Calculation of positron states and annihilation in solids: A density-gradient-correction scheme

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Author(s): Barbiellini, B. & Puska, M. J. & Korhonen, T. & Harju, A. & Torsti, T. &
Nieminen, Risto M.
Title: Calculation of positron states and annihilation in solids: A
density-gradient-correction scheme
Year: 1996
Version: Final published version
Please cite the original version:
Barbiellini, B. & Puska, M. J. & Korhonen, T. & Harju, A. & Torsti, T. & Nieminen, Risto
M. 1996. Calculation of positron states and annihilation in solids: A
density-gradient-correction scheme. Physical Review B. Volume 53, Issue 24.
16201-16213. ISSN 1550-235X (electronic). DOI: 10.1103/physrevb.53.16201.
Rights: © 1996 American Physical Society (APS). This is the accepted version of the following article: Barbiellini, B.
& Puska, M. J. & Korhonen, T. & Harju, A. & Torsti, T. & Nieminen, Risto M. 1996. Calculation of positron
states and annihilation in solids: A density-gradient-correction scheme. Physical Review B. Volume 53, Issue
24. 16201-16213. ISSN 1550-235X (electronic). DOI: 10.1103/physrevb.53.16201, which has been
published in final form at http://journals.aps.org/prb/abstract/10.1103/PhysRevB.53.16201.
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PHYSICALREVIEWB VOLUME53,NUMBER24 15JUNE1996-II
Calculation of positron states and annihilation in solids: A density-gradient-correction scheme
B. Barbiellini, M. J. Puska, T. Korhonen, A. Harju, T. Torsti, and R. M. Nieminen
LaboratoryofPhysics,HelsinkiUniversityofTechnology,02150Espoo,Finland
(cid:126)Received 22June1995;revisedmanuscriptreceived20September1995(cid:33)
Thegeneralizedgradientcorrectionmethodforpositron-electroncorrelationeffectsinsolids(cid:64)B.Barbiellini
etal., Phys. Rev. B 51, 7341 (cid:126)1995(cid:33)(cid:35) is applied in several test cases. The positron lifetime, energetics, and
momentumdistributionoftheannihilatingelectron-positronpairsareconsidered.Thecomparisonwithexperi-
ments shows systematic improvement in the predictive power of the theory compared to the local-density
approximationresultsforpositronstatesandannihilationcharacteristics.(cid:64)S0163-1829(cid:126)96(cid:33)03824-6(cid:35)
I.INTRODUCTION theeffectofthelocalizedpositron,givesresultsverysimilar
to the full two-component calculations. The conventional
Experimentalmethodsbasedonpositronannihilationgive scheme can be justified by stating that the positron with its
valuableinformationontheelectronicandionicstructuresof screening cloud forms a neutral quasiparticle that does not
condensedmedia,especiallydefectsinsolids.1–3Theexperi- affecttheremainingelectronicsystem,i.e.,thetotalelectron
mental output is, however, indirect, e.g., in the form of the density of the system is a superposition of the ‘‘clean’’ sys-
lifetime of the positron or data related to the momentum tem density and the positron-induced screening cloud. We
contentoftheannihilatingelectron-positronpairinaspecific will adopt the conventional scheme in this paper also when
environment. Clearly, the interpretation of these data calls calculating positron states at vacancy defects.
for theoretical methods with quantitative predicting power.4 The shortcomings of the LDA in the electronic-structure
On the other hand, the positron annihilation measurements calculations are well known. They include too diffuse elec-
give unique experimental data to be used in comparing the tron densities of atoms, the overbinding in molecules and in
results of many-body theories for electron-electron and solids, and the too narrow band gaps of semiconductors and
electron-positron interactions. insulators.5 Therefore, it is not surprising that the LDA for
The modern ab initio electronic-structure calculations for positron calculations also has problems.8 First, the LDA
the properties of different types of materials including per- electron-positron correlation potential fails clearly for posi-
fect crystal lattices, defect and surface systems, and finite tronsoutsidesolidsurfacesandininsulatorssuchasrare-gas
clusters of atoms are usually based on the density-functional solids, in which cases the screening electron cloud cannot
theory (cid:126)DFT(cid:33).5 The success of the DFT stems from the fact follow the positron. To remedy this deficiency one has to
thattheelectron-electroninteractionscanbehandledsimply, employanonlocalconstructionsuchastheweighteddensity
butwithoftensufficientaccuracyusingthelocal-densityap- approximation9 or use some ad hoc approach.10 Second,
proximation (cid:126)LDA(cid:33) for the exchange and correlation effects. when calculating the positron annihilation characteristics for
Thecalculationofpositronstatesandannihilationcharac- solids the LDA has shown a clear tendency to overestimate
teristics can also be based on the DFT. For delocalized pos- therateofpositronannihilation.Onehastriedtocorrectthis
itronstatesinperfectlatticestheusualDFTforanelectronic deficiency by omitting the electron-positron correlation (cid:126)en-
system is the sufficient starting point, because the locally hancement(cid:33) when calculating the annihilation with core
vanishing positron density does not affect the electronic electrons.11,12 In the case of semiconductors one can argue
structure. The DFT calculations for positron states usually thatthepositronscreeningbyvalenceelectronsis,duetothe
employ the LDA. This means that for the effective positron bandgap,weakerthaninanelectrongas.Thisideahasbeen
potential the attractive part due to the electron-positron cor- successfully used in a semiempirical approach, which ac-
relation effects is obtained from the local electron density counts for the reduced screening ability by using the (cid:126)finite(cid:33)
and using the results of the many-body calulations for a de- high-frequency dielectric constant as a parameter. The two-
localized positron in a homogeneous electron gas. The LDA dimensional angular correlation of the annihilation radiation
also means that the results of these many-body calculations (cid:126)2D ACAR(cid:33) measurements for several transition metals3,13
are used also for the contact electron density at the positron indicate that the enhancement for d electrons is smaller than
site that determines the positron annihilation rate. predicted by the LDA calculations. Finally, it has been
Inthecaseofapositrontrappedbyadefectthemaximum pointedoutthatthemoreaccuratethemany-bodycalculation
ofthepositrondensitymaybeofthesameorderasthelocal for the homogeneous gas, the higher the annihilation rate
electron density. Then one should base the calculations on increasing the discrepancy between the theoretical and ex-
the two-component density functional theory (cid:126)TCDFT(cid:33),6 perimental positron lifetimes.14
which solves for the mutually self-consistent electron and In the electronic-structure calculations, several attempts
positron densities. However, practical calculations6,7 have andsuggestionstogobeyondtheLDAhavebeenstudied.5,15
shown that due to canceling effects, the ‘‘conventional’’ Over the past years especially the generalized gradient ap-
scheme, in which the electron density is calculated without proximation(cid:126)GGA(cid:33)hasattractedwideinterest.16IntheGGA
0163-1829/96/53(cid:126)24(cid:33)/16201(cid:126)13(cid:33)/$10.00 53 16201 ©1996TheAmericanPhysicalSociety
16202 B.BARBIELLINIetal. 53
thegradientoftheelectrondensityisincluded,inadditionto is obtained from the single-electron wave functions (cid:99)(cid:50)(r)
i
the local density. The GGA is able to improve the descrip-
tion of some of the cohesive properties compared to the (cid:40)
LDA,17 but a general improvement in different kinds of sys- n(cid:50)(cid:126)r(cid:33)(cid:53) (cid:117)(cid:99)i(cid:50)(cid:126)r(cid:33)(cid:117)2, (cid:126)1(cid:33)
temshasnotyetresulted(cid:126)see,forRef.18(cid:33).Anexampleofa i
physical situation in which the use of the GGA for the where the sum is over the occupied states i. Then the effec-
electronic-structure calculation provides a significant im- tive potential V(cid:49)(r) for a positron is constructed as the sum
provement over the LDA is the H dissociation on the of Coulomb V (r) and electron-positron correlation parts
2 Coul
Cu(cid:126)111(cid:33) surface.19 It has also been showed that the GGA V (r)
corr
gives results in better agreement with the experiments than
the LDA for finite systems (cid:126)atoms and molecules(cid:33) and for V(cid:49)(cid:126)r(cid:33)(cid:53)V (cid:126)r(cid:33)(cid:49)V (cid:126)r(cid:33). (cid:126)2(cid:33)
Coul corr
metallic surfaces.20,21 One reason is that the interactions in
The Coulomb part arises from the nuclei and from the Har-
these systems are related to the tails of the electronic wave (cid:50)
tree potential of the electron density n (r). The correlation
functions, for which the GGA should, in principle, give a
part describes the energy lowering due to the pileup of the
better description than the LDA.
