category due to theirvery large unit cells: examples include In
2O3and the majority
of SiC polytypes (see the discussion in Sec. IV.A.3 ). Positron
annihilation has been employed to investigate defects in these
kinds of materials; see, e.g., Niki et al. (2001) ,Uedono et al.
(2002) ,Cheung et al. (2007) ,Keeble et al. (2007) ,Kilanski
et al. (2009) ,Mackie et al. (2009) ,Gentils et al. (2010) ,Islam
et al. (2011) ,Guagliardo et al. (2012) , and Korhonen et al.
(2012) . While the results are promising and pave the way for
future studies, systematic studies are still missing.
The main challenge in identifying vacancy defects in these
multielement compounds is the large number of possibleFilip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1619
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vacancy-type defects already for monovacancies. This in
contrast to elemental semiconductors (for which the number
is 1) and simple compounds such as GaAs (for which the
number is 2). Often for these kinds of simple systems one
kind of vacancy defect dominates, making the identification
with positron annihilation methods relatively straightforward,
as seen in the discussions in the previous section. For multi-
element compounds, the definitions of stoichiometry and
chemical potential are much more complicated, and it is
less probable that in a given sample there would be onedominant kind of defect—this is seen, in particular, in the
studies on chalcogenide-structured materials ( Niki et al. ,
2001 ;Kilanski et al. , 2009 ;Islam et al. , 2011 ;Korhonen
et al. , 2012 ). In addition to the three possible monovacancies
in such materials, the number of possible binary complexes of
intrinsic defects only is strongly increased compared to sim-
ple structures (e.g., divacancies, vacancy-antisite complexes).
In ‘‘simple’’ complex oxides such as SnO
2orZrO 2
(Guagliardo et al. , 2012 ) the situation is not as bad as in
the multielement compounds, but the possibility of metal
vacancy–oxygen vacancy complexes with multiple oxygen
vacancies makes detailed defect identification difficult. For
multimetal complex oxides ( Uedono et al. , 2002 ;Cheung
et al. , 2007 ;Keeble et al. , 2007 ;Mackie et al. , 2009 ;Gentils
et al. , 2010 ) these two challenges are combined, making
identification even more difficult. These challenges should,
however, be overcome by systematic studies where state-of-
the-art theoretical calculations and careful experiments arecombined. Significant efforts by several actors in the field can
be anticipated, and the challenge of detecting O vacancies in
complex oxides should be highlighted.
Semiconductor alloys, such as Si
1/C0xGexandInxGa1/C0xN,
possess another complication that is harder to tackle, espe-
cially when xis significantly different from either 0 or 1.
Combining theoretical calculations and experiments is con-
ceptually more complicated. As an example, in a calculation
one knows whether an In or Ga atom has been removed to
make a vacancy on the cation sublattice in InxGa1/C0xN, but
in experiments the positrons are primarily sensitive to the
‘‘cation vacancy’’ ( Chichibu et al. , 2006 ,2011 ,2013 ;Uedono
et al. , 2009 ,2012 ). Hence the second-nearest-neighbor envi-
ronment becomes very important in defect identification. This
phenomenon is yet to be studied in detail. There is significant
room for improvement, but here it might not be sufficient to
apply existing, even if state-of-the-art, theoretical methods to
account for the randomness. The importance of the effects of
the local environment of the vacancy defects being identified
compared to the long- range disorder need to be elucidated
(Kuitunen, Tuomisto, and Slotte, 2007 ;Kilpela ¨inen et al. ,
2010 ,2011 ). Systematic studies in all these materials are
required in order to fully elucidate the vacancy defect iden-
tities and roles.
B. Positron states at interfaces and surfaces
In order to understand positron annihilation parameters
measured for such complex systems as semiconductor alloys
or heterostructures, the first question one has to address,
preferably with the help of computational modeling, is which
kinds of regions the positron will be likely to sample. In otherwords, what is the ‘‘affinity’’ of positrons for, for instance,
GaN or InN clusters in InGaN alloys [see, e.g., Chichibu et al.
