Makkonen, Ilja Defect identification in semiconductors with positron annihilation

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Tuomisto, F.; Makkonen, I.
Defect identification in semiconductors with positron annihilation: Experiment and theory
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Reviews of Modern Physics
DOI:
10.1103/RevModPhys.85.1583
Published: 01/11/2013
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Tuomisto, F., & Makkonen, I. (2013). Defect identification in semiconductors with positron annihilation:
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REVIEWS OF MODERN PHYSICS, VOLUME 85, OCTOBER–DECEMBER 2013
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†
COMPCentreofExcellence,HelsinkiInstituteofPhysicsandDepartmentofAppliedPhysics,
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 methods
allows for detailed identification of the defects and their chemical surroundings. Also charge
statesanddefectlevelsinthebandgapareaccessible.Inthisreviewthemainexperimentaland
theoreticalanalysistechniquesaredescribed.Theusageofthesemethodsisillustratedthrough
examples in technologically important elemental and compound semiconductors. Future
challenges include the analysis of noncrystalline materials and of transient defect-related
phenomena.
DOI: 10.1103/RevModPhys.85.1583 PACS numbers: 61.72.J(cid:1), 78.70.Bj, 71.60.+z, 81.05.(cid:1)t
CONTENTS D. Functionals for electron-positron correlation
effects 1602
I. Introduction 1584
E. The atomic superposition method 1603
A. Defects in semiconductors 1584
F. Numerical approaches for self-consistent
1. Role and formation of defects in
calculations 1603
semiconductors 1584
IV. Results 1605
2. Studying defects in semiconductors 1585
A. An overview of results obtained in thepast
B. Positron annihilation spectroscopy 1586
two decades 1605
1. Background 1586
1. Elemental semiconductors Si, Ge, and C 1606
2. Positron annihilation methods 1587
2. Traditional III-Vand II-VI semiconductors 1606
II. Experimental Techniques 1588
3. Novel semiconductors: III-N, SiC, and ZnO 1607
A. Positrons in solids 1588
B. Vacancy-(multi)donor complexes in highly n-type
1. Implantation, thermalization, and diffusion 1588
doped silicon 1608
2. Positron states and trapping 1589
C. Thevacancy-fluorine complexin silicon and
3. Trapping model 1590 silicon-germanium alloys 1610
B. Positron lifetime spectroscopy 1591 D. The EL2 defect in gallium arsenide 1611
1. Experimental details 1591 E. Thegallium vacancy–tellurium complexin
2. Data analysis 1593 gallium arsenide 1613
3. Information revealedby the positron lifetime 1594 F. Thegallium vacancyand its complexes in
C. Doppler broadening spectroscopy 1595 gallium nitride 1614
1. Experimental details 1595 G. Metal vacancy–nitrogenvacancy complexes in
2. Data analysis 1596 III-nitrides and their alloys 1615
3. Chemical information contained in H. The substitutional lithium-on-zinc-site defect
Doppler spectra 1597 in zinc oxide 1618
III. Theory and Computational Methods 1598 V. Future Challenges 1619
A. Two-component electron-positron A. Materials with complexcrystal structures 1619
density-functional theory 1598 B. Positron states at interfaces and surfaces 1620
B. Modeling localized positrons 1599 C. Positron thermalization and trapping in
C. Positron annihilation parameters 1600 nanocrystalline, amorphous, and molecular
1. Annihilation rate and lifetime 1600 systems 1621
2. Momentum density of annihilating D. Pump-probe experiments with positron
electron-positron pairs 1601 annihilation spectroscopy 1622
E. Toward higher slow-positron beam intensity 1623
VI. Summary 1624
Acknowledgments 1624
*filip.tuomisto@aalto.fi References 1624
†ilja.makkonen@aalto.fi
0034-6861=2013=85(4)=1583(49) 1583 (cid:1) 2013 American Physical Society
1584 FilipTuomisto and Ilja Makkonen: Defect identificationin semiconductors with ...
on Slow Positron Beams), and PSSD/PSD (Positron Studies
I. INTRODUCTION
on Semiconductors and Defects) and to references therein.
Positron annihilation spectroscopy has been widely used
forstudyingdefectsinsemiconductorssincetheearly1980s, A. Defects in semiconductors
whilethefirstreportsdealingwithradiationdamageinsilicon
and germanium had been published already in the 1970s Lattice defects in semiconductors are like spices in your
(Cheng and Yeh, 1973; Arifov, Arutyunov, and Ilyasov, food:toomuchisdisgusting,toolittleisworthless,whilejust
1977). The early developments of both experimental and the right kind and amount makes the day. Another common
theoretical approaches applicable to semiconductor studies feature is that both are typically present in amounts much
were reviewed by Schultz and Lynn (1988) and Puska and smaller than the host. There exists a wide variety of review
Nieminen (1994). An introductory book on positron annihi- articles and books on defects in semiconductors. For a de-
lation studies of defects in semiconductors has also been tailedpictureofthefieldoneisstronglyadvisedtobrowsethe
written by Krause-Rehberg and Leipner (1999). Our aim in proceedingsvolumesoftheICDS(InternationalConferences
writing this review is twofold. First, we want to introduce onDefectsinSemiconductors).Atheoreticalperspectivecan
the basic concepts behind the experimental and theoretical befoundinthebookbyLannooandBourgoin(1981),whilea
methodsofpositronannihilationandreviewthelatestdevel- recentvolumecoversmanypracticalissueseasilyaccessible
opmentsthathaveledtothepossibilityofidentifyingdefects tothenewcomerstothefield(McCluskeyandHaller,2012).
in semiconductors with a high level of detail. Second, by
going through a variety of examples in both elemental and 1. Role and formation of defects in semiconductors
compound semiconductors, we want to illustrate how these
Defects in crystalline solids are static interruptions to the
methods can beapplied toimproveourunderstanding ofthe
periodicity of the crystal. They can be classified by their
physics of defects in semiconductors.
spatial extent into point defects that are zero dimensional
Theorganizationofthisreviewisasfollows.First,wegive
and extended defects that can be one dimensional (e.g.,
an introduction to defects in semiconductors and the history
dislocations), two dimensional (e.g., stacking faults), and
andmethodsofpositronannihilation.Inthesecondpart,after
three dimensional (e.g., aggregates of impurities). It is not
briefly explaining the necessary concepts related to the be-
unusualtohaveimportantdensitiesofmorethanoneofthese
havior of positrons in solids, we delveinto the details of the
kinds of defects in a given crystalline material, such as an
experimental methods most used in semiconductor studies:
elemental (e.g., silicon or germanium) or a compound (e.g.,
positronlifetimespectroscopyandDopplerbroadeningspec-
gallium arsenide or zinc oxide) semiconductor. Quite typi-
troscopy. Here our aim is to give a frank account of the
cally they also affect each other’s properties and presence,
strengths and weaknesses of the experimental setups and e.g.,the formationofstackingfaultsinacrystalmayinduce
analysismethods,hopingtoprovideusefulreferencematerial vacancy defects. In this review, the emphasis is on point
forthespecialistandatthesametimeprovidethenonpracti- defects in general and on vacancy defects, in particular, as
tioner additional means to assess positron results and inter- thepositronmethodsaremostsensitivetodefectswithextra
pretations. The same approach is applied in the third part openvolume.