screening charge around the positron. In the LDA for the
Recently, we have proposed a GGA method for positron
positron states V (r) for a given point is calculated from
states in solids.22 Clearly, the screening of the positron corr
the electron density at that point. That is,
shoulddependnotonlyonthelocalelectrondensitybutalso
on its gradient as well. In the case of positrons, the GGA VLDA(cid:126)r(cid:33)(cid:53)(cid:101)EG(cid:64)n(cid:50)(cid:126)r(cid:33)(cid:35), (cid:126)3(cid:33)
should be done consistently for both the electron-positron corr corr
correlation potential and the contact electron density at the where (cid:101)EG(n(cid:50)) is the electron-positron correlation energy
corr
positron. The latter is an aspect which is not met in the pure per a delocalized positron in a homogeneous electron gas
electronic-structurecalculations.Becausethecontactdensity with density n(cid:50). After the positron potential is constructed
can be directly monitored by different types of positron an- its wave function (cid:99)(cid:49)(r) and energy eigenvalue are solved
nihilation measurements, the comparison of the theoretical from the corresponding one-particle Schro¨dinger equation.
and experimental positron annihilation parameters is a In the DFT the Kohn-Sham wave functions are actually
unique method for testing many-body theories such as DFT auxiliary functions used to construct the total density, which
and different approximations within them. In order to use is the quantity with a real physical meaning in the theory.
consistent data for the electron-positron correlation energy The wave functions are, however, customarily used as real
and the contact density, we introduce for the latter an inter- single-particlewavefunctionsinmanydifferentcontexts.5In
polation form based on the many-body calculations of Ar- this work we will also start the discussion of the positron
ponen and Pajanne.23 annihilation characteristics by using the single-particle wave
The results obtained for the properties of solids using the functions for electrons and we will actually employ the
GGAfortheelectronicstructureshowsomescatter.24,25This Kohn-Sham wave functions in calculating the momentum
ispresumablyaresultofcomputationalapproximationssuch distribution of the annihilating electron-positron pairs.
as the pseudopotential approximation or the shape approxi- Thepositronannihilationrateasafunctionofthemomen-
mations for the electron density or potential. In the case of tumpoftheannihilatingpositron-electronpairisdetermined
GGAforpositronstateswethereforestudydifferentmethods from the electron and positron wave functions as28,4
for constructing the electron density. The methods include (cid:85)(cid:69) (cid:85)
athtoemliicn-esaprhemreusffianp-ptrinoxiomrbaittiaoln2(cid:126)6LM(cid:126)ATSOA(cid:33)(cid:33) amnedthtohde awtoitmhiinc tshue- (cid:114)(cid:126)p(cid:33)(cid:53)(cid:112)re2c(cid:40) eip(cid:149)r(cid:99)(cid:49)(cid:126)r(cid:33)(cid:99)i(cid:50)(cid:126)r(cid:33)(cid:65)(cid:103)i(cid:126)r(cid:33)dr 2, (cid:126)4(cid:33)
i
perposition method.27 In the LMTO ASA method the elec-
tron density and the potentials are self-consistent, but they wherer istheclassicalelectronradiusandc isthespeedof
e
are assumed to be spherical. In the atomic superposition light. The summation is over all occupied electron states.
methodtherearenoshapeapproximations,buttheelectronic (cid:103)(r) istheenhancementfactorfortheithstate,i.e.,theratio
i
structure is non-self-consistent. Moreover, we compare sev- between the contact electron density at the positron and un-
eralcalculatedannihilationparameters,i.e.,thepositronlife- perturbed (cid:126)host(cid:33) electron density at that point.
time, positron affinity, and the 2D ACAR maps, with their The shape of ACAR spectra of the s and p valence elec-
experimentalcounterparts.Thecomparisonofthetheoretical tronsinsimplemetalsisalreadywelldescribedbyaconstant
andexperimentalresultsiseasiestinthecaseofperfectcrys- state-independent enhancement factor, i.e., within the inde-
tallattices,whichhavedelocalizedpositronstates.However, pendent particle model (cid:126)IPM(cid:33) in which (cid:103)(r)(cid:91)1.3 Kahana29
i
we also consider localized positron states trapped at vacan- has applied the Bethe-Golstone formalism to a positron in
cies of the crystal lattice. the homogeneous electron gas and found that the enhance-
ment factor has a momentum dependence. Kahana proposed
the parametrization29
II.THEORY
(cid:103)(cid:53)(cid:101)(cid:126)p(cid:33)(cid:53)a(cid:126)r (cid:33)(cid:49)b(cid:126)r (cid:33)(cid:126)p/p (cid:33)2(cid:49)c(cid:126)r (cid:33)(cid:126)p/p (cid:33)4, p(cid:60)p ,
Thedeterminationofelectronandpositronstatesinsolids i s s f s f F(cid:126)5(cid:33)
is possible on the basis of the TCDFT.6 In the conventional
scheme, which is exact for delocalized positron states, the where r is the electron density parameter
s
electrondensityn(cid:50)(r) isfirstcalculatedwithouttheeffectof (cid:64)rs(cid:53)(3/4(cid:112)n(cid:50))1/3(cid:35) and pF is the Fermi momentum (cid:126)in
the positron. In the Kohn-Sham method the electron density atomic units p (cid:53)1.92/r ). The 2D ACAR measurements
F s
53 CALCULATIONOFPOSITRONSTATESANDANNIHILATION ... 16203
performed for the alkali metals have yielded experimental
values for the ratios b/a (cid:126)0.0 for Li and Na, 0.2 for K(cid:33) and
c/a (cid:126)0.2 for Li, 0.4 for Na and K(cid:33).31 In inhomogenous sys-
tems the position dependence of (cid:103)(r) is much more impor-
i
tant than the state dependence. In that case the enhancement
factor (cid:103)(r) in the momentum density equation (cid:126)4(cid:33) can be
i
approximated by the average over the states i,
(cid:103)(r)(cid:53)(cid:94)(cid:103)(r)(cid:38).32 In this approximation the effect of the en-
i
hancement factor to the momentum content is similar to that
of including the positron wave function.