(2006) ], or for different layers in quantum wells or super-
lattices formed of these materials? Further, what is the effect
of polarization in polar semiconductor heterostructures from
the point of view of positron studies?
On the other hand, a requirement for studying surfaces
with positrons or for the construction of efficient positronmoderators (Sec. V.E) is an understanding of the positron
surface state. For instance, one wants to know how to best
modify the moderator’s surface in order to obtain high
emission efficiency from the bound state into the vacuum.The interaction between the positron and the surface is noteasy to model. In addition, the image potential sensed by thepositron above a conducting surface is a highly nonlocalcorrelation effect, especially when positronium formationis also expected and the van der Waals interaction plays
a role. Currently effects such as this can be modeled
using only simple models [see, e.g., Saniz et al. (2007 ,
2008) and Mukherjee et al. (2010) ]. More understanding
and quantitative modeling is needed in this area as well.
The concepts of material-specific positron affinity and
the positron affinity difference determining the separation
of positron energy levels between two solids in contact
(Boev, Puska, and Nieminen, 1987 ;Puska, Lanki, and
Nieminen, 1989 ) are based on a model strictly speaking valid
only for metals. In the model, it is assumed that the Fermilevels equalize themselves via charge transfer and formationof an interface dipole /C1¼/C22
A/C0/C0/C22B/C0, where /C22A;B/C0are the
electron chemical potentials for materials AandBwhile
separated. Then, the difference between positron energy
levels Ețon the different sides of the interface is ( Puska,
Lanki, and Nieminen, 1989 )
/C1EA;B
ț¼EA
ț/C0EB
ț¼/C1ț/C22A
ț/C0/C22B
ț
¼/C22A/C0/C0/C22B/C0ț/C22A
ț/C0/C22B
ț; (58)
where /C22A;B
țare the chemical potentials for the positron.
As a consequence, it is useful to define a bulk property
Aț¼/C22/C0ț/C22ț, the positron affinity, and calculate the
difference in positron energies using the difference of thepositron affinities. However, if one or both of the materialsare semiconductors, and the Fermi level is aligned within theband gap, there are no extended electronic states to accom-modate the charge on the semiconductor side. However, theremay exist localized states at the interface but these cannot be
predicted using bulk properties only. First-principles model-
ing of the interfaces themselves is needed [see the relateddiscussion by Van de Walle, Lyons, and Janotti (2010) ].
Further, creation of an interface always involves strain whichalso affects the positron energy ( Boev, Puska, and Nieminen,
1987 ;Puska, Lanki, and Nieminen, 1989 ) levels as well as
electron band alignments [see, e.g., Moses et al. (2011) ].
Another reason why first-principles calculations (beyond
the atomic superposition method) should be used throughoutfor modeling positron states and annihilation in semiconduc-tor structures is that in addition to the band gap modulationand variation of positron energy levels, there may exist hugemacroscopic electric fields in polar semiconductor hetero-
structures and superlattices ( Bernardini and Fiorentini, 1998 ;
Fiorentini et al. , 1999 ;Lefebvre et al. , 2001 ). In a classical1620 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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model ( Fiorentini et al. , 1999 ), a discontinuity of the trans-
verse polarization at an interface corresponds to an interfacecharge. Once again, the detailed electronic structure of the
interface determines how much charge it actually can accom-
modate, and first-principles calculations are a necessity.Similarly, the model does not account for the screening ofthe macroscopic electric field when the potential differenceacross a heterostructure’s layer becomes larger than theenergy band gap of the material.
