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Puska, M. J.; Mäkinen, S.; Manninen, Matti; Nieminen, RistoScreening of positrons in semiconductors and insulators

Published in:Physical Review B

DOI:10.1103/PhysRevB.39.7666

Published: 15/04/1989

Document VersionPublisher's PDF, also known as Version of record

Please cite the original version:Puska, M. J., Mäkinen, S., Manninen, M., & Nieminen, R. (1989). Screening of positrons in semiconductors andinsulators. Physical Review B, 39(11), 7666-7679. https://doi.org/10.1103/PhysRevB.39.7666

PHYSICAL REVIEW B VOLUME 39, NUMBER 11 15 APRIL 1989-I

Screening of positrons in semiconductors and insulators

M. J. Puska* and S. MakinenDepartment of Physics, Uniuersity of Jyvaskylii, SF 401-00 Jyvaskylii, Finland

M. Manninen and R. M. NieminenLaboratory ofPhysics, Helsinki Uniuersity of Technology, SF 021-50 Espoo, Finland

(Received 8 June 1988)

Theoretical models are presented for the enhancement of the electron density at a positron in asemiconductor or insulator host. The model better suited for typical semiconductors is based on themany-body theory for the screening of a positron in electron gas. The starting point of the modelfor insulators is the atomic polarizability. The common parameter in both models is the high-frequency dielectric constant. Moreover, the enhancement depends on the ambient electron densityin the semiconductor model and on the unit-cell volume in the insulator model. With use of themodels developed, positron lifetimes in perfect semiconductor and insulator crystals have been cal-culated. In the calculations, three-dimensional electron densities and electrostatic potentials are ob-tained by atomic superposition and the fully three-dimensional positron wave functions are solved

by a relaxation method. The calculated positron lifetimes agree with the experimental ones within afew picoseconds. Moreover, we have used the model to predict lifetimes of positrons trapped by lat-tice defects such as vacancies and vacancy clusters.

I. INTRODUCTION

The electronic screening of a positron in condensedmatter manifests itself in a very pronounced way: due tothe electron pileup the observed positron lifetimes aremuch shorter than the ones predicted by theindependent-particle model (IPM), which omits theelectron-positron correlation effects. ' The screening of apositron in a metal is well understood on the basis of themany-body theory for a positive delocalized particle in ahomogeneous electron gas. The enhancement of theelectron density at the positron depends on the density ofthe unperturbed system. For instance, the enhancementfactor for a high-density electron gas corresponding tothe density parameter r, =2 (atomic units) is about 4 andincreases rapidly to the value of about 40 when the elec-tron density decreases to r, =6. The diverging behaviorat the vanishing electron density is in fact due to the for-mation of a negative positronium ion (Ps ) whichguarantees a finite contact density also at this limit. Theelectron gas picture cannot be as such applied in the caseof semiconductors or insulators because the polarizabilityor the screening efficiency of the electrons in the filledvalence bands is, due to the lack of the low-energy excita-tions over the Fermi surface, less than the polarizabilityof the conduction electrons in metallic systems.

In this paper we introduce two simple models whichdescribe the screening of positrons in semiconductors andinsulators. The starting point of the first model is theabove-mentioned theory for the positron in a homogene-ous electron gas whereas the second model is based onthe "atomic" polarizability of the constituents of thesolid. It turns out that the former suits for the typicalgroup-IV elemental and III-V and II-VI compound seini-conductors while the latter works in the case of large-

band-gap insulators. The borderline between semicon-ductors and insulators is, of course, not sharp and oneshould try to find a more general theory which coversboth regions and which gives our two models at the ap-propriate limits. However, it has been found in this workthat the two models overlap so that some of the host sys-tems can be described equally well by both the models.

The two models not only give insight into the mecha-nisms of screening but also a quantitative description ofits effect on the positron annihilation rate and the corre-lation energy. This is most useful for practical calcula-tions for positron states in perfect and defected crystallattices. The practical calculational schemes use thescreening properties of positrons in the sense of theKohn-Sham local-density approximation (LDA): thecorrelation energy and the annihilation rate at a givenpoint in the host system depend only on the electron den-sity at that point. Moreover, in the conventional scheme,which does not employ the two-component density-functional theory, these quantities are calculated for allstates, i.e., including the localized ones, using the unper-turbed host electron density (the zero positron densitylimit of the two-component theory).

The basis of the present work lies in the accurate rnea-surements of positron lifetimes in bulk semiconductorsand insulators. The trends seen in these provide guide-lines for the models and the absolute values are the neces-sary test data to be reproduced in calculations. However,the extraction of the positron bulk lifetimes from themeasured lifetime spectra is ambiguous due to the life-time components connected with annihilation at defectsin the samples, at sample surfaces, or in the positronsource. For example, two values, 220 ps (Ref. 5) and-230 ps, ' have been suggested for the bulk positronlifetime in GaAs. The difference is only 10 ps but i.s very

39 7666 1989 The American Physical Society

39 SCREENiNG OF POSITRONS IN SEMICONDUCTORS AND. . . 7667

important because it afFects the interpretation of the ex-perimental data in defect-containing simples and the cru-cial identificatiori of defects present. Our calculations,free from adjustable parameters, will help in decompos-ing and interpreting the lifetime spectra.

The assignment of difFerent lifetime components withdifFerent defects is an even more difficult task than the ex-traction of the positron bulk lifetimes as there can beseveral types of defects present simultaneously and in ad-dition they can exist in difFerent charge states. ' '

Theoretical predictions will help the analysis of the corn-plicated data. ' Therefore we have applied the models incalculations of positron states in vacancies and divacan-cies in semiconductors and insulators.

Bn(r) = —2Zn(0) .Br

(5)

Let us assuine that the screening charge decays exponen-tially as

proportional to Z/er, where e is the (static) dielectricconstant of the medium. This means that the screeningcloud of a point charge contains efFectively Z(1 —I/e)electrons. This fact is used in the following simple pic-ture as a sum rule for the electron density screening thepositron in a dielectric medium.