The total annihilation rate is obtained from Eq. (cid:126)4(cid:33) by
integrating over the momentum. The result is
(cid:69)
(cid:108)(cid:53)(cid:112)r2c n(cid:49)(cid:126)r(cid:33)n(cid:50)(cid:126)r(cid:33)(cid:103)(cid:126)r(cid:33)dr. (cid:126)6(cid:33)
e
Thepositronlifetime(cid:116)isthentheinverseoftheannihilation
FIG. 1. Positron lifetime in a homogeneous electron gas as a
rate (cid:108). Equation (cid:126)6(cid:33) is in the spirit of the DFT in the sense functionofthedensityparameterr . Theresultsofthemany-body
s
that the total densities, not the wave functions, are involved. calculationsbyArponenandPajanne(cid:126)Ref.23(cid:33)andthosebyLantto
In the LDA for the positron states the enhancement factor (cid:126)Ref. 34(cid:33) are shown as filled and open circles, respectively. The
(cid:103)(r) is calculated as lifetimesobtainedinthescaledprotonapproximation(cid:126)Ref.37(cid:33)are
denoted by a dotted line. The Stachowiak-Lach (cid:126)Ref. 14(cid:33) result is
(cid:103)LDA(cid:126)r(cid:33)(cid:53)(cid:103)EG(cid:64)n(cid:50)(cid:126)r(cid:33)(cid:35), (cid:126)7(cid:33) drawnbyadash-dottedline.TheinterpolationfunctionbyBoron´ski
where (cid:103)EG(n(cid:50)) is the enhancement factor for the homoge- andNieminen(cid:126)Ref.6(cid:33)andthepresentone(cid:64)Eq.(cid:126)10(cid:33)(cid:35)aredrawnby
adashedandbyasolidline,respectively.
neouselectrongas.Ouraiminthisworkistogobeyondthe
LDA by introducing into Eqs. (cid:126)7(cid:33) and (cid:126)3(cid:33) corrections de-
the momentum distribution along a certain direction. This
pending on the gradient of the electron density. However,
fact reduces the detailed information content of the data.
firstweconsiderinSec.IIAtheelectron-positroncorrelation
in the homogeneous electron gas.
The 2D ACAR experiments do not give directly the mo- A.Positroninahomogeneouselectrongas
mentumdistributionofEq.(cid:126)4(cid:33)butitsintegralalongacertain The DFT LDA calculations for positron states in solids
direction3 are based on many-body calculations for a delocalized posi-
(cid:69) troninahomogeneouselectrongas.Severalapproachesexist
N(cid:126)p ,p (cid:33)(cid:53) (cid:114)(cid:126)p(cid:33)dp . (cid:126)8(cid:33) for solving this model system. The scheme of Arponen and
x y z Pajanne23 is based on correcting the results of the random-
phase approximation (cid:126)RPA(cid:33) in a boson formalism. Lantto34
Actually the experimental information is the integral
has used the hypernetted-chain approximation. The calcula-
N(p ,p ) convoluted by the experimental resolution func-
x y
tion. The Lock-Crisp-West (cid:126)LCW(cid:33) folding of the 2D ACAR tions by Arponen and Pajanne are generally considered as
the most accurate ones.34,14 The situation is, however, much
data allows one to put in evidence the Fermi surface breaks
less satisfactory than in the case of the clean homogeneous
by giving the partial positron annihilation rates for occupied
electron gas for which accurate Monte Carlo calculations
electronic states at given k points within the first Brillouin
exists.35 The difficulty in the Monte Carlo calculations for a
zone.TheLCWfoldingofthemomentumdistributionofEq.
(cid:126)4(cid:33) is defined as positron in a homogeneous electron gas arise from the fact
that very high statistics are needed for an accurate descrip-
(cid:69)
tion of the electron cusp at the positron.36
(cid:40)
(cid:108)(cid:126)k(cid:33)(cid:53)const (cid:114)(cid:126)k(cid:49)G(cid:33)(cid:53)(cid:112)r2c n(cid:49)(cid:126)r(cid:33)n(cid:50)(cid:126)r,k(cid:33)(cid:103)(cid:126)r(cid:33)dr, For the correlation energy, Boron´ski and Nieminen6 have
e
G
given an interpolation form, which for the metallic densities
(cid:126)9(cid:33)
followsthedatacalculatedbyArponenandPajanne.Wewill
(cid:50)
where G is a reciprocal lattice vector and n (r,k) is the adopt this interpolation form for the present calculations. In
density of the occupied electronic states at k. If the positron the case of the enhancement factor the situation is not so
density n(cid:49) and the enhancement factor (cid:103)are constants, one straightforward.Figure1showstheensuingpositronlifetime
obtains the electron momentum distribution in the first Bril- in a homogeneous electron gas as a function of the density
louin zone (cid:126)the Lock-Crisp-West theorem33(cid:33). In the case of parameter r . The results of the many-body calculations by
s
metals, this distribution consists of occupied valence-state Arponen and Pajanne23 as well as those by Lantto34 are
regions with a certain constant density separated by the so- shown as discrete points. Figure 1 shows also the result of
called Fermi-surface breaks from the unoccupied valence- the theory by Stachowiak and Lach.14 The lifetimes esti-
stateregionswithalowerdensity.Accordingtothemeasure- mated by scaling37 the contact density for a proton in a ho-
ments, however, (cid:108)(k) varies for different k and thus the mogeneous electron gas are added. They give a rigorous
Fermi-surface breaks are modulated by the positron wave lowerboundforthepositronlifetime.Atlowdensitiesallthe
function and the enhancement effects. As the unfolded 2D results approach the value of 500 ps, which is the average
ACAR data, also its LCW folding is actually an integral of lifetime for a positronium atom. The scatter in the results of
16204 B.BARBIELLINIetal. 53
the many-body calculations is surprisingly large. In the re- latterlimitleadstotheIPMresultwithnoenhancement.We
gion, important for the transition metals, i.e., slightly above interpolate between these limits by using for the induced
100 ps, the scatter is about 15 ps. Around 230 ps, which is contact screening charge (cid:68)n the form
typical for semiconductors, the scatter is close to 30 ps.