Makkonen et al. (2010) modeled the behavior of positrons
in nonpolar and polar superlattices composed of GaN, AlN,and InN using first-principles methods. For nonpolar super-lattice models it is observed that the separation of the positronenergy levels depends on strain. In the case of polar super-lattice models, the macroscopic electric field drives the posi-
tron density toward one of the two inequivalent interfaces in
the models. Although cation vacancies situated at the nitridelayers are expected to be energetically more favorable thanthis positron’s interface state, the trapping rate to the vacan-cies is expected to decrease with increasing distance of thevacancy from the preferred interface. This is expected to
provide interface sensitivity in positron measurements made
for such structures.
In conclusion, understanding positron annihilation results
measured for semiconductor heterostructures requires con-sideration and ideally supporting first-principles modelingof positron states and annihilation to understand where the
annihilation signal is coming from and the indirect informa-
tion contained in the data. The electronic structure has to bemodeled self-consistently and models should include detailedstructures of the interfaces. In the case of systems in whichthe common LDA and GGA exchange and correlation
functionals underestimate the energy band gap, one benefits
from the use of hybrid exchange-correlation functionals(Becke, 1993 ;Perdew, Ernzerhof, and Burke, 1996 ;Adamo
and Barone, 1999 ;Heyd, Scuseria, and Ernzerhof, 2003 )
for the description of the electronic structure, although thecomputational cost is then significantly higher.
C. Positron thermalization and trapping in nanocrystalline,
amorphous, and molecular systems
Positron annihilation spectroscopy can be used to study
properties of nanoscale structures and their surfaces, eithernanocrystalline [see, e.g., Weber et al. (2002) andEijtet al.
(2006) ] or as embedded in the bulk of the material
(Nagai et al. , 2000 ;Chichibu et al. , 2006 ). The sizes of
nanocrystals can be correlated with smearing effects ofvarious origins observed in measured momentum densities(Saniz, Barbiellini, and Denison, 2002 ;Weber et al. , 2002 ;
Toyama et al. , 2012 ).
In a typical positron experiment, it is assumed that the
positrons thermalize very rapidly within a few picoseconds.
Then the time-dependent diffusion equation and the conven-tional trapping model (see Sec. II.A.3 ) can be applied to
describe the diffusion and trapping kinetics of positrons.According to Jensen and Walker (1990) , measurable devia-
tions from the conventional trapping model will happen only
if the trapping rate and/or annihilation rate differ from thethermal rate for a sufficient fraction of the thermalizationperiod. Effects of incomplete thermalization of positrons have
been observed experimentally for semiconductors and insu-
lators ( Mills and Crane, 1985 ;Gullikson and Mills, 1986 ;
Lynn and Nielsen, 1987 ;Nissila ¨, Saarinen, and Hautoja ¨rvi,
2001 ) and even for metals ( Nielsen, Lynn, and Chen, 1986 ;
Huomo et al. , 1987 ). In bulk studies with energetic positrons,
it is usually safe to neglect nonthermal effects. However, if
one is measuring nanocrystalline samples, it might happenthat the positrons leave the crystal already at nonthermalenergies. If the nonthermal trapping rate differs from the
thermal rate, the conventional trapping model is not appli-
cable. A resonant-trapping mechanism effective at nonther-mal energies has been proposed for metals ( McMullen and
Stott, 1986 ;Puska and Manninen, 1987 ;Jensen and Walker,
1990 ) and similar mechanisms also play a role in semicon-
ductors ( Puska, Corbel, and Nieminen, 1990 ), and especially
in trapping to vacancy clusters and voids in metals ( Puska and
Manninen, 1987 ;Jensen and Walker, 1992 ). Also, even if the
positron does end up thermalized within a nanocrystal itmight not possess a well-defined delocalized ‘‘bulk state’’if the material has a negative positron work function and
the nanocrystal’s dimensions are smaller than the thermal
wavelength
4of the positron.