The electron density n(r) screening a positive pointcharge obeys the cusp condition' at the origin, where thepoint charge Z is located, i.e.,

II. MODKI S FOR POSITRON SCREENINGn (r) =no+ Ce (6)

A. Semiconductor model

The screening of a positron by conduction electrons inmetals can be described by the many-body calculationsfor a delocaliaed positron in a homogeneous electrongas. The roost important results from these calculationsare the positron arinihilation rate and the positron corre-lation energy. The former depends on the contact elec-tron density n (0), which is a function of the ambientelectron density n„=3/4n r, (given in a.u. ):

A, „=mrocn(0) =AD(n„)[1+kg (n, )],where

A,o(nU ) = rrrocn„

=16mn„=1.2/r, ns

is the IPM result, in which ro is the classical electron ra-dius and c is the speed of light. Above, in Eq. (1) b,g isthe enhancement factor, which gives the relative pileup ofthe electron density at the positron. In the metallic den-sity range, 2 & r, & 6, an accurate interpolation formula"for the enhancemerit factor is

r, +10Ag(n, )-= (3)

or

A,,(n„)=2+ 134n„ns (4)

The latter form shows that the low-density limit for theannihilation rate is 2 ns ', which corresponds to thespin-averaged lifetime of a Ps atom. ' The positroncorrelation energy is defined as the lowering of the totalenergy of the system due to the electron relaxation rela-tive to the uniform distribution. It is also determined bymany-body calculations for a positron in a homogeneouselectron gas.

In semiconductors and insulators the existence of aband gap prevents the valence electrons from respondingto external perturbations as efFectively as the conductionelectrons in metals. The screening is not perfect, whichmeans that a stationary point charge Z produces a long-range Coulomb potential with an asymptotic behavior

where no is the unperturbed constant density. C and aare parameters which are eliminated using the cusp con-dition and the sum rule

Ce 4m.r dr =Z 1 ———2ar 2 1

0 E'

In the case of the light positron in the electronic systemthe cusp condition Eq. (5) has to be modified. At lowelectron densities the screening corresponds to the two-body (electron-positron) scattering, ' and it can be de-scribed by scattering of particles with chaige —e (elec-tron charge) and mass p=m /2 (m =electron mass) froma point charge +e of infinite mass. As a result, theright-hand side of Eq. (5) has to be divided by two.Moreover, due to the light mass (strong recoil) of the pos-itron it is more appropriate to use the high-frequencydielectric constant e„. In other words, ions have no timeto relax according to the instantaneous position of thepositron. However, the static and high-frequency dielec-tric constants are not very difFerent, so that both wouldgive rather similar results for positron states and annihi-lation rates in semiconductors. With these substitutionsthe contact density calculated to the linear order in rioreads as

n (0)=4no+ 1

8m

+201no ns (9)

In order to make the result of this simple calculationagree with Eq. (4) at the metallic liinit e = ~ we have tosubstitute the numeric factor of 201 by 134. As a matterof fact, Eq. (9) is the low electron density limit due to theapproxiinations made in Eq. (8). Higher-order correc-tions to Eq. (9) bring it at finite densities (r, -2. . . 6)closer to the result of Eq. (4), but then the annihilationrate is no longer linear in no Therefore .we prefer the

Thereafter, the positron annihilation rate can be calculat-ed from the contact electron density as

A,, =mrocn(0)

7668 PUSKA, MAKINEN, MANNINEN, AND NIEMINEN 39

A, , =2 1 — + 134no ns1 —j.

(1—1/e )r, +10=31+r

ns (10)

above simple substitution and get the following par-ameter-free model for the positron annihilation rate insemiconductors:

12 r, +101+f(r„s ) ns (14)

states. They fitted E so that the dielectric function cal-culated in the model approaches at the zero-frequencylimit the experimental static dielectric constant. Thewidth of the valence band is simply the Fermi energy (EF)in the free-electron model with the average valence elec-tron density of the semiconductor. They found that thepositron annihilation rate can be expressed as

B. Insulator model

=12 11+ A +BR ns

p3 E~+2

(12)

where A and B are parameters which can be determinedby fitting the calculated positron lifetimes to the experi-mental ones. The parameter 3 will make the enhance-ment factor always larger than 1.5, which is the acceptedvalue for tightly bound core electrons. ' %'e have found(see below) that the following values reproduce best theexisting experimental data:

A =0.684, B =0.0240 . (13)

Due to the two parameters introduced, the insulatormodel can be considered rather as an interpolationscheme, contrary to the semiconductor model Eq. (10),which has no adjustable parameters. The insulator modelis applied in this work mainly to solids with covalentbonds or to solids with partial ionic-bond character. Theapplication to positron states in rare-gas solids, whichpreviously have been studied extensively, would require amore careful treatment' of the correlatiori effects due toatomic polarizatiori, because e is very close to unity forthese solids.

In insulators a more realistic approach than startingfrom the free-electron-gas model is based on the atomicpolarizabilities of the constituents of the solid. Theenhancement of the electron density at the positron is as-sumed in the lowest order to be proportional to the atom-ic polarizability a„, which in practical calculations canbe estimated from the Clausius-Mossotti relation

3fl (e—1)&a~=

4m. (@+2)

where 0 is the unit-cell volume. The enhancement factoris now constant contrary to Eq. (10), which dependsthrough the term r, on the local electron density. (How-ever, as will be clearly demonstrated below, the unit-cellvolume 0 and cube of the eQeetive r, value affecting thepositron in the solid are linearly related to each other. )

The corresponding positron annihilation rate is

(15)

Brandt and Reinheimer estimated [within the Bohm-Pines random-phase approximation (RPA)] the reductionfactor f for a set of r, and E„values. A reasonable fit tothe data reads'

0.37'1+0.18r,

(16)

When inserted in Eq. (14) this does not give the presentform in Eq. (10) and the resulting enhancement factor isgenerally too small. However, if c& is used as a free pa-rameter, a single value, c.z

=0.2, reproduces well the ex-perimental bulk lifetimes for several semiconductors.

Chiba et al. have developed a theoretical model fordescribing the positron correlation with tightly boundelectrons such as those in insulators. They derived anenhancement factor which depends on the binding energyand on the electron momentum of the core level. Forpractical calculations they made the Thomas-Fermi ap-proximation in which the electron binding energy and thelocal Fermi momentum are expressed in terms of the lo-cal electron density. It follows that the enhancement fac-tor then is a function of the electron density only. Theweak point of this approximation is, as will be seenbelow, that the observed variations in the enhancementfactor are so large that they cannot be described as aris-ing only from the density variations.