The widely used function by Boron´ski and Nieminen6 in- (cid:68)n (cid:53)(cid:68)n exp(cid:126)(cid:50)(cid:97)(cid:101)(cid:33). (cid:126)12(cid:33)
GGA LDA
terpolating Lantto’s results is also shown in Fig. 1. It repro-
The corresponding enhancement factor reads
duces quite faithfully Lantto’s points, but at low electron
densities it gives unphysically low lifetimes. We are not sat-
(cid:103) (cid:53)1(cid:49)(cid:126)(cid:103) (cid:50)1(cid:33)exp(cid:126)(cid:50)(cid:97)(cid:101)(cid:33). (cid:126)13(cid:33)
isfied with this parametrization and for the practical calcula- GGA LDA
tions we have made another interpolation function. We use Above,(cid:97)isanadjustableparameter.Itwillbedeterminedso
the results by Arponen and Pajanne23 because we want the that the calculated and experimental lifetimes agree as well
lifetimecalculationstobeconsistentwiththecorrelationen- as possible for a large number of different types of solids.
ergy used. Our enhancement function reads as Wehavefoundthatthe(cid:97)(cid:53)0.22usedwiththeinterpolation
form of Eq. (cid:126)10(cid:33) gives lifetimes for different types of metals
(cid:103)(cid:53)1(cid:49)1.23rs(cid:50)0.0742rs2(cid:49) 61 rs3. (cid:126)10(cid:33) and semiconductors, in good agreement with experiment.
Inourpreviouswork22weshowedthattheGGAstrongly
This function has the same form as that used by Stachowiak
and Lach.14 As a matter of fact, the only fitting parameter in reduces the enhancement factor in the ion core region. The
resulting enhancement factor reflects sensitively the shell
thisequationisthefactorinthefrontofthesquareterm.The
structureoftheatominquestioninthesensethatitisnearly
first two terms are fixed to reproduce the high-density RPA
constant over the spatial region dominated by a given shell.
limit and the last term the low-density positronium limit. In
This means that the old scheme,40,27 in which the annihila-
thefittingprocedurewehaveusedtheArponen-Pajannedata
points only up to r (cid:53) 5 because the data at lower densities tions with different atomic shells are separated and constant
islessreliable(cid:64)thesFriedelsumruleisnotobeyedforr (cid:53)6 enhancementfactorsareusedforthed andcoreshells,finds
and8(cid:126)Ref.23(cid:33)andther (cid:53)8pointreacheseventhesscaled partial justification within the DFT, i.e., using the total elec-
proton limit in Fig. 1(cid:35). Ass we will see below, the positron tron density as the starting point.
ThepresentGGAmodelforpositronsreducesthecontact
lifetimes calculated in the LDA with this interpolation for-
density at the positron. This means that the screening de-
mula are systematically lower than the experimental ones.
creases, approaching zero at the IPM limit of (cid:101)!(cid:96). In this
sense the GGA model is related to the model by Puska
B.Positroninaninhomogeneouselectrongas etal.41 for positron annihilation in semiconductors. In that
1.Gradientcorrectionforthepositronannihilationrate model the contact density decreases because the normaliza-
tion of the screening cloud is changed to correspond to the
The LDA shows in electronic-structure calculations a
reduced screening in semiconductors relative to the perfect
clear tendency to overestimate the magnitude of the correla-
screeninginafree-electron-gasmodelrelevantformetals.In
tion energy. This is seen, for example, in the case of free
the semiconductor model by Puska etal. the norm of the
atoms, for which comparison with experiments is possible.38
screening cloud depends on the high-frequency dielectric
Theoverestimationofthecorrelationenergyhasbeentraced
constant of the material. Puska etal. showed that the model
backtotheshapeofthecorrelationholeclosetotheelectron.
usedwiththeBoron´ski-Niemineninterpolationschemegives
In the GGA for electrons the correlation energy is improved
positron lifetimes for group-IV and III-V semiconductors in
by reducing the charge redistributed by the correlation hole
good agreement with experiments and that experimental
near the fixed electron (cid:126)for a definition of this redistributed
variationsseeninthepositronlifetimecanbecorrelatedwith
charge, see Ref. 39(cid:33). Similarly, the gradient correction for
the variations in the high-frequency dielectric constant.
the electron-positron correlation should reduce the electron
density near the positron and thereby decrease the enhance-
2.Gradientcorrectionfortheelectron-positroncorrelation
ment factor and increase the positron lifetime. This will also
potential
reducethemagnitudeoftheelectron-positroncorrelationen-
ergy. Let us suppose that the electron-positron correlation for
IntheGGAtheeffectsofthenonuniformelectrondensity an electron gas with a relevant density is mainly character-
are described in terms of the ratio between the local length izedbyonelengtha, asisthecaseofthescaledpositronium
scalen/(cid:117)(cid:185)n(cid:117) ofthedensityvariationsandthelocalThomas- approximation.9 Then for the electron-positron correlation
Fermi screening length (q )(cid:50)1 (cid:64)in atomic units q energydEcorr/d(1/a) isconstantandthenormalizationfactor
(cid:53)(cid:65)(4/(cid:112))p (cid:35). The lowest-orTdFer gradient correction to thTeF ofthescreeningcloudscalesasad withd(cid:53)3 forthedimen-
F
LDA correlation hole density is proportional to the sion of space. Compared to the IPM result, the electron-
parameter38 positron correlation increases the annihilation rate as
(cid:108)(cid:50)(cid:108) (cid:53)((cid:103)(cid:50)1)(cid:108) , which is proportional to the density
IPM IPM
(cid:101)(cid:53)(cid:117)(cid:185)n(cid:117)2/(cid:126)nq (cid:33)2(cid:53)(cid:117)(cid:185)lnn(cid:117)2/q2 . (cid:126)11(cid:33) of the screening cloud at the positron. Consequently, we
TF TF
have the scaling law42
We use this parameter in describing the reduction of the
screening cloud close to the positron. In the case of a uni- Ecorr(cid:53)c (cid:126)(cid:108)(cid:50)(cid:108) (cid:33)1/d(cid:49)c . (cid:126)14(cid:33)
formelectrongas(cid:101)(cid:53)0,whereaswhenthedensityvariations 1 IPM 2
are rapid (cid:101)approaches infinity. At the former limit the LDA In the scaled positronium approximation, the second coeffi-
resultfortheinducedscreeningchargeismaintainedandthe cient c (cid:53)0 and the correlation energy in rydbergs reads as
2
53 CALCULATIONOFPOSITRONSTATESANDANNIHILATION ... 16205
(cid:64)3/4(cid:126)(cid:103)(cid:50)1(cid:33)(cid:35)1/3 theLMTOASAcalculationsweuseabasissetconsistingof
Ecorr(cid:53)(cid:50) . (cid:126)15(cid:33) s partial waves for delocalized positron states, whereas for
r
s states at vacancies we enlarge the basis by including also p
The values of the correlation energy calculated by Arponen and d waves. The lattice constants used in the calculations
and Pajanne23 obey the form of Eq. (cid:126)15(cid:33) quite well and the aretheexperimentalonesfortheverylowtemperatures.The
coefficientc hasarelativelysmallvalueof0.11Ry.There- lattice constant used are quoted in Refs. 43 and 41.