In order to fully understand positron thermalization and
trapping into various kinds of states (vacancy, void, interface,or surface state) in nanocrystals, one needs to addresstransition rates between different states including also the
possibility of nonthermal and resonant effects, and model the
thermalization and trapping processes using a model includ-ing both spatial and momentum transport and having realistic
geometries for the nanocrystals. Further, more understanding
of thermalization and trapping mechanisms and their effec-tiveness in not only nanocrystalline but also amorphous andmolecular matter is needed on a general level. In soft and
molecular matter, the most important information in the
annihilation signal is often contained in the pick-off annihi-lation of ortho-Ps (the positron within triplet positroniumannihilating with outside electrons); see, e.g., Tao (1972) ,
Eldrup, Lightbody, and Sherwood (1981) ,Jean (1990) ,
Hirata, Kobayashi, and Ujihira (1996) ,Dong et al. (2009) ,
Sane et al. (2009) , and Quinn et al. (2012) . This correlated
state involving also nonlocal correlations (van der Waals
interaction) with the surroundings is extremely challenging
to model for any realistic system [see Barbiellini and
Platzman (2009) and Zubiaga, Tuomisto, and Puska
(2012) ].
Related to the above we note that the trapping of positrons
in vacancy defects even in crystalline semiconductor systems
poses challenges for the present theoretical models when theenergetics of trapping is considered ( Makkonen and Puska,
2007 ). An important difference between vacancies in metals
and semiconductors is the lower positron binding energy
expected for the latter. This is because of the stronger repul-sive interaction between a delocalized positron and nuclei inthe denser structures of metals. Consequently, the lowering of
4The thermal wavelength can be estimated as
/C21ț
th¼hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3m/C3kBTp /C2550/C18300 K
T/C191=2/C23A:Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1621
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the positron’s energy eigenvalue between the bulk state and
the trapped state at a semiconductor vacancy is not neces-sarily much larger in magnitude than the energy stored in theaccompanying ionic relaxation [see the terms in Eq. ( 40)].
Calculations even suggest that the bound state resembles thecase of an electron or a hole trapped into a small polaron
state in ionic crystals. According to present theoretical
models, the trapping could not even be energetically favor-able in some important materials systems (Si, Ge, GaAs) orthen an energy minimum (even a metastable one) with atrapped positron would not exist ( Makkonen and Puska,
2007 ). These predictions are not supported by the experi-
mental observation in which trapping to the same defects isobserved, and possible temperature dependences are due toother mechanisms. When interpreting experiments especiallyin new materials, the starting point is often in trying tounderstand which defects can trap positrons. Potential bor-
derline cases continue to pose an important challenge for
theoretical models.
D. Pump-probe experiments with positron annihilation
spectroscopy
The state of the art of positron annihilation spectroscopy
experiments in studying defects in semiconductors is basedon the control of sample temperature during measurements intypical ranges of 10–600 K, providing information on theequilibrium charge states of the detected defects, and makingit possible to identify the electronically important defects.
Sample illumination with sub-band-gap monochromatic light
during experiments brings additional data on optical chargetransitions of these defects, relating them to optoelectronicproperties of semiconductors. However, in order to directlyrelate the vacancy defects that can be identified with positronannihilation spectroscopy to carrier dynamics (e.g., nonradia-tive defect-related recombination processes) directly affect-ing the function of optoelectronic devices, more sophisticatedexperiments need to be thought of. One step in this directionis to time modulate the illumination: Fig. 26shows a recent
result obtained in natural diamond with modulated illumina-
tion, where the recombination process happens to be slow
enough (seconds to hundreds of seconds) for the effects tobe monitored by usual experiments ( Ma¨ki, Tuomisto et al. ,
2011 ;Ma¨kiet al. , 2012 ). Typically in semiconductor
materials the recombination rates are several orders of mag-
nitude shorter (down to the nanosecond scale); hence further
development is required.