D. Correlation potential and the practical calculations

When applying Eqs. (10) and (12) for calculations ofpositron states and annihilation rates the correlation en-ergy used in the LDA potential should be consistent withthem. A scaling argument' asserts for the correlation en-ergy

where f (r„E ) is a factor describing the reduction ofelectron enhancement in semiconductors relative to met-als for which f =1 gives Eq. (4). Above, e is the "gapparameter" related to the widths of the band gap and thevalence band as

C. Previous models

Brandt and Reinheimer' studied in a pioneering workthe screening of a positive point charge in semiconduc-tors and insulators. They used the Penn model, ' inwhich the host is mimicked by a homogeneous electrongas with an energy gap (E ) in the parabolic density of

where A, is the actual annihilation rate and Xo is the IPMresult Eq. (2). The correlation energy has been calculat-ed for the case of a homogeneous electron gas and apractical interpolation formula" is available. This can beused for semiconductor and insulator systems described

39 SCREENING OF POSITRONS IN SEMICONDUCTORS AND. . .

by Eqs. (10) and (12) by scaling the electron-gas resultEEG

1/3EEG g

corr corr~gEG

(18)0

~ I&+I

V

where hgEG and hg are the enhancement factors for theelectron gas and for the present models, respectively.

In actual calculations in which the discrete latticestructure with the inhornogeneous electron density hasbeen included the above formulas have to be used in thesense of the local approximation. In the conventionalscheme for calculating positron states the electron struc-ture undisturbed by the presence of the positron is used.The potential affecting the positron at point r in the hostis then obtained as

&+(r)= &c,„i, b(r)+&„„(n(r)), (19)

where Vc,„&, b is the Coulomb potential rising from thehost charge distribution and n (r) is the host electrondensity. Next, the corresponding positron wave function++ is solved. The total positron annihilation rate k andthe positron lifetime ~ are calculated then also in the lo-cal approximation as

A, = 1/r= fdr ~V+~ [A,, (n„(r))+A,,(n, (r))] . (20)

Above, the annihilation rates with valence (n, ) and core(n, ) electrons are taken into account separately. Equa-tion (4), (10), or (12) should be used for A, „whereas theannihilation rate with core electrons can be calculatedstarting from the IPM result and using a constantenhancement factor of y, = 1.5, i.e.

,(n, ) =. 16m y, n, ns (21)

In this work we have performed the calculations usingthe practical method introduced by Puska and Niem-inen. ' The host electron density and the Coulomb po-tential are constructed by superimposing free atoms,solved by a computer program using the local-density ap-proximation ' for the electron exchange and correlation.The fully three-dimensional Schrodinger equation issolved for the positron wave function and eigenenergy bya numerical relaxation method. The mesh points, whichare used to describe the potential, electron density, andthe positron wave function in the crystal form anorthorombic Bravais lattice. Translation and point syrn-metries are used to reduce the integration volume asmuch as possible. The calculation scheme is known to bepowerful for describing positron states in perfect metallattices, in point defects, and on metal surfaces. ' Re-cently, it has been successfully used for semiconductors Siand GaAs.

[112] Direction

FIG. 1. Total electron density in InP. Atomic superpositionis used. The value for the contour nearest to the interstitial re-gion is 0.01ao . The values increase monotonically towards theion cores with the spacing of 0.005ao

electron densities. In this work we even use sphericalatomic densities. The resulting electron density in thezinc-blende lattice structure is shown in Fig. 1 in the caseof InP. The superimposed electron density does not showthe pileup at the bonds between the neighboring atoms.The charge transfer towards ionicity cannot be takenproperly into account either. However, when compazedwith self-consistent electron structure calculations, thesuperpos&tion describes surprisingly well the electrondensity at the interstitial regions and its rise close to theatom chains. In fact, these are the most important as-pects determining the positron state and the annihilationrate because the positron has a strong tendency to seekout into open interstitial reg&ons. This tendency is clearlyseen in Fig. 2, which shows the delocalized positron wavefunction in the ideal InP lattice. The superposition ofatomic densities may be most questionable for the most

0~ W

Q~ W

III. POSITRONS IN PERFECT LATTICES

A. Results

The present calculations rely on the fact that the elec-tron density in a semiconductor can quickly and reason-ably well be obtained by the superposition of free-atogn

(112] Direction

FIG. 2. Delocalized positron wave function in perfect InP.The ion positions are denoted by solid circles. The lower of thenearest neighbors in the (111)direction is In and the upper oneP. The wave function is at maximum in the interstitial regionsbetween the ion chains and vanishes at the nuclei. The contourspacing is —,

' of the maximum value.

7670 PUSKA, MAKINEN, MANNINEN, AND NIEMINEN 39

ionic compounds treated in this work. For this kind ofcases self-consistent electronic structures could give morereliable results, but their calculation is enormously morecomplicated than the atorgic superposition.

The lattice constants and high-frequency dielectricfunctions used in the present calculations for semicon-ductors and insulators are listed in Table I. Moreover,several experimental values for the positron lifetime ipthe perfect lattices are given and we have chosen an"average" value ~,„,for the experiment-theory compar-isons. The positron annihilation characteristics calculat-ed according to the semiconductor model [Eqs. (10) and(18)] are collected in Table II for several perfect semicon-ductor and insulator lattices. The corresponding dataarising from the insulator model [Eqs. (12), (13), and (18)]are shown in Table III. The calculated positron bulk life-times are in good agreement with the experimental valuesw,*„,given in Table I, if C, GaN, MgO, SiC, and GaP aretreated as insulators and the rest of the hosts as semicon-ductors. The good agreement between calculated andmeasured results is demonstrated also in Figs. 3(a) and

3(b), which show the bulk lifetimes as a function of theunit-cell volume of the lattice. This agreement givescredence to the simple models for the positron screeningused and to the interpretat;ion of experimental lifetimespectra.

In the insulator model the dependence on the dielectricconstant e is much stronger than in the case of thesemiconductor model. This leads to a larger scatter inthe bulk lifetimes when they are plotted as a function ofthe unit-cell volume in Fig. 3(a). For the typical insula-tors the agreement between the theoretical and experi-mental lifetimes is not as good as for semiconductors.This may be partly due to the fact that the experimentaldata in Fig. 3 are taken from several sources. Systematicexperimental work is called for to further clarify positronannihilation in insulators.

The core annihilation rates for the hosts studied aregenerally rather small, typically -0.3 ns '; for C, BP,and Si as small as -0. 1 ns '. Only in the case of insula-tors GaN and MgO do they make a rather large contribu-tion to the total annihilation rates. The core annihilation

TABLE I. Data for semiconductors and insulators. In the calculations the zinc-blende or diamondstructures are used, except in the case of MgO, which has the sodium chloride structure. The latticeconstants a (Ref. 24), high-frequency dielectric constants e„(Ref. 25), and experimental positron bulklifetimes 1 pt are given. The lattice parameter for GaN in the zinc-blende structure has been calculatedby assuming the density of the actual wurtzite lattice structure. ~,*„pt is the "average" experimental pos-itron bulk lifetime used in experiment-theory comparisons in this work.