2
fore, one can use in the practical GGA calculations the cor-
relation energy Ecorr , which is obtained from the A.Positronlifetimesandaffinitiesinperfectcrystallattices
GGA
homogeneous-electron-gas result (ELcoDrrA) by the scaling Thepositronlifetimesandaffinitiescalculatedforseveral
(cid:83) (cid:68)
types of perfect solids in the LDA and GGA models are
(cid:108) (cid:50)(cid:108) 1/3
EGcoGrrA(cid:126)r(cid:33)(cid:53)ELcoDrrA(cid:64)n(cid:50)(cid:126)r(cid:33)(cid:35) (cid:108)GGA(cid:50)(cid:108)IPM ctiommepraerleadtivine tToatbhleeLI.DTAheboGthGAaccionrcdreinagsetsotthheepLoMsitTroOnAliSfeA-
LDA IPM
and the atomic superposition method. This is mainly due to
(cid:53)ELcoDrrA(cid:64)n(cid:50)(cid:126)r(cid:33)(cid:35)e(cid:50)(cid:97)(cid:101)/3, (cid:126)16(cid:33) the change in the enhancement factor; the differences in the
correlation potentials influence through the changes in posi-
where (cid:108) and (cid:108) are the annihilation rates in a homo-
LDA GGA tron wave function, but this effect is small. Among the el-
geneous electron gas, i.e., in the LDA model, and in the
ementalmaterialstheincreaseofthepositronlifetimeislarg-
GGA model, respectively. We use for the correlation energy est, about 20%, for the alkali metals (cid:126)except Li(cid:33), late
Ecorr the interpolation form of Ref. 6. Our formula (cid:126)16(cid:33) ne-
LDA transition metals, and noble metals. Similar increases are
glectsthesmalltermc2(1(cid:50)e(cid:50)(cid:97)(cid:101)/3). Moreover,withtheres- seen also for elemental solids (cid:126)Zn and Ge(cid:33) and compounds
caling of the potential we are not able to describe the image (cid:126)III-VandII-IVcompounds(cid:33)withrelativelyhigh-lyingfilled
potential at a metallic surface seen by the positron.9
d bands.Thecommonfeatureforthesematerialsistherela-
The resulting positron potential is more repulsive in the tively high annihilation rate with core electrons (cid:126)including
GGA than in the LDA. The open volume for the positron the uppermost d electrons(cid:33). The GGA suppresses very effi-
state decreases in the GGA, raising the positron zero-point
cientlytheenhancementfactorforthecoreand d-bandelec-
energy. This effect can be seen in the calculated positron
trons. For the earth-alkali and the early transition metals the
affinities A(cid:49). The affinity is defined as43 increaseoflifetimeduetoGGAissmaller,around10%,and
for the Al with a very compact core electron structure the
A(cid:49)(cid:53)(cid:109)(cid:50)(cid:49)(cid:109)(cid:49), (cid:126)17(cid:33) increase is only about 5%.
where (cid:109)(cid:50) and (cid:109)(cid:49) are the electron and positron chemical A set of the calculated LDA and GGA positron lifetimes
potentials measured with respect to a common electrostatic of Table I are compared with experimental values in Fig. 2.
crystal zero, i.e., (cid:109)(cid:50) and (cid:109)(cid:49) are the position of the Fermi We have tried to make a collection that represents many
level and the bottom of the lowest positron band, respec- types of solids, but, on the other hand, we have tried to use
tively. The positron affinity can be measured using the slow as few experimental sources as possible. This is because we
positron beam techniques.44,45 The comparison of the theo- want to get reliable trends between different materials also
retical and experimental positron affinities directly tests the whenthelifetimedifferencesaresmall.TheLDAresultsare
validity of the positron potential construction. consistentlybelowtheexperimentalpositronlifetimes.How-
ever, it is interesting to note that the LDA results fall quite
accurately on the same line and that good agreement with
III.RESULTSANDDISCUSSION
experiment can be achieved simply by multiplying the LDA
In order to test the GGA scheme we perform calculations results by a constant. A least-squares fit with the theoretical
forpositronstatesandannihilationcharacteristicsforperfect values corresponding to the enhancement factor of Eq. (cid:126)10(cid:33)
crystal lattices as well as for lattices containing vacancies. gives the value of 1.21 for this constant. A similar behavior
WeusetheLMTOmethodwithintheASA(cid:126)Ref.26(cid:33)andthe is valid also for the LDA results obtained using the
atomic superposition method.27 In the LMTO ASA the self- Boronski-Nieminen enhancement form,6 but the constant
consistent potentials and charge densities are spherical factorissomewhatsmaller,about1.1.Withtheexceptionof
around the nuclei and in the case of diamond-type lattices Al,theGGAresultsagreewellwiththeexperiment,showing
also around interstitial tetrahedral sites. In the atomic super- thattheGGAisabletointroduceacorrectionthatispropor-
positionmethodtheelectrondensityandtheCoulombpoten- tional to the calculated positron lifetime itself. For the mate-
tialareconstructednon-self-consistentlybyoverlappingfree rials in Fig. 2 (cid:126)with the striking exception of Al(cid:33) the GGA
atomdensitiesandCoulombpotentials,respectively.Thefull correctionstotheLDAarequitesimilar,about20%.There-
three-dimensional geometry of the problem is retained and fore the good agreement seen in Fig. 2 may be to some
the resulting three-dimensional Schro¨dinger equation is extent fortuitous. In order to see if the finer details of GGA
solved in a real-space point mesh.27,30 We calculate the mo- are also physical, it would be very interesting to compare
mentum distribution (cid:126)4(cid:33) with the scheme by Singh and experimentalandGGAlifetimesfortheearlytransitionmet-
Jarlborg46 using the LMTO ASA method and including the als for which the relative GGA corrections are clearly
corrections for the effects of the overlapping spheres. Va- smaller. Unfortunately, it is hard to find a consistent and
cancy calculations are performed with the LMTO ASA or comprehensive set of experimental lifetimes for transition
the atomic superposition method within the supercell ap- metals.
proximation. Periodic boundary conditions have been em- Positron lifetimes calculated for several materials using
ployed for the positron state (cid:126)positron wave vector k(cid:91)0). In the atomic superposition method are given in Table I. The
16206 B.BARBIELLINIetal. 53
TABLE I. Theoretical positron bulk lifetimes (cid:116)for different
typesofsolids.TheresultsareobtainedwiththeLMTOASAorthe
atomicsuperpositionmethodusingtheLDAenhancementfactorof
Eq. (cid:126)2(cid:33) and the corresponding GGA with (cid:97)(cid:53)0.22. In the atomic
superpositionmethodtheground-stateelectronicconfigurationsare
used for the free atoms. For example, the configuration 3d104s is
used for Cu, whereas the results obtained with the configuration
3d94s2 aregivenwithinparenthesisforcomparison.