where the theoretical methods are presented. In Secs. II and In contrast to metals, in semiconductors very dilute con-
III we go through examples where both experimental and centrations (e.g., less than ppm) of defects may have impor-
theoretical positron methods have been applied to study tanteffectsontheelectricalandopticalproperties.Thisisdue
various semiconductor materials and defects therein. The totheelectronicstatescreatedbythedefectsinthetypically
focus of these sections is, in addition to showing how the (0.5–5 eV) wide band gap of the semiconductor. Depending
positronmethodsworkinpractice,ontheinterpretationsthat onthepositioninthegap,electronscanbeexcitedtoorfrom
canbemadeaboutthedefectsidentifiedinthesetechnologi- these states (from or to the bands or other states in the gap)
cally relevant materials. The results are systematically com- thermally, electrically, oroptically. In practice, the electrical
pared to the knowledge obtained by other experimental and and optical properties of semiconductors are defined by
theoretical methods in order to give a frame of reference. controlled introduction of impurities in the host lattice, but
Finallywediscussthepresentchallengesandpossiblefuture oftenitisnotpossibletocompletelyeliminatetheformation
directions in semiconductor research with positrons. ofotherdefects,suchasvacancydefects,atomsoninterstitial
It is important to note that we do not attempt to make an lattice sites, extra impurities, or antisite defects (the latter
exhaustive review of all positron work on defects in semi- existonlyincompoundsemiconductors).Eitherthesedefects
conductors.Tocovermostofthepublishedworksonpositron have a detrimental effect on the targeted property or some-
annihilation in solids, we refer the interested reader to the timestheycanassistinobtainingthedesiredfunctionalityof
reviews by Berko and Hereford (1956), Ferrell (1956), thematerial.Itisalsopossibleforsomedefectstobeneutral
Schultz and Lynn (1988), Asoka-Kumar, Lynn, and Welch from the point of view of the property to be controlled.
(1994), Puska and Nieminen (1994), Krause-Rehberg et al. Understanding of the properties of a semiconductor
(1998), and Saarinen, Hautoja¨rvi, and Corbel (1998). requires(i)identificationofthedefectspresentinthelattice,
A number of books have been published on the subject of (ii) their quantification, and (iii) knowledge of the nature of
positronannihilation insolids, aswell as chaptersinvarious the states they introduce in the band gap (i.e., their effects
edited volumes. For detailed accounts see the conference on the properties). Examples of defect properties are sub-
proceedings of the ICPA (International Conferences on band-gap light absorption and emission, and introduction or
Positron Annihilation), SLOPOS (International Workshops removalofelectronstoorfromtheconductionorthevalence
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013
Filip Tuomisto and Ilja Makkonen: Defect identificationin semiconductorswith ... 1585
band. Control of the semiconductor properties requires in transient spectroscopy (DLTS) [see, e.g., Svensson, Ryden,
addition that the formation and introduction mechanisms of andLewerentz(1989),Dobaczewskietal.(1994),andLook,
these defects are understood, as well as their other physical Hemsky, and Sizelove (1999)]. Optical spectroscopies give
propertiessuchashowtheyinteractwithotherdefectsinthe access to another set of basic properties of semiconductors,
latticeandwhether theycan bemadetomovewith thehope namely,theopticalabsorptionandemissionthatareparticu-
of them getting trapped at a neutralizing location or driven larly important in optoelectronic device applications such as
out of the region of interest. It should not be a surprise that light-emitting diodes or laser diodes. Absorption and lumi-
manydifferentexperimentalandtheoreticalmethodsneedto nescence spectroscopies provide detailed information on the
be employed in order to obtain even a small part of the opticaltransitionsbetweenthevalenceandconductionbands
required knowledge. Finally, after all this understanding, and on the positions and nature of defect-induced electronic
one needs to be able to manufacture the semiconductor states in the band gap.
material in such a way that desired defects are introduced Theabovetechniques providedetailed information onthe
but the harmful ones are not. Quite often this is very electrical and optical properties generated by the defects,
challenging. but usually they do not allow for direct identification of the
Usuallyitisratherstraightforwardtocontroltheintroduc- defects in question, and in optical spectroscopy the determi-
tion of the desired impurities in the semiconductor matrix. nation of defect concentrations is challenging (Reshchikov
Dopants can be added to the growth environment invarious and Morkoc¸, 2005). Optical absorption by local vibrational
waysortheycanbediffusedinorimplantedafterthegrowth modes in the infrared (IR) wavelengths can be used to
process. The most important limitations are solubility in the identify defects through their vibrational frequency finger-
case of in situ or diffusion doping, while implantation is prints(Bergmanetal.,1988;Gotzetal.,1996).Thismethod
mostly applicable to close-to-the-surface doping profiles. is particularly useful in the case of hydrogen-related defects
However, while introduction of dopants is controllable and
in semiconductors because of the very distinct frequencies
requires active measures, other kinds of point defects are
originatingfromthelowatomicmassofhydrogen.Thereisa
formed either thermally, due to kinetic or chemical effects,
set of techniques based on photon spectroscopy in the pres-
orasradiationdamageinthecaseofimplantationprocessing.
enceofamagneticfieldthatareverysensitivetothedetailed
In addition, the growth environment may contain some
atomic structure thanks to the hyperfine interactions. These
unwanted impurities that are difficult to control: typical
methods employ the electron spin resonances (ESRs), and
omnipresent elements are oxygen and hydrogen. Further,
require the defect to be studied to have a paramagnetic ion-
for example, in the case of wide-band-gap semiconductors
ized state that can be excitedby anexternal field. Variations
suchastheIII-nitrides,wherenativesubstratesarenoteasily
of these experiments include electron paramagnetic reso-
available, the lattice mismatch between the thin film and
nance (EPR), optically detected magnetic resonance, and
substratecausesinitialstressesandstrainthataremostoften
electron nuclear double resonance (Watkins and Corbett,
relaxed through the generation of dislocations and other
1964). The ESR methods are sensitive to the number of the
extended defects. There are many ways to try to avoid the
active centers (instead of the concentration) and give a very
formation of the unwanted defects or to try to remove them
detailed atomic structure of the defects that are detected.
by postprocessing, such as thermal treatments. Even though
Challenges are encountered with samples with high free-
manydefect-relatedproblemshavebeenidentifiedandsolved
carrierconcentrationsduetoefficientmicrowaveabsorption,
overthepast60yearsofsemiconductorresearch,theconstant
whilethinfilmsoftenhavetoofewactivecentersintotaleven
quest for faster, cheaper, less power consuming, and new
if their concentration is high.
kinds of electronics generates the need for new materials
Theelectricalandopticaldefectspectroscopymethodsare
properties and hence creates new defect-related challenges.
intrinsically nondestructive, i.e., the semiconductor samples
andtheirpropertiesarenotalteredduringthemeasurements.