Host

SiGe

a(a.u. )

10.2610.69

12.016.0

+expt

(ps)

218,' 219 222'228 c 230d

+expt

(ps)

220229

A1PA1AsA1SbGaPGaAsGaSbInPInAsInSb

10.3010.6111.6110.3010.6911.5711.0911.4312.26

7.68.2

10.29.1

10.914.49.6

12.315.7

223,' 225220,g 230" 231,' 232, ' 235'

247 f 260 k 2601

235, 242, ' 244, 247"247, 257'

258,f 280" 282"

224232260244257280

CdTeHg Te

12.2612.22

7.214.0

289,' 2912740

BeoBPCGawMgOSiC

'Reference 26.Reference 27.

'Reference 7.Reference 28.

'Reference 29.Reference 30.gReference 5."Reference 8.

7.208.606.728.477.968.21

3.08.25.665.43.06.6

'Reference 6."Reference 31."Reference 32.'Reference 33.

Reference 34."Reference 35.'Reference 36.~Reference 37.

180"166P157'

115180166157

39 SCREENING OF POSITRONS IN SEMICONDUCTORS AND. . . 7671

Host

*Si*Ge

(ns '}

4.4714.129

(ns ')

0.1030.301

T

(ps)

219226

effS

(a.u. )

2.2982.421

3.5203.884

AlpA1As

*A1SbGaP*GaAs*GaSb*InP*InAs*InSb

4.2744.0723.5454.2714.0213.6003.7333.5923.291

0.1200.1830.1700.3150.3400.2870.3390.3460.283

228235269218229257246254280

2.3282.4002.6392.3422.4392.6402.5452.6322.825

3.4923.6884.4293.5723.8644.5214.1274.4585.186

BeOBPCGaNMgOSiC

8.8346.497

11.886.3267.5867.283

0.2080.1230.1450.7670.7900.198

11115183

141119134

1.6221.8901.4621.8961.7231.789

2.1412.6542.0962.5922.2352.477

TABLE II. Positron annihilation characteristics in perfectlattices calculated using the semiconductor model [Eq. (7)] forscreening. A,, and A., are the annihilation rates due to thevalence and core electrons, respectively. v is the positron bulklifetime, r,' corresponds to the average electron densityaffecting the positron, and Ag' is the enhancement factordefined in Eq. (19). The hosts which are well described by thesemiconductor model are indicated by an asterisk.

250

QJ

.E 200—

~ 150—

4/lc)

CL

1000

300

l/lCL

275ECP~ 250

~ 225Vlc)

CL

Insulator model

MQO ~ Ga~

0 SiC~ BP

I

Semiconductor model

AISb

gGaSbInAs

InPAlAsAIP OGaAs~ Gesi

C~ BeO

I I

50 100Unit-cell volume {a.u.}

GaP

InSb

150

TABLE III. Positron annihilation characteristics in perfectlattices calculated using the insulator model [Eq. (9)] for screen-ing. A, , and A,, are the annihilation rates due to the valence andcore electrons, respectively. ~ is the positron bulk lifetime, r,'corresponds to the average electron density affecting the posi-tron, and hg is the enhancement factor defined in Eq. (19). Thehosts which are well described by the insulator model are indi-cated by an asterisk.

200100

I I

150 200Unit-cell volume {a.u}

(b,'i

250

FIG. 3. The positron lifetimes in perfect insulator and semi-conductor lattices as a function of the unit-cell volume. Thetheoretical values (solid circles) are obtained in (a) by using theinsulator model [Eq. (12)] and in (b) by using the semiconductormodel [Eq. (10)]. The experimental values (open circles) are the7 p$ values given in Table I.

Host

SiGe

AlpA1AsAlSb

GaPGaAsGaSbInPInAsInSb

*BeO*BPtie C*GaN*MgO*SiC

(ns ')

4.4054.328

3.9593.8983.7404.0694.0343.9773.7723.8483,846

8.8346.5838.6174.9.475.1905.907

(ns ')

0.1280.395

0.1470.2320.2400.3880.4410.4130.4430.4810.443

0.2090.1340.1430.8210.7910.211

(ps)

221212

244242251224223228237231233

111149114173167163

effS

(a.u. )

2.2592.361

2.2862.3502.5602.2922.3752.5482.4672.5372.697

1.4261.7681.4611.8871.7431.778

3.2313.745

2.9413.2164.2303.0803.5034.4823.7194.2345.287

1.1332.0321.2391.7691.2901.765

rate depends on the core enhancement factor. For exam-ple, in the case of semiconductors, if the IPM [with thevalue of 1.0 for the enhancement factor instead of 1.5(Ref. 15)] is used the positron lifetimes increase typicallyby 5 ps. Thus the choice of the core enhancement factormay cause small systematic errors in the calculated posi-tron lifetimes.

B. Discussion

In order to achieve criteria for which of the two mod-els should be chosen in a specific case we analyze the pos-itron annihilation in bulk semiconductors and insulatorsin more detail. We define a density parameter r,' whichcorresponds to the e6'ective valence electron density withrespect to the positron annihilation. This means that r,'inserted in Eq. (10) or (12) gives the valence annihilationrates in Tables II and III, i.e., rates which are obtained byintegrating over the lattice volume using the three-dimensional positron and electron densities. Another im-

7672 PUSKA, MAKINEN, MANNINEN, AND NIEMINEN

I I

Semiconductor model

10

00 50 100 150

Q (a.u.)8~+2

200

FIG. 4. The enhancement factors Ag, (solid circles) repro-ducing the experimental positron lifetimes 7

p&in the insulator

model. For the definition, see Eq. (23). The abscissa is propor-tional to the atomic polarizability in Eq. (11). The line shown isthe linear least-squares fit to the data for C, GaN, MgO, SiC,and GaP. The slope and the vertical axis intercept give the pa-rameters A and B of the insulator model Eq. (12). For compar-ison, the enhancement factors Ag' (solid squares) calculated byEq. (22) in the semiconductor model are also shown.

00 5 10 15

Dielectric function E

FIG. 6. The dielectric constants e, calculated from the posi-tron bulk lifetimes ~,*„p,. The values correspond to the semicon-ductor model in Eq. (25). The abscissa is the measured high-frequency dielectric constant. The line shown corresponds toe,=e„. The error bars correspond to an uncertainty of +2 ps inthe positron lifetime.

lAg = d +BQ, IN .6~+2

(22)

portant parameter is the enhancement Ag which for semi-conductor (SC) and insulator (IN) models is, according toEqs. (1), (10), and (12),

(1—1/e„)(r,' ) +10ea, SC,

When the semiconductor model is applied in the realthree-dimensional calculations the enhancement at agiven point depends on the local electron density. In theupper part of Eq. (22) the efFective electron density r,' isused and therefore the enhancement factor obtained isalso an effective one, bg

' . On the other hand, in the in-sulator model the enhancement is a density-independentconstant as indicated also in the notation.