LMTOASA Atomicsuperposition
(cid:116)LDA (cid:116)GGA (cid:116)LDA (cid:116)GGA
Material Lattice (cid:126)ps(cid:33) (cid:126)ps(cid:33) (cid:126)ps(cid:33) (cid:126)ps(cid:33)
Li bcc 257 282 259 284
C diamond 86 96 84 93
Na bcc 279 329 281 337
Al fcc 144 153 149 160 FIG.2. Positronlifetimesinperfectlattices.Thesolidandopen
Si diamond 186 210 184 207 circlesgivetheGGAandLDAresultsasafunctionoftheexperi-
K bcc 329 392 332 402 mental ones (cid:126)Ref. 48(cid:33), respectively. The solid line corresponds to
Ca fcc 245 276 250 281 theperfectagreementbetweenthetheoreticalandexperimentalre-
Sc fcc 167 189 173 198 sults,whereasthedashedlineisalinearleast-squaresfittotheLDA
Ti fcc 127 145 132 153 data.
V bcc 103 119 107 125
Cr bcc 91 105 96 118 the self-consistent density of the solid. The positron density
Mn fcc 93 108 97 114 iscalculatedineachcaseaccordingtotheelectrondensityin
Fe bcc 91 108 94 111 question.Therebythepositrondensityfollowsthetransferof
Co fcc 89 106 91 109 the electron density in such a way that the overlap and the
Ni fcc 88 107 90 109 positronlifetimechangeonlyslightly.Theshapeapproxima-
Cu fcc 96 118 101(cid:126)98(cid:33) 130(cid:126)120(cid:33) tion (cid:126)spherical charge distributions and potentials(cid:33) made in
Zn fcc 120 144 124 156 the LMTO ASA can partly be considered as such a charge
Ge diamond 191 228 190 229 transferanditseffectthepositronlifetimeisalsoquitesmall.
Rb bcc 341 409 343 420 In contrast with the LDA, the positron lifetimes obtained
Nb bcc 109 122 114 135 in the GGA using the atomic superposition method differ in
Mo bcc 101 112 106 126 some cases quite strongly from their LMTO ASA counter-
Pd fcc 94 114 99 131 parts. The reason is that the gradient is sensitive to the self-
Ag fcc 109 136 113 148 consistency and the shape of the electron density. This is
Cs bcc 356 430 358 439 seen clearly in the case of Cu for which we have used two
Ta bcc 108 117 108 124 different atomic configurations, 3d104s and 3d94s2 (cid:126)see
W bcc 93 101 95 109 TableI(cid:33).ThelatterconfigurationgivesintheGGAalifetime
Pt fcc 88 101 92 116 close to the LMTO ASA value. When the configuration is
Au fcc 98 118 102 130 changed to 3d104s the positron lifetime increases by 10 ps.
SiC zincblende 124 139 121 134 For comparison, in the LDA the difference is only 3 ps due
GaAs zincblende 190 231 190 232 to the feedback effect. In the case of II-VI compound semi-
InP zincblende 201 248 200 247 conductors the strong charge transfer between the atoms is
ZnS zincblende 179 223 179 232 not taken properly into account in the atomic superposition
CdTe zincblende 228 290 228 310 calculations. This is also seen to affect remarkably the posi-
tronlifetimescalculatedwithintheGGA.Theshapeapproxi-
HgTe zincblende 222 285 222 310
mationoftheASAisnotexpectedtoinfluencethecompari-
TiC rocksalt 94 110 101 116
son of the GGA results between the LMTO ASA and the
YBa Cu O 120 157 143 188
2 3 7 atomic superposition method, because the GGA correction
arises mainly in the ion core region where the true electron
LDAresultsforthebccandfccmetalsaretypicallylessthan density is spherical. Therefore, one can conclude that in or-
5 ps longer than those obtained by the LMTO ASA method. der to get accurate positron lifetimes in the GGA, self-
In the case of the diamond and zinc-blende structures the consistent electron densities should be used. However, it
atomic superposition gives slightly, of the order of 2 ps, couldbepossibletoimprovetheatomicsuperpositionresults
shorter lifetimes than or very similar lifetimes to the LMTO by using for free atoms electronic configurations that corre-
ASA method. In the LDA there exists an efficient feedback spond to the partial-wave expansions of the LMTO ASA
effect, which tries to keep the positron lifetime unchanged method.47
irrespective of the small transfers of electron density.27,7 In The GGA reduces the percentages of the core electron
thepresentcomparisontheelectrondensitychangesfromthe contributions relative to the total annihilation rates. In the
non-self-consistent superposition of free atom densities to case of the alkali metals (cid:126)with the exception of Li(cid:33) the core
53 CALCULATIONOFPOSITRONSTATESANDANNIHILATION ... 16207
the case of electronic structure calculations for the cohesive
properties such as the lattice constant and bulk moduli.18
B.Positronlifetimesatvacanciesinsolids
WehaveemployedthepresentGGAmodelalsoincalcu-
lating annihilation characteristics for positrons trapped at
ideal vacancies in solids. This means that the ions neighbor-
ing the vacancy are not allowed to relax from their ideal
lattice positions. In the case of metal vacancies this is ex-
pected to be a good approximation because first-principles
calculations indicate only very small relaxations.49 For va-
cancies in semiconductors with more open lattice structures
the ionic relaxations are more important (cid:126)see, for example,
Ref. 50 for the vacancy in Si and Refs. 51 and 52 for the
FIG. 3. Positron affinities in perfect lattices. The solid and vacancies in GaAs(cid:33) and the relaxation may depend strongly
dashedlinesaredrawnthroughtheGGAandLDAresults,respec-
onthechargestateofthevacancy.Inthesecasesthetrapping
tively. The open and filled circles are the experimental results by
of the positron has a tendency to compensate the inward
Jibalyetal.(cid:126)Ref.44(cid:33)andbyGidleyandFrieze(cid:126)Ref.45(cid:33).
relaxationofthevacancy.53,52,7Moreover,ithasbeenshown
that the charge state does not strongly affect the positron
contributionsarelarge,about25–35%intheLDA,asinthe lifetime if the ionic relaxation is omitted.54 As a result, the
calculation by Daniuk etal.47 The GGA reduces them by ideal (cid:126)neutral(cid:33) vacancy is a relevant reference system also in
about one-third. For Al and Ca, the core contributions are the case of semiconductors.