In the example shown in Fig. 26, the sample-source-
sample sandwich was illuminated with high-power light-
emitting diodes (LEDs), as the illumination intensity can be
a bottleneck when maximal ionization efficiency is needed.In steady state (during illumination) the fraction of ionizeddefects will depend on their optical absorption cross section,
and the relaxation rate of the latter may be quite fast
especially at elevated temperatures. By controlling theLEDs with a fast electrical switch, the illumination can bechanged from steady state to transient mode, allowingpump-probe experiments. In these experiments the collec-
tion of positron annihilation data can be divided into time
slots (1–10 s in the example experiment), both during andafter the illumination pulse. By repeating the measurementseveral hundred times, the data collected in each time slot
can be summed and time-dependent positron lifetime spec-
tra obtained. The experimental data in Fig. 26show that
in a case when both the rising time and the relaxation timeof the illumination effect are slow (LED response is in the
microsecond regime, so the effect comes from the studied
material), the positron experiments can be used to followthe population and depopulation of vacancy defect levels.Combining optical absorption experiments, illumination-
power-dependent steady-state positron experiments and
these pump-probe positron experiments allow one to self-consistently determine the optical absorption cross sectionand the vacancy concentrat ions, without pr ior knowledge of
the positron trapping coefficient ( Ma¨ki, Tuomisto et al. ,
2011 ).
These experiments are naturally not restricted to diamond
or a specific wavelength of light. It is sufficient that thevacancy defects detected with positrons have electron levels
in the band gap of the semiconductors and can be optically
ionized. A good example is given in Sec. IV.D where the EL2
defect studies in GaAs are reviewed. In addition, electronscan be excited to deep levels in the gap also by electrical
(bias) pulses, in a manner similar to DLTS. Here only near-
surface phenomena can be monitored, so the method wouldbe applicable only with slow-positron beams. Also thin metalcontacts on the samples to be studied are necessary. There is
no obstacle in the realization of these kinds of experiments,
as there are several reports of measurements of biasedsamples for determining positron mobility in semiconductors(Simpson et al. ,1 9 9 0 ;Ma¨kinen et al. , 1991 ). With simple
setups for pump-probe experiments [such as the one de-
scribed by Ma¨kiet al. (2012) ], one can easily go down to
the microsecond regime in time scales, although new dataanalysis techniques are needed: Continuous measurements
with varying illumination pulse lengths and periods should
be performed as the collection of data in microsecond-rangetime slots is not feasible. Further reduction of the time scaledown to the nanosecond range is possible by synchronizingthe excitation with the time modulation of a pulsed positron
beam, where the pulsing periods are typically in tens of
nanoseconds.
200
190
180
170
160
150
140Average positron lifetime (ps)
80 60 40 20 0 -20
Illumination modulation time (s)Illumination on (400 nm LED) High photon flux (1016 cm-2s-1 )
Low photon flux (1014 cm-2s-1 )
FIG. 26. Example of average positron lifetime measured with
time-modulated illumination in a natural diamond sample.1622 Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with …
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Application of a strong (varying) magnetic field to samples
during positron experiments is also an option, but in manycases it has severe effects on the detection of the signal itself,as (i) the positrons are magnetically guided to and focused on
the sample in a slow-positron beam, and (ii) the photomulti-
plier tube(s) in positron lifetime experiments are extremelysensitive to magnetic fields. Performing Doppler broadening(and ACAR) experiments in bulk crystals in varying magneticfields has been realized and effects on the positron signalsreported ( Kawasuso et al. , 2011 ,2012 ). Theoretical calcula-
tions also predict observable changes for certain types of
defects in semiconductors ( Alatalo, Puska, and Nieminen,
1993 ). Hence developments in this direction should also be
pursued. In the case of lifetime experiments, technology alsoposes a challenge: the time resolution and efficiency require-ments are such that present-day avalanche photodiodes(APDs, the reasonable alternative for photomultiplier tubes)
are not applicable. Recent results on AlGaN APDs are,
however, encouraging ( Sunet al. , 2010 ).