Figure 4 illustrates the determination of the parame-ters A and B appearing in the formulas for the annihila-

20— I I

1nsulator model

15

'0 12060 180Unit-cell volume (a.u. )

240

FIG. 5. The cube of the effective electron density parameterseen by the positron as a function of the unit-cell volume. Forthe definition, see text. The line shown is the linear least-squares fit [Eq. (24}]to the data.

5 10 15Dielectric function 6

FIG. 7. Same as Fig. 6 but for the insulator model in Eq. (25).

39 SCREENING OF POSITRONS IN SEMICONDUCTORS AND. . . 7673

tion rate [Eq. (12)] and enhancement bg [Eq. (22)] in theinsulator model. The enhancement factors hg„whichreproduce the experimental positron lifetimes 7 pt givenin Table I, are plotted as a function of the atomic polari-zability Eq. (11}. As a matter of fact, this determinationshould be done in a self-consistent way, because Ag, 's arecalculated as

eff)3

hg, =expt

—1C (23)

(r,' ) -0.104Q —1.58 . (24)

This means that the "open volume" for the positron in

where r,' (given in a.u. ) and A,, (in ns ') are the theoreti-cal results depending on the values of A and B (r,"„,isgiven in ns). However, this dependence comes throughthe correlation potential [Eq. (18)] and is very weak.Therefore only one improvement iteration is needed. Thepoints for C, GaN, MgO, SiC, GaP, and Si fall nicely onthe same straight line determining the parameters 3 andB as in Eq. (13). The semiconductors to the right of Sifall below this line, indicating that the linear dependenceassumed in the insulator model is no longer adequate. Inorder to fully assess the model for insulators more data isclearly needed, especially to fill the gap between SiC andGaP.

From Tables II and III we can see that Si can be de-scribed equally well in both models, i.e., the ranges of va-

lidity of the models overlap. The fact that the models forinsulators and semiconductors seem to join rathersmoothly means that the borderline between these modelsis not sharp. In practice, one may choose the model to beused by calculating the value of Q[(e —1)/(e +2)] forthe system in question and comparing it to the borderlineof the models where it is around 100—110a.u.

The values of r,' and hg for the "borderline" case Siclarify the differences between the two models. hg' 'scalculated in the semmiconductor model for several hostsare added as squares in Fig. 4, in which the straight linegives the insulator model result, and the circles denotethe constant enhancements Ag reproducing the experi-mental positron lifetimes. The high electron density re-gions near the ion cores contribute to the positron annihi-lation more in the insulator model than in the semicon-ductor model, because the enhancement is constant in theformer whereas in the latter model the enhancement de-creases when the electron density increases. Thereforethe r,' 's are smaller in the insulator model. In the an-nihilation rate this difference is compensated by a smallerenhancement in the insulator model. Therefore bothmodels give similar rates for Si. However, to the right ofSi, e.g. , in the cases of GaAs and Ge, the compensation inenhancement should be larger than given by the insulatormodel (the bg, 's fall below the straight line). As a conse-quence, the insulator model gives, e.g., for GaAs and Ge,a shorter positron lifetime than the semiconductor model.

The cubes (r,' ) are plotted in Fig. 5 as a function ofthe unit-cell volume. They fall in a more or less straightline

BR+2+expt

expt

( &eff )3gtheor s, theor

C 12

( &eff )3gtheor s, theor

C 12

SC,(25)

, IN .

Thereafter, as indicated already above, we use the calcu-lated core electron annihilation rates A, ',

""' (given inns '), the calculated etfective electron densities n,'h„„(ina.u.} or r,'th„, 's (in a.u.), and the experimental positron

CdT

m -10C

-15

CcI 4cI

Te Ss

FIG. 8. Self-consistent electron energy-band structure forCdTe. The calculation is performed by the LMTO-ASAmethod. Relativistic effects, except the spin-orbit coupling, areincluded (scalar relativistic bands). The zero of the energy scaleis the top of the valence band.

the interstitial regions of the host is directly proportionalto the unit-cell volume. The r,' values are remarkablylarger than the values resulting if the valence electronswere uniformly distributed over the whole unit-cellvolume. For example, r,' =2.30 for Si whereas the con-ventionally calculated r, =2.01. The difference rejectsthe fact that in semiconductors positrons reside in the in-terstitial regions whereas the valence electron density re-sides in bonds between the neighboring atoms. This ten-dency would be even clearer if the electron density wereself-consistent instead of the superposition of free atoms.For comparison, for the fcc metal Al the atomic superpo-sition calculation' gives r,' =2. 10, which is very close tothe conventional r, =2.07.

The dependence of the annihilation rate on the dielec-tric constant e becomes weaker when e increases inthe insulator model [Eq. (12)] and this tendency continueseven more clearly in the semiconductor model [Eq. (10)].In order to demonstrate the importance of the e depen-dence in the annihilation rate we have solved Eqs. (10)and (12) for the dielectric constant:

PUSKA, MAKINEN, MANNINEN, AND NIEMINEN 39

lifetimes r,„~„(in ns) and obtain a dielectric constant e,corresponding to the measured positron lifetime. All thevalues needed are given in Tables I, II, and III. In fact,this is a very good test for the models proposed, becausethe theoretical results for k',"' ' and r,',z„, in the givenmodel depend only weakly on the dielectric constant e„through the correlation potential [Eq. (18); note thecube-root dependence]. Moreover, a rigorous test wouldbe to use the e, value obtained from Eq. (25) in a new pos-itron state calculation and thereby to make an iterationtowards "self-consistency. " The results calculated usingthe data in Tables I, II, and III are shown in Figs. 6 and 7for the semiconductor and insulator models, respectively.The error bars correspond to an uncertainty of +2 ps inthe positron lifetime. The semiconductor model in Fig. 6gives reasonable results when e„ is greater than -9, butfails drastically for the typical insulators with sma11 evalues. The error bars are rather large especially for thehigh-e„semiconductors. The insulator model worksvery well when e„ is less than -9, but it is interesting tonote that it can predict dielectric constants also for InP,GaAs, and Si. For higher e the e values seem to de-cline. Thus the regions of validity of the two modelsoverlap also in this sense. The strong correlations seen inFigs. 6 and 7 imply that the positron lifetime measure-ments could be used in a sense to determine the high-frequency dielectric constants for semiconductors and in-sulators.