2. Studying defects in semiconductors There is a wide variety of methods based on the use of ion
Astheexistenceofdefectsiswhatmakessemiconductors beamsthatinturnaredestructive,butprovidecrucialdataon
such useful materials, defects in semiconductors have been the defect properties. Most common of these are Rutherford
studiedforaslongassemiconductorshavebeenknown.The backscattering and nuclear reaction analysis which are very
widevarietyofmethodscanberoughlydividedintoelectrical efficient for detecting and identifying atoms that are not in
measurements, optical spectroscopy, particle beam methods, correct lattice positions (Wahl et al., 1997; Yu et al., 2002).
microscopy,andtheoreticalcalculations.Detailedreviewson Nondestructive particle beam methods include muon spin
thesemethodscanbefoundintheliterature[see,e.g.,Stavola rotation and positron annihilation spectroscopies, of which
(1998)]. In the following we briefly go through the defect the first is particularly useful for modeling behavior of
detection,identification,andquantificationcapabilitiesofthe hydrogeninsemiconductors(Stavola,1998),whilethelatter
most used methods in semiconductor defect studies. is selectively sensitive to vacancy-type defects. Electron
Measuring electrical properties from the defect point microscopy methods have already reached (sub-)atomic
of view typically leads to the determination of resistivity resolution; this holds especially for transmission electron
(conductivity), free-carrier concentration and mobility, con- microscopy (TEM). In addition, the latest advances in
centrationsofionizeddonorsandacceptors,anddeepcarrier the so-called Z contrast allow the identification of atomic
traps. By definition, these properties can be considered the speciesaswell (Pennycook,2012).Hence exactpositionsof
most basic properties of a semiconductor. The most popular atoms can be imaged in sample cross sections, providing
methodsemployedareHalleffectexperimentsanddeep-level direct experimental identification of extended defects and
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013
1586 FilipTuomisto and Ilja Makkonen: Defect identificationin semiconductors with ...
impurities, given the fact that the concentrations (densities) observed at high concentrations in many semiconductors,
are high enough, as the typical size of atomic-resolution and their role in doping and compensation can be quantita-
images is of the order of 10(cid:2)10 nm2. Another challenge tively discussed.
inimagingintrinsicpointdefectsisthattheymaybecreated
in the preparation of cross-sectional samples.
1. Background
Calculations of the electronic structure of semiconductors
and their defects is possible from first principles. By far The existence of the antiparticle of the electron, the posi-
the most popular method is the density-functional theory tron,waspredictedbyDirac(1928),anditsfirstexperimental
(supercell calculations) with the electron-electron exchange observation came in 1932 (Anderson, 1933). Positron-
and correlation described through the local-density ap- electron annihilation was eagerly studied throughout the
proximation (LDA) or semilocal generalized-gradient 1940sand1950s,andexperimentalmethodsweredeveloped.
approximations (GGA). The computing power of modern In the late 1960s it was understood that positrons were
supercomputers allows for efficient calculations with rela- sensitive to lattice defects in metals (MacKenzie et al.,
tively large supercells (up to 1000 atoms) of the formation 1967; Bergersen and Stott, 1969; Connors and West, 1969;
enthalpies and charge transition levels of point defects with Hodges, 1970). The development of variable-energy slow-
thesemethods[forreviews,seeVandeWalleandNeugebauer positron beams (Schultz and Lynn, 1988) and of the theory
(2004), Drabold and Estreicher (2007), Janotti and Van de of positrons in semiconductors and defects in the 1980s
Walle(2009),andVandeWalle,Lyons,andJanotti,(2010)]. (Puska and Nieminen, 1994) made research on thin films
However, both the LDA and GGA suffer from predicting and coatings accessible to positron spectroscopy and led
incorrectbandgapsandhencethereliabilityofthepredicted to an ongoing growth in interest in these methods for mate-
defectlevelsisoftendebated.Atomicstructuresofthedefects rialsresearchsincetheearly1990s(Fig.1).Positronannihi-
seem to be less affected by the different approximations. lation spectroscopy is nowadays applied in (cid:3)200 research
Rather recently so-called hybrid functionals (Becke, 1993; laboratories worldwide, while there are (cid:3)40 operational
Perdew, Ernzerhof, and Burke, 1996; Adamo and Barone, slow-positron beams in (cid:3)30 research laboratories.
1999; Heyd, Scuseria, and Ernzerhof, 2003) have been ap- As described above, many techniques are applied to
plied,wherepartoftheexchangeandcorrelationiscalculated identify defects in semiconductors on the atomic scale. The
withintheHartree-Fockapproachtoimprovethedescription advantage of the positron annihilation method lies in its
of nonlocal effects. This approximation has significantly ability to selectively detect vacancy-type defects. This is
improved the predicted band gaps for semiconductors and based on two special properties of the positron: it has a
allowedfornewinterpretationsforsomedefectlevels.Atthe positivechargeanditannihilateswithelectrons.Anenergetic
time ofwriting thisreview, the computational complexityof positron which has penetrated into a solid rapidly loses its
the hybrid functionals limits the supercell sizes to roughly energy and then lives for a few hundred picoseconds in
100 atoms; hence especially in the case of charged defects thermalequilibriumwiththeenvironment.Duringitsthermal
so-calledsupercellcorrectionsneedtobeconsidered(Makov motionthepositroninteractswithdefects,whichmayleadto
andPayne, 1995;Schultz, 2000;Freysoldt,Neugebauer,and trapping into a localized state. Thus the final positron anni-
Van de Walle, 2009). With constantly improving computing hilation with an electron can happen from various states.
power, more accurateapproaches, such asthe GW quasipar- Energy and momentum are conserved in the annihilation
ticle approximation and quantum Monte Carlo methods, are process, where two photons of about 511 keV are emitted
becoming more and more applicable in defect calculations intooppositedirections.Thesephotonscarryinformationon
[see, e.g., Ertekin et al. (2012) and Rinke et al. (2012)]. the state of the annihilated positron. The positron lifetime is
inverselyproportionaltotheelectrondensityencounteredby
B. Positronannihilation spectroscopy
Positron annihilation spectroscopy is a characterization 1000 theoDrye voef lpoopsmiteronnt so fi ns lsoewm picoosnitdrounc tboersa mansd a dnedfects
method for probing the local electron density and atomic
structure at the site chosen by the electrostatic interaction ers 800
ofthepositronwith itsenvironment.Theinformation onthe ap
p
structure can be measured in the time and energy spectra of ed 600
h
the positron annihilation radiation. It is thus possible to blis
Pu 400
investigate experimentally local structures embedded in the
bulk of the material, such as missing atoms (vacancies), 200
clusteringofatoms,superlatticesanddevicestructures,quan-
tumdots,aswellasfreevolume,andvoidsizesinpolymers 0
or even biological materials. These imperfections often de- 1950 1960 1970 1980 1990 2000 2010
termine the crucial properties of the materials, such as me- Year
chanical properties, electrical conductivity, diffusivity, or
FIG. 1. Thenumberofpaperspublishedperyearonpositronsin
light emission. The positron annihilation methods have had condensedmatterphysicsandmaterialsresearch.Thedevelopment
a significant impact on defect spectroscopy in solids by of slow positron beams and of the theory of positrons in semi-
introducing an experimental technique for the unambiguous conductors and defects during the 1970s and 1980s predates the
identification of vacancies. Native vacancies have been strongincreaseinresearchactivity.DatafromISIWebofScience.