E. Toward higher slow-positron beam intensity
In the field of condensed matter and materials physics,
the vast majority of slow-positron beams is used for experi-ments on the Doppler broadening of the positron-electronannihilation radiation. Other techniques, i.e., positron life-
time spectroscopy, ACAR, PAES, and RHEPD, require either
sophisticated beam pulsing electronics (or other timing tech-nology), or high-intensity sources, or both. Hence these othertechniques are currently restricted to a handful of large-scalefacilities, although pulsed positron beams have beenconstructed at laboratory-scale facilities, too.
The laboratory-scale facilities typically employ radioac-
tive ( /C12
ț) isotopes, such as22Na that has relatively
low intensity (up to 109positrons =s). The low intensity is
balanced by the practical half-life of 2.6 years allowingreasonable use of the same source for 6–10 years. On theother hand, large-scale facilities with high-intensity sourcesare able to provide up to 10
12positrons =s, making use of
pair production with the high-energy gamma flux created by
a nuclear reactor or a particle accelerator ( Cassidy et al. ,
2009 ;Krause-Rehberg et al. , 2011 ). For more details about
intense positron sources, see Hugenschmidt et al. (2004) ,
Schut et al. (2004) , and Hawari et al. (2009) . High-intensity
positron sources indeed represent the state of the art ofexperimental development, but require significant resources
due to their large scale.
The present-day moderator solutions are using either a thin
W foil (efficiency /C2410
/C04) or a solid Ne moderator (efficiency
/C2410/C03). The thin W foil moderator has relatively poor
efficiency, but produces slow positrons with a narrow energyspectrum. The solid Ne moderator is better in efficiency, but
produces positrons with a wider energy spectrum that makes
it impractical for positron lifetime beams ( Mills and
Gullikson, 1986 ;Mills and Platzman, 2001 ). In addition, in
the case of high-voltage floated positron sources (groundedsample stages for ease of sample state manipulation), therequirement of weekly regeneration and cryogenics onhigh-voltage platforms is not particularly appealing in the
case of solid Ne moderators. Further energy selection and
beam formation techniques reduce the beam intensity by anadditional order of magnitude, hence resulting in actual
(maximum) slow-positron beam intensities of 10
4and
108eț=sfor laboratory-scale and large-scale facilities, re-
spectively, when using the more common passive thin W foil
moderation. Often the beam intensities are roughly an order
of magnitude lower. Beam bunching and chopping further
reduces the intensity in the case of pulsed positron beams.
There is significant room for improvement in the positron
moderators in both their efficiency and the directional dis-
persion (exit cone of positrons) that both limit the beam
intensity. The maximum realistic beam intensity for defectstudies is of the order of 10
8eț=s(requirement of no
positron-positron interactions), so the large-scale facilities
are already close in that respect, but there is no practical
upper limit for PAES and RHEPD beam intensities.
Limitations are imposed also from the signal detection point
of view: measurement times of much less than 100 s per
spectrum are not feasible in the case of Doppler broadening,
as the peak stabilization requires some time. Also the capa-
bility of lifetime experiments going above 1000 counts =sis
limited (here the detection efficiency is much lower than in
Doppler broadening experiments: in typical experimentalconfigurations about 1% of all the annihilations are detected).
The wide directional dispersion of the moderated positrons is
partly responsible for the small number of SPMs, comparable
to a SEM, as the focusing of the beam with reasonable
intensity even at a large-scale facility results in spot sizes
of the order of 5/C22m.