9.3% and 23.0% in the LDA, respectively, and 5.9% and Another important issue for the calculations of positron
16.3% in the GGA, respectively. In the case of semiconduc- states at vacancies is how the effects of the finite positron
tors, the relative core contributions are small in magnitude, density on the electronic structure should be taken into ac-
e.g., 3.1% for Si and 11.0% for GaAs in the LDA. This count. As discussed in the Introduction, the conventional
reflects the large interstitial open volume. The reduction due scheme used in the present calculations is a good starting
to the GGA is typically smaller for semiconductors than for point.TotestthisweperformedaLDAtwo-componentDFT
(cid:126)Ref.6(cid:33)calculationforapositrontrappedatavacancyinCu.
metals.ThevaluesforSiandGaAsare2.3%and9.3%inthe
We used the LMTO ASA supercell method described below
GGA, respectively. The relative core contribution can be es-
andtheparametrizationfortheenhancementgiveninRef.7.
timated from the ACAR or from the coincidence Doppler
Thepositrondistributionobtainedisclosetoandthepositron
line-shape data. An accurate estimation may, however, be
lifetime is only about 2 ps shorter than that in the conven-
difficult due to superposition of the core and the umklapp
tional LDA scheme. A similar agreement between the full
components of the valence annihilation at high momentum
two-component and the conventional scheme has been ob-
values. On the other hand, in the theory an unambiguous
tained previously for metal vacancies within the jellium
distinction between the core and valence electrons is diffi-
model6 and recently for the Ga vacancy in GaAs in a calcu-
cult. For instance, in the case of the transition, noble, and
lation using the pseudo-potential plane-wave description for
Zn-column metals, as well as the II-VI compound semicon- electronicstructure.7Encouragedbythisexperience,wewill
ductors,theuppermostd electronshavetobetreatedasband
in this work compare the LDA and GGA models for the
states, which can hybridize with the s valence states. positronstatesbyusingonlythecomputationallymuchmore
The positron affinities calculated within the LDA and efficient conventional scheme.
GGA models are compared with those measured by the re- In the vacancy calculations we use the supercell scheme,
emitted positron spectroscopy44,45 in Fig. 3. The electron in which a large cell containing one vacancy is repeated pe-
chemical potentials needed are calculated in all cases within riodically so that a regular superlattice of vacancies is
the LDA (cid:126)Ref. 43(cid:33) in order to study the effects of the GGA formed. In order to get converged results for the positron
for positron states only. However, we have calculated that if lifetime, it is necessary to use such a large supercell that the
the effect of the GGA (cid:126)Ref. 38(cid:33) on electron states is taken positronwavefunctionisvanishinglysmallatitsboundaries
intoaccount,theFermilevelraisesandthemagnitudeofthe whenavacancyiscreatedatthecenter.Intheatomicsuper-
positron affinity decreases by (cid:59) 0.3 eV. Compared to the positioncalculationswehaveusedsupercellsupto255,249,
LDA results the GGA for the positron states reduces the and 215 atoms for vacancies in fcc, bcc, and zinc-blende (cid:126)or
magnitude of the affinity by 0.3–0.7 eV due to the increase diamond(cid:33) structures, respectively. In the LMTO ASA calcu-
of the positron zero-point energy. The LDA is seen to over- lations the electronic structures of the vacancies are calcu-
estimate the magnitude of the affinity within the 3d series lated using smaller super cells, which in the fcc, bcc, and
andtheretheGGAisaclearimprovement.Inthecaseofthe zinc-blende structures contain 63, 63, and 31 atoms, respec-
4d and5d transitionmetalsstudiedtheLDAresultsare,with tively. In both methods the results are then extrapolated to
the exception of Mo, in quite good agreement with experi- the infinite supercell size. The slow convergence of the pos-
mentsandtheGGAleadstoaffinitieswithtoosmallmagni- itron lifetime with the size of the supercell can be seen, for
tudes in comparison with the experimental data. The trend example, in the case of the vacancy in Cu. In the LMTO
that the GGA improves the LDA results for the 3d series ASA calculations the converged lifetime is about 20 ps
while worsens them for the 4d and 5d series is also seen in longer than that obtained with the small supercell of eight
16208 B.BARBIELLINIetal. 53
TABLEII. Positronstrappedbyvacanciesinsolids.Thelifetimes(cid:116)andpositronbindingenergiesE are
b
obtainedwiththeLMTOASAortheatomicsuperpositionmethodusingtheLDAenhancementfactorofEq.
(cid:126)2(cid:33)andthecorrespondingGGAwith(cid:97)(cid:53)0.22.
LMTOASA Atomicsuperposition
(cid:116)LDA E (cid:116)GGA E (cid:116)LDA E (cid:116)GGA E
b b b b
Lattice Material (cid:126)ps(cid:33) (cid:126)eV(cid:33) (cid:126)ps(cid:33) (cid:126)eV(cid:33) (cid:126)ps(cid:33) (cid:126)eV(cid:33) (cid:126)ps(cid:33) (cid:126)eV(cid:33)
fcc Al 215 2.1 237 1.9 212 2.1 231 2.1
Cu 158 1.7 193 1.9 153 1.3 200 1.5
bcc Fe 159 2.6 186 2.7 158 3.4 183 3.7
Nb 197 3.1 218 3.2 195 3.7 225 3.9
dia Si 215 0.7 244 0.7 209 0.4 240 0.3
zb GaAs,V 214 0.3 264 0.2
Ga
GaAs,V 212 0.2 261 0.2
As
atomic sites. The atomic superposition method gives a simi- of the self-consistency of the electronic density. As in the
lartrendwithapositronlifetimeincreaseofabout10ps.We caseofpositronbulklifetimes,theseeffectsaremoreimpor-
conclude that positron lifetimes obtained with supercells of tant in the GGA than in the LDA.
four and eight atomic sites, which are used, for example, in The theoretical positron lifetimes for vacancies are com-
the LMTO ASA calculations of Ref. 12, are not converged pared with experiment in Table III. The first column gives
with respect to the positron distribution. This kind of super- the experimental lifetimes for positrons trapped at different
cell gives, however, reasonable estimates for positron life- vacancies. For the experiment-theory comparison the ratios
times in comparison with experiments. between the vacancy and bulk lifetimes are calculated from
The results of our calculations for positron states at va- the experimental works and from the results of Tables I and
cancies in solids are collected in Table II. For the vacancies II. It can be seen that within the calculation scheme, LMTO
in the diamond or zinc-blende structure the supercell calcu- ASA, or atomic superposition, the LDA and the GGA mod-
lationswiththeLMTOASAmethoddemandrelativelymore els give similar ratios. For the metal vacancies the lifetime
computerresourcesthanthecalculationsforfccandbccmet- ratiosobtainedusingtheatomicsuperpositionmethodareof
als, because empty spheres are needed. Therefore we have the same order as or slightly smaller than the experimental
performed atomic superposition calculations in the case of ones. The ratios from the LMTO ASA calculations are sys-
vacancies in Si and GaAs and the LMTO ASA calculation tematically larger than those from the atomic superposition
onlyforthevacancyinSi.WithintheLDAtheLMTO-ASA method, reflecting a stronger localization of the positron at
method gives slightly longer positron lifetimes than the the vacancy. The reason for the differences between the
atomic superposition method. In the GGA the LMTO ASA LMTO ASA and the atomic superposition methods can be
lifetimes are either shorter or longer than the atomic super- understood as follows. In the former self-consistent calcula-
position ones. The positron trapping energies are quite simi- tion a dipole-type potential arises so that electron density is
lar in both methods. The differences between the LMTO transferredfromthevacancyregion(cid:126)thevacancysphereand
ASA and the atomic superposition results reflect the effects its 12 nearest-neighbor atomic spheres(cid:33) further away to the
TABLEIII. Positronstrappedbyvacanciesinsolids.Theexperimentalpositronlifetimes(cid:116)vac aregiven.