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013
Filip Tuomisto and Ilja Makkonen: Defect identificationin semiconductorswith ... 1587
the positron. The momentum of the annihilated electron Krause-Rehberg et al., 2011). In both cases, the positrons
causes an angular deviation from the 180(cid:4) straight angle have a wide and continuous energy spectrum with mean
betweenthetwo511keVphotonsandcreatesaDopplershift energies in the hundreds of keV. These fast positrons can be
intheirenergy.Thustheobservationofpositronannihilation useddirectlytoprobethebulk(severalhundredsofmicrons)
radiation gives experimental information on the electronic of a material. In order to study thin films and coatings, the
and defect structures of solids. For more detailed accounts positronsneedtobesloweddownandifpossiblemonochro-
on the positron annihilation in solids in general, see, e.g., mated. Many crystal surfaces, such as those of heavy metal
West (1973) and Hautoja¨rvi (1979). elements (e.g., W), have a negative work function for posi-
The sensitivity of positron annihilation spectroscopy to trons, resulting in the fact that thermalized positrons within
vacancy-typedefectsiseasytounderstand.Thefreepositron the solid can be emitted to the vacuum with an energy of a
in a crystal lattice experiences strong repulsion from the few eV if they reach the surface (Tong, 1972). These slow
positive ion cores. An open-volume defect like a vacant positrons can be magnetically guided and electrostatically
latticesiteisthereforeanattractivecenterwherethepositron accelerated to form a variable-energy beam allowing for,
gets trapped. The reduced electron density at the vacant site e.g., depth profiling.
increases the positron lifetime. In addition, the missing Thetwomostusedmethodsindefectstudieswithpositron
valence and core electrons cause substantial changes in the annihilation are the positron lifetime spectroscopy and
momentum distribution of the annihilated electrons. Two Doppler broadening (of the positron-electron annihilation
positron techniques have been efficiently used in defect radiation) spectroscopy. These techniques are very efficient
studies in semiconductors, namely, the positron lifetime and ingivingimportantinformationonvacancydefectsinmetals
the Doppler broadeningofthe511keV line.There arethree and semiconductors: the vacancy defects can be identified
mainadvantagesofpositronannihilationspectroscopywhich (sublatticeincompounds,sizeinthecaseofvacancyclusters,
can be listed as follows. First, the identification of vacancy- anddecorationbyimpurities),theirchargestates(inthecase
type defects is straightforward. Second, the technique is of semiconductors) can be determined, and their concentra-
stronglysupportedbytheory,sincetheannihilationcharacter- tions1canbeevaluatedinthetechnologicallyimportantrange
isticscanbecalculatedfromfirstprinciples.Finally,positron from 1015–1019 cm(cid:1)3. Thanks to recent developments in
annihilationcanbeappliedtobulkcrystalsandthinlayersof theoreticalcalculations,computationalstudiescanbedirectly
anyelectrical conduction type. compared with positron experiments, providing possibilities
Aspecialfeatureisthatthepositroncanformaboundstate fordetailedinterpretationsofexperimentaldata.Thepositron
withanelectroninasystemwithlowenough(local)electron lifetime and Doppler broadening techniques are also widely
density (Mohorovicic, 1934; Deutsch, 1951; Mogensen, used in free-volume studies of molecular materials. Angular
1995; Charlton and Humberston, 2001). This hydrogenlike correlation of annihilation radiation (ACAR) (Beringer and
quasiatom is called positronium (Ps), with a mass of Montgomery, 1942; Berko, Haghgooie, and Mader, 1977) is
1:022 MeV=c2 and a diameter of 1:06 A(cid:1) in its ground state used to detect essentially the same phenomenon as Doppler
in vacuum. The binding energy of Ps is 6.8 eV in vacuum, broadening, namely, the momentum distribution of electron-
i.e., half of the ionization energy of the hydrogen atom.
positron annihilation radiation. The resolution of this tech-
Dependingonthespinofthepositronrelativetotheelectron,
nique is superior to Doppler broadening, but count rates are
Psisineitherthesinglet(antiparallelspins,parapositronium,
correspondinglylower.ThebetterresolutionofACARallows
p-Ps) or triplet (parallel spins, orthopositronium, o-Ps) state.
for detailed studies of the electronic structure (e.g., Fermi
The self-annihilation properties of these two states are very
surfaces in metals), but does not bring significant improve-
different(CharltonandHumberston,2001),andtheso-called
ments in the case of defect studies in semiconductors. It
pick-off annihilation of o-Ps, in which the positron in Ps
should be noted that these three techniques are easily used
annihilates with an electron from the surroundings, prevails
in bulk materials employing fast positrons and can be used
inmatter(Brandt,Berko,andWalker,1960).Importantly,the
with slow-positron beams, although requiring high intensity
interactionbetweenPsandmatterispredominantlyrepulsive
(except for Doppler broadening).
due to the electron-electron repulsion. These properties are
Positron-induced Auger electron spectroscopy (PAES)
useful when porous media or soft condensed matter are
(Weiss et al., 1989; Soininen, Schwab, and Lynn, 1991)
studied with positron annihilation (Mogensen, 1995; Jean,
and reflection high-energy positron diffraction (RHEPD)
Mallon, and Schrader, 2003).