Three factors determine the efficiency of a positron mod-
erator: (i) the fraction of positrons stopped within the mod-erator, (ii) the fraction of stopped positrons reaching the
moderator surface, and (iii) the fraction of surface-reaching
positrons emitted from the surface (see Fig. 27). For conven-
tional thin-film W moderators in the transmission geometry,
the first is roughly 5% and the second roughly 20%; (these are
fully determined by the thickness) and the quality (crystalline
is best) of the film optimum is about 1/C22m. The emission
efficiency can be optimized by careful surface preparation
through thermal treatments in high vacuum and through
maintaining a good vacuum in the beam. However, it is
very difficult to have an emission (extraction) efficiencyabove 1%—hence the total efficiency is at best 10
/C04. The
first two limiting factors can be significantly improved by
using a semiconductor material (an order of magnitude
thicker than W, hence stopping more positrons) and applying
a voltage across to improve the diffusion to the surface
FIG. 27. Schematic of a thin-film moderator in transmission
geometry.Filip Tuomisto and Ilja Makkonen: Defect identification in semiconductors with … 1623
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013

(Shan et al. , 1994 ). By choosing, e.g., GaN, SiC, or diamond,
the fraction of positrons reaching the extraction surface could
be increased up to 50%. These three materials also have a
negative positron affinity ( Coleman, 2000 ), and hence posi-
tron extraction from the surface should be possible. The
remaining, important, challenge is in the nature of semicon-ductors: an electrical dipole is formed at the semiconductor-
vacuum interface, creating a strong positron trap preventing
positron emission. The emphasis on the development in this
area should be in the surface processing and passivation
techniques (from the positron point of view) that will allowefficient positron extraction. The radiation hardness and
mature semiconductor device technology in III-nitrides sug-
gests that this material family could be the key to improved
positron moderators.
Future improvements in moderator efficiencies would
have important consequences: the significant reduction of
measurement times significantly is important for both
laboratory-scale and large-scale facilities. Of more generalimportance is that improved moderator efficiency will make
the construction of a reasonable (i.e., reasonable intensity)
isotope-source-based slow-positron beam much more acces-
sible for a great number of laboratories. This is because one
could achieve the present intensities (which are satisfactoryfor standard experiments) with a source whose activity is only
2 MBq, similar to widely used common fast-positron sources.
Finally, there are direct experimental benefits from lower
source activity, such as less background and noise in the
experiments, and the beams can be constructed with muchmore compact dimensions (downscaling from /C243m beam
line length to less than 1 m). Improved beam intensities
would clearly also benefit the pump-probe experiments de-
scribed in Sec. V.D, as well as enable the efficient use of
PAES and RHEPD. Also SPM could be used more efficientlyin a scanning mode, mimicking the depth profiling typical of
spreading resistance measurements ( Krause-Rehberg et al. ,
2001 ).
VI. SUMMARY
Positron annihilation spectroscopy is a characterization
method for probing the local electron density and atomic
structure at the site chosen by the electrostatic interaction
of the positron with its environment. The positron annihila-
tion methods have had a significant impact on defect spec-troscopy in solids by introducing an experimental technique
for the unambiguous identification of vacancies. Native va-
cancies have been identified and found to be present at high
concentrations in many semiconductors, and their role in
doping and compensation can be quantitatively discussed.Defect charge states and transitions, as well as their formation
and migration processes, can be studied with positron
methods.
We summarized the basic concepts behind the experi-
mental and theoretical methods of positron annihilation and
reviewed the latest developments that have led to the
possibility of identifying d efects in semiconductors with a
high level of detail. We hope to provide useful referencematerial for the specialist and at the same time provide the
nonpractitioner additional mea ns to assess positron resultsand interpretations through a frank account of the strengths
and weaknesses of the experimental and theoreticalmethods. The examples from various technologically
important semiconductors illustrate the important combi-
nation of experiment and theo ry in detailed defect identi-
fication with positrons. Fut ure challenges include the
development of quantitative th eoretical models for non-
crystalline systems and the development of experimental
arrangements enabling the analysis of defect-related
transient phenomena.
ACKNOWLEDGMENTS
We thank the Academy of Finland, Helsinki Institute of
Physics and Aalto University for financial support. We are
particularly thankful to B. Barbiellini, Z. Q. Chen, P. G.
Coleman, R. Krause-Rehberg, M. D. McCluskey, M. J.Puska, Z. Tang, and A. Uedono for their valuable suggestionsand comments during the writing of this manuscript. Weacknowledge the computational resources provided by the
Aalto Science-IT project.
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