In the case of GaAs, there is a debate concerning thebulk lifetime value. Dannefaer et al. have strongly sug-gested that it should be -220 ps, which is shorter thanthe value of -230 ps found by other experimentalists (seeTable I). The value of 230 ps for GaAs as well as the ex-perimental bulk lifetimes shown in Fig. 3 are obtainedfrom measurements for p-type and semi-insulating semi-conductors for which one-component lifetime spectra areobtained after subtraction of small source terms. Thissingle component is then interpreted as the positron bulklifetime. On the other hand, the result of Dannefaeret a/. is obtained by 6tting the trapping model to thedecompositions of experimental spectra, i.e., not directlyas an observed lifetime component. Dannefaer has been

able to decompose the measured spectra by gathering arelatively large number of counts in the lifetime spectra.Theoretically, the lifetime of 220 ps can be reproduced byusing the metallic limit (e = ao ) in Eq. (10). However, inorder to be consistent one should then observe, e.g., forSi, a bulk lifetime substantially lower than 220 ps, whichis the value accepted also by Dannefaer. On the otherhand, if the insulator model described above were validfor GaAs, the predicted positron bulk lifetime would be,according to Table III, 223 ps. But, because GaAs is onthe right-hand side of Si on the polarizability scale (Fig.4), we prefer to use the semiconductor model, whichworks already for Si.

Previously, ' positron bulk lifetimes in semiconductorshave been calculated with the hnear muftin-tin orbitals(LMTO) method within the atomic-spheres approxima-tion (LMTO-ASA). The electron structures in thesecalculations are self-consistent although the actual three-dimensional geometry is not exactly taken into accountdue to the ASA. In these calculations the form (11) byBrandt and Reinheimer' is used for the annihilation rate,and the value of e =0.2 is chosen in order to reproducethe experimental bulk lifetimes in average. The lifetimeresults agree well with the present ones showing that theelectronic structure can well be approximated for thispurpose by the superposition of free atoms.

C. II-VI compound semiconductors

A prominent feature in the electron band structure ofII-VI compound semiconductors is the narrow d bands atrather high energies. For example, Fig. 8 shows the bandstructure for CdTe, which has been calculated using theLMTO-ASA method. The Cd 4d bands are in the hetero-polar gap, slightly separated from the valence band. TheTe 5s bands lie below the Cd 4d bands, but their width isstill considerable. The problem in calculating positronannihilation rates in these materials is the description ofthe enhancement for the d bands. It should be largerthan the enhancement for the tightly bound core but lessthan that for the more delocalized valence electrons. Wehave used the same scheme as in the context' of the tran-sition metals, i.e., the enhancement factor of the d elec-

TABLE IV. Positron annihilation characteristics in perfect CdTe and in HgTe lattices. The calcula-tions are performed using the semiconductor model [Eq. (7)] for positron annihilation with valence elec-trons, which include Cd 5s, Hg 6s, and Te 5s and 5p electrons in the atomic superposition. Cd 4d and

Hg 5d electrons are treated with a constant enhancement factor yz. For yz either the value of 1.5 cor-responding to the tightly bound core electrons or the values appropriate for d electrons in Ag (1.95) andAu (2.35) metals are used. A,„k„and A,z are the annihilation rates due to the valence, core, and d elec-trons, respectively. r is the positron bulk lifetime, r,' corresponds to the average electron densityaft'ecting the positron, and hg' is the enhancement factor defined in Eq. (19).

Host

CdTe 1.51.95

(ns ')

3.054

(ns ')

0.141

(ns ')

0.2340.304

7

(ps)

292286

jefS

(a.u. )

2.885

gg eft'

5.114

Hg Te 1.52.35

3.206 0.150 0.2810.440

275263

2.873 5.338

39 SCREENING OF POSITRONS IN SEMICONDUCTORS AND. . . 7675

trons is a density-independent constant (this should belarger than 1.5 used for the core). For the valence elec-trons the above semiconductor model has been used. Wehave included the outermost s electrons of the column-VIcomponent to the valence electrons. This is justi6ed, asin the solid they form a band of more delocal&zed elec-trons than the electrons in the highest d band of thecolumn-II component. For example, according to ourI.MTO-ASA calculations about 96% of the Cd 4d elec-trons in CdTe are localized in the atomic spheres (radius3.0156 a.u. ) surrounding the Cd nuclei, whereas the local-ization of the Te Ss electrons in the similar spheresaround the Te nuclei is about 75%.

The positron annihilation characteristics calculated forperfect CdTe and HgTe lattices are given in Table IV.Two limits are considered. The use of the core enhance-ment factor of 1.5 also for uppermost d electrons givesthe upper limit for the positron lifetime. The lower limitis obtained by using the d-enhancement factors deter-mined in the case of nearby Ag and Au metals for Cd andHg, respectively. These enhancement factors are certain-ly too high, because in the present cases the localizationof the d electrons is stronger due to the larger atomicnumbers and the larger distances between the metalatoms. For CdTe the two lifetime estimates are ratherclose to each other and to the values obtained experimen-tally (Table I). The scatter is much larger for HgTe,where the higher limit is nearer the experimental estimate(Table I). As a matter of fact, the experimental values aresomewhat uncertain because the extraction of positronbulk lifetimes from the many-component spectra hasbeen difticult. However, the present calculationsconfirm the simple picture that from these two semicon-ductors, which have nearly the same lattice constants,HgTe should have the smaller positron lifetime, becausethe Hg atom is larger than the Cd atom. The calculatedr,' and Ag' values for CdTe and HgTe are also shown inTable IV. They lie within the values typical for semicon-ductors and would be situated near InSb ip Figs. 4 and 5.To conclude, the semiconductor model presented above

Vp

[112] Direction

FIG. 10. Localized positron wave function in the P vacancyin InP. The ion positions are denoted by solid circles. The con-tour spacing is

8of the maximum value.

can be used with minor modifications to include d-electron enhancement descriptions of positron states andannihilation rates in II-VI compound semiconductors aswell.

IV. POSITRONS TRAPPEDBY VACANCIES AND DIVACANCIES

A. Results

The insulator and semiconductor models presentedform a basis for calculations of positron states and an-nihilation rates also in defected lattices. Here we showresults for idea/ vacancies and nearest-neighbor divacan-cies in insulators and semj. conductors, i.e., the lattice re-laxation around the defect is not taken into account.Moreover, the atomic superposition used does not allowdescription of differences in the electron structures be-tween the diferent charge states of the defect.