(Ichimiya, 1992; Kawasuso and Okada, 1998) are extremely
surface-sensitive positron-using techniques that require the
2. Positron annihilation methods useofaslow-positronbeam,butdonotrequirethemeasure-
ment of positron-electron annihilation radiation. These
Positronscanbecreatedinseveralways,ofwhichthemost
techniques have important benefits compared to their ‘‘tradi-
common, in the case of laboratory-scale facilities, is using
radioactive((cid:1)þ)isotopes,suchas22Na,whichhasrelatively tional’’electroncounterparts.InPAESthesecondary-electron
lowintensity(upto109 positrons=s),butpracticalhalf-lifeof background is completely suppressed and the sensitivity is
enhanced to only the first atomic layer (Weiss et al., 1989;
2.6 years allowing reasonable use of the same source for
6–10 years. High-intensity sources (up to 1012 positrons=s)
at large-scale facilities make use of pair production with 1Hereanoteonvocabularyiswarranted:inthecaseofdefectsin
the high-energy gamma flux created by a nuclear reactor semiconductors, it is typical to speak about concentrations when
(Hugenschmidt et al., 2004; Schut et al., 2004; Hawari densitiesaremeant.Henceconcentrationsaregivenintheunitsof
et al., 2009) or a particle accelerator (Cassidy et al., 2009; cm(cid:1)3 instead of ppm or ppb.
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013
1588 FilipTuomisto and Ilja Makkonen: Defect identificationin semiconductors with ...
Jensen and Weiss, 1990; Soininen, Schwab, and Lynn, positron-positron interactions or molecular positronium
1991; Hugenschmidt, Mayer, and Schreckenbach, 2010). In (Cassidy and Mills, 2007). It should be noted that the
RHEPDthepositroncrystalpotentialandthepositivecharge defect-spectroscopic techniques based on the detection of
of the positron give rise to total reflection below a critical positron-electron annihilation radiation rely on the nonexis-
angle, again resulting in enhanced sensitivity to only the tence of positron-positron interactions. In these measure-
topmost atomic layer at the surface (Kawasuso and Okada, ments, there is only one positron in the sample at any given
1998; Kawasuso et al., 2003; Fukaya, Mochizuki, and time. As positrons in crystalline solids annihilate within a
Kawasuso, 2012). The drawback of these techniques is the time frame of a few nanoseconds, a maximum intensity of
necessity for a high-intensity positron beam as the measure- about 108 positrons=s is imposed.
ment times with laboratory-scale low-intensity beams are far
toolong.
II. EXPERIMENTALTECHNIQUES
The main technological issues limiting further develop-
ment of laboratory-scale experimental techniques are the A. Positrons in solids
poor efficiency of the moderation process when creating a
slow-positronbeam(resultinginthenecessityforlarge-scale 1. Implantation, thermalization, and diffusion
facilities with high-intensity sources) and the directional
Forafulldescriptionofthephysicsofpositronsinsolids,
dispersion of the moderated positrons. The latter is partly
seeSchultzandLynn(1988)andPuskaandNieminen(1994).
responsible for the small number of scanning positron mi-
In the following we briefly describe the necessary concepts
croprobes(SPMs),comparabletoascanningelectronmicro-
andmodelsneededtoanalyzeandinterprettheexperimental
scope (SEM), as the focusing of the beam with reasonable
data.
intensityevenat a large-scalefacility resultsin spot sizesof
The stopping profile of energetic positrons emitted by a
the order of 5 (cid:2)m (Greif et al., 1997; Triftsha¨user et al., radioactive ((cid:1)þ) source is exponential (Brandt and Paulin,
1997).AnotherlimitationfortheSPMisthelateralstraggling
1977):
and the positron diffusion length of several hundreds of
nanometers in a perfect crystal that will limit the spot size (cid:4)½g=cm3(cid:6)
PðxÞ¼(cid:3)expð(cid:1)(cid:3)xÞ; (cid:3)(cid:5)16 cm(cid:1)1; (1)
evenifthefocusisimproved.Themoderatorefficienciesare E1:4 ½MeV(cid:6)
max
in the 10(cid:1)5–10(cid:1)4 range for passive crystalline heavy metal
where(cid:4)isthedensityofthesolidandE isthemaximum
(e.g., W) moderators and in the 10(cid:1)3 range for solid Ne max
energy of the continuous (cid:1)þ radiation spectrum. The most
moderators (Mills and Gullikson, 1986). As the latter need
common isotope for positron experiments is 22Na with
toberegenerated weekly,theyare somewhatcomplicated to
E ¼0:54 MeV. Hence for this isotope the characteristic
use.Thisholds,inparticular,whenthesourceandmoderator max
penetration depth 1=(cid:3) is, e.g., 110 (cid:2)m in Si and 40 (cid:2)m in
are floated at a high voltage in order to have the sample
GaN.Thismeansthatpositronsimplanteddirectlyasemitted
grounded for easy manipulation. Sample manipulation
fromthe source probe thebulkofasolid.It shouldbenoted
(temperature, illumination, and bias control) is essential for
thattheaverageenergyofpositronsemittedby22NaisE ¼
sophisticated thin-film studies, and hence in many cases the av
0:18 MeV. The (electronic) interactions during the stopping
W moderator is better suited.
are the same for positrons as for electrons (Lennard et al.,
Further technological limitations arise when the most
1995), and hence in the case of semiconductors there is
powerful technique, positron lifetime spectroscopy, is used
essentially no lattice damage caused by positron implanta-
withslow-positronbeams.Thetraditionaltechniquedepends
tion. Additionally, a typical total fluence of at most
on the existence of a start signal (given by the 22Na source 1012 cm(cid:1)2 isimplantedinthesamplesduringanexperiment.
which emits practically simultaneously with the positron a
Thisisseveralordersofmagnitudelessthantypicallyusedin
high-energy 1.27 MeV photon). The moderation process
electronirradiationexperimentswiththepurposeofcreating
strongly limits the usefulness of this start signal. Hence, latticedamage(Saarinenetal.,1995;Tuomisto,Rankietal.,
either a positron-fly-by-detecting sensor must be installed 2007; Chen, Betsuyaku, and Kawasuso, 2008).
or the beam must be tagged (e.g., by detecting secondary For monoenergetic positrons obtained from a low-energy
electronsejectedfromthesamplesurfacebypositronimpact) positronbeam(typicalenergiesarebelow50keV),thestop-
or modulated in time in order to retrieve a timing signal. ping profile can be described by (Valkealahti and Nieminen,
The modulation of the positron beam has been shown to be 1984)
the approach of choice, but requires radio-frequency beam
d
bunching and chopping that have their own complications PðxÞ¼(cid:1) exp½(cid:1)ðx=x Þ2(cid:6); (2)
(Mills, 1980; Scho¨dlbauer et al., 1987; Suzuki et al., 1992; dx 0
Tashiroetal.,2001;ReuringsandLaakso,2007).Theadvan- wherex givesthepeakpositionoftheprofilewhilethemean
0
tage is a time resolution good enough for studying semi- stopping depth is x(cid:2) (cid:5)0:886x due to the asymmetry of the
0
conductors and metals where the positron lifetimes are an profile.Themeanstoppingdepthisgivenasx(cid:2) ¼AEn [keV],
order of magnitude shorter than in molecular matter. In where A(cid:5)4(cid:2)10(cid:1)6=(cid:4) ½(cid:2)g=cm2(cid:6) and n(cid:5)1:6. This de-
principle, the beam could be modulated also by trapping scriptionisverycloselymatchedtotheMonteCarlosimula-
thepositronsintoamagnetictrapandreleasingthematgiven tions for stopping profiles of low-energy electrons (Shimizu
time intervals. This approach is, however, better suited for and Ze-Jun, 1992; Ghosh and Aers, 1995; Dapor, 1996;
applications where bunches containing a large amount of Denison and Farrell, 2004; Nyka¨nen et al., 2012). The
positrons are required, such as in experiments studying mean stopping depth varies from a few nanometers up to a
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013
Filip Tuomisto and Ilja Makkonen: Defect identificationin semiconductorswith ... 1589
fsetuwdymniceraorm-seutrefrasc.eHleanyceersloawnd-etnheinrgfiylmposs.itronscanbeusedto (cid:4)ðpÞ¼(cid:7)r2ecX(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)Z dre(cid:1)ip(cid:8)rcþðrÞciðrÞ(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)2; (5)
i
The stopping and the ensuing thermalization of the
positrons are fast processes, taking only 1–3 ps at room where the summation goes over occupied electron states.