expt
The ratios between the vacancy and bulk lifetimes are calculated from the experimental results (cid:126)Expt.(cid:33) and
fromthetheoreticalresultsobtainedwiththeLMTOASAortheatomicsuperpositionmethodusingtheLDA
enhancementfactorofEq.(cid:126)2(cid:33)andthecorrespondingGGAwith(cid:97)(cid:53)0.22.
(cid:116)vac/(cid:116)bulk
(cid:116)vac LMTOASA Atomicsuperposition
expt
Material (cid:126)ps(cid:33) Expt. LDA GGA LDA GGA
Al 251a 1.54a 1.49 1.55 1.42 1.44
Cu 179b 1.63b 1.65 1.64 1.51 1.54
Fe 175c 1.59c 1.75 1.72 1.68 1.65
Nb 210d 1.72d 1.81 1.79 1.71 1.67
Si 1.16 1.16 1.14 1.16
GaAs,V3(cid:50) 260e 1.13e 1.13 1.14
Ga
GaAs,V(cid:50) 257f 1.11f 1.12 1.13
As
GaAs,V0 295f 1.28f
As
aFromRef.55. dFromRef.58.
bFromRef.56 eFromRef.59.
cFromRef.57. fFromRef.60.
53 CALCULATIONOFPOSITRONSTATESANDANNIHILATION ... 16209
region of the perfect lattice. The dipole, reversed for the
positron potential, results in a localization of the positron
wave function that is stronger than in the atomic superposi-
tion calculations. This strong positron localization increases
thepositronlifetime.Notethatthefeedbackeffectdiscussed
above in the context of the bulk lifetimes is not effective for
this kind of longe-range field.
Thevacancy-bulklifetimeratiosobtainedwiththeLMTO
ASA and the atomic superposition method for Si are very
similar. The experimental vacancy-bulk lifetime ratios for
thetriplynegativeGavacancyandthesinglynegativeAsin
GaAsagreewellwiththeatomicsuperpositionresultsforthe
ideal vacancies. The much larger experimental ratio for the
neutral As vacancy may be a result of a large lattice relax-
ation occurring between the negative and the neutral charge
states.51 FIG. 4. (cid:126)a(cid:33) Momentum density and (cid:126)b(cid:33) the corresponding en-
hancement factor for annihilating positron-electron pairs for Cu
C.Angularcorrelationofannihilationradiation alongthedirection(cid:71)-Xinthemomentumspace.Thedensitiesare
normalizedtogivetherelativetotalvalenceannihilationrates.The
1.2DACARspectra solid, dashed, and dash-dotted lines are the results in the GGA,
The2DACARtechniquestandsoutinitsuniquecapabil- LDA,andIPMcalculations,respectively.
itytoyieldhigh-resolutioninformationaboutthemomentum
density and the Fermi surface (cid:126)FS(cid:33) of metals. Moreover, the theorem.66 At low momenta mainly the sp bands contribute.
LCWtheorem33proposesthatwhenthe2DACARspectrum The relative contribution of the localized d electrons in-
is folded into the first Brillouin zone, mainly the structures creases towards higher Brillouin zones with the maximum
relatedtotheFSremain.Insomecases,theFSbreakscanbe around 11–13 mrad. Wakoh etal.63 concluded that the IPM
inferred just by studying the occupation of the bands cross- overestimates the contributions of the d electrons. Figure
ing the Fermi energy. However, the presence of the positron 4(cid:126)b(cid:33)showsthatboththeLDAandtheGGApredictareduc-
makes the procedure inexact, because the background is tion of the enhancement in the momentum region where the
modulated by positron annihilation with the electronic states d electrons dominate and should improve the theoretical fit
that do not contribute to the FS. These positron wave func- to the experimental data.
tion effects are included within the IPM. However, again In order to find out whether the LDA or the GGA agrees
enhancement has to be considered, although it is not of such better with the experiment, we have compared the shapes of
crucial importance as in describing the positron lifetime. the 2D ACAR distributions in Fig. 5. The figure shows the
For example, in the case of Cu a careful comparison of calculated 2D ACAR spectra containing the contributions
the 2D ACAR distributions shows a considerable deviation from both the core and valence electrons and corresponding
between the IPM and the experiment.63,64 The deviation is to the momentum distribution (cid:126)4(cid:33) integrated in the (cid:64)111(cid:35) di-
attributed to many-body effects. In order to compare theory rection. The corresponding experimental data64 are also
with experiment, Wakoh etal.63 considered the ratio shown. A cut of the two-dimensional map along the (cid:64)1–1 0(cid:35)
Nexpt(cid:126)p ,p (cid:33)(cid:50)NIPM(cid:126)p ,p (cid:33)
r(cid:126)p ,p (cid:33)(cid:53) x y x y , (cid:126)18(cid:33)
x y NIPM(cid:126)p ,p (cid:33)
x y
where Nexpt and NIPM are normalized to the same volume.
The shape of r(p ,p ) indicates that the many-body en-
x y
hancement increases as the first Brillouin zone is ap-
proached; outside this zone the enhancement gradually di-
minished, indicating that for tightly bound electrons the
enhancement is smaller than for the sp conduction electrons
in the first zone. We have performed a detailed study on
copper to check whether the LDA and the GGA explain
these trends.
The momentum density of annihilating positron-electron
pairsforthevalencestatesinCuisshowninFig.4(cid:126)a(cid:33)along
the direction (cid:71)-X (cid:126)at the X point, p(cid:39)6.7 mrad(cid:33). Due to a
symmetry-induced selection rule,65 only two bands, which
FIG. 5. Two-dimensional ACAR spectra for Cu. The momen-
are hybridizations of an sp-like and a d-like conduction tumdensitiesareintegratedinthe(cid:64)111(cid:35)directionandnormalizedto
band, contribute along this direction. The FS breaks are the same value at the zero momentum. The cuts along the
clearly visible within the first and second Brillouin zones. (cid:64)1(cid:50)10(cid:35) direction are given. The solid, dashed, and dash-dotted
These discontinuities are not shifted by the electron-positron lines are the results in the GGA, LDA, and IPM calculations, re-
correlation, which is in agreement with the Majumdar spectively.Theexperimentalpointsaregivenbydots.