The wave function of a positron trapped by the two

VI„Vg„Vp

Q4 libel

VQ

[yg2] Direction

FIG. 9. Localized positron wave function in the In vacancyin InP. The ion positions are denoted by solid circles. The con-tour spacing is —' of the maximum value.

[].12] Direction

FIG. 11. Localized positron wave function in the nearest-neighbor divacancy in InP. The ion positions are denoted bysolid circles. The contour spacing is 8 of the maximum value.

7676 PUSKA, MAKINEN, MANNINEN, AND NIEMINEN 39

kinds of vacancies and the nearest-neighbor divacancy inInp are shown in Figs. 9—11. The localization in the va-cancies is weaker than in typical metal vacancies. Espe-cially, in the smaller P vacancy (Fig. 10) the wave func-tion has a tendency to Bow over into the open interstitialregions. It is seen that the open volume corresponding tothe divacancy is needed to maintain localization similar

to that at metal vacancies. The difFerence between thetwo kinds of vacancies with respect to the positronresponse is much clearer in the insulators studied than inthe III-V semiconductors. In GaN, SiC, and MgG, wefind that N, C, and 0 vacancies cannot bind positrons atall. This is interesting, because, for example in the caseof MgO, one expects that the 0 vacancies are positively

TABLE V. Properties of positrons trapped at vacancies and nearest-neighbor divacancies in insula-tors and semiconductors. For C, SiC, GaN, MgO, BeO, BP, and GaP the calculations are performedusing the insulator model [Eq. (9)] whereas in tlM other cases the semiconductor model [Eq. (7)] is used.Cd 4d and Hg Sd electrons are treated as tightly bound core electrons with the enhancement factoryz =1.5. A,„and A,, are the annihilation rates due to the valence and core electrons, respectively. ~ isthe positron lifetime, r,' corresponds to the average electron density affecting the positron, and hg isthe enhancement factor defined in Eq. (19). E~ is the positron binding energy in the defect.

Host

Si

Defect

SiDivac.

(ns ')

3.8593.204

{ns ')

O.OS30.029

7

(ps)

256309

effS

(a.u. )

2.5092.858

4.0805.233

(eV)

0.421.29

GeDivac.

3.6243.059

0.1730.102

263316

2.6353.001

4.5265.891

0.361.10

A1P AlPDivac.

3.7273.7493.101

0.0500.0770.034

265261319

2.5242.5152.862

3.9933.9685.0&1

0.590.161.19

AlAs AlAsDivac.

3.5933 ~ 5422.257

0.1020.1030.023

271274439

2.5922.6173.997

4.2154.288

11.01

0.500.221.57

Alsb AlSbDivac.

3.2403.0502.170

0.1140.0840.026

298319455

2.8142.9504.$38

5.0155.526

14.80

0.260,351.26

GaP GaPDivac. 2.949

0.233

0.212 316

2.396

2.S51 3.080

0.08&0

0.27

GaAs GaAsDivac.

3.6003.4962.994

0.1730.2290.118

265268321

2.6182.6713.006

4.3824.5515.779

0.440.241.03

GaSb GaSbDivac.

3.3183.0872.760

0.1710.1690.100

287307350

2.8002.9673.290

5.0725.7167.188

0.210.380.93

InPDivac.

3.2633.3802.829

0.1290.2830.110

295273340

2.7912.7213.136

4.9144.6756.273

0.600.061.05

InAs InAsDivac.

3.1943.2412.769

0.1550.2650.117

299285347

2.8682.8353.251

5.2795.1576.928

0.490.110.98

InSb

CdTe

InSbDivac.

CdTeDivac.

3.0192.9152.608

2.8932.6682.444

0.1520.1940.099

0.2260.2860.157

315322369

321339384

3.0343.1313.517

3.0123.2343.540

6.0246.4558.456

5.5886.5218.033

0.300.26

. 0.88

0.110.210.60

39 SCREENING OF POSITRONS IN SEMICONDUCTORS AND. . . 7677

TABLE V. (Continued}.

Host Defect (ns ') (ns ')7

(ps)

jefS

(a.u. ) (eV)

HgTe HgTeDivac.

3.0332.8202.578

0.2520.3550.184

304315362

3.0083.2153.542

5.8806.8098.542

0.160.220.66

BeO Be0Divac.

7.719

6.156

0.067

0.075 160

1.491

- 1.608

1.133

1.133

0.73&0

1.46

BP BpDivac.

5.3674.712

0.0690.055

184210

1.8931.977

2.0322.032

&00.390.89

CDivac.

6.7964.799

0.0740.048

1462Q6

1.5811.776

1.2391.239

0.602.17

GaN GaNDivac.

3.457

2.666

0.211

0.205

273

348

2.126

2.319

1.769

1.769

1.51&0

2.00

MgO Mg0Divac.

3.571

3.622

0.182

0.136

266 1.974

1.965

1.290

1.290

1.82&0

2.46

SiC SiCDivac.

3.961

3.026

0.056

0.046

249

326

2.031

2.222

1.765

1.765

1.44(0

2.50

charged and therefore likely to repel positrons. Thepresent calculations show that already the atomic sizeeffect strongly opposes localization.

The numerical results for posj.tron states in vacanciesand divacancies are collected in Table V and the positronlifetimes in semiconductors are shown also in Fig. 12.The increase of the positron lifetime between the bulkstate and the vacancy state in semiconductors is rathersmall. The ratio of the lifetimes is only about 1.12-1.16.For metals the ratio is typically around 1.5 —1.6,rejecting a more localized positron wave function in aclose-packed lattice. In the insulators studied the valuesare near those for metals when a bound state is found.For divacancies the relative increase of the positron life-time is of the same order as for vacancies in metals. Fig-ure 12 demonstrates that the positron lifetimes corre-sponding to bulk, single vacancies, and divacancies in-crease linearly as a function of the unit-cell volume. Inthe case of vacancies several groups have to be separatedaccording to the size (or the row in the Periodic Table ofthe elements) of the atom missing. The lifetime simplymeasures the amount of "open volume" available for thepositron.