temperature in both metals (Jensen and Walker, 1990) and It should be noted that the momentum distribution (cid:4)ðpÞ of
semiconductors (Jorch, Lynn, and McMullen, 1984). This is the annihilation radiation is mainly that of the annihilating
considerably less than typical positron lifetimes of the order electrons ‘‘seen by the positron,’’ because the momentum of
of 150–300 ps. ACAR experiments have shown that the the thermalized positron is negligible. The theoretical meth-
momentum distribution of annihilating positrons follows the ods used to determine the positron’s ground state and the
sample temperature down to 10 K (Hyodo, McMullen, and annihilation parameters are reviewed in Sec. III.
Stewart,1986).Inafewcasesincompletethermalizationmay In analogy to free carriers, the positron also has localized
be important, e.g., positrons implanted at a very low energy statesatlatticeimperfections.Atvacancy-typedefectswhere
canescapethesamplenonthermallythroughthesurface[see ions are missing, the repulsion sensed by the positron is
GulliksonandMills(1986),Nielsen,Lynn,andChen(1986), lowered and the positron experiences these kinds of defects
Huomoetal.(1987),andLynnandNielsen(1987)andthere as potential wells. As a result, localized positron states at
are also related problematics in measurements made for open-volumedefectsareformed.Thepositrongroundstateat
nanocrystallinematter;seeSec.V.C].Thisneedstobetaken a vacancy-type defect is generally deep; the binding energy
into consideration when interpreting data from near-surface is about 1 eV or more (Makkonen and Puska, 2007). In a
layers.Afterimplantationandstoppingthetransportofther- vacancydefecttheelectrondensityislocallyreduced.Thisis
malizedpositronscanbequiteefficientlydescribedbydiffu- reflected in the positron lifetime which is longer than in the
sion theory developed for free carriers (Bergersen et al., defect-free lattice. Hence the positron lifetime measurement
1974). The positron diffusion coefficient in semiconductors isaprobeofvacancydefectsinmaterials.Positronannihila-
at room temperature is typically Dþ ¼1–2 cm2=Vs. The tion at a vacancy-type defect also leads to changes in the
characteristic diffusion length during the positron lifetime (cid:5) momentum distribution probed by the Doppler broadening
is Lþ ¼ðDþ(cid:5)Þ1=2 ¼100–200 nm. experiment.Themomentumdistributionarisingfromvalence
electron annihilation becomes slightly narrower due to a
2. Positron states and trapping lower electron density. In addition, the localized positron at
a vacancy has a reduced overlap with ion cores leading to a
After implantation and thermalization the positron in a
considerable decrease in annihilation with high-momentum
semiconductor is in a Bloch-like state in a perfect periodic
core electrons. The localized positron has time to interact
crystal lattice. The thermalized positron at its ground state
with the host lattice during its lifetime of >150 ps and
canbedescribedtoagoodapproximation(seeSec.III)bya
enlargetheaverageopenvolumeofthevacancybyrepelling
single-particle Schro¨dinger equation
neighboring positiveion cores.
A negatively charged impurity atom or an intrinsic point
1
(cid:1) r2c ðrÞþVðrÞc ðrÞ¼E c ðrÞ; (3) defect can bind positrons at shallow states even if these
2m(cid:7) þ þ þ þ
defectsdonotcontainopenvolume.Beingapositiveparticle,
where the positron potential consists of an electrostatic the positron can be localized at the hydrogenic (Rydberg)
Coulomb potential and a term that takes into account the stateoftheCoulombfieldaroundanegativelychargedcenter.
electron-positroncorrelationeffects.BecauseoftheCoulomb The situation is analogous to the binding of an electron
repulsionfrompositiveioncores,thepositronwavefunctionis to a shallow donor atom. The positron binding energy at the
concentratedintheinterstitialspacebetweentheatomsinthe negative ion can be estimated from simple effective-mass
lattice.Thepositronenergybandisparabolicandfreeparticle theory:
like(Boev,Puska,andNieminen,1987).Theeffectivemassof
cthloeupdoosiftreolnecitsromn(cid:7)s((cid:5)Be1r:g5emrs0ednuaentdoPpahjaonnnoen,s1a9n6d9)th.escreening Eion ¼13:6eVmm0(cid:7)(cid:8) Zn22 ¼10–100 meV; (6)
The positron lifetime and the Doppler broadening of the
where(cid:8)isthedielectricconstant,m(cid:7) istheeffectivemassof
annihilation radiation can also be calculated once the corre-
thepositron,Z is the charge ofthenegativeion, andn is the
sponding electronic structure of the solid system is known.
quantum number. With Z¼1–3 and n¼1–4, Eq. (6) typi-
The positron annihilation rate (cid:6), the inverse of the positron
callyyieldsE ¼10–100 meV,indicatingthatpositronscan
lifetime(cid:5),canbethoughttobeproportionaltotheoverlapof ion
bethermallyexcitedfromtheRydberg statesat100–300K.
the electron and positron densities:
The hydrogenic positron state around a negative ion
1 Z has a typical extension of 10–100 A(cid:1) and thus positrons
¼(cid:6)¼(cid:7)r2c jc ðrÞj2n ðrÞdr; (4)
(cid:5) e þ (cid:1) probe the same electron density as in the defect-free lattice.
As a consequence, the annihilation characteristics (positron
where r is the classical electron radius, c is the velocity of lifetime, momentum density of annihilating pairs) are not
e
light, and n ðrÞ is the electron density. The momentum different from those in the lattice. Although the negative
(cid:1)
distribution (cid:4)ðpÞ of the annihilation radiation is a nonlocal ions cannot be identified with the experimental parameters,
quantity and requires knowledge of all electron wave information on their concentration can be obtained in the
functions c overlapping with the positron. In the simplest positron lifetime and Doppler broadening experiments when
i
approximation it can bewritten in the form they competewith vacancies in positron trapping.
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013
1590 FilipTuomisto and Ilja Makkonen: Defect identificationin semiconductors with ...
The positron transition from a free Bloch state to a local- 3. Trapping model
izedstateatadefectiscalledpositrontrapping.Thetrapping
The practical situation during a measurement, where
is analogous to carrier capture. However, it must be fast
onlyonepositronatatimeisinthesample,canbedescribed
enough to compete with annihilation. The positron trapping
by a relatively simple kinetic rate model (time-dependent
rate (cid:9) into a defect D is proportional to the defect concen-
D diffusion equation):
tration ½D(cid:6), (cid:9) ¼(cid:2) ½D(cid:6)=N , where N is the atomic den-
D D at at
sity of the host lattice. The trapping coefficient (cid:2) depends @nðr;tÞ
D ¼Dr2nðr;tÞ(cid:1)(cid:2) r(cid:8)½nðr;tÞE(cid:6)(cid:1)(cid:6)nðr;tÞþS;
on the defect and the host lattice. Since the positron binding @t þ
energy at vacancies is typically >1 eV, the thermal escape
(8)
(detrapping) of positrons from the vacancies can usually be
neglected. Because of the Coulomb repulsion, the trapping where nðr;tÞ is the probability density of finding a delocal-
coefficientatpositivelychargedvacanciesissosmallthatthe ized (free) positron at the position r and time t, D is the
trapping does not occur during the short positron lifetime diffusion coefficient, and (cid:2) is the positron mobility. In the
þ
ofafewhundredpicoseconds(Puska,Corbel,andNieminen, sinkterm(cid:6)isthesumofthe‘‘free’’-positronannihilationrate
1990). Therefore, the positron technique does not detect (cid:6)B and tPhe trapping rates to the defects in the system (cid:6)¼
vacancies or other defects in their positive charge states. (cid:6) ðrÞþ (cid:9)ðrÞ.InprincipleDand(cid:2) canalsobefunctions
B i i þ
The trapping coefficient at neutral vacancies is typically ofthepositionr,butonlyinlayeredsystems wheretheyare
(cid:2) (cid:5)1014–1015 s(cid:1)1, independentof temperature. The posi- constant throughout the layer (similar to (cid:6) ), while (cid:9)ðrÞ is
D B i
tron trapping coefficient at negative vacancies is typically smoothandfollowsthedefectprofile.Inthesimplestcasethe
(cid:2) (cid:5)1015–1016 s(cid:1)1 at 300 K. The experimental fingerprint source term SðrÞ vanishes, but if positrons are allowed to
D
of a negative vacancy is the increase of (cid:2)D with decreas- escapefromthetrappedPstatesatthedefects,thesourceterm
ing temperature. The T(cid:1)1=2 dependence of (cid:2) is simply isnon-negligible(S¼ (cid:10)n ).Inpracticethethreespatial
D i i D;i
due to the increase of the amplitude of the free-positron dimensionscanbereducedtojustone(x),asboththelateral
Coulomb wave in the presence of a negative defect as the stragglingandthespotsizeoftheimplantedpositronsessen-
thermal velocity of the positron decreases. The temperature tiallyresultintheexperimentaldatabeingspatiallyaveraged
dependence of (cid:2) allows one to experimentally distinguish intheplaneperpendiculartothemainimplantationdirection.
D
negativevacancy defects from neutral ones. The initial condition nðx;0Þ is given by the positron im-
The positron trapping coefficient (cid:2) at the hydrogenic plantation profile PðxÞ (initially the positron is free, so the
ion
statesaroundnegativeionsisofthesameorderofmagnitude probabilityoffindingapositroninatrappedstateatadefect
as that at negative vacancies. Furthermore, the trapping co- is initially(cid:9)0), while the boundary conditions can be usu-
efficient exhibits a similar T(cid:1)1=2 temperature dependence. ally thought of as being those of a semi-infinite system
Unlike in the case of vacancy defects, the thermal escape of assuming thick enough samples so that positrons implanted
positronsfromthe negativeionsplaysacrucialroleat usual from one side do not reach the other:
experimentaltemperatures.Theprincipleofdetailedbalance
nðx;0Þ¼PðxÞ; (9)
yieldsthefollowingexpressionfortheratioofdetrappingand
trapping rates (Manninen and Nieminen, 1981):
limnðx;tÞ¼0; (10)
(cid:2) (cid:3) x!1
(cid:10)(cid:9)¼c1ion m2(cid:7)(cid:7)kℏB2T 3=2e(cid:1)Eion=kBT; (7) D@@nx(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:2)þEð0Þ¼(cid:11)nð0;tÞ: (11)
x¼0
wherec istheconcentrationofthenegativeions.Typically
ion Here (cid:11) represents the positron transition rate to states at
thenegativeions(shallowtraps) influence positronannihila-
the sample surface. The simplest kind of experiment, where
tion at low temperatures (T<100 K), but the ions are not
positronsemittedbyaradioactivesourcearedirectlyinjected
observedathightemperatures(T>300 K),wheretheescape
intoasample,furthersimplifiestheaboveexpressions.Thisis
rate is high.
becausethedataareessentiallyaveragedoverawidespread
It is worth noting that the determination of absolute
of implantation depths, effectively removing the spatial di-
defectconcentrationsbasedonpositronexperimentsdepends
mensionfromEq.(8).Thesituationcanthenbedescribedby
directly on the knowledge of the trapping coefficient, while
a set of rate equations (N different defects):
the comparison of defect concentrations (say, in two differ-
(cid:2) (cid:3)
X X
ently doped semiconductor samples) gives accurate propor- dn
B ¼(cid:1) (cid:6) þ (cid:9) n þ (cid:10) n ; (12)
tions even when the trapping coefficient is not known. In dt B j B j D;j
j j
extensivelystudiedsemiconductors,suchasGaAs,GaN,and
ZnO (Saarinen et al., 1995; Oila et al., 2003; Tuomisto,
dn
Saarinen, Look, and Farlow, 2005), the cross correlation of D;j¼(cid:9) n (cid:1)ð(cid:6) þ(cid:10) Þn ðj¼1;...;NÞ: (13)
dt j B D;j j D;j
optical, electrical, and positron experiments has narrowed
down the trapping coefficient of negatively charged cation Here the probability of a positron being in the free state is
vacancies to (cid:2)V(cid:1) (cid:5)ð2–3Þ(cid:2)1015 s(cid:1)1. Theory (Puska and nBðtÞandtheprobabilityofitbeinginatrappedstateatdefect
Nieminen, 1994) predicts that the neutral-vacancy trapping j is nD;jðtÞ. The corresponding annihilation, trapping, and
coefficient should be a factor of 2–3 lower than that escaperatesaregivenby(cid:6)B,(cid:6)D;j,(cid:9)j,and(cid:10)j,respectively.In
of negatively charged vacancies; hence often the value of practice(cid:10) (cid:1)0onlyforshallowpositrontraps(negative-ion-
j
(cid:2) (cid:5)1(cid:2)1015 s(cid:1)1 is used for neutral vacancies. typedefects)atsufficientlyhightemperatures.Theboundary
V
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013
Filip Tuomisto and Ilja Makkonen: Defect identificationin semiconductorswith ... 1591
and initial conditions in Eq. (9) are simplified to n ð0Þ¼1 positronandaneutrino,leavinganexcited22Nenucleusthat
B
andn ð0Þ¼0.As anexample,inthe casewhere positrons rapidly decays through the emission of a 1.2745 MeV
D;j
aretrappedatonekindofvacancydefect(V)andonekindof (cid:13) photon. This photon can be used as a start signal for the
shallow trap (st), the above set of equations becomes positronlifetimemeasurement,whilethestopsignalisgiven
by one of the two 511 keVannihilation (cid:13) photons.