According to Table V positron binding energies in va-cancies are small, only a few tenths of an eV. Thisrejects the rather delocalized character of the bound pos-itron state. In a compound system the binding energy ishigher in the larger vacancy, i.e., corresponding to the

340—InP

~r

Ga5b .rInAs

jr'r Insb—

GaAs rGaPrlA Ge

r Si+ 300—E

CV rrAs rr +V

V ~VGe& 260 —stO VGa

CL r Vp

Vsb

VEn IrVIn r VGa r

VAS~ ir

GaSbr

Inp InAs

VSbrr

Insbr

GaAs r

220 —e~ GaPrsiI I I

150 175 200Unit-ceil volume (a.u. l

125 225

FIG. 12. The calculated lifetimes for positrons delocalized inperfect semiconductor lattices (circles), trapped at vacancies(triangles), and trapped at nearest-neighbor divacancies(squares), as a function of the un~t-cell volume. The dashedlines are drawn in order to guide the eye.

PUSKA, MAKINEN, MANNINEN, AND NIEMINEN

missing atom on the lower row in the Periodic Table, orif the elements are on the same row, the element on theright-hand side.

B. Discussion

Experimental data for positron lifetimes in vacanciesand divacancies exist for Si and GaAs. For the Si vacan-cy (Vs;) the experimental values are 266 —270 ps (Refs. 26and 27) and for the Si divacancy the values 320—325 ps(Refs. 26 and 39) have been suggested. The presenttheoretical predictions are smaller, 256 and 309 ps. Thismay be due to the omission of the outward lattice relaxa-tion. In the case of GaAs the experimental findings aremore complex. The positron lifetime of 260 ps is con-nected with Vz, . For VA, two positron lifetime com-ponents of 260 and 295 ps depending on sample condi-tions have been suggested. Our theoretical result forV&, is 265 ps, i.e., slightly above the experimental value.In the case of VA, the experimental results are on bothsides of the calculated value of 268 ps. According to themodel presented by Corbel et al. the two experimentalpositron lifetimes correspond to different charge states ofthe VA, . The change in the positron lifetime can be, atleast partly, explained by the change of lattice relaxation.The present model for the screening of positrons in semi-conductors opens an additiopa1 possibility. Namely, thescreening may depend on the charge state of the defect.One obtains an order of magnitude estimate for this effectby using the metallic limit for the screening. In the caseof VA, the metallic screening gives a lifetime 12 ps short-er than in the semiconductor model. Dannefaer andKerr ' have proposed that the lifetime component of 295ps should be connected with divacancies in GaAs. How-ever, this assignment seems improbable according topresent theory, which predicts for the defect a remark-ab1y longer lifetime of 317 ps.

Previously the positron states in vacancies and smallvacancy clusters have been calculated by using theBrandt-Reinheimer enhancement model Eq. (14) withcg =0.2. ' The previous atomic superposition calcula-tions give essentially the same results as the present cal-culations. The LMTO-ASA Green's-function calcula-tions, ' which use self-consistent electron densities, differfrom the present treatment because they can deal withthe different charge states. The LMTO-ASA Green's-function calculations showed that the positron lifetime ina vacancy does not depend strongly on the charge state ifthe electron-positron correlation is treated similarly ineach case, and no relaxation occurs. The positron life-times in vacancies calculated by the LMTO-ASAGreen's-function method are similar to the present re-sults. According to the earlier calculations the positronlifetime in VA, is —10 ps longer than in the V~, whereasthe present calculations give only a difference of 3 ps.

This small disagreement rejects the effects of the self-consistency of the electron density. The positron bindingenergy in a defect is more sensitive to the details of theelectron structure than the lifetime. For example, theLMTO-ASA Green's-function calculations predict forvacancies binding energies which are larger than thepresent values approximately by a factor of 2. Moreover,in that scheme the binding energy depends on the chargestate of the defect. The actual magnitude of the bindingenergy has an important effect on the positron trappingrate into the defect.

V. CONCLUSIONS

We have developed two models which describe theelectron-positron correlation in insulators and semicon-ductors, in particular the enhancement of the electrondensity at the positron. The high-frequency dielectricconstant has been found to be the relevant quantity fordetermining the effect of the band gap on the screeningresponse of the valence electrons. The first model, whichstarts from the properties of electron gas, is better suitedto typical semiconductors. This model is free from ad-justable parameters and is therefore of great value in sup-porting the interpretation of measurements, first of all thedetermination of positron bulk lifetimes. The basis of thesecond remodel is to treat the polarizability of the valenceelectrons as that of more tightly bound core electrons,and the C1ausius-Mossotti relation is used. Therefore themodel describes better insulators. The model containstwo adjustable parameters common for all materials, andit can be considered as an interpolation scheme betweendifferent insulator hosts.

We have used the models to predict positron lifetimesat vacancy-type defects in semiconductors and insulators.The main conclusion from the defect calculations is thatthe lifetime measures faithfully the amount of the openvolume available for the positron. This is seen already inthe results for the perfect lattices. However, some of theexperimental findings are difFicult to understand using thepresent models. Firstly, one should take the 1attice relax-ation into account at least in the calculation of the elec-tron density for the defect, but the relaxed positions ofthe atoms are not accurately known. Secondly, thechange of the charge state of the defect affects the relaxa-tion, and it might also affect elect;ron-positron correla-tion. Thirdly, ip the present calculations the averageelectron density at the defect is not affected by the local-ized positron as it in principle should be. Calculationsperformed with the two-component density-functionaltheory cou1d answer this question.

ACKNOWLEPGMENT

We are grateful to P. Hautojarvi for many helpful dis-cussions.

*Permanent address: Laboratory of Physics, Helsinki Universi-ty of Technology, SF-02150 Espoo, Finland.

~Permanent address: Department of Physics, University ofJyvaskyli, SF-40100 Jyvaskyla, Finland.

For reviews, see, e.g., R. M. Nieminen and M. Manninen, inPositrons in Solids, edited by P. Hautojarvi (Springer-Verlag,Heidelberg, 1979); R. M. Nieminen, in Positron SolEd StatePhysics, edited by W. Brandt and A. Dupasquier (North-

39 SCREENING OF POSIT+.ONS IN SEMICONDUCTORS AND . -- 7679

Holland, Amsterdam, 1983).2J. Arponen and E. Pajanne, Ann. Phys. {N.Y.) 121, 343 (1979};

J. Phys. F 9, 2359 (1979).For a review, see M. J. Puska, Phys. Status Solidi A 102, 11

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C. Corbel, M. Stucky, P. Hautojirvi, K. Saarinen, and P.Moser, Phys. Rev. B 38, 8192 (1988); see also C. Corbel, M.Stucky, P. Hautojarvi, K. Saarinen, and P. Moser, Defects inElectronic Materials, edited by M. Stavola, S. J. Pearton, andG. Davies, Mater. Res. Soc. Symp. Proc. 104, 475 (1988).

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