dn
dtB ¼(cid:1)ð(cid:6)Bþ(cid:9)V þ(cid:9)stÞnBþ(cid:10)stnst; (14) In practice the positron source material is in the form of
NaCl which is typically stored as a water solution. The
dnV ¼(cid:9) n (cid:1)(cid:6) n ; (15) experimentisperformedbysandwichingthepositronsource
dt V B V V betweentwoidenticalsamplepieces.Thiscanbedoneeither
dn by depositing some of the NaCl directly on one of the
dtst ¼(cid:9)stnB(cid:1)ð(cid:6)stþ(cid:10)stÞnst: (16) samples, and then placing the other on top of it, or by first
making a sealed positron source through packaging some of
Applying the initial condition, the above equations can be
the NaCl in thin foil. Common foil choices are Al, Ni, and
solvedandtheprobabilityofapositrontobealiveattimetis
sometimesapolymersuchasKapton.Thepackagingsolution
obtained as
X ispreferableforreuseofsourcematerial,whilethemetalfoils
nðtÞ¼nBðtÞþnVðtÞþnstðtÞ¼ Iiexpð(cid:6)itÞ: (17) allow for a wider range of measurement temperatures. The
i sourcepackageneedstobemadeasthinaspossibleinorder
This means that exponential decay should be observed in toensurethatamaximalfractionofpositronsemittedbythe
experiments. The experimental lifetime spectrum in fact source enters the samples and that as few positrons as pos-
measures the probability of positron annihilation in the time sible annihilate in the source itself: e.g., typical Al-foil
interval t(cid:8)(cid:8)(cid:8)tþdt, and hence the lifetime spectrum thickness that is used is 1:5 (cid:2)m, and at most two layers of
(cid:1)dnðtÞ=dt is in this case composed of a sum of three com- foil are on each side of the deposited NaCl. Typical activity
ponents. The fractions of positron annihilations at various of such a source is 10–30 (cid:2)Ci [ð0:3–1Þ(cid:2)106 Bq]. Such a
states are in this examplegivenby
sample-source-sample sandwich can then be placed on a
(cid:12) ¼1(cid:1)(cid:12) (cid:1)(cid:12) ; (18) sample holder connected to a temperature control system,
B V st
andtheexperimentcanbedesignedinsuchawayastoallow
(cid:9) for, e.g., sample illumination.
(cid:12) ¼ V ; (19)
V (cid:6) þ(cid:9) þð(cid:9) =1þ(cid:10) =(cid:6) Þ The lifetime experiment itself is performed by detecting
B V st st st
the (cid:13) photons serving as start and stop signals with two
(cid:12) ¼ (cid:9)st : (20) relatively large scintillating detectors (lateral dimensions
st ð1þ(cid:10) =(cid:6) Þ½(cid:6) þ(cid:9) þð(cid:9) =1þ(cid:10) =(cid:6) Þ(cid:6) and thicknesses of the order of centimeters) coupled with
st st B V st st st
photomultipliertubes[see,e.g.,Nissila¨etal.(2001)],eachof
These expressions are useful as they can be compared
which is tuned and optimized for one of the two photons.
with experimentally determined time-averaged quantities
Detector geometry is optimized with respect to overall
such as the average positron lifetime and parameters
efficiency (covering as wide a fraction of the solid angle
describing the Doppler broadening of annihilation radia-
as possible), taking into account restrictions imposed by
tion, as these parameters (P) measure the superposition of
different scintillator materials. As an example, plastic
the annihilations over all positron states: P¼(cid:12) P þ
P B B scintillatorscanbeusedinasimplecollineargeometrywhere
(cid:12) P .Depth-resolvedanalysisofthelatterispossible
j D;j D;j the sample-source-sample sandwich is placed on the axis
whenusingaconventionalslowpositronbeam(Schultzand
defined by the two detectors and the detectors are put as
Lynn, 1988). Then one can employ the steady-stateversion
closetoeachotheraspossible.Ontheotherhand,specialcare
of Eq. (8). Often, especially in the case of thin semicon-
needstobetakenwhenusingBaF scintillators,whichbenefit
ductor epilayers where the vacancy defect concentrations 2
from a significantly higher detection efficiency and enable
tend to be high, the diffusion can be neglected altogether
better resolution, but whose high-Z Ba causes strong (cid:13)
and the positron implantation profile in Eq. (2) provides a
scattering (Becvaret al., 2000).
sufficient approximation of the depth distribution of the
In a conventional lifetime measurement the two detector
positronsignal.However,inmanycasessolvingthesteady-
signals are analyzed with analog nuclear instrumentation
stateversionofEq.(8)andfittingit(vanVeenetal.,1991)
electronics: constant fraction discriminators to choose pho-
to data measured in layered structures provides additional
tons ofcorrect energy, atime-to-amplitude converter, letting
insight to the experimental observations. For more detailed
through only pulses spaced close enough in time, and a
discussions about the trapping model, see, e.g., Saarinen,
multichannel analyzer (MCA) connected to a measurement
Hautoja¨rvi, and Corbel (1998) and Krause-Rehberg and
computer.Theresultisahistogramofannihilationeventsasa
Leipner (1999).
function of time differences between the start and stop sig-
nals, i.e., the positron lifetime spectrum. Typical time inter-
B. Positronlifetime spectroscopy vals for each MCA channel are of the order of 25 ps. The
modern and considerably cheaper way of doing the same is
1. Experimental details
throughdirectdigitization(fastanalog-to-digitalconversion)
A positron lifetime experiment can be performed in a of the detector pulses and performing the signal analysis by
relatively simple way by using a radioactive 22Na positron software(Rytso¨la¨etal.,2002;Saitoetal.,2002;Nissila¨etal.,
source. 22Na decays through the (cid:1)þ process, producing a 2005;Becvar,Cizek,andProchazka,2008).Theadvantageof
Rev. Mod. Phys., Vol. 85, No. 4, October–December 2013